3.4.60 \(\int x (a+b x)^n (c+d x^2)^3 \, dx\) [360]

Optimal. Leaf size=282 \[ -\frac {a \left (b^2 c+a^2 d\right )^3 (a+b x)^{1+n}}{b^8 (1+n)}+\frac {\left (b^2 c+a^2 d\right )^2 \left (b^2 c+7 a^2 d\right ) (a+b x)^{2+n}}{b^8 (2+n)}-\frac {3 a d \left (b^2 c+a^2 d\right ) \left (3 b^2 c+7 a^2 d\right ) (a+b x)^{3+n}}{b^8 (3+n)}+\frac {d \left (3 b^4 c^2+30 a^2 b^2 c d+35 a^4 d^2\right ) (a+b x)^{4+n}}{b^8 (4+n)}-\frac {5 a d^2 \left (3 b^2 c+7 a^2 d\right ) (a+b x)^{5+n}}{b^8 (5+n)}+\frac {3 d^2 \left (b^2 c+7 a^2 d\right ) (a+b x)^{6+n}}{b^8 (6+n)}-\frac {7 a d^3 (a+b x)^{7+n}}{b^8 (7+n)}+\frac {d^3 (a+b x)^{8+n}}{b^8 (8+n)} \]

[Out]

-a*(a^2*d+b^2*c)^3*(b*x+a)^(1+n)/b^8/(1+n)+(a^2*d+b^2*c)^2*(7*a^2*d+b^2*c)*(b*x+a)^(2+n)/b^8/(2+n)-3*a*d*(a^2*
d+b^2*c)*(7*a^2*d+3*b^2*c)*(b*x+a)^(3+n)/b^8/(3+n)+d*(35*a^4*d^2+30*a^2*b^2*c*d+3*b^4*c^2)*(b*x+a)^(4+n)/b^8/(
4+n)-5*a*d^2*(7*a^2*d+3*b^2*c)*(b*x+a)^(5+n)/b^8/(5+n)+3*d^2*(7*a^2*d+b^2*c)*(b*x+a)^(6+n)/b^8/(6+n)-7*a*d^3*(
b*x+a)^(7+n)/b^8/(7+n)+d^3*(b*x+a)^(8+n)/b^8/(8+n)

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Rubi [A]
time = 0.11, antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {786} \begin {gather*} -\frac {5 a d^2 \left (7 a^2 d+3 b^2 c\right ) (a+b x)^{n+5}}{b^8 (n+5)}+\frac {3 d^2 \left (7 a^2 d+b^2 c\right ) (a+b x)^{n+6}}{b^8 (n+6)}-\frac {a \left (a^2 d+b^2 c\right )^3 (a+b x)^{n+1}}{b^8 (n+1)}+\frac {\left (a^2 d+b^2 c\right )^2 \left (7 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^8 (n+2)}-\frac {3 a d \left (a^2 d+b^2 c\right ) \left (7 a^2 d+3 b^2 c\right ) (a+b x)^{n+3}}{b^8 (n+3)}+\frac {d \left (35 a^4 d^2+30 a^2 b^2 c d+3 b^4 c^2\right ) (a+b x)^{n+4}}{b^8 (n+4)}-\frac {7 a d^3 (a+b x)^{n+7}}{b^8 (n+7)}+\frac {d^3 (a+b x)^{n+8}}{b^8 (n+8)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x)^n*(c + d*x^2)^3,x]

[Out]

-((a*(b^2*c + a^2*d)^3*(a + b*x)^(1 + n))/(b^8*(1 + n))) + ((b^2*c + a^2*d)^2*(b^2*c + 7*a^2*d)*(a + b*x)^(2 +
n))/(b^8*(2 + n)) - (3*a*d*(b^2*c + a^2*d)*(3*b^2*c + 7*a^2*d)*(a + b*x)^(3 + n))/(b^8*(3 + n)) + (d*(3*b^4*c
^2 + 30*a^2*b^2*c*d + 35*a^4*d^2)*(a + b*x)^(4 + n))/(b^8*(4 + n)) - (5*a*d^2*(3*b^2*c + 7*a^2*d)*(a + b*x)^(5
+ n))/(b^8*(5 + n)) + (3*d^2*(b^2*c + 7*a^2*d)*(a + b*x)^(6 + n))/(b^8*(6 + n)) - (7*a*d^3*(a + b*x)^(7 + n))
/(b^8*(7 + n)) + (d^3*(a + b*x)^(8 + n))/(b^8*(8 + n))

Rule 786

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int x (a+b x)^n \left (c+d x^2\right )^3 \, dx &=\int \left (-\frac {a \left (b^2 c+a^2 d\right )^3 (a+b x)^n}{b^7}+\frac {\left (b^2 c+a^2 d\right )^2 \left (b^2 c+7 a^2 d\right ) (a+b x)^{1+n}}{b^7}+\frac {3 a d \left (-3 b^2 c-7 a^2 d\right ) \left (b^2 c+a^2 d\right ) (a+b x)^{2+n}}{b^7}+\frac {d \left (3 b^4 c^2+30 a^2 b^2 c d+35 a^4 d^2\right ) (a+b x)^{3+n}}{b^7}-\frac {5 a d^2 \left (3 b^2 c+7 a^2 d\right ) (a+b x)^{4+n}}{b^7}+\frac {3 d^2 \left (b^2 c+7 a^2 d\right ) (a+b x)^{5+n}}{b^7}-\frac {7 a d^3 (a+b x)^{6+n}}{b^7}+\frac {d^3 (a+b x)^{7+n}}{b^7}\right ) \, dx\\ &=-\frac {a \left (b^2 c+a^2 d\right )^3 (a+b x)^{1+n}}{b^8 (1+n)}+\frac {\left (b^2 c+a^2 d\right )^2 \left (b^2 c+7 a^2 d\right ) (a+b x)^{2+n}}{b^8 (2+n)}-\frac {3 a d \left (b^2 c+a^2 d\right ) \left (3 b^2 c+7 a^2 d\right ) (a+b x)^{3+n}}{b^8 (3+n)}+\frac {d \left (3 b^4 c^2+30 a^2 b^2 c d+35 a^4 d^2\right ) (a+b x)^{4+n}}{b^8 (4+n)}-\frac {5 a d^2 \left (3 b^2 c+7 a^2 d\right ) (a+b x)^{5+n}}{b^8 (5+n)}+\frac {3 d^2 \left (b^2 c+7 a^2 d\right ) (a+b x)^{6+n}}{b^8 (6+n)}-\frac {7 a d^3 (a+b x)^{7+n}}{b^8 (7+n)}+\frac {d^3 (a+b x)^{8+n}}{b^8 (8+n)}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(578\) vs. \(2(282)=564\).
time = 0.38, size = 578, normalized size = 2.05 \begin {gather*} \frac {(a+b x)^{1+n} \left (-5040 a^7 d^3+5040 a^6 b d^3 (1+n) x-360 a^5 b^2 d^2 \left (c \left (56+15 n+n^2\right )+7 d \left (2+3 n+n^2\right ) x^2\right )+120 a^4 b^3 d^2 (1+n) x \left (3 c \left (56+15 n+n^2\right )+7 d \left (6+5 n+n^2\right ) x^2\right )-6 a^3 b^4 d \left (3 c^2 \left (1680+1066 n+251 n^2+26 n^3+n^4\right )+30 c d \left (112+198 n+103 n^2+18 n^3+n^4\right ) x^2+35 d^2 \left (24+50 n+35 n^2+10 n^3+n^4\right ) x^4\right )+6 a^2 b^5 d (1+n) x \left (3 c^2 \left (1680+1066 n+251 n^2+26 n^3+n^4\right )+10 c d \left (336+370 n+137 n^2+20 n^3+n^4\right ) x^2+7 d^2 \left (120+154 n+71 n^2+14 n^3+n^4\right ) x^4\right )+b^7 \left (105+176 n+86 n^2+16 n^3+n^4\right ) x \left (c^3 \left (192+104 n+18 n^2+n^3\right )+3 c^2 d \left (96+76 n+16 n^2+n^3\right ) x^2+3 c d^2 \left (64+56 n+14 n^2+n^3\right ) x^4+d^3 \left (48+44 n+12 n^2+n^3\right ) x^6\right )-a b^6 \left (c^3 \left (20160+24552 n+12154 n^2+3135 n^3+445 n^4+33 n^5+n^6\right )+9 c^2 d \left (3360+7172 n+5380 n^2+1871 n^3+331 n^4+29 n^5+n^6\right ) x^2+15 c d^2 \left (1344+3160 n+2734 n^2+1135 n^3+241 n^4+25 n^5+n^6\right ) x^4+7 d^3 \left (720+1764 n+1624 n^2+735 n^3+175 n^4+21 n^5+n^6\right ) x^6\right )\right )}{b^8 (1+n) (2+n) (3+n) (4+n) (5+n) (6+n) (7+n) (8+n)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x)^n*(c + d*x^2)^3,x]

[Out]

((a + b*x)^(1 + n)*(-5040*a^7*d^3 + 5040*a^6*b*d^3*(1 + n)*x - 360*a^5*b^2*d^2*(c*(56 + 15*n + n^2) + 7*d*(2 +
3*n + n^2)*x^2) + 120*a^4*b^3*d^2*(1 + n)*x*(3*c*(56 + 15*n + n^2) + 7*d*(6 + 5*n + n^2)*x^2) - 6*a^3*b^4*d*(
3*c^2*(1680 + 1066*n + 251*n^2 + 26*n^3 + n^4) + 30*c*d*(112 + 198*n + 103*n^2 + 18*n^3 + n^4)*x^2 + 35*d^2*(2
4 + 50*n + 35*n^2 + 10*n^3 + n^4)*x^4) + 6*a^2*b^5*d*(1 + n)*x*(3*c^2*(1680 + 1066*n + 251*n^2 + 26*n^3 + n^4)
+ 10*c*d*(336 + 370*n + 137*n^2 + 20*n^3 + n^4)*x^2 + 7*d^2*(120 + 154*n + 71*n^2 + 14*n^3 + n^4)*x^4) + b^7*
(105 + 176*n + 86*n^2 + 16*n^3 + n^4)*x*(c^3*(192 + 104*n + 18*n^2 + n^3) + 3*c^2*d*(96 + 76*n + 16*n^2 + n^3)
*x^2 + 3*c*d^2*(64 + 56*n + 14*n^2 + n^3)*x^4 + d^3*(48 + 44*n + 12*n^2 + n^3)*x^6) - a*b^6*(c^3*(20160 + 2455
2*n + 12154*n^2 + 3135*n^3 + 445*n^4 + 33*n^5 + n^6) + 9*c^2*d*(3360 + 7172*n + 5380*n^2 + 1871*n^3 + 331*n^4
+ 29*n^5 + n^6)*x^2 + 15*c*d^2*(1344 + 3160*n + 2734*n^2 + 1135*n^3 + 241*n^4 + 25*n^5 + n^6)*x^4 + 7*d^3*(720
+ 1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6)*x^6)))/(b^8*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(
6 + n)*(7 + n)*(8 + n))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1213\) vs. \(2(282)=564\).
time = 0.09, size = 1214, normalized size = 4.30

method	result	size
norman	\(\frac {d^{3} x^{8} {\mathrm e}^{n \ln \left (b x +a \right )}}{8+n}+\frac {n a \left (b^{6} c^{3} n^{6}+33 b^{6} c^{3} n^{5}+18 a^{2} b^{4} c^{2} d \,n^{4}+445 b^{6} c^{3} n^{4}+468 a^{2} b^{4} c^{2} d \,n^{3}+3135 b^{6} c^{3} n^{3}+360 a^{4} b^{2} c \,d^{2} n^{2}+4518 a^{2} b^{4} c^{2} d \,n^{2}+12154 b^{6} c^{3} n^{2}+5400 a^{4} b^{2} c \,d^{2} n +19188 a^{2} b^{4} c^{2} d n +24552 b^{6} c^{3} n +5040 a^{6} d^{3}+20160 a^{4} b^{2} c \,d^{2}+30240 a^{2} b^{4} c^{2} d +20160 b^{6} c^{3}\right ) x \,{\mathrm e}^{n \ln \left (b x +a \right )}}{b^{7} \left (n^{8}+36 n^{7}+546 n^{6}+4536 n^{5}+22449 n^{4}+67284 n^{3}+118124 n^{2}+109584 n +40320\right )}+\frac {n a \,d^{3} x^{7} {\mathrm e}^{n \ln \left (b x +a \right )}}{b \left (n^{2}+15 n +56\right )}-\frac {a^{2} \left (b^{6} c^{3} n^{6}+33 b^{6} c^{3} n^{5}+18 a^{2} b^{4} c^{2} d \,n^{4}+445 b^{6} c^{3} n^{4}+468 a^{2} b^{4} c^{2} d \,n^{3}+3135 b^{6} c^{3} n^{3}+360 a^{4} b^{2} c \,d^{2} n^{2}+4518 a^{2} b^{4} c^{2} d \,n^{2}+12154 b^{6} c^{3} n^{2}+5400 a^{4} b^{2} c \,d^{2} n +19188 a^{2} b^{4} c^{2} d n +24552 b^{6} c^{3} n +5040 a^{6} d^{3}+20160 a^{4} b^{2} c \,d^{2}+30240 a^{2} b^{4} c^{2} d +20160 b^{6} c^{3}\right ) {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{8} \left (n^{8}+36 n^{7}+546 n^{6}+4536 n^{5}+22449 n^{4}+67284 n^{3}+118124 n^{2}+109584 n +40320\right )}-\frac {\left (-b^{6} c^{3} n^{6}+9 a^{2} b^{4} c^{2} d \,n^{5}-33 b^{6} c^{3} n^{5}+234 a^{2} b^{4} c^{2} d \,n^{4}-445 b^{6} c^{3} n^{4}+180 a^{4} b^{2} c \,d^{2} n^{3}+2259 a^{2} b^{4} c^{2} d \,n^{3}-3135 b^{6} c^{3} n^{3}+2700 a^{4} b^{2} c \,d^{2} n^{2}+9594 a^{2} b^{4} c^{2} d \,n^{2}-12154 b^{6} c^{3} n^{2}+2520 a^{6} d^{3} n +10080 a^{4} b^{2} c \,d^{2} n +15120 a^{2} b^{4} c^{2} d n -24552 b^{6} c^{3} n -20160 b^{6} c^{3}\right ) x^{2} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{6} \left (n^{7}+35 n^{6}+511 n^{5}+4025 n^{4}+18424 n^{3}+48860 n^{2}+69264 n +40320\right )}-\frac {\left (-3 b^{2} c \,n^{2}+7 a^{2} d n -45 b^{2} c n -168 b^{2} c \right ) d^{2} x^{6} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{2} \left (n^{3}+21 n^{2}+146 n +336\right )}-\frac {3 \left (-b^{4} c^{2} n^{4}+5 a^{2} b^{2} c d \,n^{3}-26 b^{4} c^{2} n^{3}+75 a^{2} b^{2} c d \,n^{2}-251 b^{4} c^{2} n^{2}+70 a^{4} d^{2} n +280 a^{2} b^{2} c d n -1066 b^{4} c^{2} n -1680 b^{4} c^{2}\right ) d \,x^{4} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{4} \left (n^{5}+30 n^{4}+355 n^{3}+2070 n^{2}+5944 n +6720\right )}+\frac {3 \left (b^{2} c \,n^{2}+15 b^{2} c n +14 a^{2} d +56 b^{2} c \right ) n a \,d^{2} x^{5} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{3} \left (n^{4}+26 n^{3}+251 n^{2}+1066 n +1680\right )}+\frac {3 \left (b^{4} c^{2} n^{4}+26 b^{4} c^{2} n^{3}+20 a^{2} b^{2} c d \,n^{2}+251 b^{4} c^{2} n^{2}+300 a^{2} b^{2} c d n +1066 b^{4} c^{2} n +280 a^{4} d^{2}+1120 a^{2} b^{2} c d +1680 b^{4} c^{2}\right ) a d n \,x^{3} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{5} \left (n^{6}+33 n^{5}+445 n^{4}+3135 n^{3}+12154 n^{2}+24552 n +20160\right )}\)	\(1214\)
gosper	\(\text {Expression too large to display}\)	\(1639\)
risch	\(\text {Expression too large to display}\)	\(1955\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x+a)^n*(d*x^2+c)^3,x,method=_RETURNVERBOSE)

[Out]

d^3/(8+n)*x^8*exp(n*ln(b*x+a))+1/b^7*n*a*(b^6*c^3*n^6+33*b^6*c^3*n^5+18*a^2*b^4*c^2*d*n^4+445*b^6*c^3*n^4+468*
a^2*b^4*c^2*d*n^3+3135*b^6*c^3*n^3+360*a^4*b^2*c*d^2*n^2+4518*a^2*b^4*c^2*d*n^2+12154*b^6*c^3*n^2+5400*a^4*b^2
*c*d^2*n+19188*a^2*b^4*c^2*d*n+24552*b^6*c^3*n+5040*a^6*d^3+20160*a^4*b^2*c*d^2+30240*a^2*b^4*c^2*d+20160*b^6*
c^3)/(n^8+36*n^7+546*n^6+4536*n^5+22449*n^4+67284*n^3+118124*n^2+109584*n+40320)*x*exp(n*ln(b*x+a))+n*a/b*d^3/
(n^2+15*n+56)*x^7*exp(n*ln(b*x+a))-a^2*(b^6*c^3*n^6+33*b^6*c^3*n^5+18*a^2*b^4*c^2*d*n^4+445*b^6*c^3*n^4+468*a^
2*b^4*c^2*d*n^3+3135*b^6*c^3*n^3+360*a^4*b^2*c*d^2*n^2+4518*a^2*b^4*c^2*d*n^2+12154*b^6*c^3*n^2+5400*a^4*b^2*c
*d^2*n+19188*a^2*b^4*c^2*d*n+24552*b^6*c^3*n+5040*a^6*d^3+20160*a^4*b^2*c*d^2+30240*a^2*b^4*c^2*d+20160*b^6*c^
3)/b^8/(n^8+36*n^7+546*n^6+4536*n^5+22449*n^4+67284*n^3+118124*n^2+109584*n+40320)*exp(n*ln(b*x+a))-(-b^6*c^3*
n^6+9*a^2*b^4*c^2*d*n^5-33*b^6*c^3*n^5+234*a^2*b^4*c^2*d*n^4-445*b^6*c^3*n^4+180*a^4*b^2*c*d^2*n^3+2259*a^2*b^
4*c^2*d*n^3-3135*b^6*c^3*n^3+2700*a^4*b^2*c*d^2*n^2+9594*a^2*b^4*c^2*d*n^2-12154*b^6*c^3*n^2+2520*a^6*d^3*n+10
080*a^4*b^2*c*d^2*n+15120*a^2*b^4*c^2*d*n-24552*b^6*c^3*n-20160*b^6*c^3)/b^6/(n^7+35*n^6+511*n^5+4025*n^4+1842
4*n^3+48860*n^2+69264*n+40320)*x^2*exp(n*ln(b*x+a))-(-3*b^2*c*n^2+7*a^2*d*n-45*b^2*c*n-168*b^2*c)/b^2*d^2/(n^3
+21*n^2+146*n+336)*x^6*exp(n*ln(b*x+a))-3*(-b^4*c^2*n^4+5*a^2*b^2*c*d*n^3-26*b^4*c^2*n^3+75*a^2*b^2*c*d*n^2-25
1*b^4*c^2*n^2+70*a^4*d^2*n+280*a^2*b^2*c*d*n-1066*b^4*c^2*n-1680*b^4*c^2)*d/b^4/(n^5+30*n^4+355*n^3+2070*n^2+5
944*n+6720)*x^4*exp(n*ln(b*x+a))+3*(b^2*c*n^2+15*b^2*c*n+14*a^2*d+56*b^2*c)*n*a/b^3*d^2/(n^4+26*n^3+251*n^2+10
66*n+1680)*x^5*exp(n*ln(b*x+a))+3*(b^4*c^2*n^4+26*b^4*c^2*n^3+20*a^2*b^2*c*d*n^2+251*b^4*c^2*n^2+300*a^2*b^2*c
*d*n+1066*b^4*c^2*n+280*a^4*d^2+1120*a^2*b^2*c*d+1680*b^4*c^2)*a/b^5*d*n/(n^6+33*n^5+445*n^4+3135*n^3+12154*n^
2+24552*n+20160)*x^3*exp(n*ln(b*x+a))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 625 vs. \(2 (282) = 564\).
time = 0.30, size = 625, normalized size = 2.22 \begin {gather*} \frac {{\left (b^{2} {\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )} {\left (b x + a\right )}^{n} c^{3}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac {3 \, {\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )} {\left (b x + a\right )}^{n} c^{2} d}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} + \frac {3 \, {\left ({\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{6} x^{6} + {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a b^{5} x^{5} - 5 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{2} b^{4} x^{4} + 20 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{3} b^{3} x^{3} - 60 \, {\left (n^{2} + n\right )} a^{4} b^{2} x^{2} + 120 \, a^{5} b n x - 120 \, a^{6}\right )} {\left (b x + a\right )}^{n} c d^{2}}{{\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} b^{6}} + \frac {{\left ({\left (n^{7} + 28 \, n^{6} + 322 \, n^{5} + 1960 \, n^{4} + 6769 \, n^{3} + 13132 \, n^{2} + 13068 \, n + 5040\right )} b^{8} x^{8} + {\left (n^{7} + 21 \, n^{6} + 175 \, n^{5} + 735 \, n^{4} + 1624 \, n^{3} + 1764 \, n^{2} + 720 \, n\right )} a b^{7} x^{7} - 7 \, {\left (n^{6} + 15 \, n^{5} + 85 \, n^{4} + 225 \, n^{3} + 274 \, n^{2} + 120 \, n\right )} a^{2} b^{6} x^{6} + 42 \, {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a^{3} b^{5} x^{5} - 210 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{4} b^{4} x^{4} + 840 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{5} b^{3} x^{3} - 2520 \, {\left (n^{2} + n\right )} a^{6} b^{2} x^{2} + 5040 \, a^{7} b n x - 5040 \, a^{8}\right )} {\left (b x + a\right )}^{n} d^{3}}{{\left (n^{8} + 36 \, n^{7} + 546 \, n^{6} + 4536 \, n^{5} + 22449 \, n^{4} + 67284 \, n^{3} + 118124 \, n^{2} + 109584 \, n + 40320\right )} b^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^n*(d*x^2+c)^3,x, algorithm="maxima")

[Out]

(b^2*(n + 1)*x^2 + a*b*n*x - a^2)*(b*x + a)^n*c^3/((n^2 + 3*n + 2)*b^2) + 3*((n^3 + 6*n^2 + 11*n + 6)*b^4*x^4
+ (n^3 + 3*n^2 + 2*n)*a*b^3*x^3 - 3*(n^2 + n)*a^2*b^2*x^2 + 6*a^3*b*n*x - 6*a^4)*(b*x + a)^n*c^2*d/((n^4 + 10*
n^3 + 35*n^2 + 50*n + 24)*b^4) + 3*((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*b^6*x^6 + (n^5 + 10*n^4 +
35*n^3 + 50*n^2 + 24*n)*a*b^5*x^5 - 5*(n^4 + 6*n^3 + 11*n^2 + 6*n)*a^2*b^4*x^4 + 20*(n^3 + 3*n^2 + 2*n)*a^3*b^
3*x^3 - 60*(n^2 + n)*a^4*b^2*x^2 + 120*a^5*b*n*x - 120*a^6)*(b*x + a)^n*c*d^2/((n^6 + 21*n^5 + 175*n^4 + 735*n
^3 + 1624*n^2 + 1764*n + 720)*b^6) + ((n^7 + 28*n^6 + 322*n^5 + 1960*n^4 + 6769*n^3 + 13132*n^2 + 13068*n + 50
40)*b^8*x^8 + (n^7 + 21*n^6 + 175*n^5 + 735*n^4 + 1624*n^3 + 1764*n^2 + 720*n)*a*b^7*x^7 - 7*(n^6 + 15*n^5 + 8
5*n^4 + 225*n^3 + 274*n^2 + 120*n)*a^2*b^6*x^6 + 42*(n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*a^3*b^5*x^5 - 210*
(n^4 + 6*n^3 + 11*n^2 + 6*n)*a^4*b^4*x^4 + 840*(n^3 + 3*n^2 + 2*n)*a^5*b^3*x^3 - 2520*(n^2 + n)*a^6*b^2*x^2 +
5040*a^7*b*n*x - 5040*a^8)*(b*x + a)^n*d^3/((n^8 + 36*n^7 + 546*n^6 + 4536*n^5 + 22449*n^4 + 67284*n^3 + 11812
4*n^2 + 109584*n + 40320)*b^8)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1675 vs. \(2 (282) = 564\).
time = 3.31, size = 1675, normalized size = 5.94 \begin {gather*} -\frac {{\left (a^{2} b^{6} c^{3} n^{6} + 33 \, a^{2} b^{6} c^{3} n^{5} + 20160 \, a^{2} b^{6} c^{3} + 30240 \, a^{4} b^{4} c^{2} d + 20160 \, a^{6} b^{2} c d^{2} + 5040 \, a^{8} d^{3} - {\left (b^{8} d^{3} n^{7} + 28 \, b^{8} d^{3} n^{6} + 322 \, b^{8} d^{3} n^{5} + 1960 \, b^{8} d^{3} n^{4} + 6769 \, b^{8} d^{3} n^{3} + 13132 \, b^{8} d^{3} n^{2} + 13068 \, b^{8} d^{3} n + 5040 \, b^{8} d^{3}\right )} x^{8} - {\left (a b^{7} d^{3} n^{7} + 21 \, a b^{7} d^{3} n^{6} + 175 \, a b^{7} d^{3} n^{5} + 735 \, a b^{7} d^{3} n^{4} + 1624 \, a b^{7} d^{3} n^{3} + 1764 \, a b^{7} d^{3} n^{2} + 720 \, a b^{7} d^{3} n\right )} x^{7} - {\left (3 \, b^{8} c d^{2} n^{7} + 20160 \, b^{8} c d^{2} + {\left (90 \, b^{8} c d^{2} - 7 \, a^{2} b^{6} d^{3}\right )} n^{6} + 3 \, {\left (366 \, b^{8} c d^{2} - 35 \, a^{2} b^{6} d^{3}\right )} n^{5} + 5 \, {\left (1404 \, b^{8} c d^{2} - 119 \, a^{2} b^{6} d^{3}\right )} n^{4} + 9 \, {\left (2803 \, b^{8} c d^{2} - 175 \, a^{2} b^{6} d^{3}\right )} n^{3} + 2 \, {\left (25245 \, b^{8} c d^{2} - 959 \, a^{2} b^{6} d^{3}\right )} n^{2} + 24 \, {\left (2143 \, b^{8} c d^{2} - 35 \, a^{2} b^{6} d^{3}\right )} n\right )} x^{6} - 3 \, {\left (a b^{7} c d^{2} n^{7} + 25 \, a b^{7} c d^{2} n^{6} + {\left (241 \, a b^{7} c d^{2} + 14 \, a^{3} b^{5} d^{3}\right )} n^{5} + 5 \, {\left (227 \, a b^{7} c d^{2} + 28 \, a^{3} b^{5} d^{3}\right )} n^{4} + 2 \, {\left (1367 \, a b^{7} c d^{2} + 245 \, a^{3} b^{5} d^{3}\right )} n^{3} + 20 \, {\left (158 \, a b^{7} c d^{2} + 35 \, a^{3} b^{5} d^{3}\right )} n^{2} + 336 \, {\left (4 \, a b^{7} c d^{2} + a^{3} b^{5} d^{3}\right )} n\right )} x^{5} + {\left (445 \, a^{2} b^{6} c^{3} + 18 \, a^{4} b^{4} c^{2} d\right )} n^{4} - 3 \, {\left (b^{8} c^{2} d n^{7} + 10080 \, b^{8} c^{2} d + {\left (32 \, b^{8} c^{2} d - 5 \, a^{2} b^{6} c d^{2}\right )} n^{6} + {\left (418 \, b^{8} c^{2} d - 105 \, a^{2} b^{6} c d^{2}\right )} n^{5} + {\left (2864 \, b^{8} c^{2} d - 785 \, a^{2} b^{6} c d^{2} - 70 \, a^{4} b^{4} d^{3}\right )} n^{4} + {\left (10993 \, b^{8} c^{2} d - 2535 \, a^{2} b^{6} c d^{2} - 420 \, a^{4} b^{4} d^{3}\right )} n^{3} + 2 \, {\left (11656 \, b^{8} c^{2} d - 1765 \, a^{2} b^{6} c d^{2} - 385 \, a^{4} b^{4} d^{3}\right )} n^{2} + 12 \, {\left (2073 \, b^{8} c^{2} d - 140 \, a^{2} b^{6} c d^{2} - 35 \, a^{4} b^{4} d^{3}\right )} n\right )} x^{4} + 3 \, {\left (1045 \, a^{2} b^{6} c^{3} + 156 \, a^{4} b^{4} c^{2} d\right )} n^{3} - 3 \, {\left (a b^{7} c^{2} d n^{7} + 29 \, a b^{7} c^{2} d n^{6} + {\left (331 \, a b^{7} c^{2} d + 20 \, a^{3} b^{5} c d^{2}\right )} n^{5} + {\left (1871 \, a b^{7} c^{2} d + 360 \, a^{3} b^{5} c d^{2}\right )} n^{4} + 20 \, {\left (269 \, a b^{7} c^{2} d + 103 \, a^{3} b^{5} c d^{2} + 14 \, a^{5} b^{3} d^{3}\right )} n^{3} + 4 \, {\left (1793 \, a b^{7} c^{2} d + 990 \, a^{3} b^{5} c d^{2} + 210 \, a^{5} b^{3} d^{3}\right )} n^{2} + 560 \, {\left (6 \, a b^{7} c^{2} d + 4 \, a^{3} b^{5} c d^{2} + a^{5} b^{3} d^{3}\right )} n\right )} x^{3} + 2 \, {\left (6077 \, a^{2} b^{6} c^{3} + 2259 \, a^{4} b^{4} c^{2} d + 180 \, a^{6} b^{2} c d^{2}\right )} n^{2} - {\left (b^{8} c^{3} n^{7} + 20160 \, b^{8} c^{3} + {\left (34 \, b^{8} c^{3} - 9 \, a^{2} b^{6} c^{2} d\right )} n^{6} + {\left (478 \, b^{8} c^{3} - 243 \, a^{2} b^{6} c^{2} d\right )} n^{5} + {\left (3580 \, b^{8} c^{3} - 2493 \, a^{2} b^{6} c^{2} d - 180 \, a^{4} b^{4} c d^{2}\right )} n^{4} + {\left (15289 \, b^{8} c^{3} - 11853 \, a^{2} b^{6} c^{2} d - 2880 \, a^{4} b^{4} c d^{2}\right )} n^{3} + 2 \, {\left (18353 \, b^{8} c^{3} - 12357 \, a^{2} b^{6} c^{2} d - 6390 \, a^{4} b^{4} c d^{2} - 1260 \, a^{6} b^{2} d^{3}\right )} n^{2} + 72 \, {\left (621 \, b^{8} c^{3} - 210 \, a^{2} b^{6} c^{2} d - 140 \, a^{4} b^{4} c d^{2} - 35 \, a^{6} b^{2} d^{3}\right )} n\right )} x^{2} + 36 \, {\left (682 \, a^{2} b^{6} c^{3} + 533 \, a^{4} b^{4} c^{2} d + 150 \, a^{6} b^{2} c d^{2}\right )} n - {\left (a b^{7} c^{3} n^{7} + 33 \, a b^{7} c^{3} n^{6} + {\left (445 \, a b^{7} c^{3} + 18 \, a^{3} b^{5} c^{2} d\right )} n^{5} + 3 \, {\left (1045 \, a b^{7} c^{3} + 156 \, a^{3} b^{5} c^{2} d\right )} n^{4} + 2 \, {\left (6077 \, a b^{7} c^{3} + 2259 \, a^{3} b^{5} c^{2} d + 180 \, a^{5} b^{3} c d^{2}\right )} n^{3} + 36 \, {\left (682 \, a b^{7} c^{3} + 533 \, a^{3} b^{5} c^{2} d + 150 \, a^{5} b^{3} c d^{2}\right )} n^{2} + 5040 \, {\left (4 \, a b^{7} c^{3} + 6 \, a^{3} b^{5} c^{2} d + 4 \, a^{5} b^{3} c d^{2} + a^{7} b d^{3}\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{8} n^{8} + 36 \, b^{8} n^{7} + 546 \, b^{8} n^{6} + 4536 \, b^{8} n^{5} + 22449 \, b^{8} n^{4} + 67284 \, b^{8} n^{3} + 118124 \, b^{8} n^{2} + 109584 \, b^{8} n + 40320 \, b^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^n*(d*x^2+c)^3,x, algorithm="fricas")

[Out]

-(a^2*b^6*c^3*n^6 + 33*a^2*b^6*c^3*n^5 + 20160*a^2*b^6*c^3 + 30240*a^4*b^4*c^2*d + 20160*a^6*b^2*c*d^2 + 5040*
a^8*d^3 - (b^8*d^3*n^7 + 28*b^8*d^3*n^6 + 322*b^8*d^3*n^5 + 1960*b^8*d^3*n^4 + 6769*b^8*d^3*n^3 + 13132*b^8*d^
3*n^2 + 13068*b^8*d^3*n + 5040*b^8*d^3)*x^8 - (a*b^7*d^3*n^7 + 21*a*b^7*d^3*n^6 + 175*a*b^7*d^3*n^5 + 735*a*b^
7*d^3*n^4 + 1624*a*b^7*d^3*n^3 + 1764*a*b^7*d^3*n^2 + 720*a*b^7*d^3*n)*x^7 - (3*b^8*c*d^2*n^7 + 20160*b^8*c*d^
2 + (90*b^8*c*d^2 - 7*a^2*b^6*d^3)*n^6 + 3*(366*b^8*c*d^2 - 35*a^2*b^6*d^3)*n^5 + 5*(1404*b^8*c*d^2 - 119*a^2*
b^6*d^3)*n^4 + 9*(2803*b^8*c*d^2 - 175*a^2*b^6*d^3)*n^3 + 2*(25245*b^8*c*d^2 - 959*a^2*b^6*d^3)*n^2 + 24*(2143
*b^8*c*d^2 - 35*a^2*b^6*d^3)*n)*x^6 - 3*(a*b^7*c*d^2*n^7 + 25*a*b^7*c*d^2*n^6 + (241*a*b^7*c*d^2 + 14*a^3*b^5*
d^3)*n^5 + 5*(227*a*b^7*c*d^2 + 28*a^3*b^5*d^3)*n^4 + 2*(1367*a*b^7*c*d^2 + 245*a^3*b^5*d^3)*n^3 + 20*(158*a*b
^7*c*d^2 + 35*a^3*b^5*d^3)*n^2 + 336*(4*a*b^7*c*d^2 + a^3*b^5*d^3)*n)*x^5 + (445*a^2*b^6*c^3 + 18*a^4*b^4*c^2*
d)*n^4 - 3*(b^8*c^2*d*n^7 + 10080*b^8*c^2*d + (32*b^8*c^2*d - 5*a^2*b^6*c*d^2)*n^6 + (418*b^8*c^2*d - 105*a^2*
b^6*c*d^2)*n^5 + (2864*b^8*c^2*d - 785*a^2*b^6*c*d^2 - 70*a^4*b^4*d^3)*n^4 + (10993*b^8*c^2*d - 2535*a^2*b^6*c
*d^2 - 420*a^4*b^4*d^3)*n^3 + 2*(11656*b^8*c^2*d - 1765*a^2*b^6*c*d^2 - 385*a^4*b^4*d^3)*n^2 + 12*(2073*b^8*c^
2*d - 140*a^2*b^6*c*d^2 - 35*a^4*b^4*d^3)*n)*x^4 + 3*(1045*a^2*b^6*c^3 + 156*a^4*b^4*c^2*d)*n^3 - 3*(a*b^7*c^2
*d*n^7 + 29*a*b^7*c^2*d*n^6 + (331*a*b^7*c^2*d + 20*a^3*b^5*c*d^2)*n^5 + (1871*a*b^7*c^2*d + 360*a^3*b^5*c*d^2
)*n^4 + 20*(269*a*b^7*c^2*d + 103*a^3*b^5*c*d^2 + 14*a^5*b^3*d^3)*n^3 + 4*(1793*a*b^7*c^2*d + 990*a^3*b^5*c*d^
2 + 210*a^5*b^3*d^3)*n^2 + 560*(6*a*b^7*c^2*d + 4*a^3*b^5*c*d^2 + a^5*b^3*d^3)*n)*x^3 + 2*(6077*a^2*b^6*c^3 +
2259*a^4*b^4*c^2*d + 180*a^6*b^2*c*d^2)*n^2 - (b^8*c^3*n^7 + 20160*b^8*c^3 + (34*b^8*c^3 - 9*a^2*b^6*c^2*d)*n^
6 + (478*b^8*c^3 - 243*a^2*b^6*c^2*d)*n^5 + (3580*b^8*c^3 - 2493*a^2*b^6*c^2*d - 180*a^4*b^4*c*d^2)*n^4 + (152
89*b^8*c^3 - 11853*a^2*b^6*c^2*d - 2880*a^4*b^4*c*d^2)*n^3 + 2*(18353*b^8*c^3 - 12357*a^2*b^6*c^2*d - 6390*a^4
*b^4*c*d^2 - 1260*a^6*b^2*d^3)*n^2 + 72*(621*b^8*c^3 - 210*a^2*b^6*c^2*d - 140*a^4*b^4*c*d^2 - 35*a^6*b^2*d^3)
*n)*x^2 + 36*(682*a^2*b^6*c^3 + 533*a^4*b^4*c^2*d + 150*a^6*b^2*c*d^2)*n - (a*b^7*c^3*n^7 + 33*a*b^7*c^3*n^6 +
(445*a*b^7*c^3 + 18*a^3*b^5*c^2*d)*n^5 + 3*(1045*a*b^7*c^3 + 156*a^3*b^5*c^2*d)*n^4 + 2*(6077*a*b^7*c^3 + 225
9*a^3*b^5*c^2*d + 180*a^5*b^3*c*d^2)*n^3 + 36*(682*a*b^7*c^3 + 533*a^3*b^5*c^2*d + 150*a^5*b^3*c*d^2)*n^2 + 50
40*(4*a*b^7*c^3 + 6*a^3*b^5*c^2*d + 4*a^5*b^3*c*d^2 + a^7*b*d^3)*n)*x)*(b*x + a)^n/(b^8*n^8 + 36*b^8*n^7 + 546
*b^8*n^6 + 4536*b^8*n^5 + 22449*b^8*n^4 + 67284*b^8*n^3 + 118124*b^8*n^2 + 109584*b^8*n + 40320*b^8)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 24687 vs. \(2 (265) = 530\).
time = 7.11, size = 24687, normalized size = 87.54 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)**n*(d*x**2+c)**3,x)

[Out]

Piecewise((a**n*(c**3*x**2/2 + 3*c**2*d*x**4/4 + c*d**2*x**6/2 + d**3*x**8/8), Eq(b, 0)), (420*a**7*d**3*log(a
/b + x)/(420*a**7*b**8 + 2940*a**6*b**9*x + 8820*a**5*b**10*x**2 + 14700*a**4*b**11*x**3 + 14700*a**3*b**12*x*
*4 + 8820*a**2*b**13*x**5 + 2940*a*b**14*x**6 + 420*b**15*x**7) + 1089*a**7*d**3/(420*a**7*b**8 + 2940*a**6*b*
*9*x + 8820*a**5*b**10*x**2 + 14700*a**4*b**11*x**3 + 14700*a**3*b**12*x**4 + 8820*a**2*b**13*x**5 + 2940*a*b*
*14*x**6 + 420*b**15*x**7) + 2940*a**6*b*d**3*x*log(a/b + x)/(420*a**7*b**8 + 2940*a**6*b**9*x + 8820*a**5*b**
10*x**2 + 14700*a**4*b**11*x**3 + 14700*a**3*b**12*x**4 + 8820*a**2*b**13*x**5 + 2940*a*b**14*x**6 + 420*b**15
*x**7) + 7203*a**6*b*d**3*x/(420*a**7*b**8 + 2940*a**6*b**9*x + 8820*a**5*b**10*x**2 + 14700*a**4*b**11*x**3 +
14700*a**3*b**12*x**4 + 8820*a**2*b**13*x**5 + 2940*a*b**14*x**6 + 420*b**15*x**7) - 30*a**5*b**2*c*d**2/(420
*a**7*b**8 + 2940*a**6*b**9*x + 8820*a**5*b**10*x**2 + 14700*a**4*b**11*x**3 + 14700*a**3*b**12*x**4 + 8820*a*
*2*b**13*x**5 + 2940*a*b**14*x**6 + 420*b**15*x**7) + 8820*a**5*b**2*d**3*x**2*log(a/b + x)/(420*a**7*b**8 + 2
940*a**6*b**9*x + 8820*a**5*b**10*x**2 + 14700*a**4*b**11*x**3 + 14700*a**3*b**12*x**4 + 8820*a**2*b**13*x**5
+ 2940*a*b**14*x**6 + 420*b**15*x**7) + 20139*a**5*b**2*d**3*x**2/(420*a**7*b**8 + 2940*a**6*b**9*x + 8820*a**
5*b**10*x**2 + 14700*a**4*b**11*x**3 + 14700*a**3*b**12*x**4 + 8820*a**2*b**13*x**5 + 2940*a*b**14*x**6 + 420*
b**15*x**7) - 210*a**4*b**3*c*d**2*x/(420*a**7*b**8 + 2940*a**6*b**9*x + 8820*a**5*b**10*x**2 + 14700*a**4*b**
11*x**3 + 14700*a**3*b**12*x**4 + 8820*a**2*b**13*x**5 + 2940*a*b**14*x**6 + 420*b**15*x**7) + 14700*a**4*b**3
*d**3*x**3*log(a/b + x)/(420*a**7*b**8 + 2940*a**6*b**9*x + 8820*a**5*b**10*x**2 + 14700*a**4*b**11*x**3 + 147
00*a**3*b**12*x**4 + 8820*a**2*b**13*x**5 + 2940*a*b**14*x**6 + 420*b**15*x**7) + 30625*a**4*b**3*d**3*x**3/(4
20*a**7*b**8 + 2940*a**6*b**9*x + 8820*a**5*b**10*x**2 + 14700*a**4*b**11*x**3 + 14700*a**3*b**12*x**4 + 8820*
a**2*b**13*x**5 + 2940*a*b**14*x**6 + 420*b**15*x**7) - 9*a**3*b**4*c**2*d/(420*a**7*b**8 + 2940*a**6*b**9*x +
8820*a**5*b**10*x**2 + 14700*a**4*b**11*x**3 + 14700*a**3*b**12*x**4 + 8820*a**2*b**13*x**5 + 2940*a*b**14*x*
*6 + 420*b**15*x**7) - 630*a**3*b**4*c*d**2*x**2/(420*a**7*b**8 + 2940*a**6*b**9*x + 8820*a**5*b**10*x**2 + 14
700*a**4*b**11*x**3 + 14700*a**3*b**12*x**4 + 8820*a**2*b**13*x**5 + 2940*a*b**14*x**6 + 420*b**15*x**7) + 147
00*a**3*b**4*d**3*x**4*log(a/b + x)/(420*a**7*b**8 + 2940*a**6*b**9*x + 8820*a**5*b**10*x**2 + 14700*a**4*b**1
1*x**3 + 14700*a**3*b**12*x**4 + 8820*a**2*b**13*x**5 + 2940*a*b**14*x**6 + 420*b**15*x**7) + 26950*a**3*b**4*
d**3*x**4/(420*a**7*b**8 + 2940*a**6*b**9*x + 8820*a**5*b**10*x**2 + 14700*a**4*b**11*x**3 + 14700*a**3*b**12*
x**4 + 8820*a**2*b**13*x**5 + 2940*a*b**14*x**6 + 420*b**15*x**7) - 63*a**2*b**5*c**2*d*x/(420*a**7*b**8 + 294
0*a**6*b**9*x + 8820*a**5*b**10*x**2 + 14700*a**4*b**11*x**3 + 14700*a**3*b**12*x**4 + 8820*a**2*b**13*x**5 +
2940*a*b**14*x**6 + 420*b**15*x**7) - 1050*a**2*b**5*c*d**2*x**3/(420*a**7*b**8 + 2940*a**6*b**9*x + 8820*a**5
*b**10*x**2 + 14700*a**4*b**11*x**3 + 14700*a**3*b**12*x**4 + 8820*a**2*b**13*x**5 + 2940*a*b**14*x**6 + 420*b
**15*x**7) + 8820*a**2*b**5*d**3*x**5*log(a/b + x)/(420*a**7*b**8 + 2940*a**6*b**9*x + 8820*a**5*b**10*x**2 +
14700*a**4*b**11*x**3 + 14700*a**3*b**12*x**4 + 8820*a**2*b**13*x**5 + 2940*a*b**14*x**6 + 420*b**15*x**7) + 1
3230*a**2*b**5*d**3*x**5/(420*a**7*b**8 + 2940*a**6*b**9*x + 8820*a**5*b**10*x**2 + 14700*a**4*b**11*x**3 + 14
700*a**3*b**12*x**4 + 8820*a**2*b**13*x**5 + 2940*a*b**14*x**6 + 420*b**15*x**7) - 10*a*b**6*c**3/(420*a**7*b*
*8 + 2940*a**6*b**9*x + 8820*a**5*b**10*x**2 + 14700*a**4*b**11*x**3 + 14700*a**3*b**12*x**4 + 8820*a**2*b**13
*x**5 + 2940*a*b**14*x**6 + 420*b**15*x**7) - 189*a*b**6*c**2*d*x**2/(420*a**7*b**8 + 2940*a**6*b**9*x + 8820*
a**5*b**10*x**2 + 14700*a**4*b**11*x**3 + 14700*a**3*b**12*x**4 + 8820*a**2*b**13*x**5 + 2940*a*b**14*x**6 + 4
20*b**15*x**7) - 1050*a*b**6*c*d**2*x**4/(420*a**7*b**8 + 2940*a**6*b**9*x + 8820*a**5*b**10*x**2 + 14700*a**4
*b**11*x**3 + 14700*a**3*b**12*x**4 + 8820*a**2*b**13*x**5 + 2940*a*b**14*x**6 + 420*b**15*x**7) + 2940*a*b**6
*d**3*x**6*log(a/b + x)/(420*a**7*b**8 + 2940*a**6*b**9*x + 8820*a**5*b**10*x**2 + 14700*a**4*b**11*x**3 + 147
00*a**3*b**12*x**4 + 8820*a**2*b**13*x**5 + 2940*a*b**14*x**6 + 420*b**15*x**7) + 2940*a*b**6*d**3*x**6/(420*a
**7*b**8 + 2940*a**6*b**9*x + 8820*a**5*b**10*x**2 + 14700*a**4*b**11*x**3 + 14700*a**3*b**12*x**4 + 8820*a**2
*b**13*x**5 + 2940*a*b**14*x**6 + 420*b**15*x**7) - 70*b**7*c**3*x/(420*a**7*b**8 + 2940*a**6*b**9*x + 8820*a*
*5*b**10*x**2 + 14700*a**4*b**11*x**3 + 14700*a**3*b**12*x**4 + 8820*a**2*b**13*x**5 + 2940*a*b**14*x**6 + 420
*b**15*x**7) - 315*b**7*c**2*d*x**3/(420*a**7*b**8 + 2940*a**6*b**9*x + 8820*a**5*b**10*x**2 + 14700*a**4*b**1
1*x**3 + 14700*a**3*b**12*x**4 + 8820*a**2*b**13*x**5 + 2940*a*b**14*x**6 + 420*b**15*x**7) - 630*b**7*c*d**2*
x**5/(420*a**7*b**8 + 2940*a**6*b**9*x + 8820*a...

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2851 vs. \(2 (282) = 564\).
time = 1.60, size = 2851, normalized size = 10.11 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^n*(d*x^2+c)^3,x, algorithm="giac")

[Out]

((b*x + a)^n*b^8*d^3*n^7*x^8 + (b*x + a)^n*a*b^7*d^3*n^7*x^7 + 28*(b*x + a)^n*b^8*d^3*n^6*x^8 + 3*(b*x + a)^n*
b^8*c*d^2*n^7*x^6 + 21*(b*x + a)^n*a*b^7*d^3*n^6*x^7 + 322*(b*x + a)^n*b^8*d^3*n^5*x^8 + 3*(b*x + a)^n*a*b^7*c
*d^2*n^7*x^5 + 90*(b*x + a)^n*b^8*c*d^2*n^6*x^6 - 7*(b*x + a)^n*a^2*b^6*d^3*n^6*x^6 + 175*(b*x + a)^n*a*b^7*d^
3*n^5*x^7 + 1960*(b*x + a)^n*b^8*d^3*n^4*x^8 + 3*(b*x + a)^n*b^8*c^2*d*n^7*x^4 + 75*(b*x + a)^n*a*b^7*c*d^2*n^
6*x^5 + 1098*(b*x + a)^n*b^8*c*d^2*n^5*x^6 - 105*(b*x + a)^n*a^2*b^6*d^3*n^5*x^6 + 735*(b*x + a)^n*a*b^7*d^3*n
^4*x^7 + 6769*(b*x + a)^n*b^8*d^3*n^3*x^8 + 3*(b*x + a)^n*a*b^7*c^2*d*n^7*x^3 + 96*(b*x + a)^n*b^8*c^2*d*n^6*x
^4 - 15*(b*x + a)^n*a^2*b^6*c*d^2*n^6*x^4 + 723*(b*x + a)^n*a*b^7*c*d^2*n^5*x^5 + 42*(b*x + a)^n*a^3*b^5*d^3*n
^5*x^5 + 7020*(b*x + a)^n*b^8*c*d^2*n^4*x^6 - 595*(b*x + a)^n*a^2*b^6*d^3*n^4*x^6 + 1624*(b*x + a)^n*a*b^7*d^3
*n^3*x^7 + 13132*(b*x + a)^n*b^8*d^3*n^2*x^8 + (b*x + a)^n*b^8*c^3*n^7*x^2 + 87*(b*x + a)^n*a*b^7*c^2*d*n^6*x^
3 + 1254*(b*x + a)^n*b^8*c^2*d*n^5*x^4 - 315*(b*x + a)^n*a^2*b^6*c*d^2*n^5*x^4 + 3405*(b*x + a)^n*a*b^7*c*d^2*
n^4*x^5 + 420*(b*x + a)^n*a^3*b^5*d^3*n^4*x^5 + 25227*(b*x + a)^n*b^8*c*d^2*n^3*x^6 - 1575*(b*x + a)^n*a^2*b^6
*d^3*n^3*x^6 + 1764*(b*x + a)^n*a*b^7*d^3*n^2*x^7 + 13068*(b*x + a)^n*b^8*d^3*n*x^8 + (b*x + a)^n*a*b^7*c^3*n^
7*x + 34*(b*x + a)^n*b^8*c^3*n^6*x^2 - 9*(b*x + a)^n*a^2*b^6*c^2*d*n^6*x^2 + 993*(b*x + a)^n*a*b^7*c^2*d*n^5*x
^3 + 60*(b*x + a)^n*a^3*b^5*c*d^2*n^5*x^3 + 8592*(b*x + a)^n*b^8*c^2*d*n^4*x^4 - 2355*(b*x + a)^n*a^2*b^6*c*d^
2*n^4*x^4 - 210*(b*x + a)^n*a^4*b^4*d^3*n^4*x^4 + 8202*(b*x + a)^n*a*b^7*c*d^2*n^3*x^5 + 1470*(b*x + a)^n*a^3*
b^5*d^3*n^3*x^5 + 50490*(b*x + a)^n*b^8*c*d^2*n^2*x^6 - 1918*(b*x + a)^n*a^2*b^6*d^3*n^2*x^6 + 720*(b*x + a)^n
*a*b^7*d^3*n*x^7 + 5040*(b*x + a)^n*b^8*d^3*x^8 + 33*(b*x + a)^n*a*b^7*c^3*n^6*x + 478*(b*x + a)^n*b^8*c^3*n^5
*x^2 - 243*(b*x + a)^n*a^2*b^6*c^2*d*n^5*x^2 + 5613*(b*x + a)^n*a*b^7*c^2*d*n^4*x^3 + 1080*(b*x + a)^n*a^3*b^5
*c*d^2*n^4*x^3 + 32979*(b*x + a)^n*b^8*c^2*d*n^3*x^4 - 7605*(b*x + a)^n*a^2*b^6*c*d^2*n^3*x^4 - 1260*(b*x + a)
^n*a^4*b^4*d^3*n^3*x^4 + 9480*(b*x + a)^n*a*b^7*c*d^2*n^2*x^5 + 2100*(b*x + a)^n*a^3*b^5*d^3*n^2*x^5 + 51432*(
b*x + a)^n*b^8*c*d^2*n*x^6 - 840*(b*x + a)^n*a^2*b^6*d^3*n*x^6 - (b*x + a)^n*a^2*b^6*c^3*n^6 + 445*(b*x + a)^n
*a*b^7*c^3*n^5*x + 18*(b*x + a)^n*a^3*b^5*c^2*d*n^5*x + 3580*(b*x + a)^n*b^8*c^3*n^4*x^2 - 2493*(b*x + a)^n*a^
2*b^6*c^2*d*n^4*x^2 - 180*(b*x + a)^n*a^4*b^4*c*d^2*n^4*x^2 + 16140*(b*x + a)^n*a*b^7*c^2*d*n^3*x^3 + 6180*(b*
x + a)^n*a^3*b^5*c*d^2*n^3*x^3 + 840*(b*x + a)^n*a^5*b^3*d^3*n^3*x^3 + 69936*(b*x + a)^n*b^8*c^2*d*n^2*x^4 - 1
0590*(b*x + a)^n*a^2*b^6*c*d^2*n^2*x^4 - 2310*(b*x + a)^n*a^4*b^4*d^3*n^2*x^4 + 4032*(b*x + a)^n*a*b^7*c*d^2*n
*x^5 + 1008*(b*x + a)^n*a^3*b^5*d^3*n*x^5 + 20160*(b*x + a)^n*b^8*c*d^2*x^6 - 33*(b*x + a)^n*a^2*b^6*c^3*n^5 +
3135*(b*x + a)^n*a*b^7*c^3*n^4*x + 468*(b*x + a)^n*a^3*b^5*c^2*d*n^4*x + 15289*(b*x + a)^n*b^8*c^3*n^3*x^2 -
11853*(b*x + a)^n*a^2*b^6*c^2*d*n^3*x^2 - 2880*(b*x + a)^n*a^4*b^4*c*d^2*n^3*x^2 + 21516*(b*x + a)^n*a*b^7*c^2
*d*n^2*x^3 + 11880*(b*x + a)^n*a^3*b^5*c*d^2*n^2*x^3 + 2520*(b*x + a)^n*a^5*b^3*d^3*n^2*x^3 + 74628*(b*x + a)^
n*b^8*c^2*d*n*x^4 - 5040*(b*x + a)^n*a^2*b^6*c*d^2*n*x^4 - 1260*(b*x + a)^n*a^4*b^4*d^3*n*x^4 - 445*(b*x + a)^
n*a^2*b^6*c^3*n^4 - 18*(b*x + a)^n*a^4*b^4*c^2*d*n^4 + 12154*(b*x + a)^n*a*b^7*c^3*n^3*x + 4518*(b*x + a)^n*a^
3*b^5*c^2*d*n^3*x + 360*(b*x + a)^n*a^5*b^3*c*d^2*n^3*x + 36706*(b*x + a)^n*b^8*c^3*n^2*x^2 - 24714*(b*x + a)^
n*a^2*b^6*c^2*d*n^2*x^2 - 12780*(b*x + a)^n*a^4*b^4*c*d^2*n^2*x^2 - 2520*(b*x + a)^n*a^6*b^2*d^3*n^2*x^2 + 100
80*(b*x + a)^n*a*b^7*c^2*d*n*x^3 + 6720*(b*x + a)^n*a^3*b^5*c*d^2*n*x^3 + 1680*(b*x + a)^n*a^5*b^3*d^3*n*x^3 +
30240*(b*x + a)^n*b^8*c^2*d*x^4 - 3135*(b*x + a)^n*a^2*b^6*c^3*n^3 - 468*(b*x + a)^n*a^4*b^4*c^2*d*n^3 + 2455
2*(b*x + a)^n*a*b^7*c^3*n^2*x + 19188*(b*x + a)^n*a^3*b^5*c^2*d*n^2*x + 5400*(b*x + a)^n*a^5*b^3*c*d^2*n^2*x +
44712*(b*x + a)^n*b^8*c^3*n*x^2 - 15120*(b*x + a)^n*a^2*b^6*c^2*d*n*x^2 - 10080*(b*x + a)^n*a^4*b^4*c*d^2*n*x
^2 - 2520*(b*x + a)^n*a^6*b^2*d^3*n*x^2 - 12154*(b*x + a)^n*a^2*b^6*c^3*n^2 - 4518*(b*x + a)^n*a^4*b^4*c^2*d*n
^2 - 360*(b*x + a)^n*a^6*b^2*c*d^2*n^2 + 20160*(b*x + a)^n*a*b^7*c^3*n*x + 30240*(b*x + a)^n*a^3*b^5*c^2*d*n*x
+ 20160*(b*x + a)^n*a^5*b^3*c*d^2*n*x + 5040*(b*x + a)^n*a^7*b*d^3*n*x + 20160*(b*x + a)^n*b^8*c^3*x^2 - 2455
2*(b*x + a)^n*a^2*b^6*c^3*n - 19188*(b*x + a)^n*a^4*b^4*c^2*d*n - 5400*(b*x + a)^n*a^6*b^2*c*d^2*n - 20160*(b*
x + a)^n*a^2*b^6*c^3 - 30240*(b*x + a)^n*a^4*b^4*c^2*d - 20160*(b*x + a)^n*a^6*b^2*c*d^2 - 5040*(b*x + a)^n*a^
8*d^3)/(b^8*n^8 + 36*b^8*n^7 + 546*b^8*n^6 + 4536*b^8*n^5 + 22449*b^8*n^4 + 67284*b^8*n^3 + 118124*b^8*n^2 + 1
09584*b^8*n + 40320*b^8)

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Mupad [B]
time = 3.48, size = 1459, normalized size = 5.17 \begin {gather*} \frac {d^3\,x^8\,{\left (a+b\,x\right )}^n\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}{n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320}-\frac {a^2\,{\left (a+b\,x\right )}^n\,\left (5040\,a^6\,d^3+360\,a^4\,b^2\,c\,d^2\,n^2+5400\,a^4\,b^2\,c\,d^2\,n+20160\,a^4\,b^2\,c\,d^2+18\,a^2\,b^4\,c^2\,d\,n^4+468\,a^2\,b^4\,c^2\,d\,n^3+4518\,a^2\,b^4\,c^2\,d\,n^2+19188\,a^2\,b^4\,c^2\,d\,n+30240\,a^2\,b^4\,c^2\,d+b^6\,c^3\,n^6+33\,b^6\,c^3\,n^5+445\,b^6\,c^3\,n^4+3135\,b^6\,c^3\,n^3+12154\,b^6\,c^3\,n^2+24552\,b^6\,c^3\,n+20160\,b^6\,c^3\right )}{b^8\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )}-\frac {x^2\,\left (n+1\right )\,{\left (a+b\,x\right )}^n\,\left (2520\,a^6\,d^3\,n+180\,a^4\,b^2\,c\,d^2\,n^3+2700\,a^4\,b^2\,c\,d^2\,n^2+10080\,a^4\,b^2\,c\,d^2\,n+9\,a^2\,b^4\,c^2\,d\,n^5+234\,a^2\,b^4\,c^2\,d\,n^4+2259\,a^2\,b^4\,c^2\,d\,n^3+9594\,a^2\,b^4\,c^2\,d\,n^2+15120\,a^2\,b^4\,c^2\,d\,n-b^6\,c^3\,n^6-33\,b^6\,c^3\,n^5-445\,b^6\,c^3\,n^4-3135\,b^6\,c^3\,n^3-12154\,b^6\,c^3\,n^2-24552\,b^6\,c^3\,n-20160\,b^6\,c^3\right )}{b^6\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )}+\frac {d^2\,x^6\,{\left (a+b\,x\right )}^n\,\left (-7\,d\,a^2\,n+3\,c\,b^2\,n^2+45\,c\,b^2\,n+168\,c\,b^2\right )\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}{b^2\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )}+\frac {3\,d\,x^4\,{\left (a+b\,x\right )}^n\,\left (n^3+6\,n^2+11\,n+6\right )\,\left (-70\,a^4\,d^2\,n-5\,a^2\,b^2\,c\,d\,n^3-75\,a^2\,b^2\,c\,d\,n^2-280\,a^2\,b^2\,c\,d\,n+b^4\,c^2\,n^4+26\,b^4\,c^2\,n^3+251\,b^4\,c^2\,n^2+1066\,b^4\,c^2\,n+1680\,b^4\,c^2\right )}{b^4\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )}+\frac {a\,n\,x\,{\left (a+b\,x\right )}^n\,\left (5040\,a^6\,d^3+360\,a^4\,b^2\,c\,d^2\,n^2+5400\,a^4\,b^2\,c\,d^2\,n+20160\,a^4\,b^2\,c\,d^2+18\,a^2\,b^4\,c^2\,d\,n^4+468\,a^2\,b^4\,c^2\,d\,n^3+4518\,a^2\,b^4\,c^2\,d\,n^2+19188\,a^2\,b^4\,c^2\,d\,n+30240\,a^2\,b^4\,c^2\,d+b^6\,c^3\,n^6+33\,b^6\,c^3\,n^5+445\,b^6\,c^3\,n^4+3135\,b^6\,c^3\,n^3+12154\,b^6\,c^3\,n^2+24552\,b^6\,c^3\,n+20160\,b^6\,c^3\right )}{b^7\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )}+\frac {a\,d^3\,n\,x^7\,{\left (a+b\,x\right )}^n\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}{b\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )}+\frac {3\,a\,d\,n\,x^3\,{\left (a+b\,x\right )}^n\,\left (n^2+3\,n+2\right )\,\left (280\,a^4\,d^2+20\,a^2\,b^2\,c\,d\,n^2+300\,a^2\,b^2\,c\,d\,n+1120\,a^2\,b^2\,c\,d+b^4\,c^2\,n^4+26\,b^4\,c^2\,n^3+251\,b^4\,c^2\,n^2+1066\,b^4\,c^2\,n+1680\,b^4\,c^2\right )}{b^5\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )}+\frac {3\,a\,d^2\,n\,x^5\,{\left (a+b\,x\right )}^n\,\left (14\,d\,a^2+c\,b^2\,n^2+15\,c\,b^2\,n+56\,c\,b^2\right )\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}{b^3\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(c + d*x^2)^3*(a + b*x)^n,x)

[Out]

(d^3*x^8*(a + b*x)^n*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040))/(109584*n +
118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320) - (a^2*(a + b*x)^n*(5040*a^6*d
^3 + 20160*b^6*c^3 + 24552*b^6*c^3*n + 12154*b^6*c^3*n^2 + 3135*b^6*c^3*n^3 + 445*b^6*c^3*n^4 + 33*b^6*c^3*n^5
+ b^6*c^3*n^6 + 30240*a^2*b^4*c^2*d + 20160*a^4*b^2*c*d^2 + 19188*a^2*b^4*c^2*d*n + 5400*a^4*b^2*c*d^2*n + 45
18*a^2*b^4*c^2*d*n^2 + 360*a^4*b^2*c*d^2*n^2 + 468*a^2*b^4*c^2*d*n^3 + 18*a^2*b^4*c^2*d*n^4))/(b^8*(109584*n +
118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320)) - (x^2*(n + 1)*(a + b*x)^n*(
2520*a^6*d^3*n - 20160*b^6*c^3 - 24552*b^6*c^3*n - 12154*b^6*c^3*n^2 - 3135*b^6*c^3*n^3 - 445*b^6*c^3*n^4 - 33
*b^6*c^3*n^5 - b^6*c^3*n^6 + 15120*a^2*b^4*c^2*d*n + 10080*a^4*b^2*c*d^2*n + 9594*a^2*b^4*c^2*d*n^2 + 2700*a^4
*b^2*c*d^2*n^2 + 2259*a^2*b^4*c^2*d*n^3 + 180*a^4*b^2*c*d^2*n^3 + 234*a^2*b^4*c^2*d*n^4 + 9*a^2*b^4*c^2*d*n^5)
)/(b^6*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320)) + (d^2*x^6
*(a + b*x)^n*(168*b^2*c + 3*b^2*c*n^2 - 7*a^2*d*n + 45*b^2*c*n)*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120
))/(b^2*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320)) + (3*d*x^
4*(a + b*x)^n*(11*n + 6*n^2 + n^3 + 6)*(1680*b^4*c^2 - 70*a^4*d^2*n + 1066*b^4*c^2*n + 251*b^4*c^2*n^2 + 26*b^
4*c^2*n^3 + b^4*c^2*n^4 - 280*a^2*b^2*c*d*n - 75*a^2*b^2*c*d*n^2 - 5*a^2*b^2*c*d*n^3))/(b^4*(109584*n + 118124
*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320)) + (a*n*x*(a + b*x)^n*(5040*a^6*d^3
+ 20160*b^6*c^3 + 24552*b^6*c^3*n + 12154*b^6*c^3*n^2 + 3135*b^6*c^3*n^3 + 445*b^6*c^3*n^4 + 33*b^6*c^3*n^5 +
b^6*c^3*n^6 + 30240*a^2*b^4*c^2*d + 20160*a^4*b^2*c*d^2 + 19188*a^2*b^4*c^2*d*n + 5400*a^4*b^2*c*d^2*n + 4518*
a^2*b^4*c^2*d*n^2 + 360*a^4*b^2*c*d^2*n^2 + 468*a^2*b^4*c^2*d*n^3 + 18*a^2*b^4*c^2*d*n^4))/(b^7*(109584*n + 11
8124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320)) + (a*d^3*n*x^7*(a + b*x)^n*(176
4*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720))/(b*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 +
4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320)) + (3*a*d*n*x^3*(a + b*x)^n*(3*n + n^2 + 2)*(280*a^4*d^2 + 1680*b^4
*c^2 + 1066*b^4*c^2*n + 251*b^4*c^2*n^2 + 26*b^4*c^2*n^3 + b^4*c^2*n^4 + 1120*a^2*b^2*c*d + 300*a^2*b^2*c*d*n
+ 20*a^2*b^2*c*d*n^2))/(b^5*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8
+ 40320)) + (3*a*d^2*n*x^5*(a + b*x)^n*(14*a^2*d + 56*b^2*c + b^2*c*n^2 + 15*b^2*c*n)*(50*n + 35*n^2 + 10*n^3
+ n^4 + 24))/(b^3*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320)
)

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