3.2.79 \(\int x (a+b x)^n (c+d x^3)^2 \, dx\) [179]

Optimal. Leaf size=248 \[ -\frac {a \left (b^3 c-a^3 d\right )^2 (a+b x)^{1+n}}{b^8 (1+n)}+\frac {\left (b^3 c-7 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{2+n}}{b^8 (2+n)}+\frac {3 a^2 d \left (4 b^3 c-7 a^3 d\right ) (a+b x)^{3+n}}{b^8 (3+n)}-\frac {a d \left (8 b^3 c-35 a^3 d\right ) (a+b x)^{4+n}}{b^8 (4+n)}+\frac {d \left (2 b^3 c-35 a^3 d\right ) (a+b x)^{5+n}}{b^8 (5+n)}+\frac {21 a^2 d^2 (a+b x)^{6+n}}{b^8 (6+n)}-\frac {7 a d^2 (a+b x)^{7+n}}{b^8 (7+n)}+\frac {d^2 (a+b x)^{8+n}}{b^8 (8+n)} \]

[Out]

-a*(-a^3*d+b^3*c)^2*(b*x+a)^(1+n)/b^8/(1+n)+(-7*a^3*d+b^3*c)*(-a^3*d+b^3*c)*(b*x+a)^(2+n)/b^8/(2+n)+3*a^2*d*(-
7*a^3*d+4*b^3*c)*(b*x+a)^(3+n)/b^8/(3+n)-a*d*(-35*a^3*d+8*b^3*c)*(b*x+a)^(4+n)/b^8/(4+n)+d*(-35*a^3*d+2*b^3*c)
*(b*x+a)^(5+n)/b^8/(5+n)+21*a^2*d^2*(b*x+a)^(6+n)/b^8/(6+n)-7*a*d^2*(b*x+a)^(7+n)/b^8/(7+n)+d^2*(b*x+a)^(8+n)/
b^8/(8+n)

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Rubi [A]
time = 0.11, antiderivative size = 248, 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 = {1634} \begin {gather*} -\frac {a \left (b^3 c-a^3 d\right )^2 (a+b x)^{n+1}}{b^8 (n+1)}+\frac {\left (b^3 c-7 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{n+2}}{b^8 (n+2)}-\frac {a d \left (8 b^3 c-35 a^3 d\right ) (a+b x)^{n+4}}{b^8 (n+4)}+\frac {d \left (2 b^3 c-35 a^3 d\right ) (a+b x)^{n+5}}{b^8 (n+5)}+\frac {21 a^2 d^2 (a+b x)^{n+6}}{b^8 (n+6)}+\frac {3 a^2 d \left (4 b^3 c-7 a^3 d\right ) (a+b x)^{n+3}}{b^8 (n+3)}-\frac {7 a d^2 (a+b x)^{n+7}}{b^8 (n+7)}+\frac {d^2 (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^3)^2,x]

[Out]

-((a*(b^3*c - a^3*d)^2*(a + b*x)^(1 + n))/(b^8*(1 + n))) + ((b^3*c - 7*a^3*d)*(b^3*c - a^3*d)*(a + b*x)^(2 + n
))/(b^8*(2 + n)) + (3*a^2*d*(4*b^3*c - 7*a^3*d)*(a + b*x)^(3 + n))/(b^8*(3 + n)) - (a*d*(8*b^3*c - 35*a^3*d)*(
a + b*x)^(4 + n))/(b^8*(4 + n)) + (d*(2*b^3*c - 35*a^3*d)*(a + b*x)^(5 + n))/(b^8*(5 + n)) + (21*a^2*d^2*(a +
b*x)^(6 + n))/(b^8*(6 + n)) - (7*a*d^2*(a + b*x)^(7 + n))/(b^8*(7 + n)) + (d^2*(a + b*x)^(8 + n))/(b^8*(8 + n)
)

Rule 1634

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin {align*} \int x (a+b x)^n \left (c+d x^3\right )^2 \, dx &=\int \left (-\frac {a \left (-b^3 c+a^3 d\right )^2 (a+b x)^n}{b^7}+\frac {\left (b^3 c-7 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{1+n}}{b^7}-\frac {3 a^2 d \left (-4 b^3 c+7 a^3 d\right ) (a+b x)^{2+n}}{b^7}+\frac {a d \left (-8 b^3 c+35 a^3 d\right ) (a+b x)^{3+n}}{b^7}+\frac {d \left (2 b^3 c-35 a^3 d\right ) (a+b x)^{4+n}}{b^7}+\frac {21 a^2 d^2 (a+b x)^{5+n}}{b^7}-\frac {7 a d^2 (a+b x)^{6+n}}{b^7}+\frac {d^2 (a+b x)^{7+n}}{b^7}\right ) \, dx\\ &=-\frac {a \left (b^3 c-a^3 d\right )^2 (a+b x)^{1+n}}{b^8 (1+n)}+\frac {\left (b^3 c-7 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{2+n}}{b^8 (2+n)}+\frac {3 a^2 d \left (4 b^3 c-7 a^3 d\right ) (a+b x)^{3+n}}{b^8 (3+n)}-\frac {a d \left (8 b^3 c-35 a^3 d\right ) (a+b x)^{4+n}}{b^8 (4+n)}+\frac {d \left (2 b^3 c-35 a^3 d\right ) (a+b x)^{5+n}}{b^8 (5+n)}+\frac {21 a^2 d^2 (a+b x)^{6+n}}{b^8 (6+n)}-\frac {7 a d^2 (a+b x)^{7+n}}{b^8 (7+n)}+\frac {d^2 (a+b x)^{8+n}}{b^8 (8+n)}\\ \end {align*}

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Mathematica [A]
time = 0.19, size = 211, normalized size = 0.85 \begin {gather*} \frac {(a+b x)^{1+n} \left (-\frac {a \left (b^3 c-a^3 d\right )^2}{1+n}+\frac {\left (b^3 c-7 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)}{2+n}+\frac {3 a^2 d \left (4 b^3 c-7 a^3 d\right ) (a+b x)^2}{3+n}+\frac {a d \left (-8 b^3 c+35 a^3 d\right ) (a+b x)^3}{4+n}+\frac {d \left (2 b^3 c-35 a^3 d\right ) (a+b x)^4}{5+n}+\frac {21 a^2 d^2 (a+b x)^5}{6+n}-\frac {7 a d^2 (a+b x)^6}{7+n}+\frac {d^2 (a+b x)^7}{8+n}\right )}{b^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(1 + n)*(-((a*(b^3*c - a^3*d)^2)/(1 + n)) + ((b^3*c - 7*a^3*d)*(b^3*c - a^3*d)*(a + b*x))/(2 + n) +
(3*a^2*d*(4*b^3*c - 7*a^3*d)*(a + b*x)^2)/(3 + n) + (a*d*(-8*b^3*c + 35*a^3*d)*(a + b*x)^3)/(4 + n) + (d*(2*b
^3*c - 35*a^3*d)*(a + b*x)^4)/(5 + n) + (21*a^2*d^2*(a + b*x)^5)/(6 + n) - (7*a*d^2*(a + b*x)^6)/(7 + n) + (d^
2*(a + b*x)^7)/(8 + n)))/b^8

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(892\) vs. \(2(248)=496\).
time = 0.25, size = 893, normalized size = 3.60 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d^2/(8+n)*x^8*exp(n*ln(b*x+a))+1/b^7*n*a*(b^6*c^2*n^6+33*b^6*c^2*n^5+445*b^6*c^2*n^4-48*a^3*b^3*c*d*n^3+3135*b
^6*c^2*n^3-1008*a^3*b^3*c*d*n^2+12154*b^6*c^2*n^2-7008*a^3*b^3*c*d*n+24552*b^6*c^2*n+5040*a^6*d^2-16128*a^3*b^
3*c*d+20160*b^6*c^2)/(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))+d^2*a/b*n/(n^2+15*n+56)*x^7*exp(n*ln(b*x+a))-a^2*(b^6*c^2*n^6+33*b^6*c^2*n^5+445*b^6*c^2*n^4-48*a^3*b^3*
c*d*n^3+3135*b^6*c^2*n^3-1008*a^3*b^3*c*d*n^2+12154*b^6*c^2*n^2-7008*a^3*b^3*c*d*n+24552*b^6*c^2*n+5040*a^6*d^
2-16128*a^3*b^3*c*d+20160*b^6*c^2)/b^8/(n^8+36*n^7+546*n^6+4536*n^5+22449*n^4+67284*n^3+118124*n^2+109584*n+40
320)*exp(n*ln(b*x+a))-(-b^6*c^2*n^6-33*b^6*c^2*n^5-24*a^3*b^3*c*d*n^4-445*b^6*c^2*n^4-504*a^3*b^3*c*d*n^3-3135
*b^6*c^2*n^3-3504*a^3*b^3*c*d*n^2-12154*b^6*c^2*n^2+2520*a^6*d^2*n-8064*a^3*b^3*c*d*n-24552*b^6*c^2*n-20160*b^
6*c^2)/b^6/(n^7+35*n^6+511*n^5+4025*n^4+18424*n^3+48860*n^2+69264*n+40320)*x^2*exp(n*ln(b*x+a))+2*d*(b^3*c*n^3
+21*b^3*c*n^2+21*a^3*d*n+146*b^3*c*n+336*b^3*c)/b^3/(n^4+26*n^3+251*n^2+1066*n+1680)*x^5*exp(n*ln(b*x+a))-7*n*
a^2*d^2/b^2/(n^3+21*n^2+146*n+336)*x^6*exp(n*ln(b*x+a))-2*n*a*d*(-b^3*c*n^3-21*b^3*c*n^2-146*b^3*c*n+105*a^3*d
-336*b^3*c)/b^4/(n^5+30*n^4+355*n^3+2070*n^2+5944*n+6720)*x^4*exp(n*ln(b*x+a))+8*(-b^3*c*n^3-21*b^3*c*n^2-146*
b^3*c*n+105*a^3*d-336*b^3*c)*a^2/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 [A]
time = 0.30, size = 474, normalized size = 1.91 \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^{2}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac {2 \, {\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{5} x^{5} + {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a b^{4} x^{4} - 4 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{2} b^{3} x^{3} + 12 \, {\left (n^{2} + n\right )} a^{3} b^{2} x^{2} - 24 \, a^{4} b n x + 24 \, a^{5}\right )} {\left (b x + a\right )}^{n} c d}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{5}} + \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^{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 )} b^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^2*(n + 1)*x^2 + a*b*n*x - a^2)*(b*x + a)^n*c^2/((n^2 + 3*n + 2)*b^2) + 2*((n^4 + 10*n^3 + 35*n^2 + 50*n + 2
4)*b^5*x^5 + (n^4 + 6*n^3 + 11*n^2 + 6*n)*a*b^4*x^4 - 4*(n^3 + 3*n^2 + 2*n)*a^2*b^3*x^3 + 12*(n^2 + n)*a^3*b^2
*x^2 - 24*a^4*b*n*x + 24*a^5)*(b*x + a)^n*c*d/((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*b^5) + ((n^7 +
28*n^6 + 322*n^5 + 1960*n^4 + 6769*n^3 + 13132*n^2 + 13068*n + 5040)*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 + 85*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 + 84
0*(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^2/((
n^8 + 36*n^7 + 546*n^6 + 4536*n^5 + 22449*n^4 + 67284*n^3 + 118124*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. 1216 vs. \(2 (248) = 496\).
time = 0.36, size = 1216, normalized size = 4.90 \begin {gather*} -\frac {{\left (a^{2} b^{6} c^{2} n^{6} + 33 \, a^{2} b^{6} c^{2} n^{5} + 445 \, a^{2} b^{6} c^{2} n^{4} + 20160 \, a^{2} b^{6} c^{2} - 16128 \, a^{5} b^{3} c d + 5040 \, a^{8} d^{2} - {\left (b^{8} d^{2} n^{7} + 28 \, b^{8} d^{2} n^{6} + 322 \, b^{8} d^{2} n^{5} + 1960 \, b^{8} d^{2} n^{4} + 6769 \, b^{8} d^{2} n^{3} + 13132 \, b^{8} d^{2} n^{2} + 13068 \, b^{8} d^{2} n + 5040 \, b^{8} d^{2}\right )} x^{8} - {\left (a b^{7} d^{2} n^{7} + 21 \, a b^{7} d^{2} n^{6} + 175 \, a b^{7} d^{2} n^{5} + 735 \, a b^{7} d^{2} n^{4} + 1624 \, a b^{7} d^{2} n^{3} + 1764 \, a b^{7} d^{2} n^{2} + 720 \, a b^{7} d^{2} n\right )} x^{7} + 7 \, {\left (a^{2} b^{6} d^{2} n^{6} + 15 \, a^{2} b^{6} d^{2} n^{5} + 85 \, a^{2} b^{6} d^{2} n^{4} + 225 \, a^{2} b^{6} d^{2} n^{3} + 274 \, a^{2} b^{6} d^{2} n^{2} + 120 \, a^{2} b^{6} d^{2} n\right )} x^{6} - 2 \, {\left (b^{8} c d n^{7} + 31 \, b^{8} c d n^{6} + 8064 \, b^{8} c d + {\left (391 \, b^{8} c d + 21 \, a^{3} b^{5} d^{2}\right )} n^{5} + {\left (2581 \, b^{8} c d + 210 \, a^{3} b^{5} d^{2}\right )} n^{4} + {\left (9544 \, b^{8} c d + 735 \, a^{3} b^{5} d^{2}\right )} n^{3} + 2 \, {\left (9782 \, b^{8} c d + 525 \, a^{3} b^{5} d^{2}\right )} n^{2} + 72 \, {\left (282 \, b^{8} c d + 7 \, a^{3} b^{5} d^{2}\right )} n\right )} x^{5} - 2 \, {\left (a b^{7} c d n^{7} + 27 \, a b^{7} c d n^{6} + 283 \, a b^{7} c d n^{5} + 21 \, {\left (69 \, a b^{7} c d - 5 \, a^{4} b^{4} d^{2}\right )} n^{4} + 2 \, {\left (1874 \, a b^{7} c d - 315 \, a^{4} b^{4} d^{2}\right )} n^{3} + 3 \, {\left (1524 \, a b^{7} c d - 385 \, a^{4} b^{4} d^{2}\right )} n^{2} + 126 \, {\left (16 \, a b^{7} c d - 5 \, a^{4} b^{4} d^{2}\right )} n\right )} x^{4} + 3 \, {\left (1045 \, a^{2} b^{6} c^{2} - 16 \, a^{5} b^{3} c d\right )} n^{3} + 8 \, {\left (a^{2} b^{6} c d n^{6} + 24 \, a^{2} b^{6} c d n^{5} + 211 \, a^{2} b^{6} c d n^{4} + 3 \, {\left (272 \, a^{2} b^{6} c d - 35 \, a^{5} b^{3} d^{2}\right )} n^{3} + 5 \, {\left (260 \, a^{2} b^{6} c d - 63 \, a^{5} b^{3} d^{2}\right )} n^{2} + 42 \, {\left (16 \, a^{2} b^{6} c d - 5 \, a^{5} b^{3} d^{2}\right )} n\right )} x^{3} + 2 \, {\left (6077 \, a^{2} b^{6} c^{2} - 504 \, a^{5} b^{3} c d\right )} n^{2} - {\left (b^{8} c^{2} n^{7} + 34 \, b^{8} c^{2} n^{6} + 20160 \, b^{8} c^{2} + 2 \, {\left (239 \, b^{8} c^{2} + 12 \, a^{3} b^{5} c d\right )} n^{5} + 4 \, {\left (895 \, b^{8} c^{2} + 132 \, a^{3} b^{5} c d\right )} n^{4} + {\left (15289 \, b^{8} c^{2} + 4008 \, a^{3} b^{5} c d\right )} n^{3} + 2 \, {\left (18353 \, b^{8} c^{2} + 5784 \, a^{3} b^{5} c d - 1260 \, a^{6} b^{2} d^{2}\right )} n^{2} + 72 \, {\left (621 \, b^{8} c^{2} + 112 \, a^{3} b^{5} c d - 35 \, a^{6} b^{2} d^{2}\right )} n\right )} x^{2} + 24 \, {\left (1023 \, a^{2} b^{6} c^{2} - 292 \, a^{5} b^{3} c d\right )} n - {\left (a b^{7} c^{2} n^{7} + 33 \, a b^{7} c^{2} n^{6} + 445 \, a b^{7} c^{2} n^{5} + 3 \, {\left (1045 \, a b^{7} c^{2} - 16 \, a^{4} b^{4} c d\right )} n^{4} + 2 \, {\left (6077 \, a b^{7} c^{2} - 504 \, a^{4} b^{4} c d\right )} n^{3} + 24 \, {\left (1023 \, a b^{7} c^{2} - 292 \, a^{4} b^{4} c d\right )} n^{2} + 1008 \, {\left (20 \, a b^{7} c^{2} - 16 \, a^{4} b^{4} c d + 5 \, a^{7} b d^{2}\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^3+c)^2,x, algorithm="fricas")

[Out]

-(a^2*b^6*c^2*n^6 + 33*a^2*b^6*c^2*n^5 + 445*a^2*b^6*c^2*n^4 + 20160*a^2*b^6*c^2 - 16128*a^5*b^3*c*d + 5040*a^
8*d^2 - (b^8*d^2*n^7 + 28*b^8*d^2*n^6 + 322*b^8*d^2*n^5 + 1960*b^8*d^2*n^4 + 6769*b^8*d^2*n^3 + 13132*b^8*d^2*
n^2 + 13068*b^8*d^2*n + 5040*b^8*d^2)*x^8 - (a*b^7*d^2*n^7 + 21*a*b^7*d^2*n^6 + 175*a*b^7*d^2*n^5 + 735*a*b^7*
d^2*n^4 + 1624*a*b^7*d^2*n^3 + 1764*a*b^7*d^2*n^2 + 720*a*b^7*d^2*n)*x^7 + 7*(a^2*b^6*d^2*n^6 + 15*a^2*b^6*d^2
*n^5 + 85*a^2*b^6*d^2*n^4 + 225*a^2*b^6*d^2*n^3 + 274*a^2*b^6*d^2*n^2 + 120*a^2*b^6*d^2*n)*x^6 - 2*(b^8*c*d*n^
7 + 31*b^8*c*d*n^6 + 8064*b^8*c*d + (391*b^8*c*d + 21*a^3*b^5*d^2)*n^5 + (2581*b^8*c*d + 210*a^3*b^5*d^2)*n^4
+ (9544*b^8*c*d + 735*a^3*b^5*d^2)*n^3 + 2*(9782*b^8*c*d + 525*a^3*b^5*d^2)*n^2 + 72*(282*b^8*c*d + 7*a^3*b^5*
d^2)*n)*x^5 - 2*(a*b^7*c*d*n^7 + 27*a*b^7*c*d*n^6 + 283*a*b^7*c*d*n^5 + 21*(69*a*b^7*c*d - 5*a^4*b^4*d^2)*n^4
+ 2*(1874*a*b^7*c*d - 315*a^4*b^4*d^2)*n^3 + 3*(1524*a*b^7*c*d - 385*a^4*b^4*d^2)*n^2 + 126*(16*a*b^7*c*d - 5*
a^4*b^4*d^2)*n)*x^4 + 3*(1045*a^2*b^6*c^2 - 16*a^5*b^3*c*d)*n^3 + 8*(a^2*b^6*c*d*n^6 + 24*a^2*b^6*c*d*n^5 + 21
1*a^2*b^6*c*d*n^4 + 3*(272*a^2*b^6*c*d - 35*a^5*b^3*d^2)*n^3 + 5*(260*a^2*b^6*c*d - 63*a^5*b^3*d^2)*n^2 + 42*(
16*a^2*b^6*c*d - 5*a^5*b^3*d^2)*n)*x^3 + 2*(6077*a^2*b^6*c^2 - 504*a^5*b^3*c*d)*n^2 - (b^8*c^2*n^7 + 34*b^8*c^
2*n^6 + 20160*b^8*c^2 + 2*(239*b^8*c^2 + 12*a^3*b^5*c*d)*n^5 + 4*(895*b^8*c^2 + 132*a^3*b^5*c*d)*n^4 + (15289*
b^8*c^2 + 4008*a^3*b^5*c*d)*n^3 + 2*(18353*b^8*c^2 + 5784*a^3*b^5*c*d - 1260*a^6*b^2*d^2)*n^2 + 72*(621*b^8*c^
2 + 112*a^3*b^5*c*d - 35*a^6*b^2*d^2)*n)*x^2 + 24*(1023*a^2*b^6*c^2 - 292*a^5*b^3*c*d)*n - (a*b^7*c^2*n^7 + 33
*a*b^7*c^2*n^6 + 445*a*b^7*c^2*n^5 + 3*(1045*a*b^7*c^2 - 16*a^4*b^4*c*d)*n^4 + 2*(6077*a*b^7*c^2 - 504*a^4*b^4
*c*d)*n^3 + 24*(1023*a*b^7*c^2 - 292*a^4*b^4*c*d)*n^2 + 1008*(20*a*b^7*c^2 - 16*a^4*b^4*c*d + 5*a^7*b*d^2)*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. 18328 vs. \(2 (228) = 456\).
time = 6.59, size = 18328, normalized size = 73.90 \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**3+c)**2,x)

[Out]

Piecewise((a**n*(c**2*x**2/2 + 2*c*d*x**5/5 + d**2*x**8/8), Eq(b, 0)), (420*a**7*d**2*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**1
3*x**5 + 2940*a*b**14*x**6 + 420*b**15*x**7) + 1089*a**7*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**1
5*x**7) + 2940*a**6*b*d**2*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**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) + 8820*a**5*b**2*d**2*x**2*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) + 20139*a**5*b**2*d**2*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**1
4*x**6 + 420*b**15*x**7) - 8*a**4*b**3*c*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) + 14700*a**
4*b**3*d**2*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
+ 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) + 30625*a**4*b**3*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) - 56*a**3*b**4*c*d*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**3*b**4*d**2*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**11*x**3 + 14700*a**3*b**12*x**4 + 8820*a**2*b**13*x**5 + 2940*a*b**14*x**6 + 42
0*b**15*x**7) + 26950*a**3*b**4*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) - 168*a**2*b
**5*c*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 + 420*b**15*x**7) + 8820*a**2*b**5*d**2*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) + 13230*a**2*b**5*d**2*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 + 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) - 10*a*b**6*c**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) - 280*a*
b**6*c*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**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**2*x**6*log(a/b + x)/(42
0*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**2*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**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) - 280*b**7*c*d
*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) + 420*b**7*d**2*x**7*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), Eq(n, -8)), (-420*a**7*d**2*log(a/b + x)/(60*a**6*b**8 + 360*a**5*b
**9*x + 900*a**4*b**10*x**2 + 1200*a**3*b**11*x**3 + 900*a**2*b**12*x**4 + 360*a*b**13*x**5 + 60*b**14*x**6) -
1029*a**7*d**2/(60*a**6*b**8 + 360*a**5*b**9*x + 900*a**4*b**10*x**2 + 1200*a**3*b**11*x**3 + 900*a**2*b**12*
x**4 + 360*a*b**13*x**5 + 60*b**14*x**6) - 2520*a**6*b*d**2*x*log(a/b + x)/(60*a**6*b**8 + 360*a**5*b**9*x + 9
00*a**4*b**10*x**2 + 1200*a**3*b**11*x**3 + 900*a**2*b**12*x**4 + 360*a*b**13*x**5 + 60*b**14*x**6) - 5754*a**
6*b*d**2*x/(60*a**6*b**8 + 360*a**5*b**9*x + 900*a**4*b**10*x**2 + 1200*a**3*b**11*x**3 + 900*a**2*b**12*x**4
+ 360*a*b**13*x**5 + 60*b**14*x**6) - 6300*a**5...

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2034 vs. \(2 (248) = 496\).
time = 3.76, size = 2034, normalized size = 8.20 \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^3+c)^2,x, algorithm="giac")

[Out]

((b*x + a)^n*b^8*d^2*n^7*x^8 + (b*x + a)^n*a*b^7*d^2*n^7*x^7 + 28*(b*x + a)^n*b^8*d^2*n^6*x^8 + 21*(b*x + a)^n
*a*b^7*d^2*n^6*x^7 + 322*(b*x + a)^n*b^8*d^2*n^5*x^8 + 2*(b*x + a)^n*b^8*c*d*n^7*x^5 - 7*(b*x + a)^n*a^2*b^6*d
^2*n^6*x^6 + 175*(b*x + a)^n*a*b^7*d^2*n^5*x^7 + 1960*(b*x + a)^n*b^8*d^2*n^4*x^8 + 2*(b*x + a)^n*a*b^7*c*d*n^
7*x^4 + 62*(b*x + a)^n*b^8*c*d*n^6*x^5 - 105*(b*x + a)^n*a^2*b^6*d^2*n^5*x^6 + 735*(b*x + a)^n*a*b^7*d^2*n^4*x
^7 + 6769*(b*x + a)^n*b^8*d^2*n^3*x^8 + 54*(b*x + a)^n*a*b^7*c*d*n^6*x^4 + 782*(b*x + a)^n*b^8*c*d*n^5*x^5 + 4
2*(b*x + a)^n*a^3*b^5*d^2*n^5*x^5 - 595*(b*x + a)^n*a^2*b^6*d^2*n^4*x^6 + 1624*(b*x + a)^n*a*b^7*d^2*n^3*x^7 +
13132*(b*x + a)^n*b^8*d^2*n^2*x^8 + (b*x + a)^n*b^8*c^2*n^7*x^2 - 8*(b*x + a)^n*a^2*b^6*c*d*n^6*x^3 + 566*(b*
x + a)^n*a*b^7*c*d*n^5*x^4 + 5162*(b*x + a)^n*b^8*c*d*n^4*x^5 + 420*(b*x + a)^n*a^3*b^5*d^2*n^4*x^5 - 1575*(b*
x + a)^n*a^2*b^6*d^2*n^3*x^6 + 1764*(b*x + a)^n*a*b^7*d^2*n^2*x^7 + 13068*(b*x + a)^n*b^8*d^2*n*x^8 + (b*x + a
)^n*a*b^7*c^2*n^7*x + 34*(b*x + a)^n*b^8*c^2*n^6*x^2 - 192*(b*x + a)^n*a^2*b^6*c*d*n^5*x^3 + 2898*(b*x + a)^n*
a*b^7*c*d*n^4*x^4 - 210*(b*x + a)^n*a^4*b^4*d^2*n^4*x^4 + 19088*(b*x + a)^n*b^8*c*d*n^3*x^5 + 1470*(b*x + a)^n
*a^3*b^5*d^2*n^3*x^5 - 1918*(b*x + a)^n*a^2*b^6*d^2*n^2*x^6 + 720*(b*x + a)^n*a*b^7*d^2*n*x^7 + 5040*(b*x + a)
^n*b^8*d^2*x^8 + 33*(b*x + a)^n*a*b^7*c^2*n^6*x + 478*(b*x + a)^n*b^8*c^2*n^5*x^2 + 24*(b*x + a)^n*a^3*b^5*c*d
*n^5*x^2 - 1688*(b*x + a)^n*a^2*b^6*c*d*n^4*x^3 + 7496*(b*x + a)^n*a*b^7*c*d*n^3*x^4 - 1260*(b*x + a)^n*a^4*b^
4*d^2*n^3*x^4 + 39128*(b*x + a)^n*b^8*c*d*n^2*x^5 + 2100*(b*x + a)^n*a^3*b^5*d^2*n^2*x^5 - 840*(b*x + a)^n*a^2
*b^6*d^2*n*x^6 - (b*x + a)^n*a^2*b^6*c^2*n^6 + 445*(b*x + a)^n*a*b^7*c^2*n^5*x + 3580*(b*x + a)^n*b^8*c^2*n^4*
x^2 + 528*(b*x + a)^n*a^3*b^5*c*d*n^4*x^2 - 6528*(b*x + a)^n*a^2*b^6*c*d*n^3*x^3 + 840*(b*x + a)^n*a^5*b^3*d^2
*n^3*x^3 + 9144*(b*x + a)^n*a*b^7*c*d*n^2*x^4 - 2310*(b*x + a)^n*a^4*b^4*d^2*n^2*x^4 + 40608*(b*x + a)^n*b^8*c
*d*n*x^5 + 1008*(b*x + a)^n*a^3*b^5*d^2*n*x^5 - 33*(b*x + a)^n*a^2*b^6*c^2*n^5 + 3135*(b*x + a)^n*a*b^7*c^2*n^
4*x - 48*(b*x + a)^n*a^4*b^4*c*d*n^4*x + 15289*(b*x + a)^n*b^8*c^2*n^3*x^2 + 4008*(b*x + a)^n*a^3*b^5*c*d*n^3*
x^2 - 10400*(b*x + a)^n*a^2*b^6*c*d*n^2*x^3 + 2520*(b*x + a)^n*a^5*b^3*d^2*n^2*x^3 + 4032*(b*x + a)^n*a*b^7*c*
d*n*x^4 - 1260*(b*x + a)^n*a^4*b^4*d^2*n*x^4 + 16128*(b*x + a)^n*b^8*c*d*x^5 - 445*(b*x + a)^n*a^2*b^6*c^2*n^4
+ 12154*(b*x + a)^n*a*b^7*c^2*n^3*x - 1008*(b*x + a)^n*a^4*b^4*c*d*n^3*x + 36706*(b*x + a)^n*b^8*c^2*n^2*x^2
+ 11568*(b*x + a)^n*a^3*b^5*c*d*n^2*x^2 - 2520*(b*x + a)^n*a^6*b^2*d^2*n^2*x^2 - 5376*(b*x + a)^n*a^2*b^6*c*d*
n*x^3 + 1680*(b*x + a)^n*a^5*b^3*d^2*n*x^3 - 3135*(b*x + a)^n*a^2*b^6*c^2*n^3 + 48*(b*x + a)^n*a^5*b^3*c*d*n^3
+ 24552*(b*x + a)^n*a*b^7*c^2*n^2*x - 7008*(b*x + a)^n*a^4*b^4*c*d*n^2*x + 44712*(b*x + a)^n*b^8*c^2*n*x^2 +
8064*(b*x + a)^n*a^3*b^5*c*d*n*x^2 - 2520*(b*x + a)^n*a^6*b^2*d^2*n*x^2 - 12154*(b*x + a)^n*a^2*b^6*c^2*n^2 +
1008*(b*x + a)^n*a^5*b^3*c*d*n^2 + 20160*(b*x + a)^n*a*b^7*c^2*n*x - 16128*(b*x + a)^n*a^4*b^4*c*d*n*x + 5040*
(b*x + a)^n*a^7*b*d^2*n*x + 20160*(b*x + a)^n*b^8*c^2*x^2 - 24552*(b*x + a)^n*a^2*b^6*c^2*n + 7008*(b*x + a)^n
*a^5*b^3*c*d*n - 20160*(b*x + a)^n*a^2*b^6*c^2 + 16128*(b*x + a)^n*a^5*b^3*c*d - 5040*(b*x + a)^n*a^8*d^2)/(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)

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Mupad [B]
time = 3.39, size = 1136, normalized size = 4.58 \begin {gather*} \frac {d^2\,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^2-48\,a^3\,b^3\,c\,d\,n^3-1008\,a^3\,b^3\,c\,d\,n^2-7008\,a^3\,b^3\,c\,d\,n-16128\,a^3\,b^3\,c\,d+b^6\,c^2\,n^6+33\,b^6\,c^2\,n^5+445\,b^6\,c^2\,n^4+3135\,b^6\,c^2\,n^3+12154\,b^6\,c^2\,n^2+24552\,b^6\,c^2\,n+20160\,b^6\,c^2\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^2\,n+24\,a^3\,b^3\,c\,d\,n^4+504\,a^3\,b^3\,c\,d\,n^3+3504\,a^3\,b^3\,c\,d\,n^2+8064\,a^3\,b^3\,c\,d\,n+b^6\,c^2\,n^6+33\,b^6\,c^2\,n^5+445\,b^6\,c^2\,n^4+3135\,b^6\,c^2\,n^3+12154\,b^6\,c^2\,n^2+24552\,b^6\,c^2\,n+20160\,b^6\,c^2\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 {a\,n\,x\,{\left (a+b\,x\right )}^n\,\left (5040\,a^6\,d^2-48\,a^3\,b^3\,c\,d\,n^3-1008\,a^3\,b^3\,c\,d\,n^2-7008\,a^3\,b^3\,c\,d\,n-16128\,a^3\,b^3\,c\,d+b^6\,c^2\,n^6+33\,b^6\,c^2\,n^5+445\,b^6\,c^2\,n^4+3135\,b^6\,c^2\,n^3+12154\,b^6\,c^2\,n^2+24552\,b^6\,c^2\,n+20160\,b^6\,c^2\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 {2\,d\,x^5\,{\left (a+b\,x\right )}^n\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )\,\left (21\,d\,a^3\,n+c\,b^3\,n^3+21\,c\,b^3\,n^2+146\,c\,b^3\,n+336\,c\,b^3\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 )}+\frac {a\,d^2\,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 {7\,a^2\,d^2\,n\,x^6\,{\left (a+b\,x\right )}^n\,\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 {2\,a\,d\,n\,x^4\,{\left (a+b\,x\right )}^n\,\left (n^3+6\,n^2+11\,n+6\right )\,\left (-105\,d\,a^3+c\,b^3\,n^3+21\,c\,b^3\,n^2+146\,c\,b^3\,n+336\,c\,b^3\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 {8\,a^2\,d\,n\,x^3\,{\left (a+b\,x\right )}^n\,\left (n^2+3\,n+2\right )\,\left (-105\,d\,a^3+c\,b^3\,n^3+21\,c\,b^3\,n^2+146\,c\,b^3\,n+336\,c\,b^3\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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(c + d*x^3)^2*(a + b*x)^n,x)

[Out]

(d^2*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
^2 + 20160*b^6*c^2 + 24552*b^6*c^2*n + 12154*b^6*c^2*n^2 + 3135*b^6*c^2*n^3 + 445*b^6*c^2*n^4 + 33*b^6*c^2*n^5
+ b^6*c^2*n^6 - 16128*a^3*b^3*c*d - 7008*a^3*b^3*c*d*n - 1008*a^3*b^3*c*d*n^2 - 48*a^3*b^3*c*d*n^3))/(b^8*(10
9584*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*(20160*b^6*c^2 - 2520*a^6*d^2*n + 24552*b^6*c^2*n + 12154*b^6*c^2*n^2 + 3135*b^6*c^2*n^3 + 445*b^6*c^2*
n^4 + 33*b^6*c^2*n^5 + b^6*c^2*n^6 + 8064*a^3*b^3*c*d*n + 3504*a^3*b^3*c*d*n^2 + 504*a^3*b^3*c*d*n^3 + 24*a^3*
b^3*c*d*n^4))/(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)
) + (a*n*x*(a + b*x)^n*(5040*a^6*d^2 + 20160*b^6*c^2 + 24552*b^6*c^2*n + 12154*b^6*c^2*n^2 + 3135*b^6*c^2*n^3
+ 445*b^6*c^2*n^4 + 33*b^6*c^2*n^5 + b^6*c^2*n^6 - 16128*a^3*b^3*c*d - 7008*a^3*b^3*c*d*n - 1008*a^3*b^3*c*d*n
^2 - 48*a^3*b^3*c*d*n^3))/(b^7*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 +
n^8 + 40320)) + (2*d*x^5*(a + b*x)^n*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)*(336*b^3*c + 21*b^3*c*n^2 + b^3*c*n^3
+ 21*a^3*d*n + 146*b^3*c*n))/(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)) + (a*d^2*n*x^7*(a + b*x)^n*(1764*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)) - (7*a^2*d^2*n*x^
6*(a + b*x)^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)) + (2*a*d*n*x^4*(a + b*x)^n*(11*n + 6*n^2 + n^3 + 6)*(336*b^
3*c - 105*a^3*d + 21*b^3*c*n^2 + b^3*c*n^3 + 146*b^3*c*n))/(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)) - (8*a^2*d*n*x^3*(a + b*x)^n*(3*n + n^2 + 2)*(336*b^3*c - 105*a
^3*d + 21*b^3*c*n^2 + b^3*c*n^3 + 146*b^3*c*n))/(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))

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