3.2.80 \(\int (a+b x)^n (c+d x^3)^2 \, dx\) [180]

Optimal. Leaf size=203 \[ \frac {\left (b^3 c-a^3 d\right )^2 (a+b x)^{1+n}}{b^7 (1+n)}+\frac {6 a^2 d \left (b^3 c-a^3 d\right ) (a+b x)^{2+n}}{b^7 (2+n)}-\frac {3 a d \left (2 b^3 c-5 a^3 d\right ) (a+b x)^{3+n}}{b^7 (3+n)}+\frac {2 d \left (b^3 c-10 a^3 d\right ) (a+b x)^{4+n}}{b^7 (4+n)}+\frac {15 a^2 d^2 (a+b x)^{5+n}}{b^7 (5+n)}-\frac {6 a d^2 (a+b x)^{6+n}}{b^7 (6+n)}+\frac {d^2 (a+b x)^{7+n}}{b^7 (7+n)} \]

[Out]

(-a^3*d+b^3*c)^2*(b*x+a)^(1+n)/b^7/(1+n)+6*a^2*d*(-a^3*d+b^3*c)*(b*x+a)^(2+n)/b^7/(2+n)-3*a*d*(-5*a^3*d+2*b^3*
c)*(b*x+a)^(3+n)/b^7/(3+n)+2*d*(-10*a^3*d+b^3*c)*(b*x+a)^(4+n)/b^7/(4+n)+15*a^2*d^2*(b*x+a)^(5+n)/b^7/(5+n)-6*
a*d^2*(b*x+a)^(6+n)/b^7/(6+n)+d^2*(b*x+a)^(7+n)/b^7/(7+n)

________________________________________________________________________________________

Rubi [A]
time = 0.08, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {1864} \begin {gather*} \frac {\left (b^3 c-a^3 d\right )^2 (a+b x)^{n+1}}{b^7 (n+1)}-\frac {3 a d \left (2 b^3 c-5 a^3 d\right ) (a+b x)^{n+3}}{b^7 (n+3)}+\frac {2 d \left (b^3 c-10 a^3 d\right ) (a+b x)^{n+4}}{b^7 (n+4)}+\frac {15 a^2 d^2 (a+b x)^{n+5}}{b^7 (n+5)}+\frac {6 a^2 d \left (b^3 c-a^3 d\right ) (a+b x)^{n+2}}{b^7 (n+2)}-\frac {6 a d^2 (a+b x)^{n+6}}{b^7 (n+6)}+\frac {d^2 (a+b x)^{n+7}}{b^7 (n+7)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^n*(c + d*x^3)^2,x]

[Out]

((b^3*c - a^3*d)^2*(a + b*x)^(1 + n))/(b^7*(1 + n)) + (6*a^2*d*(b^3*c - a^3*d)*(a + b*x)^(2 + n))/(b^7*(2 + n)
) - (3*a*d*(2*b^3*c - 5*a^3*d)*(a + b*x)^(3 + n))/(b^7*(3 + n)) + (2*d*(b^3*c - 10*a^3*d)*(a + b*x)^(4 + n))/(
b^7*(4 + n)) + (15*a^2*d^2*(a + b*x)^(5 + n))/(b^7*(5 + n)) - (6*a*d^2*(a + b*x)^(6 + n))/(b^7*(6 + n)) + (d^2
*(a + b*x)^(7 + n))/(b^7*(7 + n))

Rule 1864

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin {align*} \int (a+b x)^n \left (c+d x^3\right )^2 \, dx &=\int \left (\frac {\left (b^3 c-a^3 d\right )^2 (a+b x)^n}{b^6}-\frac {6 a^2 d \left (-b^3 c+a^3 d\right ) (a+b x)^{1+n}}{b^6}+\frac {3 a d \left (-2 b^3 c+5 a^3 d\right ) (a+b x)^{2+n}}{b^6}+\frac {2 d \left (b^3 c-10 a^3 d\right ) (a+b x)^{3+n}}{b^6}+\frac {15 a^2 d^2 (a+b x)^{4+n}}{b^6}-\frac {6 a d^2 (a+b x)^{5+n}}{b^6}+\frac {d^2 (a+b x)^{6+n}}{b^6}\right ) \, dx\\ &=\frac {\left (b^3 c-a^3 d\right )^2 (a+b x)^{1+n}}{b^7 (1+n)}+\frac {6 a^2 d \left (b^3 c-a^3 d\right ) (a+b x)^{2+n}}{b^7 (2+n)}-\frac {3 a d \left (2 b^3 c-5 a^3 d\right ) (a+b x)^{3+n}}{b^7 (3+n)}+\frac {2 d \left (b^3 c-10 a^3 d\right ) (a+b x)^{4+n}}{b^7 (4+n)}+\frac {15 a^2 d^2 (a+b x)^{5+n}}{b^7 (5+n)}-\frac {6 a d^2 (a+b x)^{6+n}}{b^7 (6+n)}+\frac {d^2 (a+b x)^{7+n}}{b^7 (7+n)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.18, size = 297, normalized size = 1.46 \begin {gather*} \frac {(a+b x)^{1+n} \left (720 a^6 d^2-720 a^5 b d^2 (1+n) x+360 a^4 b^2 d^2 \left (2+3 n+n^2\right ) x^2-6 a b^5 d \left (10+17 n+8 n^2+n^3\right ) x^2 \left (c \left (42+13 n+n^2\right )+d \left (12+7 n+n^2\right ) x^3\right )-12 a^3 b^3 d \left (c \left (210+107 n+18 n^2+n^3\right )+10 d \left (6+11 n+6 n^2+n^3\right ) x^3\right )+6 a^2 b^4 d (1+n) x \left (2 c \left (210+107 n+18 n^2+n^3\right )+5 d \left (24+26 n+9 n^2+n^3\right ) x^3\right )+b^6 \left (180+216 n+91 n^2+16 n^3+n^4\right ) \left (c^2 \left (28+11 n+n^2\right )+2 c d \left (7+8 n+n^2\right ) x^3+d^2 \left (4+5 n+n^2\right ) x^6\right )\right )}{b^7 (1+n) (2+n) (3+n) (4+n) (5+n) (6+n) (7+n)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(1 + n)*(720*a^6*d^2 - 720*a^5*b*d^2*(1 + n)*x + 360*a^4*b^2*d^2*(2 + 3*n + n^2)*x^2 - 6*a*b^5*d*(1
0 + 17*n + 8*n^2 + n^3)*x^2*(c*(42 + 13*n + n^2) + d*(12 + 7*n + n^2)*x^3) - 12*a^3*b^3*d*(c*(210 + 107*n + 18
*n^2 + n^3) + 10*d*(6 + 11*n + 6*n^2 + n^3)*x^3) + 6*a^2*b^4*d*(1 + n)*x*(2*c*(210 + 107*n + 18*n^2 + n^3) + 5
*d*(24 + 26*n + 9*n^2 + n^3)*x^3) + b^6*(180 + 216*n + 91*n^2 + 16*n^3 + n^4)*(c^2*(28 + 11*n + n^2) + 2*c*d*(
7 + 8*n + n^2)*x^3 + d^2*(4 + 5*n + n^2)*x^6)))/(b^7*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n))

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(699\) vs. \(2(203)=406\).
time = 0.25, size = 700, normalized size = 3.45 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d^2/(7+n)*x^7*exp(n*ln(b*x+a))+a*(b^6*c^2*n^6+27*b^6*c^2*n^5+295*b^6*c^2*n^4-12*a^3*b^3*c*d*n^3+1665*b^6*c^2*n
^3-216*a^3*b^3*c*d*n^2+5104*b^6*c^2*n^2-1284*a^3*b^3*c*d*n+8028*b^6*c^2*n+720*a^6*d^2-2520*a^3*b^3*c*d+5040*b^
6*c^2)/b^7/(n^7+28*n^6+322*n^5+1960*n^4+6769*n^3+13132*n^2+13068*n+5040)*exp(n*ln(b*x+a))+d^2*a*n/b/(n^2+13*n+
42)*x^6*exp(n*ln(b*x+a))-(-b^6*c^2*n^6-27*b^6*c^2*n^5-12*a^3*b^3*c*d*n^4-295*b^6*c^2*n^4-216*a^3*b^3*c*d*n^3-1
665*b^6*c^2*n^3-1284*a^3*b^3*c*d*n^2-5104*b^6*c^2*n^2+720*a^6*d^2*n-2520*a^3*b^3*c*d*n-8028*b^6*c^2*n-5040*b^6
*c^2)/b^6/(n^7+28*n^6+322*n^5+1960*n^4+6769*n^3+13132*n^2+13068*n+5040)*x*exp(n*ln(b*x+a))+2*(b^3*c*n^3+18*b^3
*c*n^2+15*a^3*d*n+107*b^3*c*n+210*b^3*c)*d/b^3/(n^4+22*n^3+179*n^2+638*n+840)*x^4*exp(n*ln(b*x+a))-6*n*a^2*d^2
/b^2/(n^3+18*n^2+107*n+210)*x^5*exp(n*ln(b*x+a))-2*n*d*a*(-b^3*c*n^3-18*b^3*c*n^2-107*b^3*c*n+60*a^3*d-210*b^3
*c)/b^4/(n^5+25*n^4+245*n^3+1175*n^2+2754*n+2520)*x^3*exp(n*ln(b*x+a))+6*(-b^3*c*n^3-18*b^3*c*n^2-107*b^3*c*n+
60*a^3*d-210*b^3*c)*d*a^2/b^5*n/(n^6+27*n^5+295*n^4+1665*n^3+5104*n^2+8028*n+5040)*x^2*exp(n*ln(b*x+a))

________________________________________________________________________________________

Maxima [A]
time = 0.30, size = 359, normalized size = 1.77 \begin {gather*} \frac {{\left (b x + a\right )}^{n + 1} c^{2}}{b {\left (n + 1\right )}} + \frac {2 \, {\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 d}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} + \frac {{\left ({\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} b^{7} x^{7} + {\left (n^{6} + 15 \, n^{5} + 85 \, n^{4} + 225 \, n^{3} + 274 \, n^{2} + 120 \, n\right )} a b^{6} x^{6} - 6 \, {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a^{2} b^{5} x^{5} + 30 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{3} b^{4} x^{4} - 120 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{4} b^{3} x^{3} + 360 \, {\left (n^{2} + n\right )} a^{5} b^{2} x^{2} - 720 \, a^{6} b n x + 720 \, a^{7}\right )} {\left (b x + a\right )}^{n} d^{2}}{{\left (n^{7} + 28 \, n^{6} + 322 \, n^{5} + 1960 \, n^{4} + 6769 \, n^{3} + 13132 \, n^{2} + 13068 \, n + 5040\right )} b^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x + a)^(n + 1)*c^2/(b*(n + 1)) + 2*((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*d/((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*b^4) + ((n^6 +
21*n^5 + 175*n^4 + 735*n^3 + 1624*n^2 + 1764*n + 720)*b^7*x^7 + (n^6 + 15*n^5 + 85*n^4 + 225*n^3 + 274*n^2 +
120*n)*a*b^6*x^6 - 6*(n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*a^2*b^5*x^5 + 30*(n^4 + 6*n^3 + 11*n^2 + 6*n)*a^3
*b^4*x^4 - 120*(n^3 + 3*n^2 + 2*n)*a^4*b^3*x^3 + 360*(n^2 + n)*a^5*b^2*x^2 - 720*a^6*b*n*x + 720*a^7)*(b*x + a
)^n*d^2/((n^7 + 28*n^6 + 322*n^5 + 1960*n^4 + 6769*n^3 + 13132*n^2 + 13068*n + 5040)*b^7)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 893 vs. \(2 (203) = 406\).
time = 0.37, size = 893, normalized size = 4.40 \begin {gather*} \frac {{\left (a b^{6} c^{2} n^{6} + 27 \, a b^{6} c^{2} n^{5} + 295 \, a b^{6} c^{2} n^{4} + 5040 \, a b^{6} c^{2} - 2520 \, a^{4} b^{3} c d + 720 \, a^{7} d^{2} + {\left (b^{7} d^{2} n^{6} + 21 \, b^{7} d^{2} n^{5} + 175 \, b^{7} d^{2} n^{4} + 735 \, b^{7} d^{2} n^{3} + 1624 \, b^{7} d^{2} n^{2} + 1764 \, b^{7} d^{2} n + 720 \, b^{7} d^{2}\right )} x^{7} + {\left (a b^{6} d^{2} n^{6} + 15 \, a b^{6} d^{2} n^{5} + 85 \, a b^{6} d^{2} n^{4} + 225 \, a b^{6} d^{2} n^{3} + 274 \, a b^{6} d^{2} n^{2} + 120 \, a b^{6} d^{2} n\right )} x^{6} - 6 \, {\left (a^{2} b^{5} d^{2} n^{5} + 10 \, a^{2} b^{5} d^{2} n^{4} + 35 \, a^{2} b^{5} d^{2} n^{3} + 50 \, a^{2} b^{5} d^{2} n^{2} + 24 \, a^{2} b^{5} d^{2} n\right )} x^{5} + 2 \, {\left (b^{7} c d n^{6} + 24 \, b^{7} c d n^{5} + 1260 \, b^{7} c d + {\left (226 \, b^{7} c d + 15 \, a^{3} b^{4} d^{2}\right )} n^{4} + 6 \, {\left (176 \, b^{7} c d + 15 \, a^{3} b^{4} d^{2}\right )} n^{3} + 5 \, {\left (509 \, b^{7} c d + 33 \, a^{3} b^{4} d^{2}\right )} n^{2} + 18 \, {\left (164 \, b^{7} c d + 5 \, a^{3} b^{4} d^{2}\right )} n\right )} x^{4} + 3 \, {\left (555 \, a b^{6} c^{2} - 4 \, a^{4} b^{3} c d\right )} n^{3} + 2 \, {\left (a b^{6} c d n^{6} + 21 \, a b^{6} c d n^{5} + 163 \, a b^{6} c d n^{4} + 3 \, {\left (189 \, a b^{6} c d - 20 \, a^{4} b^{3} d^{2}\right )} n^{3} + 4 \, {\left (211 \, a b^{6} c d - 45 \, a^{4} b^{3} d^{2}\right )} n^{2} + 60 \, {\left (7 \, a b^{6} c d - 2 \, a^{4} b^{3} d^{2}\right )} n\right )} x^{3} + 8 \, {\left (638 \, a b^{6} c^{2} - 27 \, a^{4} b^{3} c d\right )} n^{2} - 6 \, {\left (a^{2} b^{5} c d n^{5} + 19 \, a^{2} b^{5} c d n^{4} + 125 \, a^{2} b^{5} c d n^{3} + {\left (317 \, a^{2} b^{5} c d - 60 \, a^{5} b^{2} d^{2}\right )} n^{2} + 30 \, {\left (7 \, a^{2} b^{5} c d - 2 \, a^{5} b^{2} d^{2}\right )} n\right )} x^{2} + 12 \, {\left (669 \, a b^{6} c^{2} - 107 \, a^{4} b^{3} c d\right )} n + {\left (b^{7} c^{2} n^{6} + 27 \, b^{7} c^{2} n^{5} + 5040 \, b^{7} c^{2} + {\left (295 \, b^{7} c^{2} + 12 \, a^{3} b^{4} c d\right )} n^{4} + 9 \, {\left (185 \, b^{7} c^{2} + 24 \, a^{3} b^{4} c d\right )} n^{3} + 4 \, {\left (1276 \, b^{7} c^{2} + 321 \, a^{3} b^{4} c d\right )} n^{2} + 36 \, {\left (223 \, b^{7} c^{2} + 70 \, a^{3} b^{4} c d - 20 \, a^{6} b d^{2}\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{7} n^{7} + 28 \, b^{7} n^{6} + 322 \, b^{7} n^{5} + 1960 \, b^{7} n^{4} + 6769 \, b^{7} n^{3} + 13132 \, b^{7} n^{2} + 13068 \, b^{7} n + 5040 \, b^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*b^6*c^2*n^6 + 27*a*b^6*c^2*n^5 + 295*a*b^6*c^2*n^4 + 5040*a*b^6*c^2 - 2520*a^4*b^3*c*d + 720*a^7*d^2 + (b^7
*d^2*n^6 + 21*b^7*d^2*n^5 + 175*b^7*d^2*n^4 + 735*b^7*d^2*n^3 + 1624*b^7*d^2*n^2 + 1764*b^7*d^2*n + 720*b^7*d^
2)*x^7 + (a*b^6*d^2*n^6 + 15*a*b^6*d^2*n^5 + 85*a*b^6*d^2*n^4 + 225*a*b^6*d^2*n^3 + 274*a*b^6*d^2*n^2 + 120*a*
b^6*d^2*n)*x^6 - 6*(a^2*b^5*d^2*n^5 + 10*a^2*b^5*d^2*n^4 + 35*a^2*b^5*d^2*n^3 + 50*a^2*b^5*d^2*n^2 + 24*a^2*b^
5*d^2*n)*x^5 + 2*(b^7*c*d*n^6 + 24*b^7*c*d*n^5 + 1260*b^7*c*d + (226*b^7*c*d + 15*a^3*b^4*d^2)*n^4 + 6*(176*b^
7*c*d + 15*a^3*b^4*d^2)*n^3 + 5*(509*b^7*c*d + 33*a^3*b^4*d^2)*n^2 + 18*(164*b^7*c*d + 5*a^3*b^4*d^2)*n)*x^4 +
3*(555*a*b^6*c^2 - 4*a^4*b^3*c*d)*n^3 + 2*(a*b^6*c*d*n^6 + 21*a*b^6*c*d*n^5 + 163*a*b^6*c*d*n^4 + 3*(189*a*b^
6*c*d - 20*a^4*b^3*d^2)*n^3 + 4*(211*a*b^6*c*d - 45*a^4*b^3*d^2)*n^2 + 60*(7*a*b^6*c*d - 2*a^4*b^3*d^2)*n)*x^3
+ 8*(638*a*b^6*c^2 - 27*a^4*b^3*c*d)*n^2 - 6*(a^2*b^5*c*d*n^5 + 19*a^2*b^5*c*d*n^4 + 125*a^2*b^5*c*d*n^3 + (3
17*a^2*b^5*c*d - 60*a^5*b^2*d^2)*n^2 + 30*(7*a^2*b^5*c*d - 2*a^5*b^2*d^2)*n)*x^2 + 12*(669*a*b^6*c^2 - 107*a^4
*b^3*c*d)*n + (b^7*c^2*n^6 + 27*b^7*c^2*n^5 + 5040*b^7*c^2 + (295*b^7*c^2 + 12*a^3*b^4*c*d)*n^4 + 9*(185*b^7*c
^2 + 24*a^3*b^4*c*d)*n^3 + 4*(1276*b^7*c^2 + 321*a^3*b^4*c*d)*n^2 + 36*(223*b^7*c^2 + 70*a^3*b^4*c*d - 20*a^6*
b*d^2)*n)*x)*(b*x + a)^n/(b^7*n^7 + 28*b^7*n^6 + 322*b^7*n^5 + 1960*b^7*n^4 + 6769*b^7*n^3 + 13132*b^7*n^2 + 1
3068*b^7*n + 5040*b^7)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 11851 vs. \(2 (187) = 374\).
time = 3.79, size = 11851, normalized size = 58.38 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**n*(c**2*x + c*d*x**4/2 + d**2*x**7/7), Eq(b, 0)), (60*a**6*d**2*log(a/b + x)/(60*a**6*b**7 + 360
*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x
**6) + 147*a**6*d**2/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b*
*11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) + 360*a**5*b*d**2*x*log(a/b + x)/(60*a**6*b**7 + 360*a**5*b**8*x
+ 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) + 822*a*
*5*b*d**2*x/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4
+ 360*a*b**12*x**5 + 60*b**13*x**6) + 900*a**4*b**2*d**2*x**2*log(a/b + x)/(60*a**6*b**7 + 360*a**5*b**8*x + 9
00*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) + 1875*a**4
*b**2*d**2*x**2/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x
**4 + 360*a*b**12*x**5 + 60*b**13*x**6) - 2*a**3*b**3*c*d/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2
+ 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) + 1200*a**3*b**3*d**2*x**3*l
og(a/b + x)/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4
+ 360*a*b**12*x**5 + 60*b**13*x**6) + 2200*a**3*b**3*d**2*x**3/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9
*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) - 12*a**2*b**4*c*d*x/(6
0*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*
x**5 + 60*b**13*x**6) + 900*a**2*b**4*d**2*x**4*log(a/b + x)/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x
**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) + 1350*a**2*b**4*d**2*x**
4/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b*
*12*x**5 + 60*b**13*x**6) - 30*a*b**5*c*d*x**2/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**
3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) + 360*a*b**5*d**2*x**5*log(a/b + x)/(60
*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x
**5 + 60*b**13*x**6) + 360*a*b**5*d**2*x**5/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b
**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) - 10*b**6*c**2/(60*a**6*b**7 + 360*a**5*b*
*8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) - 4
0*b**6*c*d*x**3/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x
**4 + 360*a*b**12*x**5 + 60*b**13*x**6) + 60*b**6*d**2*x**6*log(a/b + x)/(60*a**6*b**7 + 360*a**5*b**8*x + 900
*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6), Eq(n, -7)),
(-60*a**6*d**2*log(a/b + x)/(10*a**5*b**7 + 50*a**4*b**8*x + 100*a**3*b**9*x**2 + 100*a**2*b**10*x**3 + 50*a*b
**11*x**4 + 10*b**12*x**5) - 137*a**6*d**2/(10*a**5*b**7 + 50*a**4*b**8*x + 100*a**3*b**9*x**2 + 100*a**2*b**1
0*x**3 + 50*a*b**11*x**4 + 10*b**12*x**5) - 300*a**5*b*d**2*x*log(a/b + x)/(10*a**5*b**7 + 50*a**4*b**8*x + 10
0*a**3*b**9*x**2 + 100*a**2*b**10*x**3 + 50*a*b**11*x**4 + 10*b**12*x**5) - 625*a**5*b*d**2*x/(10*a**5*b**7 +
50*a**4*b**8*x + 100*a**3*b**9*x**2 + 100*a**2*b**10*x**3 + 50*a*b**11*x**4 + 10*b**12*x**5) - 600*a**4*b**2*d
**2*x**2*log(a/b + x)/(10*a**5*b**7 + 50*a**4*b**8*x + 100*a**3*b**9*x**2 + 100*a**2*b**10*x**3 + 50*a*b**11*x
**4 + 10*b**12*x**5) - 1100*a**4*b**2*d**2*x**2/(10*a**5*b**7 + 50*a**4*b**8*x + 100*a**3*b**9*x**2 + 100*a**2
*b**10*x**3 + 50*a*b**11*x**4 + 10*b**12*x**5) - a**3*b**3*c*d/(10*a**5*b**7 + 50*a**4*b**8*x + 100*a**3*b**9*
x**2 + 100*a**2*b**10*x**3 + 50*a*b**11*x**4 + 10*b**12*x**5) - 600*a**3*b**3*d**2*x**3*log(a/b + x)/(10*a**5*
b**7 + 50*a**4*b**8*x + 100*a**3*b**9*x**2 + 100*a**2*b**10*x**3 + 50*a*b**11*x**4 + 10*b**12*x**5) - 900*a**3
*b**3*d**2*x**3/(10*a**5*b**7 + 50*a**4*b**8*x + 100*a**3*b**9*x**2 + 100*a**2*b**10*x**3 + 50*a*b**11*x**4 +
10*b**12*x**5) - 5*a**2*b**4*c*d*x/(10*a**5*b**7 + 50*a**4*b**8*x + 100*a**3*b**9*x**2 + 100*a**2*b**10*x**3 +
50*a*b**11*x**4 + 10*b**12*x**5) - 300*a**2*b**4*d**2*x**4*log(a/b + x)/(10*a**5*b**7 + 50*a**4*b**8*x + 100*
a**3*b**9*x**2 + 100*a**2*b**10*x**3 + 50*a*b**11*x**4 + 10*b**12*x**5) - 300*a**2*b**4*d**2*x**4/(10*a**5*b**
7 + 50*a**4*b**8*x + 100*a**3*b**9*x**2 + 100*a**2*b**10*x**3 + 50*a*b**11*x**4 + 10*b**12*x**5) - 10*a*b**5*c
*d*x**2/(10*a**5*b**7 + 50*a**4*b**8*x + 100*a**3*b**9*x**2 + 100*a**2*b**10*x**3 + 50*a*b**11*x**4 + 10*b**12
*x**5) - 60*a*b**5*d**2*x**5*log(a/b + x)/(10*a**5*b**7 + 50*a**4*b**8*x + 100*a**3*b**9*x**2 + 100*a**2*b**10
*x**3 + 50*a*b**11*x**4 + 10*b**12*x**5) - 2*b**6*c**2/(10*a**5*b**7 + 50*a**4*b**8*x + 100*a**3*b**9*x**2 + 1
00*a**2*b**10*x**3 + 50*a*b**11*x**4 + 10*b**12...

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1477 vs. \(2 (203) = 406\).
time = 4.12, size = 1477, normalized size = 7.28 \begin {gather*} \frac {{\left (b x + a\right )}^{n} b^{7} d^{2} n^{6} x^{7} + {\left (b x + a\right )}^{n} a b^{6} d^{2} n^{6} x^{6} + 21 \, {\left (b x + a\right )}^{n} b^{7} d^{2} n^{5} x^{7} + 15 \, {\left (b x + a\right )}^{n} a b^{6} d^{2} n^{5} x^{6} + 175 \, {\left (b x + a\right )}^{n} b^{7} d^{2} n^{4} x^{7} + 2 \, {\left (b x + a\right )}^{n} b^{7} c d n^{6} x^{4} - 6 \, {\left (b x + a\right )}^{n} a^{2} b^{5} d^{2} n^{5} x^{5} + 85 \, {\left (b x + a\right )}^{n} a b^{6} d^{2} n^{4} x^{6} + 735 \, {\left (b x + a\right )}^{n} b^{7} d^{2} n^{3} x^{7} + 2 \, {\left (b x + a\right )}^{n} a b^{6} c d n^{6} x^{3} + 48 \, {\left (b x + a\right )}^{n} b^{7} c d n^{5} x^{4} - 60 \, {\left (b x + a\right )}^{n} a^{2} b^{5} d^{2} n^{4} x^{5} + 225 \, {\left (b x + a\right )}^{n} a b^{6} d^{2} n^{3} x^{6} + 1624 \, {\left (b x + a\right )}^{n} b^{7} d^{2} n^{2} x^{7} + 42 \, {\left (b x + a\right )}^{n} a b^{6} c d n^{5} x^{3} + 452 \, {\left (b x + a\right )}^{n} b^{7} c d n^{4} x^{4} + 30 \, {\left (b x + a\right )}^{n} a^{3} b^{4} d^{2} n^{4} x^{4} - 210 \, {\left (b x + a\right )}^{n} a^{2} b^{5} d^{2} n^{3} x^{5} + 274 \, {\left (b x + a\right )}^{n} a b^{6} d^{2} n^{2} x^{6} + 1764 \, {\left (b x + a\right )}^{n} b^{7} d^{2} n x^{7} + {\left (b x + a\right )}^{n} b^{7} c^{2} n^{6} x - 6 \, {\left (b x + a\right )}^{n} a^{2} b^{5} c d n^{5} x^{2} + 326 \, {\left (b x + a\right )}^{n} a b^{6} c d n^{4} x^{3} + 2112 \, {\left (b x + a\right )}^{n} b^{7} c d n^{3} x^{4} + 180 \, {\left (b x + a\right )}^{n} a^{3} b^{4} d^{2} n^{3} x^{4} - 300 \, {\left (b x + a\right )}^{n} a^{2} b^{5} d^{2} n^{2} x^{5} + 120 \, {\left (b x + a\right )}^{n} a b^{6} d^{2} n x^{6} + 720 \, {\left (b x + a\right )}^{n} b^{7} d^{2} x^{7} + {\left (b x + a\right )}^{n} a b^{6} c^{2} n^{6} + 27 \, {\left (b x + a\right )}^{n} b^{7} c^{2} n^{5} x - 114 \, {\left (b x + a\right )}^{n} a^{2} b^{5} c d n^{4} x^{2} + 1134 \, {\left (b x + a\right )}^{n} a b^{6} c d n^{3} x^{3} - 120 \, {\left (b x + a\right )}^{n} a^{4} b^{3} d^{2} n^{3} x^{3} + 5090 \, {\left (b x + a\right )}^{n} b^{7} c d n^{2} x^{4} + 330 \, {\left (b x + a\right )}^{n} a^{3} b^{4} d^{2} n^{2} x^{4} - 144 \, {\left (b x + a\right )}^{n} a^{2} b^{5} d^{2} n x^{5} + 27 \, {\left (b x + a\right )}^{n} a b^{6} c^{2} n^{5} + 295 \, {\left (b x + a\right )}^{n} b^{7} c^{2} n^{4} x + 12 \, {\left (b x + a\right )}^{n} a^{3} b^{4} c d n^{4} x - 750 \, {\left (b x + a\right )}^{n} a^{2} b^{5} c d n^{3} x^{2} + 1688 \, {\left (b x + a\right )}^{n} a b^{6} c d n^{2} x^{3} - 360 \, {\left (b x + a\right )}^{n} a^{4} b^{3} d^{2} n^{2} x^{3} + 5904 \, {\left (b x + a\right )}^{n} b^{7} c d n x^{4} + 180 \, {\left (b x + a\right )}^{n} a^{3} b^{4} d^{2} n x^{4} + 295 \, {\left (b x + a\right )}^{n} a b^{6} c^{2} n^{4} + 1665 \, {\left (b x + a\right )}^{n} b^{7} c^{2} n^{3} x + 216 \, {\left (b x + a\right )}^{n} a^{3} b^{4} c d n^{3} x - 1902 \, {\left (b x + a\right )}^{n} a^{2} b^{5} c d n^{2} x^{2} + 360 \, {\left (b x + a\right )}^{n} a^{5} b^{2} d^{2} n^{2} x^{2} + 840 \, {\left (b x + a\right )}^{n} a b^{6} c d n x^{3} - 240 \, {\left (b x + a\right )}^{n} a^{4} b^{3} d^{2} n x^{3} + 2520 \, {\left (b x + a\right )}^{n} b^{7} c d x^{4} + 1665 \, {\left (b x + a\right )}^{n} a b^{6} c^{2} n^{3} - 12 \, {\left (b x + a\right )}^{n} a^{4} b^{3} c d n^{3} + 5104 \, {\left (b x + a\right )}^{n} b^{7} c^{2} n^{2} x + 1284 \, {\left (b x + a\right )}^{n} a^{3} b^{4} c d n^{2} x - 1260 \, {\left (b x + a\right )}^{n} a^{2} b^{5} c d n x^{2} + 360 \, {\left (b x + a\right )}^{n} a^{5} b^{2} d^{2} n x^{2} + 5104 \, {\left (b x + a\right )}^{n} a b^{6} c^{2} n^{2} - 216 \, {\left (b x + a\right )}^{n} a^{4} b^{3} c d n^{2} + 8028 \, {\left (b x + a\right )}^{n} b^{7} c^{2} n x + 2520 \, {\left (b x + a\right )}^{n} a^{3} b^{4} c d n x - 720 \, {\left (b x + a\right )}^{n} a^{6} b d^{2} n x + 8028 \, {\left (b x + a\right )}^{n} a b^{6} c^{2} n - 1284 \, {\left (b x + a\right )}^{n} a^{4} b^{3} c d n + 5040 \, {\left (b x + a\right )}^{n} b^{7} c^{2} x + 5040 \, {\left (b x + a\right )}^{n} a b^{6} c^{2} - 2520 \, {\left (b x + a\right )}^{n} a^{4} b^{3} c d + 720 \, {\left (b x + a\right )}^{n} a^{7} d^{2}}{b^{7} n^{7} + 28 \, b^{7} n^{6} + 322 \, b^{7} n^{5} + 1960 \, b^{7} n^{4} + 6769 \, b^{7} n^{3} + 13132 \, b^{7} n^{2} + 13068 \, b^{7} n + 5040 \, b^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*x + a)^n*b^7*d^2*n^6*x^7 + (b*x + a)^n*a*b^6*d^2*n^6*x^6 + 21*(b*x + a)^n*b^7*d^2*n^5*x^7 + 15*(b*x + a)^n
*a*b^6*d^2*n^5*x^6 + 175*(b*x + a)^n*b^7*d^2*n^4*x^7 + 2*(b*x + a)^n*b^7*c*d*n^6*x^4 - 6*(b*x + a)^n*a^2*b^5*d
^2*n^5*x^5 + 85*(b*x + a)^n*a*b^6*d^2*n^4*x^6 + 735*(b*x + a)^n*b^7*d^2*n^3*x^7 + 2*(b*x + a)^n*a*b^6*c*d*n^6*
x^3 + 48*(b*x + a)^n*b^7*c*d*n^5*x^4 - 60*(b*x + a)^n*a^2*b^5*d^2*n^4*x^5 + 225*(b*x + a)^n*a*b^6*d^2*n^3*x^6
+ 1624*(b*x + a)^n*b^7*d^2*n^2*x^7 + 42*(b*x + a)^n*a*b^6*c*d*n^5*x^3 + 452*(b*x + a)^n*b^7*c*d*n^4*x^4 + 30*(
b*x + a)^n*a^3*b^4*d^2*n^4*x^4 - 210*(b*x + a)^n*a^2*b^5*d^2*n^3*x^5 + 274*(b*x + a)^n*a*b^6*d^2*n^2*x^6 + 176
4*(b*x + a)^n*b^7*d^2*n*x^7 + (b*x + a)^n*b^7*c^2*n^6*x - 6*(b*x + a)^n*a^2*b^5*c*d*n^5*x^2 + 326*(b*x + a)^n*
a*b^6*c*d*n^4*x^3 + 2112*(b*x + a)^n*b^7*c*d*n^3*x^4 + 180*(b*x + a)^n*a^3*b^4*d^2*n^3*x^4 - 300*(b*x + a)^n*a
^2*b^5*d^2*n^2*x^5 + 120*(b*x + a)^n*a*b^6*d^2*n*x^6 + 720*(b*x + a)^n*b^7*d^2*x^7 + (b*x + a)^n*a*b^6*c^2*n^6
+ 27*(b*x + a)^n*b^7*c^2*n^5*x - 114*(b*x + a)^n*a^2*b^5*c*d*n^4*x^2 + 1134*(b*x + a)^n*a*b^6*c*d*n^3*x^3 - 1
20*(b*x + a)^n*a^4*b^3*d^2*n^3*x^3 + 5090*(b*x + a)^n*b^7*c*d*n^2*x^4 + 330*(b*x + a)^n*a^3*b^4*d^2*n^2*x^4 -
144*(b*x + a)^n*a^2*b^5*d^2*n*x^5 + 27*(b*x + a)^n*a*b^6*c^2*n^5 + 295*(b*x + a)^n*b^7*c^2*n^4*x + 12*(b*x + a
)^n*a^3*b^4*c*d*n^4*x - 750*(b*x + a)^n*a^2*b^5*c*d*n^3*x^2 + 1688*(b*x + a)^n*a*b^6*c*d*n^2*x^3 - 360*(b*x +
a)^n*a^4*b^3*d^2*n^2*x^3 + 5904*(b*x + a)^n*b^7*c*d*n*x^4 + 180*(b*x + a)^n*a^3*b^4*d^2*n*x^4 + 295*(b*x + a)^
n*a*b^6*c^2*n^4 + 1665*(b*x + a)^n*b^7*c^2*n^3*x + 216*(b*x + a)^n*a^3*b^4*c*d*n^3*x - 1902*(b*x + a)^n*a^2*b^
5*c*d*n^2*x^2 + 360*(b*x + a)^n*a^5*b^2*d^2*n^2*x^2 + 840*(b*x + a)^n*a*b^6*c*d*n*x^3 - 240*(b*x + a)^n*a^4*b^
3*d^2*n*x^3 + 2520*(b*x + a)^n*b^7*c*d*x^4 + 1665*(b*x + a)^n*a*b^6*c^2*n^3 - 12*(b*x + a)^n*a^4*b^3*c*d*n^3 +
5104*(b*x + a)^n*b^7*c^2*n^2*x + 1284*(b*x + a)^n*a^3*b^4*c*d*n^2*x - 1260*(b*x + a)^n*a^2*b^5*c*d*n*x^2 + 36
0*(b*x + a)^n*a^5*b^2*d^2*n*x^2 + 5104*(b*x + a)^n*a*b^6*c^2*n^2 - 216*(b*x + a)^n*a^4*b^3*c*d*n^2 + 8028*(b*x
+ a)^n*b^7*c^2*n*x + 2520*(b*x + a)^n*a^3*b^4*c*d*n*x - 720*(b*x + a)^n*a^6*b*d^2*n*x + 8028*(b*x + a)^n*a*b^
6*c^2*n - 1284*(b*x + a)^n*a^4*b^3*c*d*n + 5040*(b*x + a)^n*b^7*c^2*x + 5040*(b*x + a)^n*a*b^6*c^2 - 2520*(b*x
+ a)^n*a^4*b^3*c*d + 720*(b*x + a)^n*a^7*d^2)/(b^7*n^7 + 28*b^7*n^6 + 322*b^7*n^5 + 1960*b^7*n^4 + 6769*b^7*n
^3 + 13132*b^7*n^2 + 13068*b^7*n + 5040*b^7)

________________________________________________________________________________________

Mupad [B]
time = 3.19, size = 878, normalized size = 4.33 \begin {gather*} \frac {a\,{\left (a+b\,x\right )}^n\,\left (720\,a^6\,d^2-12\,a^3\,b^3\,c\,d\,n^3-216\,a^3\,b^3\,c\,d\,n^2-1284\,a^3\,b^3\,c\,d\,n-2520\,a^3\,b^3\,c\,d+b^6\,c^2\,n^6+27\,b^6\,c^2\,n^5+295\,b^6\,c^2\,n^4+1665\,b^6\,c^2\,n^3+5104\,b^6\,c^2\,n^2+8028\,b^6\,c^2\,n+5040\,b^6\,c^2\right )}{b^7\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {d^2\,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 )}{n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040}+\frac {x\,{\left (a+b\,x\right )}^n\,\left (-720\,a^6\,b\,d^2\,n+12\,a^3\,b^4\,c\,d\,n^4+216\,a^3\,b^4\,c\,d\,n^3+1284\,a^3\,b^4\,c\,d\,n^2+2520\,a^3\,b^4\,c\,d\,n+b^7\,c^2\,n^6+27\,b^7\,c^2\,n^5+295\,b^7\,c^2\,n^4+1665\,b^7\,c^2\,n^3+5104\,b^7\,c^2\,n^2+8028\,b^7\,c^2\,n+5040\,b^7\,c^2\right )}{b^7\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {2\,d\,x^4\,{\left (a+b\,x\right )}^n\,\left (n^3+6\,n^2+11\,n+6\right )\,\left (15\,d\,a^3\,n+c\,b^3\,n^3+18\,c\,b^3\,n^2+107\,c\,b^3\,n+210\,c\,b^3\right )}{b^3\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {a\,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\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}-\frac {6\,a^2\,d^2\,n\,x^5\,{\left (a+b\,x\right )}^n\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}{b^2\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {2\,a\,d\,n\,x^3\,{\left (a+b\,x\right )}^n\,\left (n^2+3\,n+2\right )\,\left (-60\,d\,a^3+c\,b^3\,n^3+18\,c\,b^3\,n^2+107\,c\,b^3\,n+210\,c\,b^3\right )}{b^4\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}-\frac {6\,a^2\,d\,n\,x^2\,\left (n+1\right )\,{\left (a+b\,x\right )}^n\,\left (-60\,d\,a^3+c\,b^3\,n^3+18\,c\,b^3\,n^2+107\,c\,b^3\,n+210\,c\,b^3\right )}{b^5\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^3)^2*(a + b*x)^n,x)

[Out]

(a*(a + b*x)^n*(720*a^6*d^2 + 5040*b^6*c^2 + 8028*b^6*c^2*n + 5104*b^6*c^2*n^2 + 1665*b^6*c^2*n^3 + 295*b^6*c^
2*n^4 + 27*b^6*c^2*n^5 + b^6*c^2*n^6 - 2520*a^3*b^3*c*d - 1284*a^3*b^3*c*d*n - 216*a^3*b^3*c*d*n^2 - 12*a^3*b^
3*c*d*n^3))/(b^7*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040)) + (d^2*x^7*(a +
b*x)^n*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720))/(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^
4 + 322*n^5 + 28*n^6 + n^7 + 5040) + (x*(a + b*x)^n*(5040*b^7*c^2 + 8028*b^7*c^2*n + 5104*b^7*c^2*n^2 + 1665*b
^7*c^2*n^3 + 295*b^7*c^2*n^4 + 27*b^7*c^2*n^5 + b^7*c^2*n^6 - 720*a^6*b*d^2*n + 2520*a^3*b^4*c*d*n + 1284*a^3*
b^4*c*d*n^2 + 216*a^3*b^4*c*d*n^3 + 12*a^3*b^4*c*d*n^4))/(b^7*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322
*n^5 + 28*n^6 + n^7 + 5040)) + (2*d*x^4*(a + b*x)^n*(11*n + 6*n^2 + n^3 + 6)*(210*b^3*c + 18*b^3*c*n^2 + b^3*c
*n^3 + 15*a^3*d*n + 107*b^3*c*n))/(b^3*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5
040)) + (a*d^2*n*x^6*(a + b*x)^n*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120))/(b*(13068*n + 13132*n^2 + 67
69*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040)) - (6*a^2*d^2*n*x^5*(a + b*x)^n*(50*n + 35*n^2 + 10*n^3 + n
^4 + 24))/(b^2*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040)) + (2*a*d*n*x^3*(a
+ b*x)^n*(3*n + n^2 + 2)*(210*b^3*c - 60*a^3*d + 18*b^3*c*n^2 + b^3*c*n^3 + 107*b^3*c*n))/(b^4*(13068*n + 1313
2*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040)) - (6*a^2*d*n*x^2*(n + 1)*(a + b*x)^n*(210*b^3*c
- 60*a^3*d + 18*b^3*c*n^2 + b^3*c*n^3 + 107*b^3*c*n))/(b^5*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^
5 + 28*n^6 + n^7 + 5040))

________________________________________________________________________________________