3.1919 \(\int (a+b x) (d+e x)^3 (a^2+2 a b x+b^2 x^2)^3 \, dx\)

Optimal. Leaf size=92 \[ \frac{3 e^2 (a+b x)^{10} (b d-a e)}{10 b^4}+\frac{e (a+b x)^9 (b d-a e)^2}{3 b^4}+\frac{(a+b x)^8 (b d-a e)^3}{8 b^4}+\frac{e^3 (a+b x)^{11}}{11 b^4} \]

[Out]

((b*d - a*e)^3*(a + b*x)^8)/(8*b^4) + (e*(b*d - a*e)^2*(a + b*x)^9)/(3*b^4) + (3*e^2*(b*d - a*e)*(a + b*x)^10)
/(10*b^4) + (e^3*(a + b*x)^11)/(11*b^4)

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Rubi [A]  time = 0.220042, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 43} \[ \frac{3 e^2 (a+b x)^{10} (b d-a e)}{10 b^4}+\frac{e (a+b x)^9 (b d-a e)^2}{3 b^4}+\frac{(a+b x)^8 (b d-a e)^3}{8 b^4}+\frac{e^3 (a+b x)^{11}}{11 b^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

((b*d - a*e)^3*(a + b*x)^8)/(8*b^4) + (e*(b*d - a*e)^2*(a + b*x)^9)/(3*b^4) + (3*e^2*(b*d - a*e)*(a + b*x)^10)
/(10*b^4) + (e^3*(a + b*x)^11)/(11*b^4)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (a+b x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^7 (d+e x)^3 \, dx\\ &=\int \left (\frac{(b d-a e)^3 (a+b x)^7}{b^3}+\frac{3 e (b d-a e)^2 (a+b x)^8}{b^3}+\frac{3 e^2 (b d-a e) (a+b x)^9}{b^3}+\frac{e^3 (a+b x)^{10}}{b^3}\right ) \, dx\\ &=\frac{(b d-a e)^3 (a+b x)^8}{8 b^4}+\frac{e (b d-a e)^2 (a+b x)^9}{3 b^4}+\frac{3 e^2 (b d-a e) (a+b x)^{10}}{10 b^4}+\frac{e^3 (a+b x)^{11}}{11 b^4}\\ \end{align*}

Mathematica [B]  time = 0.0525224, size = 360, normalized size = 3.91 \[ \frac{1}{3} b^5 e x^9 \left (7 a^2 e^2+7 a b d e+b^2 d^2\right )+\frac{1}{8} b^4 x^8 \left (63 a^2 b d e^2+35 a^3 e^3+21 a b^2 d^2 e+b^3 d^3\right )+a b^3 x^7 \left (15 a^2 b d e^2+5 a^3 e^3+9 a b^2 d^2 e+b^3 d^3\right )+\frac{7}{2} a^2 b^2 x^6 \left (5 a^2 b d e^2+a^3 e^3+5 a b^2 d^2 e+b^3 d^3\right )+\frac{7}{5} a^3 b x^5 \left (9 a^2 b d e^2+a^3 e^3+15 a b^2 d^2 e+5 b^3 d^3\right )+\frac{1}{4} a^4 x^4 \left (21 a^2 b d e^2+a^3 e^3+63 a b^2 d^2 e+35 b^3 d^3\right )+a^5 d x^3 \left (a^2 e^2+7 a b d e+7 b^2 d^2\right )+\frac{1}{2} a^6 d^2 x^2 (3 a e+7 b d)+a^7 d^3 x+\frac{1}{10} b^6 e^2 x^{10} (7 a e+3 b d)+\frac{1}{11} b^7 e^3 x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^7*d^3*x + (a^6*d^2*(7*b*d + 3*a*e)*x^2)/2 + a^5*d*(7*b^2*d^2 + 7*a*b*d*e + a^2*e^2)*x^3 + (a^4*(35*b^3*d^3 +
63*a*b^2*d^2*e + 21*a^2*b*d*e^2 + a^3*e^3)*x^4)/4 + (7*a^3*b*(5*b^3*d^3 + 15*a*b^2*d^2*e + 9*a^2*b*d*e^2 + a^
3*e^3)*x^5)/5 + (7*a^2*b^2*(b^3*d^3 + 5*a*b^2*d^2*e + 5*a^2*b*d*e^2 + a^3*e^3)*x^6)/2 + a*b^3*(b^3*d^3 + 9*a*b
^2*d^2*e + 15*a^2*b*d*e^2 + 5*a^3*e^3)*x^7 + (b^4*(b^3*d^3 + 21*a*b^2*d^2*e + 63*a^2*b*d*e^2 + 35*a^3*e^3)*x^8
)/8 + (b^5*e*(b^2*d^2 + 7*a*b*d*e + 7*a^2*e^2)*x^9)/3 + (b^6*e^2*(3*b*d + 7*a*e)*x^10)/10 + (b^7*e^3*x^11)/11

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Maple [B]  time = 0., size = 616, normalized size = 6.7 \begin{align*}{\frac{{b}^{7}{e}^{3}{x}^{11}}{11}}+{\frac{ \left ( \left ( a{e}^{3}+3\,bd{e}^{2} \right ){b}^{6}+6\,{b}^{6}{e}^{3}a \right ){x}^{10}}{10}}+{\frac{ \left ( \left ( 3\,ad{e}^{2}+3\,b{d}^{2}e \right ){b}^{6}+6\, \left ( a{e}^{3}+3\,bd{e}^{2} \right ) a{b}^{5}+15\,{b}^{5}{e}^{3}{a}^{2} \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 3\,a{d}^{2}e+b{d}^{3} \right ){b}^{6}+6\, \left ( 3\,ad{e}^{2}+3\,b{d}^{2}e \right ) a{b}^{5}+15\, \left ( a{e}^{3}+3\,bd{e}^{2} \right ){a}^{2}{b}^{4}+20\,{b}^{4}{e}^{3}{a}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( a{d}^{3}{b}^{6}+6\, \left ( 3\,a{d}^{2}e+b{d}^{3} \right ) a{b}^{5}+15\, \left ( 3\,ad{e}^{2}+3\,b{d}^{2}e \right ){a}^{2}{b}^{4}+20\, \left ( a{e}^{3}+3\,bd{e}^{2} \right ){a}^{3}{b}^{3}+15\,{b}^{3}{e}^{3}{a}^{4} \right ){x}^{7}}{7}}+{\frac{ \left ( 6\,{a}^{2}{d}^{3}{b}^{5}+15\, \left ( 3\,a{d}^{2}e+b{d}^{3} \right ){a}^{2}{b}^{4}+20\, \left ( 3\,ad{e}^{2}+3\,b{d}^{2}e \right ){a}^{3}{b}^{3}+15\, \left ( a{e}^{3}+3\,bd{e}^{2} \right ){a}^{4}{b}^{2}+6\,{b}^{2}{e}^{3}{a}^{5} \right ){x}^{6}}{6}}+{\frac{ \left ( 15\,{a}^{3}{d}^{3}{b}^{4}+20\, \left ( 3\,a{d}^{2}e+b{d}^{3} \right ){a}^{3}{b}^{3}+15\, \left ( 3\,ad{e}^{2}+3\,b{d}^{2}e \right ){a}^{4}{b}^{2}+6\, \left ( a{e}^{3}+3\,bd{e}^{2} \right ){a}^{5}b+b{e}^{3}{a}^{6} \right ){x}^{5}}{5}}+{\frac{ \left ( 20\,{a}^{4}{d}^{3}{b}^{3}+15\, \left ( 3\,a{d}^{2}e+b{d}^{3} \right ){a}^{4}{b}^{2}+6\, \left ( 3\,ad{e}^{2}+3\,b{d}^{2}e \right ){a}^{5}b+ \left ( a{e}^{3}+3\,bd{e}^{2} \right ){a}^{6} \right ){x}^{4}}{4}}+{\frac{ \left ( 15\,{a}^{5}{d}^{3}{b}^{2}+6\, \left ( 3\,a{d}^{2}e+b{d}^{3} \right ){a}^{5}b+ \left ( 3\,ad{e}^{2}+3\,b{d}^{2}e \right ){a}^{6} \right ){x}^{3}}{3}}+{\frac{ \left ( 6\,{a}^{6}{d}^{3}b+ \left ( 3\,a{d}^{2}e+b{d}^{3} \right ){a}^{6} \right ){x}^{2}}{2}}+{a}^{7}{d}^{3}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

1/11*b^7*e^3*x^11+1/10*((a*e^3+3*b*d*e^2)*b^6+6*b^6*e^3*a)*x^10+1/9*((3*a*d*e^2+3*b*d^2*e)*b^6+6*(a*e^3+3*b*d*
e^2)*a*b^5+15*b^5*e^3*a^2)*x^9+1/8*((3*a*d^2*e+b*d^3)*b^6+6*(3*a*d*e^2+3*b*d^2*e)*a*b^5+15*(a*e^3+3*b*d*e^2)*a
^2*b^4+20*b^4*e^3*a^3)*x^8+1/7*(a*d^3*b^6+6*(3*a*d^2*e+b*d^3)*a*b^5+15*(3*a*d*e^2+3*b*d^2*e)*a^2*b^4+20*(a*e^3
+3*b*d*e^2)*a^3*b^3+15*b^3*e^3*a^4)*x^7+1/6*(6*a^2*d^3*b^5+15*(3*a*d^2*e+b*d^3)*a^2*b^4+20*(3*a*d*e^2+3*b*d^2*
e)*a^3*b^3+15*(a*e^3+3*b*d*e^2)*a^4*b^2+6*b^2*e^3*a^5)*x^6+1/5*(15*a^3*d^3*b^4+20*(3*a*d^2*e+b*d^3)*a^3*b^3+15
*(3*a*d*e^2+3*b*d^2*e)*a^4*b^2+6*(a*e^3+3*b*d*e^2)*a^5*b+b*e^3*a^6)*x^5+1/4*(20*a^4*d^3*b^3+15*(3*a*d^2*e+b*d^
3)*a^4*b^2+6*(3*a*d*e^2+3*b*d^2*e)*a^5*b+(a*e^3+3*b*d*e^2)*a^6)*x^4+1/3*(15*a^5*d^3*b^2+6*(3*a*d^2*e+b*d^3)*a^
5*b+(3*a*d*e^2+3*b*d^2*e)*a^6)*x^3+1/2*(6*a^6*d^3*b+(3*a*d^2*e+b*d^3)*a^6)*x^2+a^7*d^3*x

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Maxima [B]  time = 0.971136, size = 508, normalized size = 5.52 \begin{align*} \frac{1}{11} \, b^{7} e^{3} x^{11} + a^{7} d^{3} x + \frac{1}{10} \,{\left (3 \, b^{7} d e^{2} + 7 \, a b^{6} e^{3}\right )} x^{10} + \frac{1}{3} \,{\left (b^{7} d^{2} e + 7 \, a b^{6} d e^{2} + 7 \, a^{2} b^{5} e^{3}\right )} x^{9} + \frac{1}{8} \,{\left (b^{7} d^{3} + 21 \, a b^{6} d^{2} e + 63 \, a^{2} b^{5} d e^{2} + 35 \, a^{3} b^{4} e^{3}\right )} x^{8} +{\left (a b^{6} d^{3} + 9 \, a^{2} b^{5} d^{2} e + 15 \, a^{3} b^{4} d e^{2} + 5 \, a^{4} b^{3} e^{3}\right )} x^{7} + \frac{7}{2} \,{\left (a^{2} b^{5} d^{3} + 5 \, a^{3} b^{4} d^{2} e + 5 \, a^{4} b^{3} d e^{2} + a^{5} b^{2} e^{3}\right )} x^{6} + \frac{7}{5} \,{\left (5 \, a^{3} b^{4} d^{3} + 15 \, a^{4} b^{3} d^{2} e + 9 \, a^{5} b^{2} d e^{2} + a^{6} b e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (35 \, a^{4} b^{3} d^{3} + 63 \, a^{5} b^{2} d^{2} e + 21 \, a^{6} b d e^{2} + a^{7} e^{3}\right )} x^{4} +{\left (7 \, a^{5} b^{2} d^{3} + 7 \, a^{6} b d^{2} e + a^{7} d e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (7 \, a^{6} b d^{3} + 3 \, a^{7} d^{2} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*b^7*e^3*x^11 + a^7*d^3*x + 1/10*(3*b^7*d*e^2 + 7*a*b^6*e^3)*x^10 + 1/3*(b^7*d^2*e + 7*a*b^6*d*e^2 + 7*a^2
*b^5*e^3)*x^9 + 1/8*(b^7*d^3 + 21*a*b^6*d^2*e + 63*a^2*b^5*d*e^2 + 35*a^3*b^4*e^3)*x^8 + (a*b^6*d^3 + 9*a^2*b^
5*d^2*e + 15*a^3*b^4*d*e^2 + 5*a^4*b^3*e^3)*x^7 + 7/2*(a^2*b^5*d^3 + 5*a^3*b^4*d^2*e + 5*a^4*b^3*d*e^2 + a^5*b
^2*e^3)*x^6 + 7/5*(5*a^3*b^4*d^3 + 15*a^4*b^3*d^2*e + 9*a^5*b^2*d*e^2 + a^6*b*e^3)*x^5 + 1/4*(35*a^4*b^3*d^3 +
63*a^5*b^2*d^2*e + 21*a^6*b*d*e^2 + a^7*e^3)*x^4 + (7*a^5*b^2*d^3 + 7*a^6*b*d^2*e + a^7*d*e^2)*x^3 + 1/2*(7*a
^6*b*d^3 + 3*a^7*d^2*e)*x^2

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Fricas [B]  time = 1.33962, size = 913, normalized size = 9.92 \begin{align*} \frac{1}{11} x^{11} e^{3} b^{7} + \frac{3}{10} x^{10} e^{2} d b^{7} + \frac{7}{10} x^{10} e^{3} b^{6} a + \frac{1}{3} x^{9} e d^{2} b^{7} + \frac{7}{3} x^{9} e^{2} d b^{6} a + \frac{7}{3} x^{9} e^{3} b^{5} a^{2} + \frac{1}{8} x^{8} d^{3} b^{7} + \frac{21}{8} x^{8} e d^{2} b^{6} a + \frac{63}{8} x^{8} e^{2} d b^{5} a^{2} + \frac{35}{8} x^{8} e^{3} b^{4} a^{3} + x^{7} d^{3} b^{6} a + 9 x^{7} e d^{2} b^{5} a^{2} + 15 x^{7} e^{2} d b^{4} a^{3} + 5 x^{7} e^{3} b^{3} a^{4} + \frac{7}{2} x^{6} d^{3} b^{5} a^{2} + \frac{35}{2} x^{6} e d^{2} b^{4} a^{3} + \frac{35}{2} x^{6} e^{2} d b^{3} a^{4} + \frac{7}{2} x^{6} e^{3} b^{2} a^{5} + 7 x^{5} d^{3} b^{4} a^{3} + 21 x^{5} e d^{2} b^{3} a^{4} + \frac{63}{5} x^{5} e^{2} d b^{2} a^{5} + \frac{7}{5} x^{5} e^{3} b a^{6} + \frac{35}{4} x^{4} d^{3} b^{3} a^{4} + \frac{63}{4} x^{4} e d^{2} b^{2} a^{5} + \frac{21}{4} x^{4} e^{2} d b a^{6} + \frac{1}{4} x^{4} e^{3} a^{7} + 7 x^{3} d^{3} b^{2} a^{5} + 7 x^{3} e d^{2} b a^{6} + x^{3} e^{2} d a^{7} + \frac{7}{2} x^{2} d^{3} b a^{6} + \frac{3}{2} x^{2} e d^{2} a^{7} + x d^{3} a^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*x^11*e^3*b^7 + 3/10*x^10*e^2*d*b^7 + 7/10*x^10*e^3*b^6*a + 1/3*x^9*e*d^2*b^7 + 7/3*x^9*e^2*d*b^6*a + 7/3*
x^9*e^3*b^5*a^2 + 1/8*x^8*d^3*b^7 + 21/8*x^8*e*d^2*b^6*a + 63/8*x^8*e^2*d*b^5*a^2 + 35/8*x^8*e^3*b^4*a^3 + x^7
*d^3*b^6*a + 9*x^7*e*d^2*b^5*a^2 + 15*x^7*e^2*d*b^4*a^3 + 5*x^7*e^3*b^3*a^4 + 7/2*x^6*d^3*b^5*a^2 + 35/2*x^6*e
*d^2*b^4*a^3 + 35/2*x^6*e^2*d*b^3*a^4 + 7/2*x^6*e^3*b^2*a^5 + 7*x^5*d^3*b^4*a^3 + 21*x^5*e*d^2*b^3*a^4 + 63/5*
x^5*e^2*d*b^2*a^5 + 7/5*x^5*e^3*b*a^6 + 35/4*x^4*d^3*b^3*a^4 + 63/4*x^4*e*d^2*b^2*a^5 + 21/4*x^4*e^2*d*b*a^6 +
1/4*x^4*e^3*a^7 + 7*x^3*d^3*b^2*a^5 + 7*x^3*e*d^2*b*a^6 + x^3*e^2*d*a^7 + 7/2*x^2*d^3*b*a^6 + 3/2*x^2*e*d^2*a
^7 + x*d^3*a^7

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Sympy [B]  time = 0.123707, size = 427, normalized size = 4.64 \begin{align*} a^{7} d^{3} x + \frac{b^{7} e^{3} x^{11}}{11} + x^{10} \left (\frac{7 a b^{6} e^{3}}{10} + \frac{3 b^{7} d e^{2}}{10}\right ) + x^{9} \left (\frac{7 a^{2} b^{5} e^{3}}{3} + \frac{7 a b^{6} d e^{2}}{3} + \frac{b^{7} d^{2} e}{3}\right ) + x^{8} \left (\frac{35 a^{3} b^{4} e^{3}}{8} + \frac{63 a^{2} b^{5} d e^{2}}{8} + \frac{21 a b^{6} d^{2} e}{8} + \frac{b^{7} d^{3}}{8}\right ) + x^{7} \left (5 a^{4} b^{3} e^{3} + 15 a^{3} b^{4} d e^{2} + 9 a^{2} b^{5} d^{2} e + a b^{6} d^{3}\right ) + x^{6} \left (\frac{7 a^{5} b^{2} e^{3}}{2} + \frac{35 a^{4} b^{3} d e^{2}}{2} + \frac{35 a^{3} b^{4} d^{2} e}{2} + \frac{7 a^{2} b^{5} d^{3}}{2}\right ) + x^{5} \left (\frac{7 a^{6} b e^{3}}{5} + \frac{63 a^{5} b^{2} d e^{2}}{5} + 21 a^{4} b^{3} d^{2} e + 7 a^{3} b^{4} d^{3}\right ) + x^{4} \left (\frac{a^{7} e^{3}}{4} + \frac{21 a^{6} b d e^{2}}{4} + \frac{63 a^{5} b^{2} d^{2} e}{4} + \frac{35 a^{4} b^{3} d^{3}}{4}\right ) + x^{3} \left (a^{7} d e^{2} + 7 a^{6} b d^{2} e + 7 a^{5} b^{2} d^{3}\right ) + x^{2} \left (\frac{3 a^{7} d^{2} e}{2} + \frac{7 a^{6} b d^{3}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**7*d**3*x + b**7*e**3*x**11/11 + x**10*(7*a*b**6*e**3/10 + 3*b**7*d*e**2/10) + x**9*(7*a**2*b**5*e**3/3 + 7*
a*b**6*d*e**2/3 + b**7*d**2*e/3) + x**8*(35*a**3*b**4*e**3/8 + 63*a**2*b**5*d*e**2/8 + 21*a*b**6*d**2*e/8 + b*
*7*d**3/8) + x**7*(5*a**4*b**3*e**3 + 15*a**3*b**4*d*e**2 + 9*a**2*b**5*d**2*e + a*b**6*d**3) + x**6*(7*a**5*b
**2*e**3/2 + 35*a**4*b**3*d*e**2/2 + 35*a**3*b**4*d**2*e/2 + 7*a**2*b**5*d**3/2) + x**5*(7*a**6*b*e**3/5 + 63*
a**5*b**2*d*e**2/5 + 21*a**4*b**3*d**2*e + 7*a**3*b**4*d**3) + x**4*(a**7*e**3/4 + 21*a**6*b*d*e**2/4 + 63*a**
5*b**2*d**2*e/4 + 35*a**4*b**3*d**3/4) + x**3*(a**7*d*e**2 + 7*a**6*b*d**2*e + 7*a**5*b**2*d**3) + x**2*(3*a**
7*d**2*e/2 + 7*a**6*b*d**3/2)

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Giac [B]  time = 1.14518, size = 556, normalized size = 6.04 \begin{align*} \frac{1}{11} \, b^{7} x^{11} e^{3} + \frac{3}{10} \, b^{7} d x^{10} e^{2} + \frac{1}{3} \, b^{7} d^{2} x^{9} e + \frac{1}{8} \, b^{7} d^{3} x^{8} + \frac{7}{10} \, a b^{6} x^{10} e^{3} + \frac{7}{3} \, a b^{6} d x^{9} e^{2} + \frac{21}{8} \, a b^{6} d^{2} x^{8} e + a b^{6} d^{3} x^{7} + \frac{7}{3} \, a^{2} b^{5} x^{9} e^{3} + \frac{63}{8} \, a^{2} b^{5} d x^{8} e^{2} + 9 \, a^{2} b^{5} d^{2} x^{7} e + \frac{7}{2} \, a^{2} b^{5} d^{3} x^{6} + \frac{35}{8} \, a^{3} b^{4} x^{8} e^{3} + 15 \, a^{3} b^{4} d x^{7} e^{2} + \frac{35}{2} \, a^{3} b^{4} d^{2} x^{6} e + 7 \, a^{3} b^{4} d^{3} x^{5} + 5 \, a^{4} b^{3} x^{7} e^{3} + \frac{35}{2} \, a^{4} b^{3} d x^{6} e^{2} + 21 \, a^{4} b^{3} d^{2} x^{5} e + \frac{35}{4} \, a^{4} b^{3} d^{3} x^{4} + \frac{7}{2} \, a^{5} b^{2} x^{6} e^{3} + \frac{63}{5} \, a^{5} b^{2} d x^{5} e^{2} + \frac{63}{4} \, a^{5} b^{2} d^{2} x^{4} e + 7 \, a^{5} b^{2} d^{3} x^{3} + \frac{7}{5} \, a^{6} b x^{5} e^{3} + \frac{21}{4} \, a^{6} b d x^{4} e^{2} + 7 \, a^{6} b d^{2} x^{3} e + \frac{7}{2} \, a^{6} b d^{3} x^{2} + \frac{1}{4} \, a^{7} x^{4} e^{3} + a^{7} d x^{3} e^{2} + \frac{3}{2} \, a^{7} d^{2} x^{2} e + a^{7} d^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*b^7*x^11*e^3 + 3/10*b^7*d*x^10*e^2 + 1/3*b^7*d^2*x^9*e + 1/8*b^7*d^3*x^8 + 7/10*a*b^6*x^10*e^3 + 7/3*a*b^
6*d*x^9*e^2 + 21/8*a*b^6*d^2*x^8*e + a*b^6*d^3*x^7 + 7/3*a^2*b^5*x^9*e^3 + 63/8*a^2*b^5*d*x^8*e^2 + 9*a^2*b^5*
d^2*x^7*e + 7/2*a^2*b^5*d^3*x^6 + 35/8*a^3*b^4*x^8*e^3 + 15*a^3*b^4*d*x^7*e^2 + 35/2*a^3*b^4*d^2*x^6*e + 7*a^3
*b^4*d^3*x^5 + 5*a^4*b^3*x^7*e^3 + 35/2*a^4*b^3*d*x^6*e^2 + 21*a^4*b^3*d^2*x^5*e + 35/4*a^4*b^3*d^3*x^4 + 7/2*
a^5*b^2*x^6*e^3 + 63/5*a^5*b^2*d*x^5*e^2 + 63/4*a^5*b^2*d^2*x^4*e + 7*a^5*b^2*d^3*x^3 + 7/5*a^6*b*x^5*e^3 + 21
/4*a^6*b*d*x^4*e^2 + 7*a^6*b*d^2*x^3*e + 7/2*a^6*b*d^3*x^2 + 1/4*a^7*x^4*e^3 + a^7*d*x^3*e^2 + 3/2*a^7*d^2*x^2
*e + a^7*d^3*x