3.1463 \(\int (d+e x)^6 (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=119 \[ -\frac{2 b^3 (d+e x)^{10} (b d-a e)}{5 e^5}+\frac{2 b^2 (d+e x)^9 (b d-a e)^2}{3 e^5}-\frac{b (d+e x)^8 (b d-a e)^3}{2 e^5}+\frac{(d+e x)^7 (b d-a e)^4}{7 e^5}+\frac{b^4 (d+e x)^{11}}{11 e^5} \]

[Out]

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

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Rubi [A]  time = 0.266867, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {27, 43} \[ -\frac{2 b^3 (d+e x)^{10} (b d-a e)}{5 e^5}+\frac{2 b^2 (d+e x)^9 (b d-a e)^2}{3 e^5}-\frac{b (d+e x)^8 (b d-a e)^3}{2 e^5}+\frac{(d+e x)^7 (b d-a e)^4}{7 e^5}+\frac{b^4 (d+e x)^{11}}{11 e^5} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^6*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

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

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 (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^4 (d+e x)^6 \, dx\\ &=\int \left (\frac{(-b d+a e)^4 (d+e x)^6}{e^4}-\frac{4 b (b d-a e)^3 (d+e x)^7}{e^4}+\frac{6 b^2 (b d-a e)^2 (d+e x)^8}{e^4}-\frac{4 b^3 (b d-a e) (d+e x)^9}{e^4}+\frac{b^4 (d+e x)^{10}}{e^4}\right ) \, dx\\ &=\frac{(b d-a e)^4 (d+e x)^7}{7 e^5}-\frac{b (b d-a e)^3 (d+e x)^8}{2 e^5}+\frac{2 b^2 (b d-a e)^2 (d+e x)^9}{3 e^5}-\frac{2 b^3 (b d-a e) (d+e x)^{10}}{5 e^5}+\frac{b^4 (d+e x)^{11}}{11 e^5}\\ \end{align*}

Mathematica [B]  time = 0.0612715, size = 398, normalized size = 3.34 \[ \frac{1}{3} b^2 e^4 x^9 \left (2 a^2 e^2+8 a b d e+5 b^2 d^2\right )+\frac{1}{2} b e^3 x^8 \left (9 a^2 b d e^2+a^3 e^3+15 a b^2 d^2 e+5 b^3 d^3\right )+\frac{1}{7} e^2 x^7 \left (90 a^2 b^2 d^2 e^2+24 a^3 b d e^3+a^4 e^4+80 a b^3 d^3 e+15 b^4 d^4\right )+d e x^6 \left (20 a^2 b^2 d^2 e^2+10 a^3 b d e^3+a^4 e^4+10 a b^3 d^3 e+b^4 d^4\right )+\frac{1}{5} d^2 x^5 \left (90 a^2 b^2 d^2 e^2+80 a^3 b d e^3+15 a^4 e^4+24 a b^3 d^3 e+b^4 d^4\right )+a d^3 x^4 \left (15 a^2 b d e^2+5 a^3 e^3+9 a b^2 d^2 e+b^3 d^3\right )+a^2 d^4 x^3 \left (5 a^2 e^2+8 a b d e+2 b^2 d^2\right )+a^3 d^5 x^2 (3 a e+2 b d)+a^4 d^6 x+\frac{1}{5} b^3 e^5 x^{10} (2 a e+3 b d)+\frac{1}{11} b^4 e^6 x^{11} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^6*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

a^4*d^6*x + a^3*d^5*(2*b*d + 3*a*e)*x^2 + a^2*d^4*(2*b^2*d^2 + 8*a*b*d*e + 5*a^2*e^2)*x^3 + a*d^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^4 + (d^2*(b^4*d^4 + 24*a*b^3*d^3*e + 90*a^2*b^2*d^2*e^2 + 80*a^3*
b*d*e^3 + 15*a^4*e^4)*x^5)/5 + d*e*(b^4*d^4 + 10*a*b^3*d^3*e + 20*a^2*b^2*d^2*e^2 + 10*a^3*b*d*e^3 + a^4*e^4)*
x^6 + (e^2*(15*b^4*d^4 + 80*a*b^3*d^3*e + 90*a^2*b^2*d^2*e^2 + 24*a^3*b*d*e^3 + a^4*e^4)*x^7)/7 + (b*e^3*(5*b^
3*d^3 + 15*a*b^2*d^2*e + 9*a^2*b*d*e^2 + a^3*e^3)*x^8)/2 + (b^2*e^4*(5*b^2*d^2 + 8*a*b*d*e + 2*a^2*e^2)*x^9)/3
+ (b^3*e^5*(3*b*d + 2*a*e)*x^10)/5 + (b^4*e^6*x^11)/11

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Maple [B]  time = 0.045, size = 427, normalized size = 3.6 \begin{align*}{\frac{{e}^{6}{b}^{4}{x}^{11}}{11}}+{\frac{ \left ( 4\,{e}^{6}a{b}^{3}+6\,d{e}^{5}{b}^{4} \right ){x}^{10}}{10}}+{\frac{ \left ( 6\,{e}^{6}{b}^{2}{a}^{2}+24\,d{e}^{5}a{b}^{3}+15\,{d}^{2}{e}^{4}{b}^{4} \right ){x}^{9}}{9}}+{\frac{ \left ( 4\,{e}^{6}{a}^{3}b+36\,d{e}^{5}{b}^{2}{a}^{2}+60\,{d}^{2}{e}^{4}a{b}^{3}+20\,{d}^{3}{e}^{3}{b}^{4} \right ){x}^{8}}{8}}+{\frac{ \left ({e}^{6}{a}^{4}+24\,d{e}^{5}{a}^{3}b+90\,{d}^{2}{e}^{4}{b}^{2}{a}^{2}+80\,{d}^{3}{e}^{3}a{b}^{3}+15\,{d}^{4}{e}^{2}{b}^{4} \right ){x}^{7}}{7}}+{\frac{ \left ( 6\,d{e}^{5}{a}^{4}+60\,{d}^{2}{e}^{4}{a}^{3}b+120\,{d}^{3}{e}^{3}{b}^{2}{a}^{2}+60\,{d}^{4}{e}^{2}a{b}^{3}+6\,{d}^{5}e{b}^{4} \right ){x}^{6}}{6}}+{\frac{ \left ( 15\,{d}^{2}{e}^{4}{a}^{4}+80\,{d}^{3}{e}^{3}{a}^{3}b+90\,{d}^{4}{e}^{2}{b}^{2}{a}^{2}+24\,{d}^{5}ea{b}^{3}+{d}^{6}{b}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 20\,{d}^{3}{e}^{3}{a}^{4}+60\,{d}^{4}{e}^{2}{a}^{3}b+36\,{d}^{5}e{b}^{2}{a}^{2}+4\,{d}^{6}a{b}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 15\,{d}^{4}{e}^{2}{a}^{4}+24\,{d}^{5}e{a}^{3}b+6\,{d}^{6}{b}^{2}{a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 6\,{d}^{5}e{a}^{4}+4\,{d}^{6}{a}^{3}b \right ){x}^{2}}{2}}+{d}^{6}{a}^{4}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

1/11*e^6*b^4*x^11+1/10*(4*a*b^3*e^6+6*b^4*d*e^5)*x^10+1/9*(6*a^2*b^2*e^6+24*a*b^3*d*e^5+15*b^4*d^2*e^4)*x^9+1/
8*(4*a^3*b*e^6+36*a^2*b^2*d*e^5+60*a*b^3*d^2*e^4+20*b^4*d^3*e^3)*x^8+1/7*(a^4*e^6+24*a^3*b*d*e^5+90*a^2*b^2*d^
2*e^4+80*a*b^3*d^3*e^3+15*b^4*d^4*e^2)*x^7+1/6*(6*a^4*d*e^5+60*a^3*b*d^2*e^4+120*a^2*b^2*d^3*e^3+60*a*b^3*d^4*
e^2+6*b^4*d^5*e)*x^6+1/5*(15*a^4*d^2*e^4+80*a^3*b*d^3*e^3+90*a^2*b^2*d^4*e^2+24*a*b^3*d^5*e+b^4*d^6)*x^5+1/4*(
20*a^4*d^3*e^3+60*a^3*b*d^4*e^2+36*a^2*b^2*d^5*e+4*a*b^3*d^6)*x^4+1/3*(15*a^4*d^4*e^2+24*a^3*b*d^5*e+6*a^2*b^2
*d^6)*x^3+1/2*(6*a^4*d^5*e+4*a^3*b*d^6)*x^2+d^6*a^4*x

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Maxima [B]  time = 1.17021, size = 564, normalized size = 4.74 \begin{align*} \frac{1}{11} \, b^{4} e^{6} x^{11} + a^{4} d^{6} x + \frac{1}{5} \,{\left (3 \, b^{4} d e^{5} + 2 \, a b^{3} e^{6}\right )} x^{10} + \frac{1}{3} \,{\left (5 \, b^{4} d^{2} e^{4} + 8 \, a b^{3} d e^{5} + 2 \, a^{2} b^{2} e^{6}\right )} x^{9} + \frac{1}{2} \,{\left (5 \, b^{4} d^{3} e^{3} + 15 \, a b^{3} d^{2} e^{4} + 9 \, a^{2} b^{2} d e^{5} + a^{3} b e^{6}\right )} x^{8} + \frac{1}{7} \,{\left (15 \, b^{4} d^{4} e^{2} + 80 \, a b^{3} d^{3} e^{3} + 90 \, a^{2} b^{2} d^{2} e^{4} + 24 \, a^{3} b d e^{5} + a^{4} e^{6}\right )} x^{7} +{\left (b^{4} d^{5} e + 10 \, a b^{3} d^{4} e^{2} + 20 \, a^{2} b^{2} d^{3} e^{3} + 10 \, a^{3} b d^{2} e^{4} + a^{4} d e^{5}\right )} x^{6} + \frac{1}{5} \,{\left (b^{4} d^{6} + 24 \, a b^{3} d^{5} e + 90 \, a^{2} b^{2} d^{4} e^{2} + 80 \, a^{3} b d^{3} e^{3} + 15 \, a^{4} d^{2} e^{4}\right )} x^{5} +{\left (a b^{3} d^{6} + 9 \, a^{2} b^{2} d^{5} e + 15 \, a^{3} b d^{4} e^{2} + 5 \, a^{4} d^{3} e^{3}\right )} x^{4} +{\left (2 \, a^{2} b^{2} d^{6} + 8 \, a^{3} b d^{5} e + 5 \, a^{4} d^{4} e^{2}\right )} x^{3} +{\left (2 \, a^{3} b d^{6} + 3 \, a^{4} d^{5} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

1/11*b^4*e^6*x^11 + a^4*d^6*x + 1/5*(3*b^4*d*e^5 + 2*a*b^3*e^6)*x^10 + 1/3*(5*b^4*d^2*e^4 + 8*a*b^3*d*e^5 + 2*
a^2*b^2*e^6)*x^9 + 1/2*(5*b^4*d^3*e^3 + 15*a*b^3*d^2*e^4 + 9*a^2*b^2*d*e^5 + a^3*b*e^6)*x^8 + 1/7*(15*b^4*d^4*
e^2 + 80*a*b^3*d^3*e^3 + 90*a^2*b^2*d^2*e^4 + 24*a^3*b*d*e^5 + a^4*e^6)*x^7 + (b^4*d^5*e + 10*a*b^3*d^4*e^2 +
20*a^2*b^2*d^3*e^3 + 10*a^3*b*d^2*e^4 + a^4*d*e^5)*x^6 + 1/5*(b^4*d^6 + 24*a*b^3*d^5*e + 90*a^2*b^2*d^4*e^2 +
80*a^3*b*d^3*e^3 + 15*a^4*d^2*e^4)*x^5 + (a*b^3*d^6 + 9*a^2*b^2*d^5*e + 15*a^3*b*d^4*e^2 + 5*a^4*d^3*e^3)*x^4
+ (2*a^2*b^2*d^6 + 8*a^3*b*d^5*e + 5*a^4*d^4*e^2)*x^3 + (2*a^3*b*d^6 + 3*a^4*d^5*e)*x^2

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Fricas [B]  time = 1.47675, size = 992, normalized size = 8.34 \begin{align*} \frac{1}{11} x^{11} e^{6} b^{4} + \frac{3}{5} x^{10} e^{5} d b^{4} + \frac{2}{5} x^{10} e^{6} b^{3} a + \frac{5}{3} x^{9} e^{4} d^{2} b^{4} + \frac{8}{3} x^{9} e^{5} d b^{3} a + \frac{2}{3} x^{9} e^{6} b^{2} a^{2} + \frac{5}{2} x^{8} e^{3} d^{3} b^{4} + \frac{15}{2} x^{8} e^{4} d^{2} b^{3} a + \frac{9}{2} x^{8} e^{5} d b^{2} a^{2} + \frac{1}{2} x^{8} e^{6} b a^{3} + \frac{15}{7} x^{7} e^{2} d^{4} b^{4} + \frac{80}{7} x^{7} e^{3} d^{3} b^{3} a + \frac{90}{7} x^{7} e^{4} d^{2} b^{2} a^{2} + \frac{24}{7} x^{7} e^{5} d b a^{3} + \frac{1}{7} x^{7} e^{6} a^{4} + x^{6} e d^{5} b^{4} + 10 x^{6} e^{2} d^{4} b^{3} a + 20 x^{6} e^{3} d^{3} b^{2} a^{2} + 10 x^{6} e^{4} d^{2} b a^{3} + x^{6} e^{5} d a^{4} + \frac{1}{5} x^{5} d^{6} b^{4} + \frac{24}{5} x^{5} e d^{5} b^{3} a + 18 x^{5} e^{2} d^{4} b^{2} a^{2} + 16 x^{5} e^{3} d^{3} b a^{3} + 3 x^{5} e^{4} d^{2} a^{4} + x^{4} d^{6} b^{3} a + 9 x^{4} e d^{5} b^{2} a^{2} + 15 x^{4} e^{2} d^{4} b a^{3} + 5 x^{4} e^{3} d^{3} a^{4} + 2 x^{3} d^{6} b^{2} a^{2} + 8 x^{3} e d^{5} b a^{3} + 5 x^{3} e^{2} d^{4} a^{4} + 2 x^{2} d^{6} b a^{3} + 3 x^{2} e d^{5} a^{4} + x d^{6} a^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

1/11*x^11*e^6*b^4 + 3/5*x^10*e^5*d*b^4 + 2/5*x^10*e^6*b^3*a + 5/3*x^9*e^4*d^2*b^4 + 8/3*x^9*e^5*d*b^3*a + 2/3*
x^9*e^6*b^2*a^2 + 5/2*x^8*e^3*d^3*b^4 + 15/2*x^8*e^4*d^2*b^3*a + 9/2*x^8*e^5*d*b^2*a^2 + 1/2*x^8*e^6*b*a^3 + 1
5/7*x^7*e^2*d^4*b^4 + 80/7*x^7*e^3*d^3*b^3*a + 90/7*x^7*e^4*d^2*b^2*a^2 + 24/7*x^7*e^5*d*b*a^3 + 1/7*x^7*e^6*a
^4 + x^6*e*d^5*b^4 + 10*x^6*e^2*d^4*b^3*a + 20*x^6*e^3*d^3*b^2*a^2 + 10*x^6*e^4*d^2*b*a^3 + x^6*e^5*d*a^4 + 1/
5*x^5*d^6*b^4 + 24/5*x^5*e*d^5*b^3*a + 18*x^5*e^2*d^4*b^2*a^2 + 16*x^5*e^3*d^3*b*a^3 + 3*x^5*e^4*d^2*a^4 + x^4
*d^6*b^3*a + 9*x^4*e*d^5*b^2*a^2 + 15*x^4*e^2*d^4*b*a^3 + 5*x^4*e^3*d^3*a^4 + 2*x^3*d^6*b^2*a^2 + 8*x^3*e*d^5*
b*a^3 + 5*x^3*e^2*d^4*a^4 + 2*x^2*d^6*b*a^3 + 3*x^2*e*d^5*a^4 + x*d^6*a^4

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Sympy [B]  time = 0.131337, size = 462, normalized size = 3.88 \begin{align*} a^{4} d^{6} x + \frac{b^{4} e^{6} x^{11}}{11} + x^{10} \left (\frac{2 a b^{3} e^{6}}{5} + \frac{3 b^{4} d e^{5}}{5}\right ) + x^{9} \left (\frac{2 a^{2} b^{2} e^{6}}{3} + \frac{8 a b^{3} d e^{5}}{3} + \frac{5 b^{4} d^{2} e^{4}}{3}\right ) + x^{8} \left (\frac{a^{3} b e^{6}}{2} + \frac{9 a^{2} b^{2} d e^{5}}{2} + \frac{15 a b^{3} d^{2} e^{4}}{2} + \frac{5 b^{4} d^{3} e^{3}}{2}\right ) + x^{7} \left (\frac{a^{4} e^{6}}{7} + \frac{24 a^{3} b d e^{5}}{7} + \frac{90 a^{2} b^{2} d^{2} e^{4}}{7} + \frac{80 a b^{3} d^{3} e^{3}}{7} + \frac{15 b^{4} d^{4} e^{2}}{7}\right ) + x^{6} \left (a^{4} d e^{5} + 10 a^{3} b d^{2} e^{4} + 20 a^{2} b^{2} d^{3} e^{3} + 10 a b^{3} d^{4} e^{2} + b^{4} d^{5} e\right ) + x^{5} \left (3 a^{4} d^{2} e^{4} + 16 a^{3} b d^{3} e^{3} + 18 a^{2} b^{2} d^{4} e^{2} + \frac{24 a b^{3} d^{5} e}{5} + \frac{b^{4} d^{6}}{5}\right ) + x^{4} \left (5 a^{4} d^{3} e^{3} + 15 a^{3} b d^{4} e^{2} + 9 a^{2} b^{2} d^{5} e + a b^{3} d^{6}\right ) + x^{3} \left (5 a^{4} d^{4} e^{2} + 8 a^{3} b d^{5} e + 2 a^{2} b^{2} d^{6}\right ) + x^{2} \left (3 a^{4} d^{5} e + 2 a^{3} b d^{6}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**6*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

a**4*d**6*x + b**4*e**6*x**11/11 + x**10*(2*a*b**3*e**6/5 + 3*b**4*d*e**5/5) + x**9*(2*a**2*b**2*e**6/3 + 8*a*
b**3*d*e**5/3 + 5*b**4*d**2*e**4/3) + x**8*(a**3*b*e**6/2 + 9*a**2*b**2*d*e**5/2 + 15*a*b**3*d**2*e**4/2 + 5*b
**4*d**3*e**3/2) + x**7*(a**4*e**6/7 + 24*a**3*b*d*e**5/7 + 90*a**2*b**2*d**2*e**4/7 + 80*a*b**3*d**3*e**3/7 +
15*b**4*d**4*e**2/7) + x**6*(a**4*d*e**5 + 10*a**3*b*d**2*e**4 + 20*a**2*b**2*d**3*e**3 + 10*a*b**3*d**4*e**2
+ b**4*d**5*e) + x**5*(3*a**4*d**2*e**4 + 16*a**3*b*d**3*e**3 + 18*a**2*b**2*d**4*e**2 + 24*a*b**3*d**5*e/5 +
b**4*d**6/5) + x**4*(5*a**4*d**3*e**3 + 15*a**3*b*d**4*e**2 + 9*a**2*b**2*d**5*e + a*b**3*d**6) + x**3*(5*a**
4*d**4*e**2 + 8*a**3*b*d**5*e + 2*a**2*b**2*d**6) + x**2*(3*a**4*d**5*e + 2*a**3*b*d**6)

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Giac [B]  time = 1.09952, size = 608, normalized size = 5.11 \begin{align*} \frac{1}{11} \, b^{4} x^{11} e^{6} + \frac{3}{5} \, b^{4} d x^{10} e^{5} + \frac{5}{3} \, b^{4} d^{2} x^{9} e^{4} + \frac{5}{2} \, b^{4} d^{3} x^{8} e^{3} + \frac{15}{7} \, b^{4} d^{4} x^{7} e^{2} + b^{4} d^{5} x^{6} e + \frac{1}{5} \, b^{4} d^{6} x^{5} + \frac{2}{5} \, a b^{3} x^{10} e^{6} + \frac{8}{3} \, a b^{3} d x^{9} e^{5} + \frac{15}{2} \, a b^{3} d^{2} x^{8} e^{4} + \frac{80}{7} \, a b^{3} d^{3} x^{7} e^{3} + 10 \, a b^{3} d^{4} x^{6} e^{2} + \frac{24}{5} \, a b^{3} d^{5} x^{5} e + a b^{3} d^{6} x^{4} + \frac{2}{3} \, a^{2} b^{2} x^{9} e^{6} + \frac{9}{2} \, a^{2} b^{2} d x^{8} e^{5} + \frac{90}{7} \, a^{2} b^{2} d^{2} x^{7} e^{4} + 20 \, a^{2} b^{2} d^{3} x^{6} e^{3} + 18 \, a^{2} b^{2} d^{4} x^{5} e^{2} + 9 \, a^{2} b^{2} d^{5} x^{4} e + 2 \, a^{2} b^{2} d^{6} x^{3} + \frac{1}{2} \, a^{3} b x^{8} e^{6} + \frac{24}{7} \, a^{3} b d x^{7} e^{5} + 10 \, a^{3} b d^{2} x^{6} e^{4} + 16 \, a^{3} b d^{3} x^{5} e^{3} + 15 \, a^{3} b d^{4} x^{4} e^{2} + 8 \, a^{3} b d^{5} x^{3} e + 2 \, a^{3} b d^{6} x^{2} + \frac{1}{7} \, a^{4} x^{7} e^{6} + a^{4} d x^{6} e^{5} + 3 \, a^{4} d^{2} x^{5} e^{4} + 5 \, a^{4} d^{3} x^{4} e^{3} + 5 \, a^{4} d^{4} x^{3} e^{2} + 3 \, a^{4} d^{5} x^{2} e + a^{4} d^{6} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

1/11*b^4*x^11*e^6 + 3/5*b^4*d*x^10*e^5 + 5/3*b^4*d^2*x^9*e^4 + 5/2*b^4*d^3*x^8*e^3 + 15/7*b^4*d^4*x^7*e^2 + b^
4*d^5*x^6*e + 1/5*b^4*d^6*x^5 + 2/5*a*b^3*x^10*e^6 + 8/3*a*b^3*d*x^9*e^5 + 15/2*a*b^3*d^2*x^8*e^4 + 80/7*a*b^3
*d^3*x^7*e^3 + 10*a*b^3*d^4*x^6*e^2 + 24/5*a*b^3*d^5*x^5*e + a*b^3*d^6*x^4 + 2/3*a^2*b^2*x^9*e^6 + 9/2*a^2*b^2
*d*x^8*e^5 + 90/7*a^2*b^2*d^2*x^7*e^4 + 20*a^2*b^2*d^3*x^6*e^3 + 18*a^2*b^2*d^4*x^5*e^2 + 9*a^2*b^2*d^5*x^4*e
+ 2*a^2*b^2*d^6*x^3 + 1/2*a^3*b*x^8*e^6 + 24/7*a^3*b*d*x^7*e^5 + 10*a^3*b*d^2*x^6*e^4 + 16*a^3*b*d^3*x^5*e^3 +
15*a^3*b*d^4*x^4*e^2 + 8*a^3*b*d^5*x^3*e + 2*a^3*b*d^6*x^2 + 1/7*a^4*x^7*e^6 + a^4*d*x^6*e^5 + 3*a^4*d^2*x^5*
e^4 + 5*a^4*d^3*x^4*e^3 + 5*a^4*d^4*x^3*e^2 + 3*a^4*d^5*x^2*e + a^4*d^6*x