3.1056 \(\int (a+b x)^6 (A+B x) (d+e x)^3 \, dx\)

Optimal. Leaf size=159 \[ \frac{e^2 (a+b x)^{10} (-4 a B e+A b e+3 b B d)}{10 b^5}+\frac{e (a+b x)^9 (b d-a e) (-2 a B e+A b e+b B d)}{3 b^5}+\frac{(a+b x)^8 (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{8 b^5}+\frac{(a+b x)^7 (A b-a B) (b d-a e)^3}{7 b^5}+\frac{B e^3 (a+b x)^{11}}{11 b^5} \]

[Out]

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

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Rubi [A]  time = 0.467694, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ \frac{e^2 (a+b x)^{10} (-4 a B e+A b e+3 b B d)}{10 b^5}+\frac{e (a+b x)^9 (b d-a e) (-2 a B e+A b e+b B d)}{3 b^5}+\frac{(a+b x)^8 (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{8 b^5}+\frac{(a+b x)^7 (A b-a B) (b d-a e)^3}{7 b^5}+\frac{B e^3 (a+b x)^{11}}{11 b^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^6*(A + B*x)*(d + e*x)^3,x]

[Out]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [B]  time = 0.197028, size = 586, normalized size = 3.69 \[ \frac{1}{8} b^3 x^8 \left (15 a^2 b e^2 (A e+3 B d)+20 a^3 B e^3+18 a b^2 d e (A e+B d)+b^3 d^2 (3 A e+B d)\right )+\frac{1}{7} b^2 x^7 \left (A b \left (45 a^2 b d e^2+20 a^3 e^3+18 a b^2 d^2 e+b^3 d^3\right )+3 a B \left (20 a^2 b d e^2+5 a^3 e^3+15 a b^2 d^2 e+2 b^3 d^3\right )\right )+\frac{1}{2} a b x^6 \left (A b \left (20 a^2 b d e^2+5 a^3 e^3+15 a b^2 d^2 e+2 b^3 d^3\right )+a B \left (15 a^2 b d e^2+2 a^3 e^3+20 a b^2 d^2 e+5 b^3 d^3\right )\right )+\frac{1}{5} a^2 x^5 \left (3 A b \left (15 a^2 b d e^2+2 a^3 e^3+20 a b^2 d^2 e+5 b^3 d^3\right )+a B \left (18 a^2 b d e^2+a^3 e^3+45 a b^2 d^2 e+20 b^3 d^3\right )\right )+\frac{1}{4} a^3 x^4 \left (A \left (18 a^2 b d e^2+a^3 e^3+45 a b^2 d^2 e+20 b^3 d^3\right )+3 a B d \left (a^2 e^2+6 a b d e+5 b^2 d^2\right )\right )+a^4 d x^3 \left (A \left (a^2 e^2+6 a b d e+5 b^2 d^2\right )+a B d (a e+2 b d)\right )+\frac{1}{3} b^4 e x^9 \left (5 a^2 B e^2+2 a b e (A e+3 B d)+b^2 d (A e+B d)\right )+\frac{1}{2} a^5 d^2 x^2 (3 a A e+a B d+6 A b d)+a^6 A d^3 x+\frac{1}{10} b^5 e^2 x^{10} (6 a B e+A b e+3 b B d)+\frac{1}{11} b^6 B e^3 x^{11} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^6*(A + B*x)*(d + e*x)^3,x]

[Out]

a^6*A*d^3*x + (a^5*d^2*(6*A*b*d + a*B*d + 3*a*A*e)*x^2)/2 + a^4*d*(a*B*d*(2*b*d + a*e) + A*(5*b^2*d^2 + 6*a*b*
d*e + a^2*e^2))*x^3 + (a^3*(3*a*B*d*(5*b^2*d^2 + 6*a*b*d*e + a^2*e^2) + A*(20*b^3*d^3 + 45*a*b^2*d^2*e + 18*a^
2*b*d*e^2 + a^3*e^3))*x^4)/4 + (a^2*(a*B*(20*b^3*d^3 + 45*a*b^2*d^2*e + 18*a^2*b*d*e^2 + a^3*e^3) + 3*A*b*(5*b
^3*d^3 + 20*a*b^2*d^2*e + 15*a^2*b*d*e^2 + 2*a^3*e^3))*x^5)/5 + (a*b*(a*B*(5*b^3*d^3 + 20*a*b^2*d^2*e + 15*a^2
*b*d*e^2 + 2*a^3*e^3) + A*b*(2*b^3*d^3 + 15*a*b^2*d^2*e + 20*a^2*b*d*e^2 + 5*a^3*e^3))*x^6)/2 + (b^2*(3*a*B*(2
*b^3*d^3 + 15*a*b^2*d^2*e + 20*a^2*b*d*e^2 + 5*a^3*e^3) + A*b*(b^3*d^3 + 18*a*b^2*d^2*e + 45*a^2*b*d*e^2 + 20*
a^3*e^3))*x^7)/7 + (b^3*(20*a^3*B*e^3 + 18*a*b^2*d*e*(B*d + A*e) + 15*a^2*b*e^2*(3*B*d + A*e) + b^3*d^2*(B*d +
3*A*e))*x^8)/8 + (b^4*e*(5*a^2*B*e^2 + b^2*d*(B*d + A*e) + 2*a*b*e*(3*B*d + A*e))*x^9)/3 + (b^5*e^2*(3*b*B*d
+ A*b*e + 6*a*B*e)*x^10)/10 + (b^6*B*e^3*x^11)/11

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Maple [B]  time = 0.002, size = 645, normalized size = 4.1 \begin{align*}{\frac{{b}^{6}B{e}^{3}{x}^{11}}{11}}+{\frac{ \left ( \left ({b}^{6}A+6\,a{b}^{5}B \right ){e}^{3}+3\,{b}^{6}Bd{e}^{2} \right ){x}^{10}}{10}}+{\frac{ \left ( \left ( 6\,a{b}^{5}A+15\,{a}^{2}{b}^{4}B \right ){e}^{3}+3\, \left ({b}^{6}A+6\,a{b}^{5}B \right ) d{e}^{2}+3\,{b}^{6}B{d}^{2}e \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 15\,{a}^{2}{b}^{4}A+20\,{a}^{3}{b}^{3}B \right ){e}^{3}+3\, \left ( 6\,a{b}^{5}A+15\,{a}^{2}{b}^{4}B \right ) d{e}^{2}+3\, \left ({b}^{6}A+6\,a{b}^{5}B \right ){d}^{2}e+{b}^{6}B{d}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 20\,{a}^{3}{b}^{3}A+15\,{a}^{4}{b}^{2}B \right ){e}^{3}+3\, \left ( 15\,{a}^{2}{b}^{4}A+20\,{a}^{3}{b}^{3}B \right ) d{e}^{2}+3\, \left ( 6\,a{b}^{5}A+15\,{a}^{2}{b}^{4}B \right ){d}^{2}e+ \left ({b}^{6}A+6\,a{b}^{5}B \right ){d}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 15\,{a}^{4}{b}^{2}A+6\,{a}^{5}bB \right ){e}^{3}+3\, \left ( 20\,{a}^{3}{b}^{3}A+15\,{a}^{4}{b}^{2}B \right ) d{e}^{2}+3\, \left ( 15\,{a}^{2}{b}^{4}A+20\,{a}^{3}{b}^{3}B \right ){d}^{2}e+ \left ( 6\,a{b}^{5}A+15\,{a}^{2}{b}^{4}B \right ){d}^{3} \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 6\,{a}^{5}bA+{a}^{6}B \right ){e}^{3}+3\, \left ( 15\,{a}^{4}{b}^{2}A+6\,{a}^{5}bB \right ) d{e}^{2}+3\, \left ( 20\,{a}^{3}{b}^{3}A+15\,{a}^{4}{b}^{2}B \right ){d}^{2}e+ \left ( 15\,{a}^{2}{b}^{4}A+20\,{a}^{3}{b}^{3}B \right ){d}^{3} \right ){x}^{5}}{5}}+{\frac{ \left ({a}^{6}A{e}^{3}+3\, \left ( 6\,{a}^{5}bA+{a}^{6}B \right ) d{e}^{2}+3\, \left ( 15\,{a}^{4}{b}^{2}A+6\,{a}^{5}bB \right ){d}^{2}e+ \left ( 20\,{a}^{3}{b}^{3}A+15\,{a}^{4}{b}^{2}B \right ){d}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,{a}^{6}Ad{e}^{2}+3\, \left ( 6\,{a}^{5}bA+{a}^{6}B \right ){d}^{2}e+ \left ( 15\,{a}^{4}{b}^{2}A+6\,{a}^{5}bB \right ){d}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{a}^{6}A{d}^{2}e+ \left ( 6\,{a}^{5}bA+{a}^{6}B \right ){d}^{3} \right ){x}^{2}}{2}}+{a}^{6}A{d}^{3}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^6*(B*x+A)*(e*x+d)^3,x)

[Out]

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

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Maxima [B]  time = 1.09249, size = 868, normalized size = 5.46 \begin{align*} \frac{1}{11} \, B b^{6} e^{3} x^{11} + A a^{6} d^{3} x + \frac{1}{10} \,{\left (3 \, B b^{6} d e^{2} +{\left (6 \, B a b^{5} + A b^{6}\right )} e^{3}\right )} x^{10} + \frac{1}{3} \,{\left (B b^{6} d^{2} e +{\left (6 \, B a b^{5} + A b^{6}\right )} d e^{2} +{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{3}\right )} x^{9} + \frac{1}{8} \,{\left (B b^{6} d^{3} + 3 \,{\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e + 9 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{2} + 5 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{3}\right )} x^{8} + \frac{1}{7} \,{\left ({\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} + 9 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e + 15 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{2} + 5 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{3}\right )} x^{7} + \frac{1}{2} \,{\left ({\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} + 5 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e + 5 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{2} +{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (5 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} + 15 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e + 9 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{2} +{\left (B a^{6} + 6 \, A a^{5} b\right )} e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (A a^{6} e^{3} + 5 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} + 9 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e + 3 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{2}\right )} x^{4} +{\left (A a^{6} d e^{2} +{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{3} +{\left (B a^{6} + 6 \, A a^{5} b\right )} d^{2} e\right )} x^{3} + \frac{1}{2} \,{\left (3 \, A a^{6} d^{2} e +{\left (B a^{6} + 6 \, A a^{5} b\right )} d^{3}\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)*(e*x+d)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.6252, size = 1725, normalized size = 10.85 \begin{align*} \frac{1}{11} x^{11} e^{3} b^{6} B + \frac{3}{10} x^{10} e^{2} d b^{6} B + \frac{3}{5} x^{10} e^{3} b^{5} a B + \frac{1}{10} x^{10} e^{3} b^{6} A + \frac{1}{3} x^{9} e d^{2} b^{6} B + 2 x^{9} e^{2} d b^{5} a B + \frac{5}{3} x^{9} e^{3} b^{4} a^{2} B + \frac{1}{3} x^{9} e^{2} d b^{6} A + \frac{2}{3} x^{9} e^{3} b^{5} a A + \frac{1}{8} x^{8} d^{3} b^{6} B + \frac{9}{4} x^{8} e d^{2} b^{5} a B + \frac{45}{8} x^{8} e^{2} d b^{4} a^{2} B + \frac{5}{2} x^{8} e^{3} b^{3} a^{3} B + \frac{3}{8} x^{8} e d^{2} b^{6} A + \frac{9}{4} x^{8} e^{2} d b^{5} a A + \frac{15}{8} x^{8} e^{3} b^{4} a^{2} A + \frac{6}{7} x^{7} d^{3} b^{5} a B + \frac{45}{7} x^{7} e d^{2} b^{4} a^{2} B + \frac{60}{7} x^{7} e^{2} d b^{3} a^{3} B + \frac{15}{7} x^{7} e^{3} b^{2} a^{4} B + \frac{1}{7} x^{7} d^{3} b^{6} A + \frac{18}{7} x^{7} e d^{2} b^{5} a A + \frac{45}{7} x^{7} e^{2} d b^{4} a^{2} A + \frac{20}{7} x^{7} e^{3} b^{3} a^{3} A + \frac{5}{2} x^{6} d^{3} b^{4} a^{2} B + 10 x^{6} e d^{2} b^{3} a^{3} B + \frac{15}{2} x^{6} e^{2} d b^{2} a^{4} B + x^{6} e^{3} b a^{5} B + x^{6} d^{3} b^{5} a A + \frac{15}{2} x^{6} e d^{2} b^{4} a^{2} A + 10 x^{6} e^{2} d b^{3} a^{3} A + \frac{5}{2} x^{6} e^{3} b^{2} a^{4} A + 4 x^{5} d^{3} b^{3} a^{3} B + 9 x^{5} e d^{2} b^{2} a^{4} B + \frac{18}{5} x^{5} e^{2} d b a^{5} B + \frac{1}{5} x^{5} e^{3} a^{6} B + 3 x^{5} d^{3} b^{4} a^{2} A + 12 x^{5} e d^{2} b^{3} a^{3} A + 9 x^{5} e^{2} d b^{2} a^{4} A + \frac{6}{5} x^{5} e^{3} b a^{5} A + \frac{15}{4} x^{4} d^{3} b^{2} a^{4} B + \frac{9}{2} x^{4} e d^{2} b a^{5} B + \frac{3}{4} x^{4} e^{2} d a^{6} B + 5 x^{4} d^{3} b^{3} a^{3} A + \frac{45}{4} x^{4} e d^{2} b^{2} a^{4} A + \frac{9}{2} x^{4} e^{2} d b a^{5} A + \frac{1}{4} x^{4} e^{3} a^{6} A + 2 x^{3} d^{3} b a^{5} B + x^{3} e d^{2} a^{6} B + 5 x^{3} d^{3} b^{2} a^{4} A + 6 x^{3} e d^{2} b a^{5} A + x^{3} e^{2} d a^{6} A + \frac{1}{2} x^{2} d^{3} a^{6} B + 3 x^{2} d^{3} b a^{5} A + \frac{3}{2} x^{2} e d^{2} a^{6} A + x d^{3} a^{6} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)*(e*x+d)^3,x, algorithm="fricas")

[Out]

1/11*x^11*e^3*b^6*B + 3/10*x^10*e^2*d*b^6*B + 3/5*x^10*e^3*b^5*a*B + 1/10*x^10*e^3*b^6*A + 1/3*x^9*e*d^2*b^6*B
+ 2*x^9*e^2*d*b^5*a*B + 5/3*x^9*e^3*b^4*a^2*B + 1/3*x^9*e^2*d*b^6*A + 2/3*x^9*e^3*b^5*a*A + 1/8*x^8*d^3*b^6*B
+ 9/4*x^8*e*d^2*b^5*a*B + 45/8*x^8*e^2*d*b^4*a^2*B + 5/2*x^8*e^3*b^3*a^3*B + 3/8*x^8*e*d^2*b^6*A + 9/4*x^8*e^
2*d*b^5*a*A + 15/8*x^8*e^3*b^4*a^2*A + 6/7*x^7*d^3*b^5*a*B + 45/7*x^7*e*d^2*b^4*a^2*B + 60/7*x^7*e^2*d*b^3*a^3
*B + 15/7*x^7*e^3*b^2*a^4*B + 1/7*x^7*d^3*b^6*A + 18/7*x^7*e*d^2*b^5*a*A + 45/7*x^7*e^2*d*b^4*a^2*A + 20/7*x^7
*e^3*b^3*a^3*A + 5/2*x^6*d^3*b^4*a^2*B + 10*x^6*e*d^2*b^3*a^3*B + 15/2*x^6*e^2*d*b^2*a^4*B + x^6*e^3*b*a^5*B +
x^6*d^3*b^5*a*A + 15/2*x^6*e*d^2*b^4*a^2*A + 10*x^6*e^2*d*b^3*a^3*A + 5/2*x^6*e^3*b^2*a^4*A + 4*x^5*d^3*b^3*a
^3*B + 9*x^5*e*d^2*b^2*a^4*B + 18/5*x^5*e^2*d*b*a^5*B + 1/5*x^5*e^3*a^6*B + 3*x^5*d^3*b^4*a^2*A + 12*x^5*e*d^2
*b^3*a^3*A + 9*x^5*e^2*d*b^2*a^4*A + 6/5*x^5*e^3*b*a^5*A + 15/4*x^4*d^3*b^2*a^4*B + 9/2*x^4*e*d^2*b*a^5*B + 3/
4*x^4*e^2*d*a^6*B + 5*x^4*d^3*b^3*a^3*A + 45/4*x^4*e*d^2*b^2*a^4*A + 9/2*x^4*e^2*d*b*a^5*A + 1/4*x^4*e^3*a^6*A
+ 2*x^3*d^3*b*a^5*B + x^3*e*d^2*a^6*B + 5*x^3*d^3*b^2*a^4*A + 6*x^3*e*d^2*b*a^5*A + x^3*e^2*d*a^6*A + 1/2*x^2
*d^3*a^6*B + 3*x^2*d^3*b*a^5*A + 3/2*x^2*e*d^2*a^6*A + x*d^3*a^6*A

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Sympy [B]  time = 0.147856, size = 802, normalized size = 5.04 \begin{align*} A a^{6} d^{3} x + \frac{B b^{6} e^{3} x^{11}}{11} + x^{10} \left (\frac{A b^{6} e^{3}}{10} + \frac{3 B a b^{5} e^{3}}{5} + \frac{3 B b^{6} d e^{2}}{10}\right ) + x^{9} \left (\frac{2 A a b^{5} e^{3}}{3} + \frac{A b^{6} d e^{2}}{3} + \frac{5 B a^{2} b^{4} e^{3}}{3} + 2 B a b^{5} d e^{2} + \frac{B b^{6} d^{2} e}{3}\right ) + x^{8} \left (\frac{15 A a^{2} b^{4} e^{3}}{8} + \frac{9 A a b^{5} d e^{2}}{4} + \frac{3 A b^{6} d^{2} e}{8} + \frac{5 B a^{3} b^{3} e^{3}}{2} + \frac{45 B a^{2} b^{4} d e^{2}}{8} + \frac{9 B a b^{5} d^{2} e}{4} + \frac{B b^{6} d^{3}}{8}\right ) + x^{7} \left (\frac{20 A a^{3} b^{3} e^{3}}{7} + \frac{45 A a^{2} b^{4} d e^{2}}{7} + \frac{18 A a b^{5} d^{2} e}{7} + \frac{A b^{6} d^{3}}{7} + \frac{15 B a^{4} b^{2} e^{3}}{7} + \frac{60 B a^{3} b^{3} d e^{2}}{7} + \frac{45 B a^{2} b^{4} d^{2} e}{7} + \frac{6 B a b^{5} d^{3}}{7}\right ) + x^{6} \left (\frac{5 A a^{4} b^{2} e^{3}}{2} + 10 A a^{3} b^{3} d e^{2} + \frac{15 A a^{2} b^{4} d^{2} e}{2} + A a b^{5} d^{3} + B a^{5} b e^{3} + \frac{15 B a^{4} b^{2} d e^{2}}{2} + 10 B a^{3} b^{3} d^{2} e + \frac{5 B a^{2} b^{4} d^{3}}{2}\right ) + x^{5} \left (\frac{6 A a^{5} b e^{3}}{5} + 9 A a^{4} b^{2} d e^{2} + 12 A a^{3} b^{3} d^{2} e + 3 A a^{2} b^{4} d^{3} + \frac{B a^{6} e^{3}}{5} + \frac{18 B a^{5} b d e^{2}}{5} + 9 B a^{4} b^{2} d^{2} e + 4 B a^{3} b^{3} d^{3}\right ) + x^{4} \left (\frac{A a^{6} e^{3}}{4} + \frac{9 A a^{5} b d e^{2}}{2} + \frac{45 A a^{4} b^{2} d^{2} e}{4} + 5 A a^{3} b^{3} d^{3} + \frac{3 B a^{6} d e^{2}}{4} + \frac{9 B a^{5} b d^{2} e}{2} + \frac{15 B a^{4} b^{2} d^{3}}{4}\right ) + x^{3} \left (A a^{6} d e^{2} + 6 A a^{5} b d^{2} e + 5 A a^{4} b^{2} d^{3} + B a^{6} d^{2} e + 2 B a^{5} b d^{3}\right ) + x^{2} \left (\frac{3 A a^{6} d^{2} e}{2} + 3 A a^{5} b d^{3} + \frac{B a^{6} d^{3}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**6*(B*x+A)*(e*x+d)**3,x)

[Out]

A*a**6*d**3*x + B*b**6*e**3*x**11/11 + x**10*(A*b**6*e**3/10 + 3*B*a*b**5*e**3/5 + 3*B*b**6*d*e**2/10) + x**9*
(2*A*a*b**5*e**3/3 + A*b**6*d*e**2/3 + 5*B*a**2*b**4*e**3/3 + 2*B*a*b**5*d*e**2 + B*b**6*d**2*e/3) + x**8*(15*
A*a**2*b**4*e**3/8 + 9*A*a*b**5*d*e**2/4 + 3*A*b**6*d**2*e/8 + 5*B*a**3*b**3*e**3/2 + 45*B*a**2*b**4*d*e**2/8
+ 9*B*a*b**5*d**2*e/4 + B*b**6*d**3/8) + x**7*(20*A*a**3*b**3*e**3/7 + 45*A*a**2*b**4*d*e**2/7 + 18*A*a*b**5*d
**2*e/7 + A*b**6*d**3/7 + 15*B*a**4*b**2*e**3/7 + 60*B*a**3*b**3*d*e**2/7 + 45*B*a**2*b**4*d**2*e/7 + 6*B*a*b*
*5*d**3/7) + x**6*(5*A*a**4*b**2*e**3/2 + 10*A*a**3*b**3*d*e**2 + 15*A*a**2*b**4*d**2*e/2 + A*a*b**5*d**3 + B*
a**5*b*e**3 + 15*B*a**4*b**2*d*e**2/2 + 10*B*a**3*b**3*d**2*e + 5*B*a**2*b**4*d**3/2) + x**5*(6*A*a**5*b*e**3/
5 + 9*A*a**4*b**2*d*e**2 + 12*A*a**3*b**3*d**2*e + 3*A*a**2*b**4*d**3 + B*a**6*e**3/5 + 18*B*a**5*b*d*e**2/5 +
9*B*a**4*b**2*d**2*e + 4*B*a**3*b**3*d**3) + x**4*(A*a**6*e**3/4 + 9*A*a**5*b*d*e**2/2 + 45*A*a**4*b**2*d**2*
e/4 + 5*A*a**3*b**3*d**3 + 3*B*a**6*d*e**2/4 + 9*B*a**5*b*d**2*e/2 + 15*B*a**4*b**2*d**3/4) + x**3*(A*a**6*d*e
**2 + 6*A*a**5*b*d**2*e + 5*A*a**4*b**2*d**3 + B*a**6*d**2*e + 2*B*a**5*b*d**3) + x**2*(3*A*a**6*d**2*e/2 + 3*
A*a**5*b*d**3 + B*a**6*d**3/2)

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Giac [B]  time = 1.25385, size = 1037, normalized size = 6.52 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)*(e*x+d)^3,x, algorithm="giac")

[Out]

1/11*B*b^6*x^11*e^3 + 3/10*B*b^6*d*x^10*e^2 + 1/3*B*b^6*d^2*x^9*e + 1/8*B*b^6*d^3*x^8 + 3/5*B*a*b^5*x^10*e^3 +
1/10*A*b^6*x^10*e^3 + 2*B*a*b^5*d*x^9*e^2 + 1/3*A*b^6*d*x^9*e^2 + 9/4*B*a*b^5*d^2*x^8*e + 3/8*A*b^6*d^2*x^8*e
+ 6/7*B*a*b^5*d^3*x^7 + 1/7*A*b^6*d^3*x^7 + 5/3*B*a^2*b^4*x^9*e^3 + 2/3*A*a*b^5*x^9*e^3 + 45/8*B*a^2*b^4*d*x^
8*e^2 + 9/4*A*a*b^5*d*x^8*e^2 + 45/7*B*a^2*b^4*d^2*x^7*e + 18/7*A*a*b^5*d^2*x^7*e + 5/2*B*a^2*b^4*d^3*x^6 + A*
a*b^5*d^3*x^6 + 5/2*B*a^3*b^3*x^8*e^3 + 15/8*A*a^2*b^4*x^8*e^3 + 60/7*B*a^3*b^3*d*x^7*e^2 + 45/7*A*a^2*b^4*d*x
^7*e^2 + 10*B*a^3*b^3*d^2*x^6*e + 15/2*A*a^2*b^4*d^2*x^6*e + 4*B*a^3*b^3*d^3*x^5 + 3*A*a^2*b^4*d^3*x^5 + 15/7*
B*a^4*b^2*x^7*e^3 + 20/7*A*a^3*b^3*x^7*e^3 + 15/2*B*a^4*b^2*d*x^6*e^2 + 10*A*a^3*b^3*d*x^6*e^2 + 9*B*a^4*b^2*d
^2*x^5*e + 12*A*a^3*b^3*d^2*x^5*e + 15/4*B*a^4*b^2*d^3*x^4 + 5*A*a^3*b^3*d^3*x^4 + B*a^5*b*x^6*e^3 + 5/2*A*a^4
*b^2*x^6*e^3 + 18/5*B*a^5*b*d*x^5*e^2 + 9*A*a^4*b^2*d*x^5*e^2 + 9/2*B*a^5*b*d^2*x^4*e + 45/4*A*a^4*b^2*d^2*x^4
*e + 2*B*a^5*b*d^3*x^3 + 5*A*a^4*b^2*d^3*x^3 + 1/5*B*a^6*x^5*e^3 + 6/5*A*a^5*b*x^5*e^3 + 3/4*B*a^6*d*x^4*e^2 +
9/2*A*a^5*b*d*x^4*e^2 + B*a^6*d^2*x^3*e + 6*A*a^5*b*d^2*x^3*e + 1/2*B*a^6*d^3*x^2 + 3*A*a^5*b*d^3*x^2 + 1/4*A
*a^6*x^4*e^3 + A*a^6*d*x^3*e^2 + 3/2*A*a^6*d^2*x^2*e + A*a^6*d^3*x