3.1675 \(\int (A+B x) (d+e x)^5 (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=206 \[ -\frac{b^3 (d+e x)^{10} (-4 a B e-A b e+5 b B d)}{10 e^6}+\frac{2 b^2 (d+e x)^9 (b d-a e) (-3 a B e-2 A b e+5 b B d)}{9 e^6}-\frac{b (d+e x)^8 (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{4 e^6}+\frac{(d+e x)^7 (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{7 e^6}-\frac{(d+e x)^6 (b d-a e)^4 (B d-A e)}{6 e^6}+\frac{b^4 B (d+e x)^{11}}{11 e^6} \]

[Out]

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

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Rubi [A]  time = 0.539008, antiderivative size = 206, 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, 77} \[ -\frac{b^3 (d+e x)^{10} (-4 a B e-A b e+5 b B d)}{10 e^6}+\frac{2 b^2 (d+e x)^9 (b d-a e) (-3 a B e-2 A b e+5 b B d)}{9 e^6}-\frac{b (d+e x)^8 (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{4 e^6}+\frac{(d+e x)^7 (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{7 e^6}-\frac{(d+e x)^6 (b d-a e)^4 (B d-A e)}{6 e^6}+\frac{b^4 B (d+e x)^{11}}{11 e^6} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

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

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

Mathematica [B]  time = 0.207899, size = 615, normalized size = 2.99 \[ \frac{1}{4} b e^2 x^8 \left (3 a^2 b e^2 (A e+5 B d)+2 a^3 B e^3+10 a b^2 d e (A e+2 B d)+5 b^3 d^2 (A e+B d)\right )+\frac{1}{7} e x^7 \left (30 a^2 b^2 d e^2 (A e+2 B d)+4 a^3 b e^3 (A e+5 B d)+a^4 B e^4+40 a b^3 d^2 e (A e+B d)+5 b^4 d^3 (2 A e+B d)\right )+\frac{1}{6} x^6 \left (60 a^2 b^2 d^2 e^2 (A e+B d)+20 a^3 b d e^3 (A e+2 B d)+a^4 e^4 (A e+5 B d)+20 a b^3 d^3 e (2 A e+B d)+b^4 d^4 (5 A e+B d)\right )+\frac{1}{5} d x^5 \left (A \left (60 a^2 b^2 d^2 e^2+40 a^3 b d e^3+5 a^4 e^4+20 a b^3 d^3 e+b^4 d^4\right )+2 a B d \left (20 a^2 b d e^2+5 a^3 e^3+15 a b^2 d^2 e+2 b^3 d^3\right )\right )+\frac{1}{2} a d^2 x^4 \left (A \left (20 a^2 b d e^2+5 a^3 e^3+15 a b^2 d^2 e+2 b^3 d^3\right )+a B d \left (5 a^2 e^2+10 a b d e+3 b^2 d^2\right )\right )+\frac{1}{3} a^2 d^3 x^3 \left (2 A \left (5 a^2 e^2+10 a b d e+3 b^2 d^2\right )+a B d (5 a e+4 b d)\right )+\frac{1}{9} b^2 e^3 x^9 \left (6 a^2 B e^2+4 a b e (A e+5 B d)+5 b^2 d (A e+2 B d)\right )+\frac{1}{2} a^3 d^4 x^2 (5 a A e+a B d+4 A b d)+a^4 A d^5 x+\frac{1}{10} b^3 e^4 x^{10} (4 a B e+A b e+5 b B d)+\frac{1}{11} b^4 B e^5 x^{11} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

a^4*A*d^5*x + (a^3*d^4*(4*A*b*d + a*B*d + 5*a*A*e)*x^2)/2 + (a^2*d^3*(a*B*d*(4*b*d + 5*a*e) + 2*A*(3*b^2*d^2 +
10*a*b*d*e + 5*a^2*e^2))*x^3)/3 + (a*d^2*(a*B*d*(3*b^2*d^2 + 10*a*b*d*e + 5*a^2*e^2) + A*(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^4)/2 + (d*(2*a*B*d*(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^4*d^4 + 20*a*b^3*d^3*e + 60*a^2*b^2*d^2*e^2 + 40*a^3*b*d*e^3 + 5*a^4*e^4))*x^5)/5 + ((60*a^2*b^
2*d^2*e^2*(B*d + A*e) + 20*a^3*b*d*e^3*(2*B*d + A*e) + a^4*e^4*(5*B*d + A*e) + 20*a*b^3*d^3*e*(B*d + 2*A*e) +
b^4*d^4*(B*d + 5*A*e))*x^6)/6 + (e*(a^4*B*e^4 + 40*a*b^3*d^2*e*(B*d + A*e) + 30*a^2*b^2*d*e^2*(2*B*d + A*e) +
4*a^3*b*e^3*(5*B*d + A*e) + 5*b^4*d^3*(B*d + 2*A*e))*x^7)/7 + (b*e^2*(2*a^3*B*e^3 + 5*b^3*d^2*(B*d + A*e) + 10
*a*b^2*d*e*(2*B*d + A*e) + 3*a^2*b*e^2*(5*B*d + A*e))*x^8)/4 + (b^2*e^3*(6*a^2*B*e^2 + 5*b^2*d*(2*B*d + A*e) +
4*a*b*e*(5*B*d + A*e))*x^9)/9 + (b^3*e^4*(5*b*B*d + A*b*e + 4*a*B*e)*x^10)/10 + (b^4*B*e^5*x^11)/11

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Maple [B]  time = 0.002, size = 692, normalized size = 3.4 \begin{align*}{\frac{B{e}^{5}{b}^{4}{x}^{11}}{11}}+{\frac{ \left ( \left ( A{e}^{5}+5\,Bd{e}^{4} \right ){b}^{4}+4\,B{e}^{5}a{b}^{3} \right ){x}^{10}}{10}}+{\frac{ \left ( \left ( 5\,Ad{e}^{4}+10\,B{d}^{2}{e}^{3} \right ){b}^{4}+4\, \left ( A{e}^{5}+5\,Bd{e}^{4} \right ) a{b}^{3}+6\,B{e}^{5}{a}^{2}{b}^{2} \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 10\,A{d}^{2}{e}^{3}+10\,B{d}^{3}{e}^{2} \right ){b}^{4}+4\, \left ( 5\,Ad{e}^{4}+10\,B{d}^{2}{e}^{3} \right ) a{b}^{3}+6\, \left ( A{e}^{5}+5\,Bd{e}^{4} \right ){a}^{2}{b}^{2}+4\,B{e}^{5}{a}^{3}b \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 10\,A{d}^{3}{e}^{2}+5\,B{d}^{4}e \right ){b}^{4}+4\, \left ( 10\,A{d}^{2}{e}^{3}+10\,B{d}^{3}{e}^{2} \right ) a{b}^{3}+6\, \left ( 5\,Ad{e}^{4}+10\,B{d}^{2}{e}^{3} \right ){a}^{2}{b}^{2}+4\, \left ( A{e}^{5}+5\,Bd{e}^{4} \right ){a}^{3}b+B{e}^{5}{a}^{4} \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 5\,A{d}^{4}e+B{d}^{5} \right ){b}^{4}+4\, \left ( 10\,A{d}^{3}{e}^{2}+5\,B{d}^{4}e \right ) a{b}^{3}+6\, \left ( 10\,A{d}^{2}{e}^{3}+10\,B{d}^{3}{e}^{2} \right ){a}^{2}{b}^{2}+4\, \left ( 5\,Ad{e}^{4}+10\,B{d}^{2}{e}^{3} \right ){a}^{3}b+ \left ( A{e}^{5}+5\,Bd{e}^{4} \right ){a}^{4} \right ){x}^{6}}{6}}+{\frac{ \left ( A{d}^{5}{b}^{4}+4\, \left ( 5\,A{d}^{4}e+B{d}^{5} \right ) a{b}^{3}+6\, \left ( 10\,A{d}^{3}{e}^{2}+5\,B{d}^{4}e \right ){a}^{2}{b}^{2}+4\, \left ( 10\,A{d}^{2}{e}^{3}+10\,B{d}^{3}{e}^{2} \right ){a}^{3}b+ \left ( 5\,Ad{e}^{4}+10\,B{d}^{2}{e}^{3} \right ){a}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,A{d}^{5}a{b}^{3}+6\, \left ( 5\,A{d}^{4}e+B{d}^{5} \right ){a}^{2}{b}^{2}+4\, \left ( 10\,A{d}^{3}{e}^{2}+5\,B{d}^{4}e \right ){a}^{3}b+ \left ( 10\,A{d}^{2}{e}^{3}+10\,B{d}^{3}{e}^{2} \right ){a}^{4} \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,A{d}^{5}{a}^{2}{b}^{2}+4\, \left ( 5\,A{d}^{4}e+B{d}^{5} \right ){a}^{3}b+ \left ( 10\,A{d}^{3}{e}^{2}+5\,B{d}^{4}e \right ){a}^{4} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,A{d}^{5}{a}^{3}b+ \left ( 5\,A{d}^{4}e+B{d}^{5} \right ){a}^{4} \right ){x}^{2}}{2}}+A{d}^{5}{a}^{4}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

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

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Maxima [B]  time = 1.02713, size = 926, normalized size = 4.5 \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)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.34572, size = 1891, normalized size = 9.18 \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)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

1/11*x^11*e^5*b^4*B + 1/2*x^10*e^4*d*b^4*B + 2/5*x^10*e^5*b^3*a*B + 1/10*x^10*e^5*b^4*A + 10/9*x^9*e^3*d^2*b^4
*B + 20/9*x^9*e^4*d*b^3*a*B + 2/3*x^9*e^5*b^2*a^2*B + 5/9*x^9*e^4*d*b^4*A + 4/9*x^9*e^5*b^3*a*A + 5/4*x^8*e^2*
d^3*b^4*B + 5*x^8*e^3*d^2*b^3*a*B + 15/4*x^8*e^4*d*b^2*a^2*B + 1/2*x^8*e^5*b*a^3*B + 5/4*x^8*e^3*d^2*b^4*A + 5
/2*x^8*e^4*d*b^3*a*A + 3/4*x^8*e^5*b^2*a^2*A + 5/7*x^7*e*d^4*b^4*B + 40/7*x^7*e^2*d^3*b^3*a*B + 60/7*x^7*e^3*d
^2*b^2*a^2*B + 20/7*x^7*e^4*d*b*a^3*B + 1/7*x^7*e^5*a^4*B + 10/7*x^7*e^2*d^3*b^4*A + 40/7*x^7*e^3*d^2*b^3*a*A
+ 30/7*x^7*e^4*d*b^2*a^2*A + 4/7*x^7*e^5*b*a^3*A + 1/6*x^6*d^5*b^4*B + 10/3*x^6*e*d^4*b^3*a*B + 10*x^6*e^2*d^3
*b^2*a^2*B + 20/3*x^6*e^3*d^2*b*a^3*B + 5/6*x^6*e^4*d*a^4*B + 5/6*x^6*e*d^4*b^4*A + 20/3*x^6*e^2*d^3*b^3*a*A +
10*x^6*e^3*d^2*b^2*a^2*A + 10/3*x^6*e^4*d*b*a^3*A + 1/6*x^6*e^5*a^4*A + 4/5*x^5*d^5*b^3*a*B + 6*x^5*e*d^4*b^2
*a^2*B + 8*x^5*e^2*d^3*b*a^3*B + 2*x^5*e^3*d^2*a^4*B + 1/5*x^5*d^5*b^4*A + 4*x^5*e*d^4*b^3*a*A + 12*x^5*e^2*d^
3*b^2*a^2*A + 8*x^5*e^3*d^2*b*a^3*A + x^5*e^4*d*a^4*A + 3/2*x^4*d^5*b^2*a^2*B + 5*x^4*e*d^4*b*a^3*B + 5/2*x^4*
e^2*d^3*a^4*B + x^4*d^5*b^3*a*A + 15/2*x^4*e*d^4*b^2*a^2*A + 10*x^4*e^2*d^3*b*a^3*A + 5/2*x^4*e^3*d^2*a^4*A +
4/3*x^3*d^5*b*a^3*B + 5/3*x^3*e*d^4*a^4*B + 2*x^3*d^5*b^2*a^2*A + 20/3*x^3*e*d^4*b*a^3*A + 10/3*x^3*e^2*d^3*a^
4*A + 1/2*x^2*d^5*a^4*B + 2*x^2*d^5*b*a^3*A + 5/2*x^2*e*d^4*a^4*A + x*d^5*a^4*A

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Sympy [B]  time = 0.16516, size = 884, normalized size = 4.29 \begin{align*} A a^{4} d^{5} x + \frac{B b^{4} e^{5} x^{11}}{11} + x^{10} \left (\frac{A b^{4} e^{5}}{10} + \frac{2 B a b^{3} e^{5}}{5} + \frac{B b^{4} d e^{4}}{2}\right ) + x^{9} \left (\frac{4 A a b^{3} e^{5}}{9} + \frac{5 A b^{4} d e^{4}}{9} + \frac{2 B a^{2} b^{2} e^{5}}{3} + \frac{20 B a b^{3} d e^{4}}{9} + \frac{10 B b^{4} d^{2} e^{3}}{9}\right ) + x^{8} \left (\frac{3 A a^{2} b^{2} e^{5}}{4} + \frac{5 A a b^{3} d e^{4}}{2} + \frac{5 A b^{4} d^{2} e^{3}}{4} + \frac{B a^{3} b e^{5}}{2} + \frac{15 B a^{2} b^{2} d e^{4}}{4} + 5 B a b^{3} d^{2} e^{3} + \frac{5 B b^{4} d^{3} e^{2}}{4}\right ) + x^{7} \left (\frac{4 A a^{3} b e^{5}}{7} + \frac{30 A a^{2} b^{2} d e^{4}}{7} + \frac{40 A a b^{3} d^{2} e^{3}}{7} + \frac{10 A b^{4} d^{3} e^{2}}{7} + \frac{B a^{4} e^{5}}{7} + \frac{20 B a^{3} b d e^{4}}{7} + \frac{60 B a^{2} b^{2} d^{2} e^{3}}{7} + \frac{40 B a b^{3} d^{3} e^{2}}{7} + \frac{5 B b^{4} d^{4} e}{7}\right ) + x^{6} \left (\frac{A a^{4} e^{5}}{6} + \frac{10 A a^{3} b d e^{4}}{3} + 10 A a^{2} b^{2} d^{2} e^{3} + \frac{20 A a b^{3} d^{3} e^{2}}{3} + \frac{5 A b^{4} d^{4} e}{6} + \frac{5 B a^{4} d e^{4}}{6} + \frac{20 B a^{3} b d^{2} e^{3}}{3} + 10 B a^{2} b^{2} d^{3} e^{2} + \frac{10 B a b^{3} d^{4} e}{3} + \frac{B b^{4} d^{5}}{6}\right ) + x^{5} \left (A a^{4} d e^{4} + 8 A a^{3} b d^{2} e^{3} + 12 A a^{2} b^{2} d^{3} e^{2} + 4 A a b^{3} d^{4} e + \frac{A b^{4} d^{5}}{5} + 2 B a^{4} d^{2} e^{3} + 8 B a^{3} b d^{3} e^{2} + 6 B a^{2} b^{2} d^{4} e + \frac{4 B a b^{3} d^{5}}{5}\right ) + x^{4} \left (\frac{5 A a^{4} d^{2} e^{3}}{2} + 10 A a^{3} b d^{3} e^{2} + \frac{15 A a^{2} b^{2} d^{4} e}{2} + A a b^{3} d^{5} + \frac{5 B a^{4} d^{3} e^{2}}{2} + 5 B a^{3} b d^{4} e + \frac{3 B a^{2} b^{2} d^{5}}{2}\right ) + x^{3} \left (\frac{10 A a^{4} d^{3} e^{2}}{3} + \frac{20 A a^{3} b d^{4} e}{3} + 2 A a^{2} b^{2} d^{5} + \frac{5 B a^{4} d^{4} e}{3} + \frac{4 B a^{3} b d^{5}}{3}\right ) + x^{2} \left (\frac{5 A a^{4} d^{4} e}{2} + 2 A a^{3} b d^{5} + \frac{B a^{4} d^{5}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**5*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

A*a**4*d**5*x + B*b**4*e**5*x**11/11 + x**10*(A*b**4*e**5/10 + 2*B*a*b**3*e**5/5 + B*b**4*d*e**4/2) + x**9*(4*
A*a*b**3*e**5/9 + 5*A*b**4*d*e**4/9 + 2*B*a**2*b**2*e**5/3 + 20*B*a*b**3*d*e**4/9 + 10*B*b**4*d**2*e**3/9) + x
**8*(3*A*a**2*b**2*e**5/4 + 5*A*a*b**3*d*e**4/2 + 5*A*b**4*d**2*e**3/4 + B*a**3*b*e**5/2 + 15*B*a**2*b**2*d*e*
*4/4 + 5*B*a*b**3*d**2*e**3 + 5*B*b**4*d**3*e**2/4) + x**7*(4*A*a**3*b*e**5/7 + 30*A*a**2*b**2*d*e**4/7 + 40*A
*a*b**3*d**2*e**3/7 + 10*A*b**4*d**3*e**2/7 + B*a**4*e**5/7 + 20*B*a**3*b*d*e**4/7 + 60*B*a**2*b**2*d**2*e**3/
7 + 40*B*a*b**3*d**3*e**2/7 + 5*B*b**4*d**4*e/7) + x**6*(A*a**4*e**5/6 + 10*A*a**3*b*d*e**4/3 + 10*A*a**2*b**2
*d**2*e**3 + 20*A*a*b**3*d**3*e**2/3 + 5*A*b**4*d**4*e/6 + 5*B*a**4*d*e**4/6 + 20*B*a**3*b*d**2*e**3/3 + 10*B*
a**2*b**2*d**3*e**2 + 10*B*a*b**3*d**4*e/3 + B*b**4*d**5/6) + x**5*(A*a**4*d*e**4 + 8*A*a**3*b*d**2*e**3 + 12*
A*a**2*b**2*d**3*e**2 + 4*A*a*b**3*d**4*e + A*b**4*d**5/5 + 2*B*a**4*d**2*e**3 + 8*B*a**3*b*d**3*e**2 + 6*B*a*
*2*b**2*d**4*e + 4*B*a*b**3*d**5/5) + x**4*(5*A*a**4*d**2*e**3/2 + 10*A*a**3*b*d**3*e**2 + 15*A*a**2*b**2*d**4
*e/2 + A*a*b**3*d**5 + 5*B*a**4*d**3*e**2/2 + 5*B*a**3*b*d**4*e + 3*B*a**2*b**2*d**5/2) + x**3*(10*A*a**4*d**3
*e**2/3 + 20*A*a**3*b*d**4*e/3 + 2*A*a**2*b**2*d**5 + 5*B*a**4*d**4*e/3 + 4*B*a**3*b*d**5/3) + x**2*(5*A*a**4*
d**4*e/2 + 2*A*a**3*b*d**5 + B*a**4*d**5/2)

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Giac [B]  time = 1.161, size = 1115, normalized size = 5.41 \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)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

1/11*B*b^4*x^11*e^5 + 1/2*B*b^4*d*x^10*e^4 + 10/9*B*b^4*d^2*x^9*e^3 + 5/4*B*b^4*d^3*x^8*e^2 + 5/7*B*b^4*d^4*x^
7*e + 1/6*B*b^4*d^5*x^6 + 2/5*B*a*b^3*x^10*e^5 + 1/10*A*b^4*x^10*e^5 + 20/9*B*a*b^3*d*x^9*e^4 + 5/9*A*b^4*d*x^
9*e^4 + 5*B*a*b^3*d^2*x^8*e^3 + 5/4*A*b^4*d^2*x^8*e^3 + 40/7*B*a*b^3*d^3*x^7*e^2 + 10/7*A*b^4*d^3*x^7*e^2 + 10
/3*B*a*b^3*d^4*x^6*e + 5/6*A*b^4*d^4*x^6*e + 4/5*B*a*b^3*d^5*x^5 + 1/5*A*b^4*d^5*x^5 + 2/3*B*a^2*b^2*x^9*e^5 +
4/9*A*a*b^3*x^9*e^5 + 15/4*B*a^2*b^2*d*x^8*e^4 + 5/2*A*a*b^3*d*x^8*e^4 + 60/7*B*a^2*b^2*d^2*x^7*e^3 + 40/7*A*
a*b^3*d^2*x^7*e^3 + 10*B*a^2*b^2*d^3*x^6*e^2 + 20/3*A*a*b^3*d^3*x^6*e^2 + 6*B*a^2*b^2*d^4*x^5*e + 4*A*a*b^3*d^
4*x^5*e + 3/2*B*a^2*b^2*d^5*x^4 + A*a*b^3*d^5*x^4 + 1/2*B*a^3*b*x^8*e^5 + 3/4*A*a^2*b^2*x^8*e^5 + 20/7*B*a^3*b
*d*x^7*e^4 + 30/7*A*a^2*b^2*d*x^7*e^4 + 20/3*B*a^3*b*d^2*x^6*e^3 + 10*A*a^2*b^2*d^2*x^6*e^3 + 8*B*a^3*b*d^3*x^
5*e^2 + 12*A*a^2*b^2*d^3*x^5*e^2 + 5*B*a^3*b*d^4*x^4*e + 15/2*A*a^2*b^2*d^4*x^4*e + 4/3*B*a^3*b*d^5*x^3 + 2*A*
a^2*b^2*d^5*x^3 + 1/7*B*a^4*x^7*e^5 + 4/7*A*a^3*b*x^7*e^5 + 5/6*B*a^4*d*x^6*e^4 + 10/3*A*a^3*b*d*x^6*e^4 + 2*B
*a^4*d^2*x^5*e^3 + 8*A*a^3*b*d^2*x^5*e^3 + 5/2*B*a^4*d^3*x^4*e^2 + 10*A*a^3*b*d^3*x^4*e^2 + 5/3*B*a^4*d^4*x^3*
e + 20/3*A*a^3*b*d^4*x^3*e + 1/2*B*a^4*d^5*x^2 + 2*A*a^3*b*d^5*x^2 + 1/6*A*a^4*x^6*e^5 + A*a^4*d*x^5*e^4 + 5/2
*A*a^4*d^2*x^4*e^3 + 10/3*A*a^4*d^3*x^3*e^2 + 5/2*A*a^4*d^4*x^2*e + A*a^4*d^5*x