Optimal. Leaf size=118 \[ \frac{e (a+b x)^7 (-3 a B e+A b e+2 b B d)}{7 b^4}+\frac{(a+b x)^6 (b d-a e) (-3 a B e+2 A b e+b B d)}{6 b^4}+\frac{(a+b x)^5 (A b-a B) (b d-a e)^2}{5 b^4}+\frac{B e^2 (a+b x)^8}{8 b^4} \]
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Rubi [A] time = 0.221503, antiderivative size = 118, 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} \[ \frac{e (a+b x)^7 (-3 a B e+A b e+2 b B d)}{7 b^4}+\frac{(a+b x)^6 (b d-a e) (-3 a B e+2 A b e+b B d)}{6 b^4}+\frac{(a+b x)^5 (A b-a B) (b d-a e)^2}{5 b^4}+\frac{B e^2 (a+b x)^8}{8 b^4} \]
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin{align*} \int (A+B x) (d+e x)^2 \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)^2 \, dx\\ &=\int \left (\frac{(A b-a B) (b d-a e)^2 (a+b x)^4}{b^3}+\frac{(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^5}{b^3}+\frac{e (2 b B d+A b e-3 a B e) (a+b x)^6}{b^3}+\frac{B e^2 (a+b x)^7}{b^3}\right ) \, dx\\ &=\frac{(A b-a B) (b d-a e)^2 (a+b x)^5}{5 b^4}+\frac{(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^6}{6 b^4}+\frac{e (2 b B d+A b e-3 a B e) (a+b x)^7}{7 b^4}+\frac{B e^2 (a+b x)^8}{8 b^4}\\ \end{align*}
Mathematica [B] time = 0.102134, size = 288, normalized size = 2.44 \[ \frac{1}{5} b x^5 \left (A b \left (6 a^2 e^2+8 a b d e+b^2 d^2\right )+4 a B \left (a^2 e^2+3 a b d e+b^2 d^2\right )\right )+\frac{1}{4} a x^4 \left (4 A b \left (a^2 e^2+3 a b d e+b^2 d^2\right )+a B \left (a^2 e^2+8 a b d e+6 b^2 d^2\right )\right )+\frac{1}{3} a^2 x^3 \left (A \left (a^2 e^2+8 a b d e+6 b^2 d^2\right )+2 a B d (a e+2 b d)\right )+\frac{1}{6} b^2 x^6 \left (6 a^2 B e^2+4 a b e (A e+2 B d)+b^2 d (2 A e+B d)\right )+\frac{1}{2} a^3 d x^2 (2 a A e+a B d+4 A b d)+a^4 A d^2 x+\frac{1}{7} b^3 e x^7 (4 a B e+A b e+2 b B d)+\frac{1}{8} b^4 B e^2 x^8 \]
Antiderivative was successfully verified.
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Maple [B] time = 0., size = 305, normalized size = 2.6 \begin{align*}{\frac{B{e}^{2}{b}^{4}{x}^{8}}{8}}+{\frac{ \left ( \left ( A{e}^{2}+2\,Bde \right ){b}^{4}+4\,B{e}^{2}a{b}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 2\,Ade+B{d}^{2} \right ){b}^{4}+4\, \left ( A{e}^{2}+2\,Bde \right ) a{b}^{3}+6\,B{e}^{2}{a}^{2}{b}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( A{d}^{2}{b}^{4}+4\, \left ( 2\,Ade+B{d}^{2} \right ) a{b}^{3}+6\, \left ( A{e}^{2}+2\,Bde \right ){a}^{2}{b}^{2}+4\,B{e}^{2}{a}^{3}b \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,A{d}^{2}a{b}^{3}+6\, \left ( 2\,Ade+B{d}^{2} \right ){a}^{2}{b}^{2}+4\, \left ( A{e}^{2}+2\,Bde \right ){a}^{3}b+B{e}^{2}{a}^{4} \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,A{d}^{2}{a}^{2}{b}^{2}+4\, \left ( 2\,Ade+B{d}^{2} \right ){a}^{3}b+ \left ( A{e}^{2}+2\,Bde \right ){a}^{4} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,A{d}^{2}{a}^{3}b+ \left ( 2\,Ade+B{d}^{2} \right ){a}^{4} \right ){x}^{2}}{2}}+A{d}^{2}{a}^{4}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.988304, size = 435, normalized size = 3.69 \begin{align*} \frac{1}{8} \, B b^{4} e^{2} x^{8} + A a^{4} d^{2} x + \frac{1}{7} \,{\left (2 \, B b^{4} d e +{\left (4 \, B a b^{3} + A b^{4}\right )} e^{2}\right )} x^{7} + \frac{1}{6} \,{\left (B b^{4} d^{2} + 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{2}\right )} x^{6} + \frac{1}{5} \,{\left ({\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} + 4 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e + 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} + 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e +{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (A a^{4} e^{2} + 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} + 2 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e\right )} x^{3} + \frac{1}{2} \,{\left (2 \, A a^{4} d e +{\left (B a^{4} + 4 \, A a^{3} b\right )} d^{2}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.38025, size = 834, normalized size = 7.07 \begin{align*} \frac{1}{8} x^{8} e^{2} b^{4} B + \frac{2}{7} x^{7} e d b^{4} B + \frac{4}{7} x^{7} e^{2} b^{3} a B + \frac{1}{7} x^{7} e^{2} b^{4} A + \frac{1}{6} x^{6} d^{2} b^{4} B + \frac{4}{3} x^{6} e d b^{3} a B + x^{6} e^{2} b^{2} a^{2} B + \frac{1}{3} x^{6} e d b^{4} A + \frac{2}{3} x^{6} e^{2} b^{3} a A + \frac{4}{5} x^{5} d^{2} b^{3} a B + \frac{12}{5} x^{5} e d b^{2} a^{2} B + \frac{4}{5} x^{5} e^{2} b a^{3} B + \frac{1}{5} x^{5} d^{2} b^{4} A + \frac{8}{5} x^{5} e d b^{3} a A + \frac{6}{5} x^{5} e^{2} b^{2} a^{2} A + \frac{3}{2} x^{4} d^{2} b^{2} a^{2} B + 2 x^{4} e d b a^{3} B + \frac{1}{4} x^{4} e^{2} a^{4} B + x^{4} d^{2} b^{3} a A + 3 x^{4} e d b^{2} a^{2} A + x^{4} e^{2} b a^{3} A + \frac{4}{3} x^{3} d^{2} b a^{3} B + \frac{2}{3} x^{3} e d a^{4} B + 2 x^{3} d^{2} b^{2} a^{2} A + \frac{8}{3} x^{3} e d b a^{3} A + \frac{1}{3} x^{3} e^{2} a^{4} A + \frac{1}{2} x^{2} d^{2} a^{4} B + 2 x^{2} d^{2} b a^{3} A + x^{2} e d a^{4} A + x d^{2} a^{4} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.119334, size = 384, normalized size = 3.25 \begin{align*} A a^{4} d^{2} x + \frac{B b^{4} e^{2} x^{8}}{8} + x^{7} \left (\frac{A b^{4} e^{2}}{7} + \frac{4 B a b^{3} e^{2}}{7} + \frac{2 B b^{4} d e}{7}\right ) + x^{6} \left (\frac{2 A a b^{3} e^{2}}{3} + \frac{A b^{4} d e}{3} + B a^{2} b^{2} e^{2} + \frac{4 B a b^{3} d e}{3} + \frac{B b^{4} d^{2}}{6}\right ) + x^{5} \left (\frac{6 A a^{2} b^{2} e^{2}}{5} + \frac{8 A a b^{3} d e}{5} + \frac{A b^{4} d^{2}}{5} + \frac{4 B a^{3} b e^{2}}{5} + \frac{12 B a^{2} b^{2} d e}{5} + \frac{4 B a b^{3} d^{2}}{5}\right ) + x^{4} \left (A a^{3} b e^{2} + 3 A a^{2} b^{2} d e + A a b^{3} d^{2} + \frac{B a^{4} e^{2}}{4} + 2 B a^{3} b d e + \frac{3 B a^{2} b^{2} d^{2}}{2}\right ) + x^{3} \left (\frac{A a^{4} e^{2}}{3} + \frac{8 A a^{3} b d e}{3} + 2 A a^{2} b^{2} d^{2} + \frac{2 B a^{4} d e}{3} + \frac{4 B a^{3} b d^{2}}{3}\right ) + x^{2} \left (A a^{4} d e + 2 A a^{3} b d^{2} + \frac{B a^{4} d^{2}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1621, size = 505, normalized size = 4.28 \begin{align*} \frac{1}{8} \, B b^{4} x^{8} e^{2} + \frac{2}{7} \, B b^{4} d x^{7} e + \frac{1}{6} \, B b^{4} d^{2} x^{6} + \frac{4}{7} \, B a b^{3} x^{7} e^{2} + \frac{1}{7} \, A b^{4} x^{7} e^{2} + \frac{4}{3} \, B a b^{3} d x^{6} e + \frac{1}{3} \, A b^{4} d x^{6} e + \frac{4}{5} \, B a b^{3} d^{2} x^{5} + \frac{1}{5} \, A b^{4} d^{2} x^{5} + B a^{2} b^{2} x^{6} e^{2} + \frac{2}{3} \, A a b^{3} x^{6} e^{2} + \frac{12}{5} \, B a^{2} b^{2} d x^{5} e + \frac{8}{5} \, A a b^{3} d x^{5} e + \frac{3}{2} \, B a^{2} b^{2} d^{2} x^{4} + A a b^{3} d^{2} x^{4} + \frac{4}{5} \, B a^{3} b x^{5} e^{2} + \frac{6}{5} \, A a^{2} b^{2} x^{5} e^{2} + 2 \, B a^{3} b d x^{4} e + 3 \, A a^{2} b^{2} d x^{4} e + \frac{4}{3} \, B a^{3} b d^{2} x^{3} + 2 \, A a^{2} b^{2} d^{2} x^{3} + \frac{1}{4} \, B a^{4} x^{4} e^{2} + A a^{3} b x^{4} e^{2} + \frac{2}{3} \, B a^{4} d x^{3} e + \frac{8}{3} \, A a^{3} b d x^{3} e + \frac{1}{2} \, B a^{4} d^{2} x^{2} + 2 \, A a^{3} b d^{2} x^{2} + \frac{1}{3} \, A a^{4} x^{3} e^{2} + A a^{4} d x^{2} e + A a^{4} d^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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