Optimal. Leaf size=75 \[ \frac{(a+b x)^6 (-2 a B e+A b e+b B d)}{6 b^3}+\frac{(a+b x)^5 (A b-a B) (b d-a e)}{5 b^3}+\frac{B e (a+b x)^7}{7 b^3} \]
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Rubi [A] time = 0.137509, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {27, 77} \[ \frac{(a+b x)^6 (-2 a B e+A b e+b B d)}{6 b^3}+\frac{(a+b x)^5 (A b-a B) (b d-a e)}{5 b^3}+\frac{B e (a+b x)^7}{7 b^3} \]
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin{align*} \int (A+B x) (d+e x) \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) \, dx\\ &=\int \left (\frac{(A b-a B) (b d-a e) (a+b x)^4}{b^2}+\frac{(b B d+A b e-2 a B e) (a+b x)^5}{b^2}+\frac{B e (a+b x)^6}{b^2}\right ) \, dx\\ &=\frac{(A b-a B) (b d-a e) (a+b x)^5}{5 b^3}+\frac{(b B d+A b e-2 a B e) (a+b x)^6}{6 b^3}+\frac{B e (a+b x)^7}{7 b^3}\\ \end{align*}
Mathematica [B] time = 0.0579466, size = 172, normalized size = 2.29 \[ \frac{1}{3} a^2 x^3 (2 A b (2 a e+3 b d)+a B (a e+4 b d))+\frac{1}{2} a^3 x^2 (a A e+a B d+4 A b d)+a^4 A d x+\frac{1}{6} b^3 x^6 (4 a B e+A b e+b B d)+\frac{1}{5} b^2 x^5 (A b (4 a e+b d)+2 a B (3 a e+2 b d))+\frac{1}{2} a b x^4 (A b (3 a e+2 b d)+a B (2 a e+3 b d))+\frac{1}{7} b^4 B e x^7 \]
Antiderivative was successfully verified.
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Maple [B] time = 0.002, size = 176, normalized size = 2.4 \begin{align*}{\frac{Be{b}^{4}{x}^{7}}{7}}+{\frac{ \left ( \left ( Ae+Bd \right ){b}^{4}+4\,Bea{b}^{3} \right ){x}^{6}}{6}}+{\frac{ \left ( Ad{b}^{4}+4\, \left ( Ae+Bd \right ) a{b}^{3}+6\,Be{a}^{2}{b}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,Ada{b}^{3}+6\, \left ( Ae+Bd \right ){a}^{2}{b}^{2}+4\,Be{a}^{3}b \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,Ad{a}^{2}{b}^{2}+4\, \left ( Ae+Bd \right ){a}^{3}b+Be{a}^{4} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,Ad{a}^{3}b+ \left ( Ae+Bd \right ){a}^{4} \right ){x}^{2}}{2}}+Ad{a}^{4}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.06231, size = 267, normalized size = 3.56 \begin{align*} \frac{1}{7} \, B b^{4} e x^{7} + A a^{4} d x + \frac{1}{6} \,{\left (B b^{4} d +{\left (4 \, B a b^{3} + A b^{4}\right )} e\right )} x^{6} + \frac{1}{5} \,{\left ({\left (4 \, B a b^{3} + A b^{4}\right )} d + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e\right )} x^{5} + \frac{1}{2} \,{\left ({\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d +{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e\right )} x^{4} + \frac{1}{3} \,{\left (2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d +{\left (B a^{4} + 4 \, A a^{3} b\right )} e\right )} x^{3} + \frac{1}{2} \,{\left (A a^{4} e +{\left (B a^{4} + 4 \, A a^{3} b\right )} d\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.36694, size = 504, normalized size = 6.72 \begin{align*} \frac{1}{7} x^{7} e b^{4} B + \frac{1}{6} x^{6} d b^{4} B + \frac{2}{3} x^{6} e b^{3} a B + \frac{1}{6} x^{6} e b^{4} A + \frac{4}{5} x^{5} d b^{3} a B + \frac{6}{5} x^{5} e b^{2} a^{2} B + \frac{1}{5} x^{5} d b^{4} A + \frac{4}{5} x^{5} e b^{3} a A + \frac{3}{2} x^{4} d b^{2} a^{2} B + x^{4} e b a^{3} B + x^{4} d b^{3} a A + \frac{3}{2} x^{4} e b^{2} a^{2} A + \frac{4}{3} x^{3} d b a^{3} B + \frac{1}{3} x^{3} e a^{4} B + 2 x^{3} d b^{2} a^{2} A + \frac{4}{3} x^{3} e b a^{3} A + \frac{1}{2} x^{2} d a^{4} B + 2 x^{2} d b a^{3} A + \frac{1}{2} x^{2} e a^{4} A + x d a^{4} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.091704, size = 226, normalized size = 3.01 \begin{align*} A a^{4} d x + \frac{B b^{4} e x^{7}}{7} + x^{6} \left (\frac{A b^{4} e}{6} + \frac{2 B a b^{3} e}{3} + \frac{B b^{4} d}{6}\right ) + x^{5} \left (\frac{4 A a b^{3} e}{5} + \frac{A b^{4} d}{5} + \frac{6 B a^{2} b^{2} e}{5} + \frac{4 B a b^{3} d}{5}\right ) + x^{4} \left (\frac{3 A a^{2} b^{2} e}{2} + A a b^{3} d + B a^{3} b e + \frac{3 B a^{2} b^{2} d}{2}\right ) + x^{3} \left (\frac{4 A a^{3} b e}{3} + 2 A a^{2} b^{2} d + \frac{B a^{4} e}{3} + \frac{4 B a^{3} b d}{3}\right ) + x^{2} \left (\frac{A a^{4} e}{2} + 2 A a^{3} b d + \frac{B a^{4} d}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13598, size = 305, normalized size = 4.07 \begin{align*} \frac{1}{7} \, B b^{4} x^{7} e + \frac{1}{6} \, B b^{4} d x^{6} + \frac{2}{3} \, B a b^{3} x^{6} e + \frac{1}{6} \, A b^{4} x^{6} e + \frac{4}{5} \, B a b^{3} d x^{5} + \frac{1}{5} \, A b^{4} d x^{5} + \frac{6}{5} \, B a^{2} b^{2} x^{5} e + \frac{4}{5} \, A a b^{3} x^{5} e + \frac{3}{2} \, B a^{2} b^{2} d x^{4} + A a b^{3} d x^{4} + B a^{3} b x^{4} e + \frac{3}{2} \, A a^{2} b^{2} x^{4} e + \frac{4}{3} \, B a^{3} b d x^{3} + 2 \, A a^{2} b^{2} d x^{3} + \frac{1}{3} \, B a^{4} x^{3} e + \frac{4}{3} \, A a^{3} b x^{3} e + \frac{1}{2} \, B a^{4} d x^{2} + 2 \, A a^{3} b d x^{2} + \frac{1}{2} \, A a^{4} x^{2} e + A a^{4} d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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