Optimal. Leaf size=128 \[ \frac {e x \left (A c e (3 c d-b e)+B \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )\right )}{c^3}+\frac {(b B-A c) (c d-b e)^3 \log (b+c x)}{b c^4}+\frac {e^2 x^2 (A c e-b B e+3 B c d)}{2 c^2}+\frac {A d^3 \log (x)}{b}+\frac {B e^3 x^3}{3 c} \]
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Rubi [A] time = 0.16, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {771} \[ \frac {e x \left (A c e (3 c d-b e)+B \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )\right )}{c^3}+\frac {e^2 x^2 (A c e-b B e+3 B c d)}{2 c^2}+\frac {(b B-A c) (c d-b e)^3 \log (b+c x)}{b c^4}+\frac {A d^3 \log (x)}{b}+\frac {B e^3 x^3}{3 c} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^3}{b x+c x^2} \, dx &=\int \left (\frac {e \left (A c e (3 c d-b e)+B \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right )}{c^3}+\frac {A d^3}{b x}+\frac {e^2 (3 B c d-b B e+A c e) x}{c^2}+\frac {B e^3 x^2}{c}-\frac {(b B-A c) (-c d+b e)^3}{b c^3 (b+c x)}\right ) \, dx\\ &=\frac {e \left (A c e (3 c d-b e)+B \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right ) x}{c^3}+\frac {e^2 (3 B c d-b B e+A c e) x^2}{2 c^2}+\frac {B e^3 x^3}{3 c}+\frac {A d^3 \log (x)}{b}+\frac {(b B-A c) (c d-b e)^3 \log (b+c x)}{b c^4}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 118, normalized size = 0.92 \[ \frac {b c e x \left (3 A c e (-2 b e+6 c d+c e x)+B \left (6 b^2 e^2-3 b c e (6 d+e x)+c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )\right )-6 (b B-A c) (b e-c d)^3 \log (b+c x)+6 A c^4 d^3 \log (x)}{6 b c^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 216, normalized size = 1.69 \[ \frac {2 \, B b c^{3} e^{3} x^{3} + 6 \, A c^{4} d^{3} \log \relax (x) + 3 \, {\left (3 \, B b c^{3} d e^{2} - {\left (B b^{2} c^{2} - A b c^{3}\right )} e^{3}\right )} x^{2} + 6 \, {\left (3 \, B b c^{3} d^{2} e - 3 \, {\left (B b^{2} c^{2} - A b c^{3}\right )} d e^{2} + {\left (B b^{3} c - A b^{2} c^{2}\right )} e^{3}\right )} x + 6 \, {\left ({\left (B b c^{3} - A c^{4}\right )} d^{3} - 3 \, {\left (B b^{2} c^{2} - A b c^{3}\right )} d^{2} e + 3 \, {\left (B b^{3} c - A b^{2} c^{2}\right )} d e^{2} - {\left (B b^{4} - A b^{3} c\right )} e^{3}\right )} \log \left (c x + b\right )}{6 \, b c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 207, normalized size = 1.62 \[ \frac {A d^{3} \log \left ({\left | x \right |}\right )}{b} + \frac {2 \, B c^{2} x^{3} e^{3} + 9 \, B c^{2} d x^{2} e^{2} + 18 \, B c^{2} d^{2} x e - 3 \, B b c x^{2} e^{3} + 3 \, A c^{2} x^{2} e^{3} - 18 \, B b c d x e^{2} + 18 \, A c^{2} d x e^{2} + 6 \, B b^{2} x e^{3} - 6 \, A b c x e^{3}}{6 \, c^{3}} + \frac {{\left (B b c^{3} d^{3} - A c^{4} d^{3} - 3 \, B b^{2} c^{2} d^{2} e + 3 \, A b c^{3} d^{2} e + 3 \, B b^{3} c d e^{2} - 3 \, A b^{2} c^{2} d e^{2} - B b^{4} e^{3} + A b^{3} c e^{3}\right )} \log \left ({\left | c x + b \right |}\right )}{b c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 252, normalized size = 1.97 \[ \frac {B \,e^{3} x^{3}}{3 c}+\frac {A \,e^{3} x^{2}}{2 c}-\frac {B b \,e^{3} x^{2}}{2 c^{2}}+\frac {3 B d \,e^{2} x^{2}}{2 c}+\frac {A \,b^{2} e^{3} \ln \left (c x +b \right )}{c^{3}}-\frac {3 A b d \,e^{2} \ln \left (c x +b \right )}{c^{2}}-\frac {A b \,e^{3} x}{c^{2}}+\frac {A \,d^{3} \ln \relax (x )}{b}-\frac {A \,d^{3} \ln \left (c x +b \right )}{b}+\frac {3 A \,d^{2} e \ln \left (c x +b \right )}{c}+\frac {3 A d \,e^{2} x}{c}-\frac {B \,b^{3} e^{3} \ln \left (c x +b \right )}{c^{4}}+\frac {3 B \,b^{2} d \,e^{2} \ln \left (c x +b \right )}{c^{3}}+\frac {B \,b^{2} e^{3} x}{c^{3}}-\frac {3 B b \,d^{2} e \ln \left (c x +b \right )}{c^{2}}-\frac {3 B b d \,e^{2} x}{c^{2}}+\frac {B \,d^{3} \ln \left (c x +b \right )}{c}+\frac {3 B \,d^{2} e x}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 200, normalized size = 1.56 \[ \frac {A d^{3} \log \relax (x)}{b} + \frac {2 \, B c^{2} e^{3} x^{3} + 3 \, {\left (3 \, B c^{2} d e^{2} - {\left (B b c - A c^{2}\right )} e^{3}\right )} x^{2} + 6 \, {\left (3 \, B c^{2} d^{2} e - 3 \, {\left (B b c - A c^{2}\right )} d e^{2} + {\left (B b^{2} - A b c\right )} e^{3}\right )} x}{6 \, c^{3}} + \frac {{\left ({\left (B b c^{3} - A c^{4}\right )} d^{3} - 3 \, {\left (B b^{2} c^{2} - A b c^{3}\right )} d^{2} e + 3 \, {\left (B b^{3} c - A b^{2} c^{2}\right )} d e^{2} - {\left (B b^{4} - A b^{3} c\right )} e^{3}\right )} \log \left (c x + b\right )}{b c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.58, size = 208, normalized size = 1.62 \[ x^2\,\left (\frac {A\,e^3+3\,B\,d\,e^2}{2\,c}-\frac {B\,b\,e^3}{2\,c^2}\right )-x\,\left (\frac {b\,\left (\frac {A\,e^3+3\,B\,d\,e^2}{c}-\frac {B\,b\,e^3}{c^2}\right )}{c}-\frac {3\,d\,e\,\left (A\,e+B\,d\right )}{c}\right )-\ln \left (b+c\,x\right )\,\left (\frac {A\,d^3}{b}-\frac {c^3\,\left (B\,b\,d^3+3\,A\,b\,e\,d^2\right )-c^2\,\left (3\,B\,b^2\,d^2\,e+3\,A\,b^2\,d\,e^2\right )+c\,\left (A\,b^3\,e^3+3\,B\,d\,b^3\,e^2\right )-B\,b^4\,e^3}{b\,c^4}\right )+\frac {A\,d^3\,\ln \relax (x)}{b}+\frac {B\,e^3\,x^3}{3\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 4.59, size = 264, normalized size = 2.06 \[ \frac {A d^{3} \log {\relax (x )}}{b} + \frac {B e^{3} x^{3}}{3 c} + x^{2} \left (\frac {A e^{3}}{2 c} - \frac {B b e^{3}}{2 c^{2}} + \frac {3 B d e^{2}}{2 c}\right ) + x \left (- \frac {A b e^{3}}{c^{2}} + \frac {3 A d e^{2}}{c} + \frac {B b^{2} e^{3}}{c^{3}} - \frac {3 B b d e^{2}}{c^{2}} + \frac {3 B d^{2} e}{c}\right ) - \frac {\left (- A c + B b\right ) \left (b e - c d\right )^{3} \log {\left (x + \frac {A b c^{3} d^{3} + \frac {b \left (- A c + B b\right ) \left (b e - c d\right )^{3}}{c}}{- A b^{3} c e^{3} + 3 A b^{2} c^{2} d e^{2} - 3 A b c^{3} d^{2} e + 2 A c^{4} d^{3} + B b^{4} e^{3} - 3 B b^{3} c d e^{2} + 3 B b^{2} c^{2} d^{2} e - B b c^{3} d^{3}} \right )}}{b c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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