Optimal. Leaf size=107 \[ \frac{8 a^2 c^3 (e x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )}{e (m+1)}-\frac{7 a^2 c^3 (e x)^{m+1}}{e (m+1)}+\frac{4 a b c^3 (e x)^{m+2}}{e^2 (m+2)}-\frac{b^2 c^3 (e x)^{m+3}}{e^3 (m+3)} \]
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Rubi [A] time = 0.116608, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {88, 64, 43} \[ \frac{8 a^2 c^3 (e x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )}{e (m+1)}-\frac{7 a^2 c^3 (e x)^{m+1}}{e (m+1)}+\frac{4 a b c^3 (e x)^{m+2}}{e^2 (m+2)}-\frac{b^2 c^3 (e x)^{m+3}}{e^3 (m+3)} \]
Antiderivative was successfully verified.
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Rule 88
Rule 64
Rule 43
Rubi steps
\begin{align*} \int \frac{(e x)^m (a c-b c x)^3}{a+b x} \, dx &=\int \left (-4 a^2 c^3 (e x)^m+\frac{8 a^3 c^3 (e x)^m}{a+b x}-2 a c^2 (e x)^m (a c-b c x)-c (e x)^m (a c-b c x)^2\right ) \, dx\\ &=-\frac{4 a^2 c^3 (e x)^{1+m}}{e (1+m)}-c \int (e x)^m (a c-b c x)^2 \, dx-\left (2 a c^2\right ) \int (e x)^m (a c-b c x) \, dx+\left (8 a^3 c^3\right ) \int \frac{(e x)^m}{a+b x} \, dx\\ &=-\frac{4 a^2 c^3 (e x)^{1+m}}{e (1+m)}+\frac{8 a^2 c^3 (e x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac{b x}{a}\right )}{e (1+m)}-c \int \left (a^2 c^2 (e x)^m-\frac{2 a b c^2 (e x)^{1+m}}{e}+\frac{b^2 c^2 (e x)^{2+m}}{e^2}\right ) \, dx-\left (2 a c^2\right ) \int \left (a c (e x)^m-\frac{b c (e x)^{1+m}}{e}\right ) \, dx\\ &=-\frac{7 a^2 c^3 (e x)^{1+m}}{e (1+m)}+\frac{4 a b c^3 (e x)^{2+m}}{e^2 (2+m)}-\frac{b^2 c^3 (e x)^{3+m}}{e^3 (3+m)}+\frac{8 a^2 c^3 (e x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac{b x}{a}\right )}{e (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0690402, size = 69, normalized size = 0.64 \[ c^3 x (e x)^m \left (\frac{8 a^2 \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )}{m+1}-\frac{7 a^2}{m+1}+\frac{4 a b x}{m+2}-\frac{b^2 x^2}{m+3}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.04, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m} \left ( -bcx+ac \right ) ^{3}}{bx+a}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (b c x - a c\right )}^{3} \left (e x\right )^{m}}{b x + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (b^{3} c^{3} x^{3} - 3 \, a b^{2} c^{3} x^{2} + 3 \, a^{2} b c^{3} x - a^{3} c^{3}\right )} \left (e x\right )^{m}}{b x + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 4.88987, size = 340, normalized size = 3.18 \begin{align*} \frac{a^{2} c^{3} e^{m} m x x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{\Gamma \left (m + 2\right )} + \frac{a^{2} c^{3} e^{m} x x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{\Gamma \left (m + 2\right )} - \frac{3 a b c^{3} e^{m} m x^{2} x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{\Gamma \left (m + 3\right )} - \frac{6 a b c^{3} e^{m} x^{2} x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{\Gamma \left (m + 3\right )} + \frac{3 b^{2} c^{3} e^{m} m x^{3} x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 3\right ) \Gamma \left (m + 3\right )}{\Gamma \left (m + 4\right )} + \frac{9 b^{2} c^{3} e^{m} x^{3} x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 3\right ) \Gamma \left (m + 3\right )}{\Gamma \left (m + 4\right )} - \frac{b^{3} c^{3} e^{m} m x^{4} x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 4\right ) \Gamma \left (m + 4\right )}{a \Gamma \left (m + 5\right )} - \frac{4 b^{3} c^{3} e^{m} x^{4} x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 4\right ) \Gamma \left (m + 4\right )}{a \Gamma \left (m + 5\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (b c x - a c\right )}^{3} \left (e x\right )^{m}}{b x + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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