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 = 0.12, antiderivative size = 107, normalized size of antiderivative = 1.00, 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 43
Rule 64
Rule 88
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*}
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Mathematica [A] time = 0.09, 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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fricas [F] time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (b^{3} c^{3} x^{3} - 3 \, a b^{2} c^{3} x^{2} + 3 \, a^{2} b c^{3} x - a^{3} c^{3}\right )} \left (e x\right )^{m}}{b x + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {{\left (b c x - a c\right )}^{3} \left (e x\right )^{m}}{b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\left (-b c x +a c \right )^{3} \left (e x \right )^{m}}{b x +a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (b c x - a c\right )}^{3} \left (e x\right )^{m}}{b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a\,c-b\,c\,x\right )}^3\,{\left (e\,x\right )}^m}{a+b\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 6.52, size = 340, normalized size = 3.18 \[ \frac {a^{2} c^{3} e^{m} m x x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{\Gamma \left (m + 2\right )} + \frac {a^{2} c^{3} e^{m} x x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{\Gamma \left (m + 2\right )} - \frac {3 a b c^{3} e^{m} m x^{2} x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{\Gamma \left (m + 3\right )} - \frac {6 a b c^{3} e^{m} x^{2} x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{\Gamma \left (m + 3\right )} + \frac {3 b^{2} c^{3} e^{m} m x^{3} x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 3\right ) \Gamma \left (m + 3\right )}{\Gamma \left (m + 4\right )} + \frac {9 b^{2} c^{3} e^{m} x^{3} x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 3\right ) \Gamma \left (m + 3\right )}{\Gamma \left (m + 4\right )} - \frac {b^{3} c^{3} e^{m} m x^{4} x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 4\right ) \Gamma \left (m + 4\right )}{a \Gamma \left (m + 5\right )} - \frac {4 b^{3} c^{3} e^{m} x^{4} x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 4\right ) \Gamma \left (m + 4\right )}{a \Gamma \left (m + 5\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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