Optimal. Leaf size=98 \[ \frac {(e x)^{1+m} \, _2F_1\left (2,\frac {1+m}{2};\frac {3+m}{2};\frac {b^2 x^2}{a^2}\right )}{a^3 c^2 e (1+m)}+\frac {b (e x)^{2+m} \, _2F_1\left (2,\frac {2+m}{2};\frac {4+m}{2};\frac {b^2 x^2}{a^2}\right )}{a^4 c^2 e^2 (2+m)} \]
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Rubi [A]
time = 0.03, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {83, 74, 371}
\begin {gather*} \frac {b (e x)^{m+2} \, _2F_1\left (2,\frac {m+2}{2};\frac {m+4}{2};\frac {b^2 x^2}{a^2}\right )}{a^4 c^2 e^2 (m+2)}+\frac {(e x)^{m+1} \, _2F_1\left (2,\frac {m+1}{2};\frac {m+3}{2};\frac {b^2 x^2}{a^2}\right )}{a^3 c^2 e (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 74
Rule 83
Rule 371
Rubi steps
\begin {align*} \int \frac {(e x)^m}{(a+b x) (a c-b c x)^2} \, dx &=a \int \frac {(e x)^m}{(a+b x)^2 (a c-b c x)^2} \, dx+\frac {b \int \frac {(e x)^{1+m}}{(a+b x)^2 (a c-b c x)^2} \, dx}{e}\\ &=a \int \frac {(e x)^m}{\left (a^2 c-b^2 c x^2\right )^2} \, dx+\frac {b \int \frac {(e x)^{1+m}}{\left (a^2 c-b^2 c x^2\right )^2} \, dx}{e}\\ &=\frac {(e x)^{1+m} \, _2F_1\left (2,\frac {1+m}{2};\frac {3+m}{2};\frac {b^2 x^2}{a^2}\right )}{a^3 c^2 e (1+m)}+\frac {b (e x)^{2+m} \, _2F_1\left (2,\frac {2+m}{2};\frac {4+m}{2};\frac {b^2 x^2}{a^2}\right )}{a^4 c^2 e^2 (2+m)}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 87, normalized size = 0.89 \begin {gather*} \frac {x (e x)^m \left (b (1+m) x \, _2F_1\left (2,1+\frac {m}{2};2+\frac {m}{2};\frac {b^2 x^2}{a^2}\right )+a (2+m) \, _2F_1\left (2,\frac {1+m}{2};\frac {3+m}{2};\frac {b^2 x^2}{a^2}\right )\right )}{a^4 c^2 (1+m) (2+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{m}}{\left (b x +a \right ) \left (-b c x +a c \right )^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.50, size = 440, normalized size = 4.49 \begin {gather*} - \frac {2 a e^{m} m^{2} x^{m} \Phi \left (\frac {a}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{- 4 a^{3} b c^{2} \Gamma \left (1 - m\right ) + 4 a^{2} b^{2} c^{2} x \Gamma \left (1 - m\right )} + \frac {a e^{m} m x^{m} \Phi \left (\frac {a}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{- 4 a^{3} b c^{2} \Gamma \left (1 - m\right ) + 4 a^{2} b^{2} c^{2} x \Gamma \left (1 - m\right )} - \frac {a e^{m} m x^{m} \Phi \left (\frac {a e^{i \pi }}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{- 4 a^{3} b c^{2} \Gamma \left (1 - m\right ) + 4 a^{2} b^{2} c^{2} x \Gamma \left (1 - m\right )} + \frac {2 b e^{m} m^{2} x x^{m} \Phi \left (\frac {a}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{- 4 a^{3} b c^{2} \Gamma \left (1 - m\right ) + 4 a^{2} b^{2} c^{2} x \Gamma \left (1 - m\right )} - \frac {b e^{m} m x x^{m} \Phi \left (\frac {a}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{- 4 a^{3} b c^{2} \Gamma \left (1 - m\right ) + 4 a^{2} b^{2} c^{2} x \Gamma \left (1 - m\right )} + \frac {b e^{m} m x x^{m} \Phi \left (\frac {a e^{i \pi }}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{- 4 a^{3} b c^{2} \Gamma \left (1 - m\right ) + 4 a^{2} b^{2} c^{2} x \Gamma \left (1 - m\right )} + \frac {2 b e^{m} m x x^{m} \Gamma \left (- m\right )}{- 4 a^{3} b c^{2} \Gamma \left (1 - m\right ) + 4 a^{2} b^{2} c^{2} x \Gamma \left (1 - m\right )} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (e\,x\right )}^m}{{\left (a\,c-b\,c\,x\right )}^2\,\left (a+b\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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