Optimal. Leaf size=78 \[ -\frac {3 a c^2 (e x)^{1+m}}{e (1+m)}+\frac {b c^2 (e x)^{2+m}}{e^2 (2+m)}+\frac {4 a c^2 (e x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {b x}{a}\right )}{e (1+m)} \]
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Rubi [A]
time = 0.04, antiderivative size = 78, 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 = {90, 66, 45}
\begin {gather*} \frac {4 a c^2 (e x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac {b x}{a}\right )}{e (m+1)}-\frac {3 a c^2 (e x)^{m+1}}{e (m+1)}+\frac {b c^2 (e x)^{m+2}}{e^2 (m+2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 66
Rule 90
Rubi steps
\begin {align*} \int \frac {(e x)^m (a c-b c x)^2}{a+b x} \, dx &=\int \left (-2 a c^2 (e x)^m+\frac {4 a^2 c^2 (e x)^m}{a+b x}-c (e x)^m (a c-b c x)\right ) \, dx\\ &=-\frac {2 a c^2 (e x)^{1+m}}{e (1+m)}-c \int (e x)^m (a c-b c x) \, dx+\left (4 a^2 c^2\right ) \int \frac {(e x)^m}{a+b x} \, dx\\ &=-\frac {2 a c^2 (e x)^{1+m}}{e (1+m)}+\frac {4 a c^2 (e x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {b x}{a}\right )}{e (1+m)}-c \int \left (a c (e x)^m-\frac {b c (e x)^{1+m}}{e}\right ) \, dx\\ &=-\frac {3 a c^2 (e x)^{1+m}}{e (1+m)}+\frac {b c^2 (e x)^{2+m}}{e^2 (2+m)}+\frac {4 a c^2 (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 = 0.06, size = 54, normalized size = 0.69 \begin {gather*} \frac {c^2 x (e x)^m \left (-3 a (2+m)+b (1+m) x+4 a (2+m) \, _2F_1\left (1,1+m;2+m;-\frac {b x}{a}\right )\right )}{(1+m) (2+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{m} \left (-b c x +a c \right )^{2}}{b x +a}\, dx\]
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 \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 = 1.92, size = 246, normalized size = 3.15 \begin {gather*} \frac {a c^{2} 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 c^{2} 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 {2 b c^{2} 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 {4 b c^{2} 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 {b^{2} c^{2} 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 )}{a \Gamma \left (m + 4\right )} + \frac {3 b^{2} c^{2} e^{m} x^{3} x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 3\right ) \Gamma \left (m + 3\right )}{a \Gamma \left (m + 4\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 (a\,c-b\,c\,x\right )}^2\,{\left (e\,x\right )}^m}{a+b\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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