Optimal. Leaf size=180 \[ -\frac{c^2 (d+e x)^{m+1} (2 c d-b e (2-m)) \, _2F_1\left (1,m+1;m+2;\frac{c (d+e x)}{c d-b e}\right )}{b^3 (m+1) (c d-b e)^2}+\frac{(d+e x)^{m+1} (2 c d-b e m) \, _2F_1\left (1,m+1;m+2;\frac{e x}{d}+1\right )}{b^3 d^2 (m+1)}-\frac{(d+e x)^{m+1} (c x (2 c d-b e)+b (c d-b e))}{b^2 d \left (b x+c x^2\right ) (c d-b e)} \]
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Rubi [A] time = 0.18127, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {740, 830, 65, 68} \[ -\frac{c^2 (d+e x)^{m+1} (2 c d-b e (2-m)) \, _2F_1\left (1,m+1;m+2;\frac{c (d+e x)}{c d-b e}\right )}{b^3 (m+1) (c d-b e)^2}+\frac{(d+e x)^{m+1} (2 c d-b e m) \, _2F_1\left (1,m+1;m+2;\frac{e x}{d}+1\right )}{b^3 d^2 (m+1)}-\frac{(d+e x)^{m+1} (c x (2 c d-b e)+b (c d-b e))}{b^2 d \left (b x+c x^2\right ) (c d-b e)} \]
Antiderivative was successfully verified.
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Rule 740
Rule 830
Rule 65
Rule 68
Rubi steps
\begin{align*} \int \frac{(d+e x)^m}{\left (b x+c x^2\right )^2} \, dx &=-\frac{(d+e x)^{1+m} (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) \left (b x+c x^2\right )}-\frac{\int \frac{(d+e x)^m ((c d-b e) (2 c d-b e m)-c e (2 c d-b e) m x)}{b x+c x^2} \, dx}{b^2 d (c d-b e)}\\ &=-\frac{(d+e x)^{1+m} (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) \left (b x+c x^2\right )}-\frac{\int \left (\frac{(-c d+b e) (-2 c d+b e m) (d+e x)^m}{b x}+\frac{c^2 d (-2 c d+b e (2-m)) (d+e x)^m}{b (b+c x)}\right ) \, dx}{b^2 d (c d-b e)}\\ &=-\frac{(d+e x)^{1+m} (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) \left (b x+c x^2\right )}+\frac{\left (c^2 (2 c d-b e (2-m))\right ) \int \frac{(d+e x)^m}{b+c x} \, dx}{b^3 (c d-b e)}-\frac{(2 c d-b e m) \int \frac{(d+e x)^m}{x} \, dx}{b^3 d}\\ &=-\frac{(d+e x)^{1+m} (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) \left (b x+c x^2\right )}-\frac{c^2 (2 c d-b e (2-m)) (d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac{c (d+e x)}{c d-b e}\right )}{b^3 (c d-b e)^2 (1+m)}+\frac{(2 c d-b e m) (d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;1+\frac{e x}{d}\right )}{b^3 d^2 (1+m)}\\ \end{align*}
Mathematica [A] time = 0.147617, size = 174, normalized size = 0.97 \[ -\frac{(d+e x)^{m+1} \left (b^2 d (m+1) (c d-b e)^2+x (b+c x) \left (c^2 d^2 (b e (m-2)+2 c d) \, _2F_1\left (1,m+1;m+2;\frac{c (d+e x)}{c d-b e}\right )-(c d-b e)^2 (2 c d-b e m) \, _2F_1\left (1,m+1;m+2;\frac{e x}{d}+1\right )\right )+b c d (m+1) x (b e-2 c d) (b e-c d)\right )}{b^3 d^2 (m+1) x (b+c x) (c d-b e)^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.61, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex+d \right ) ^{m}}{ \left ( c{x}^{2}+bx \right ) ^{2}}}\, 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 (e x + d\right )}^{m}}{{\left (c x^{2} + b x\right )}^{2}}\,{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 (e x + d\right )}^{m}}{c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (e x + d\right )}^{m}}{{\left (c x^{2} + b x\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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