Optimal. Leaf size=350 \[ \frac{c^3 (d+e x)^{m+1} \left (b^2 e^2 \left (m^2-7 m+12\right )-6 b c d e (4-m)+12 c^2 d^2\right ) \, _2F_1\left (1,m+1;m+2;\frac{c (d+e x)}{c d-b e}\right )}{2 b^5 (m+1) (c d-b e)^3}-\frac{(d+e x)^{m+1} \left (-b^2 e^2 (1-m) m-6 b c d e m+12 c^2 d^2\right ) \, _2F_1\left (1,m+1;m+2;\frac{e x}{d}+1\right )}{2 b^5 d^3 (m+1)}+\frac{(d+e x)^{m+1} \left (c x (2 c d-b e) \left (-b^2 e^2 (1-m)-6 b c d e+6 c^2 d^2\right )+b (c d-b e) \left (-b^2 e^2 (1-m)-b c d e (m+4)+6 c^2 d^2\right )\right )}{2 b^4 d^2 \left (b x+c x^2\right ) (c d-b e)^2}-\frac{(d+e x)^{m+1} (c x (2 c d-b e)+b (c d-b e))}{2 b^2 d \left (b x+c x^2\right )^2 (c d-b e)} \]
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Rubi [A] time = 0.427836, antiderivative size = 350, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {740, 822, 830, 65, 68} \[ \frac{c^3 (d+e x)^{m+1} \left (b^2 e^2 \left (m^2-7 m+12\right )-6 b c d e (4-m)+12 c^2 d^2\right ) \, _2F_1\left (1,m+1;m+2;\frac{c (d+e x)}{c d-b e}\right )}{2 b^5 (m+1) (c d-b e)^3}-\frac{(d+e x)^{m+1} \left (-b^2 e^2 (1-m) m-6 b c d e m+12 c^2 d^2\right ) \, _2F_1\left (1,m+1;m+2;\frac{e x}{d}+1\right )}{2 b^5 d^3 (m+1)}+\frac{(d+e x)^{m+1} \left (c x (2 c d-b e) \left (-b^2 e^2 (1-m)-6 b c d e+6 c^2 d^2\right )+b (c d-b e) \left (-b^2 e^2 (1-m)-b c d e (m+4)+6 c^2 d^2\right )\right )}{2 b^4 d^2 \left (b x+c x^2\right ) (c d-b e)^2}-\frac{(d+e x)^{m+1} (c x (2 c d-b e)+b (c d-b e))}{2 b^2 d \left (b x+c x^2\right )^2 (c d-b e)} \]
Antiderivative was successfully verified.
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Rule 740
Rule 822
Rule 830
Rule 65
Rule 68
Rubi steps
\begin{align*} \int \frac{(d+e x)^m}{\left (b x+c x^2\right )^3} \, dx &=-\frac{(d+e x)^{1+m} (b (c d-b e)+c (2 c d-b e) x)}{2 b^2 d (c d-b e) \left (b x+c x^2\right )^2}-\frac{\int \frac{(d+e x)^m \left (6 c^2 d^2-b^2 e^2 (1-m)-b c d e (4+m)+c e (2 c d-b e) (2-m) x\right )}{\left (b x+c x^2\right )^2} \, dx}{2 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)}{2 b^2 d (c d-b e) \left (b x+c x^2\right )^2}+\frac{(d+e x)^{1+m} \left (b (c d-b e) \left (6 c^2 d^2-b^2 e^2 (1-m)-b c d e (4+m)\right )+c (2 c d-b e) \left (6 c^2 d^2-6 b c d e-b^2 e^2 (1-m)\right ) x\right )}{2 b^4 d^2 (c d-b e)^2 \left (b x+c x^2\right )}+\frac{\int \frac{(d+e x)^m \left ((c d-b e)^2 \left (12 c^2 d^2-6 b c d e m-b^2 e^2 (1-m) m\right )-c e (2 c d-b e) \left (6 c^2 d^2-6 b c d e-b^2 e^2 (1-m)\right ) m x\right )}{b x+c x^2} \, dx}{2 b^4 d^2 (c d-b e)^2}\\ &=-\frac{(d+e x)^{1+m} (b (c d-b e)+c (2 c d-b e) x)}{2 b^2 d (c d-b e) \left (b x+c x^2\right )^2}+\frac{(d+e x)^{1+m} \left (b (c d-b e) \left (6 c^2 d^2-b^2 e^2 (1-m)-b c d e (4+m)\right )+c (2 c d-b e) \left (6 c^2 d^2-6 b c d e-b^2 e^2 (1-m)\right ) x\right )}{2 b^4 d^2 (c d-b e)^2 \left (b x+c x^2\right )}+\frac{\int \left (\frac{(c d-b e)^2 \left (12 c^2 d^2-6 b c d e m-b^2 e^2 (1-m) m\right ) (d+e x)^m}{b x}+\frac{c^3 d^2 \left (-12 c^2 d^2+6 b c d e (4-m)-b^2 e^2 \left (12-7 m+m^2\right )\right ) (d+e x)^m}{b (b+c x)}\right ) \, dx}{2 b^4 d^2 (c d-b e)^2}\\ &=-\frac{(d+e x)^{1+m} (b (c d-b e)+c (2 c d-b e) x)}{2 b^2 d (c d-b e) \left (b x+c x^2\right )^2}+\frac{(d+e x)^{1+m} \left (b (c d-b e) \left (6 c^2 d^2-b^2 e^2 (1-m)-b c d e (4+m)\right )+c (2 c d-b e) \left (6 c^2 d^2-6 b c d e-b^2 e^2 (1-m)\right ) x\right )}{2 b^4 d^2 (c d-b e)^2 \left (b x+c x^2\right )}+\frac{\left (12 c^2 d^2-6 b c d e m-b^2 e^2 (1-m) m\right ) \int \frac{(d+e x)^m}{x} \, dx}{2 b^5 d^2}-\frac{\left (c^3 \left (12 c^2 d^2-6 b c d e (4-m)+b^2 e^2 \left (12-7 m+m^2\right )\right )\right ) \int \frac{(d+e x)^m}{b+c x} \, dx}{2 b^5 (c d-b e)^2}\\ &=-\frac{(d+e x)^{1+m} (b (c d-b e)+c (2 c d-b e) x)}{2 b^2 d (c d-b e) \left (b x+c x^2\right )^2}+\frac{(d+e x)^{1+m} \left (b (c d-b e) \left (6 c^2 d^2-b^2 e^2 (1-m)-b c d e (4+m)\right )+c (2 c d-b e) \left (6 c^2 d^2-6 b c d e-b^2 e^2 (1-m)\right ) x\right )}{2 b^4 d^2 (c d-b e)^2 \left (b x+c x^2\right )}+\frac{c^3 \left (12 c^2 d^2-6 b c d e (4-m)+b^2 e^2 \left (12-7 m+m^2\right )\right ) (d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac{c (d+e x)}{c d-b e}\right )}{2 b^5 (c d-b e)^3 (1+m)}-\frac{\left (12 c^2 d^2-6 b c d e m-b^2 e^2 (1-m) m\right ) (d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;1+\frac{e x}{d}\right )}{2 b^5 d^3 (1+m)}\\ \end{align*}
Mathematica [A] time = 0.531144, size = 337, normalized size = 0.96 \[ -\frac{(d+e x)^{m+1} \left (x^2 \left (-(b+c x) \left ((b+c x) \left (2 c^3 d^3 \left (b^2 e^2 \left (m^2-7 m+12\right )+6 b c d e (m-4)+12 c^2 d^2\right ) \, _2F_1\left (1,m+1;m+2;\frac{c (d+e x)}{c d-b e}\right )-2 (c d-b e)^3 \left (b^2 e^2 (m-1) m-6 b c d e m+12 c^2 d^2\right ) \, _2F_1\left (1,m+1;m+2;\frac{e x}{d}+1\right )\right )-2 b c d (m+1) (2 c d-b e) (b e-c d) \left (b^2 e^2 (m-1)-6 b c d e+6 c^2 d^2\right )\right )-2 b^2 c d (m+1) (c d-b e)^2 \left (b^2 e^2 (m-1)-b c d e (m+4)+6 c^2 d^2\right )\right )+2 b^4 d^2 (m+1) (c d-b e)^3-2 b^3 d (m+1) x (c d-b e)^3 (4 c d-b e (m-1))\right )}{4 b^5 d^3 (m+1) x^2 (b+c x)^2 (c d-b e)^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.71, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex+d \right ) ^{m}}{ \left ( c{x}^{2}+bx \right ) ^{3}}}\, 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 )}^{3}}\,{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^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \, b^{2} c x^{4} + b^{3} x^{3}}, 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 )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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