Optimal. Leaf size=224 \[ \frac{g (d+e x)^m}{c e^2 (3-m) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{(d+e x)^m \left (\frac{c (d+e x)}{2 c d-b e}\right )^{\frac{1}{2}-m} (b e g (3-2 m)-2 c (e f (3-m)-d g m)) \, _2F_1\left (-\frac{3}{2},\frac{5}{2}-m;-\frac{1}{2};\frac{c d-b e-c e x}{2 c d-b e}\right )}{3 e^2 (3-m) (2 c d-b e)^2 (-b e+c d-c e x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]
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Rubi [A] time = 0.327517, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {794, 679, 677, 70, 69} \[ \frac{g (d+e x)^m}{c e^2 (3-m) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{(d+e x)^m \left (\frac{c (d+e x)}{2 c d-b e}\right )^{\frac{1}{2}-m} (b e g (3-2 m)-2 c (e f (3-m)-d g m)) \, _2F_1\left (-\frac{3}{2},\frac{5}{2}-m;-\frac{1}{2};\frac{c d-b e-c e x}{2 c d-b e}\right )}{3 e^2 (3-m) (2 c d-b e)^2 (-b e+c d-c e x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 794
Rule 679
Rule 677
Rule 70
Rule 69
Rubi steps
\begin{align*} \int \frac{(d+e x)^m (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx &=\frac{g (d+e x)^m}{c e^2 (3-m) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{(b e g (3-2 m)-2 c (e f (3-m)-d g m)) \int \frac{(d+e x)^m}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx}{2 c e (3-m)}\\ &=\frac{g (d+e x)^m}{c e^2 (3-m) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{\left ((b e g (3-2 m)-2 c (e f (3-m)-d g m)) (d+e x)^m \left (1+\frac{e x}{d}\right )^{-m}\right ) \int \frac{\left (1+\frac{e x}{d}\right )^m}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx}{2 c e (3-m)}\\ &=\frac{g (d+e x)^m}{c e^2 (3-m) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{\left ((b e g (3-2 m)-2 c (e f (3-m)-d g m)) (d+e x)^m \left (1+\frac{e x}{d}\right )^{\frac{1}{2}-m} \sqrt{c d^2-b d e-c d e x}\right ) \int \frac{\left (1+\frac{e x}{d}\right )^{-\frac{5}{2}+m}}{\left (c d^2-b d e-c d e x\right )^{5/2}} \, dx}{2 c e (3-m) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}\\ &=\frac{g (d+e x)^m}{c e^2 (3-m) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{\left (c d^2 e (b e g (3-2 m)-2 c (e f (3-m)-d g m)) (d+e x)^m \left (-\frac{c d e \left (1+\frac{e x}{d}\right )}{-c d e-\frac{e \left (c d^2-b d e\right )}{d}}\right )^{\frac{1}{2}-m} \sqrt{c d^2-b d e-c d e x}\right ) \int \frac{\left (\frac{c d}{2 c d-b e}+\frac{c e x}{2 c d-b e}\right )^{-\frac{5}{2}+m}}{\left (c d^2-b d e-c d e x\right )^{5/2}} \, dx}{2 \left (-c d e-\frac{e \left (c d^2-b d e\right )}{d}\right )^2 (3-m) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}\\ &=\frac{g (d+e x)^m}{c e^2 (3-m) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{(b e g (3-2 m)-2 c (e f (3-m)-d g m)) (d+e x)^m \left (\frac{c (d+e x)}{2 c d-b e}\right )^{\frac{1}{2}-m} \, _2F_1\left (-\frac{3}{2},\frac{5}{2}-m;-\frac{1}{2};\frac{c d-b e-c e x}{2 c d-b e}\right )}{3 e^2 (2 c d-b e)^2 (3-m) (c d-b e-c e x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.427978, size = 185, normalized size = 0.83 \[ \frac{2 (d+e x)^m \left (-e (d+e x) (b e-c d+c e x) \left (\frac{c (d+e x)}{2 c d-b e}\right )^{\frac{1}{2}-m} (b e g (3-2 m)+2 c (d g m+e f (m-3))) \, _2F_1\left (-\frac{1}{2},\frac{5}{2}-m;\frac{1}{2};\frac{-c d+b e+c e x}{b e-2 c d}\right )-\frac{e (b e-2 c d)^2 (-b e g+c d g+c e f)}{c}\right )}{3 e^3 (b e-2 c d)^3 ((d+e x) (c (d-e x)-b e))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.07, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ex+d \right ) ^{m} \left ( gx+f \right ) \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{-{\frac{5}{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 (g x + f\right )}{\left (e x + d\right )}^{m}}{{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac{5}{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{\sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (g x + f\right )}{\left (e x + d\right )}^{m}}{c^{3} e^{6} x^{6} + 3 \, b c^{2} e^{6} x^{5} - c^{3} d^{6} + 3 \, b c^{2} d^{5} e - 3 \, b^{2} c d^{4} e^{2} + b^{3} d^{3} e^{3} - 3 \,{\left (c^{3} d^{2} e^{4} - b c^{2} d e^{5} - b^{2} c e^{6}\right )} x^{4} -{\left (6 \, b c^{2} d^{2} e^{4} - 6 \, b^{2} c d e^{5} - b^{3} e^{6}\right )} x^{3} + 3 \,{\left (c^{3} d^{4} e^{2} - 2 \, b c^{2} d^{3} e^{3} + b^{3} d e^{5}\right )} x^{2} + 3 \,{\left (b c^{2} d^{4} e^{2} - 2 \, b^{2} c d^{3} e^{3} + b^{3} d^{2} e^{4}\right )} x}, 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 (g x + f\right )}{\left (e x + d\right )}^{m}}{{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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