Optimal. Leaf size=205 \[ \frac{(d+e x)^m (-b e+c d-c e x) \left (\frac{c (d+e x)}{2 c d-b e}\right )^{\frac{1}{2}-m} (b e g (2 m+1)-2 c (d g m+e f (m+1))) \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{c d-b e-c e x}{2 c d-b e}\right )}{c^2 e^2 (m+1) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{g (d+e x)^m \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{c e^2 (m+1)} \]
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Rubi [A] time = 0.312948, antiderivative size = 205, 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} \[ \frac{(d+e x)^m (-b e+c d-c e x) \left (\frac{c (d+e x)}{2 c d-b e}\right )^{\frac{1}{2}-m} (b e g (2 m+1)-2 c (d g m+e f (m+1))) \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{c d-b e-c e x}{2 c d-b e}\right )}{c^2 e^2 (m+1) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{g (d+e x)^m \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{c e^2 (m+1)} \]
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)}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx &=-\frac{g (d+e x)^m \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{c e^2 (1+m)}-\frac{(b e g (1+2 m)-2 c (d g m+e f (1+m))) \int \frac{(d+e x)^m}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{2 c e (1+m)}\\ &=-\frac{g (d+e x)^m \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{c e^2 (1+m)}-\frac{\left ((b e g (1+2 m)-2 c (d g m+e f (1+m))) (d+e x)^m \left (1+\frac{e x}{d}\right )^{-m}\right ) \int \frac{\left (1+\frac{e x}{d}\right )^m}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{2 c e (1+m)}\\ &=-\frac{g (d+e x)^m \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{c e^2 (1+m)}-\frac{\left ((b e g (1+2 m)-2 c (d g m+e f (1+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{1}{2}+m}}{\sqrt{c d^2-b d e-c d e x}} \, dx}{2 c e (1+m) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}\\ &=-\frac{g (d+e x)^m \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{c e^2 (1+m)}-\frac{\left ((b e g (1+2 m)-2 c (d g m+e f (1+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{1}{2}+m}}{\sqrt{c d^2-b d e-c d e x}} \, dx}{2 c e (1+m) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}\\ &=-\frac{g (d+e x)^m \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{c e^2 (1+m)}+\frac{(b e g (1+2 m)-2 c (d g m+e f (1+m))) (d+e x)^m \left (\frac{c (d+e x)}{2 c d-b e}\right )^{\frac{1}{2}-m} (c d-b e-c e x) \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{c d-b e-c e x}{2 c d-b e}\right )}{c^2 e^2 (1+m) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.34664, size = 155, normalized size = 0.76 \[ -\frac{2 (d+e x)^m \sqrt{(d+e x) (c (d-e x)-b e)} \left (\frac{e \left (\frac{c (d+e x)}{2 c d-b e}\right )^{-m-\frac{1}{2}} (b e g (2 m+1)-2 c (d g m+e f (m+1))) \, _2F_1\left (\frac{1}{2},-m-\frac{1}{2};\frac{3}{2};\frac{-c d+b e+c e x}{b e-2 c d}\right )}{c}+e (e f-d g)\right )}{e^3 (2 m+1) (b e-2 c d)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.073, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ex+d \right ) ^{m} \left ( gx+f \right ){\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{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}}{\sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}\,{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 e^{2} x^{2} + b e^{2} x - c d^{2} + b d e}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{m} \left (f + g x\right )}{\sqrt{- \left (d + e x\right ) \left (b e - c d + c e x\right )}}\, dx \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}}{\sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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