Optimal. Leaf size=146 \[ \frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{d \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)}-\frac{2 e \sqrt{b x+c x^2}}{d \sqrt{d+e x} (c d-b e)} \]
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Rubi [A] time = 0.0806936, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {744, 21, 715, 112, 110} \[ \frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{d \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)}-\frac{2 e \sqrt{b x+c x^2}}{d \sqrt{d+e x} (c d-b e)} \]
Antiderivative was successfully verified.
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Rule 744
Rule 21
Rule 715
Rule 112
Rule 110
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^{3/2} \sqrt{b x+c x^2}} \, dx &=-\frac{2 e \sqrt{b x+c x^2}}{d (c d-b e) \sqrt{d+e x}}-\frac{2 \int \frac{-\frac{c d}{2}-\frac{c e x}{2}}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{d (c d-b e)}\\ &=-\frac{2 e \sqrt{b x+c x^2}}{d (c d-b e) \sqrt{d+e x}}+\frac{c \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{d (c d-b e)}\\ &=-\frac{2 e \sqrt{b x+c x^2}}{d (c d-b e) \sqrt{d+e x}}+\frac{\left (c \sqrt{x} \sqrt{b+c x}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{x} \sqrt{b+c x}} \, dx}{d (c d-b e) \sqrt{b x+c x^2}}\\ &=-\frac{2 e \sqrt{b x+c x^2}}{d (c d-b e) \sqrt{d+e x}}+\frac{\left (c \sqrt{x} \sqrt{1+\frac{c x}{b}} \sqrt{d+e x}\right ) \int \frac{\sqrt{1+\frac{e x}{d}}}{\sqrt{x} \sqrt{1+\frac{c x}{b}}} \, dx}{d (c d-b e) \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}\\ &=-\frac{2 e \sqrt{b x+c x^2}}{d (c d-b e) \sqrt{d+e x}}+\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{1+\frac{c x}{b}} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{d (c d-b e) \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.161222, size = 127, normalized size = 0.87 \[ \frac{2 \sqrt{x (b+c x)} \left (e \sqrt{x} \sqrt{-\frac{d}{e}} \sqrt{\frac{d}{e x}+1} E\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{d}{e}}}{\sqrt{x}}\right )|\frac{b e}{c d}\right )+d \sqrt{\frac{b}{c x}+1}\right )}{d x \sqrt{\frac{b}{c x}+1} \sqrt{d+e x} (c d-b e)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.291, size = 216, normalized size = 1.5 \begin{align*} 2\,{\frac{\sqrt{x \left ( cx+b \right ) }\sqrt{ex+d}}{ \left ( be-cd \right ) cdx \left ( ce{x}^{2}+bxe+cdx+bd \right ) } \left ({\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{2}e\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}-{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) bcd\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}+{x}^{2}{c}^{2}e+xbce \right ) } \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{1}{\sqrt{c x^{2} + b x}{\left (e x + d\right )}^{\frac{3}{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 x^{2} + b x} \sqrt{e x + d}}{c e^{2} x^{4} + b d^{2} x +{\left (2 \, c d e + b e^{2}\right )} x^{3} +{\left (c d^{2} + 2 \, b d e\right )} x^{2}}, 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{1}{\sqrt{x \left (b + c x\right )} \left (d + e x\right )^{\frac{3}{2}}}\, 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{1}{\sqrt{c x^{2} + b x}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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