Optimal. Leaf size=231 \[ -\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{\sqrt{c} e^2 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{4 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{e^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 \sqrt{b x+c x^2}}{e \sqrt{d+e x}} \]
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Rubi [A] time = 0.15631, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {732, 843, 715, 112, 110, 117, 116} \[ -\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{\sqrt{c} e^2 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{4 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{e^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 \sqrt{b x+c x^2}}{e \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 732
Rule 843
Rule 715
Rule 112
Rule 110
Rule 117
Rule 116
Rubi steps
\begin{align*} \int \frac{\sqrt{b x+c x^2}}{(d+e x)^{3/2}} \, dx &=-\frac{2 \sqrt{b x+c x^2}}{e \sqrt{d+e x}}+\frac{\int \frac{b+2 c x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{e}\\ &=-\frac{2 \sqrt{b x+c x^2}}{e \sqrt{d+e x}}+\frac{(2 c) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{e^2}-\frac{(2 c d-b e) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{e^2}\\ &=-\frac{2 \sqrt{b x+c x^2}}{e \sqrt{d+e x}}+\frac{\left (2 c \sqrt{x} \sqrt{b+c x}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{x} \sqrt{b+c x}} \, dx}{e^2 \sqrt{b x+c x^2}}-\frac{\left ((2 c d-b e) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x} \sqrt{d+e x}} \, dx}{e^2 \sqrt{b x+c x^2}}\\ &=-\frac{2 \sqrt{b x+c x^2}}{e \sqrt{d+e x}}+\frac{\left (2 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}{e^2 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{\left ((2 c d-b e) \sqrt{x} \sqrt{1+\frac{c x}{b}} \sqrt{1+\frac{e x}{d}}\right ) \int \frac{1}{\sqrt{x} \sqrt{1+\frac{c x}{b}} \sqrt{1+\frac{e x}{d}}} \, dx}{e^2 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=-\frac{2 \sqrt{b x+c x^2}}{e \sqrt{d+e x}}+\frac{4 \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 )}{e^2 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{2 \sqrt{-b} (2 c d-b e) \sqrt{x} \sqrt{1+\frac{c x}{b}} \sqrt{1+\frac{e x}{d}} F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{\sqrt{c} e^2 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.561032, size = 195, normalized size = 0.84 \[ \frac{2 \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )+2 i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (2 d+e x)\right )}{e^2 \sqrt{\frac{b}{c}} \sqrt{x (b+c x)} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.338, size = 353, normalized size = 1.5 \begin{align*} 2\,{\frac{\sqrt{x \left ( cx+b \right ) }\sqrt{ex+d}}{x \left ( ce{x}^{2}+bxe+cdx+bd \right ){e}^{2}c} \left ({\it EllipticF} \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}}}-2\,{\it EllipticF} \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}}}-2\,{\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}}}+2\,{\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{\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}}{e^{2} x^{2} + 2 \, d e x + d^{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{\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{\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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