Optimal. Leaf size=246 \[ \frac{4 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{3 \sqrt{c} e^2 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 \sqrt{c} e^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x}}{3 e} \]
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Rubi [A] time = 0.189719, antiderivative size = 246, 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 = {734, 843, 715, 112, 110, 117, 116} \[ \frac{4 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 \sqrt{c} e^2 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 \sqrt{c} e^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x}}{3 e} \]
Antiderivative was successfully verified.
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Rule 734
Rule 843
Rule 715
Rule 112
Rule 110
Rule 117
Rule 116
Rubi steps
\begin{align*} \int \frac{\sqrt{b x+c x^2}}{\sqrt{d+e x}} \, dx &=\frac{2 \sqrt{d+e x} \sqrt{b x+c x^2}}{3 e}-\frac{\int \frac{b d+(2 c d-b e) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{3 e}\\ &=\frac{2 \sqrt{d+e x} \sqrt{b x+c x^2}}{3 e}+\frac{(2 d (c d-b e)) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{3 e^2}-\frac{(2 c d-b e) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{3 e^2}\\ &=\frac{2 \sqrt{d+e x} \sqrt{b x+c x^2}}{3 e}+\frac{\left (2 d (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}{3 e^2 \sqrt{b x+c x^2}}-\frac{\left ((2 c d-b e) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{x} \sqrt{b+c x}} \, dx}{3 e^2 \sqrt{b x+c x^2}}\\ &=\frac{2 \sqrt{d+e x} \sqrt{b x+c x^2}}{3 e}-\frac{\left ((2 c d-b e) \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}{3 e^2 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{\left (2 d (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}{3 e^2 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=\frac{2 \sqrt{d+e x} \sqrt{b x+c x^2}}{3 e}-\frac{2 \sqrt{-b} (2 c d-b e) \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 )}{3 \sqrt{c} e^2 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{4 \sqrt{-b} d (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 )}{3 \sqrt{c} e^2 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 1.39797, size = 226, normalized size = 0.92 \[ \frac{2 \left (-i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (b e-c d) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )+i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (b e-2 c d) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+(b+c x) (d+e x) (b e-2 c d+c e x)\right )}{3 c e^2 \sqrt{x (b+c x)} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.273, size = 467, normalized size = 1.9 \begin{align*} -{\frac{2}{3\,x \left ( ce{x}^{2}+bxe+cdx+bd \right ){e}^{2}{c}^{2}}\sqrt{x \left ( cx+b \right ) }\sqrt{ex+d} \left ( 2\,{b}^{2}d\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) ec-2\,\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) b{c}^{2}{d}^{2}+\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 ){b}^{3}{e}^{2}-3\,\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 ){b}^{2}cde+2\,\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 ) b{c}^{2}{d}^{2}-{x}^{3}{c}^{3}{e}^{2}-{x}^{2}b{c}^{2}{e}^{2}-{x}^{2}{c}^{3}de-xb{c}^{2}de \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}}{\sqrt{e x + d}}\,{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}}, 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 )}}{\sqrt{d + e x}}\, 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}}{\sqrt{e x + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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