Optimal. Leaf size=241 \[ -\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{3 c^{3/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{4 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 c^{3/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 e \sqrt{b x+c x^2} \sqrt{d+e x}}{3 c} \]
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Rubi [A] time = 0.191716, antiderivative size = 241, 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 = {742, 843, 715, 112, 110, 117, 116} \[ -\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 c^{3/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{4 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 c^{3/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 e \sqrt{b x+c x^2} \sqrt{d+e x}}{3 c} \]
Antiderivative was successfully verified.
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Rule 742
Rule 843
Rule 715
Rule 112
Rule 110
Rule 117
Rule 116
Rubi steps
\begin{align*} \int \frac{(d+e x)^{3/2}}{\sqrt{b x+c x^2}} \, dx &=\frac{2 e \sqrt{d+e x} \sqrt{b x+c x^2}}{3 c}+\frac{2 \int \frac{\frac{1}{2} d (3 c d-b e)+e (2 c d-b e) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{3 c}\\ &=\frac{2 e \sqrt{d+e x} \sqrt{b x+c x^2}}{3 c}-\frac{(d (c d-b e)) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{3 c}+\frac{(2 (2 c d-b e)) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{3 c}\\ &=\frac{2 e \sqrt{d+e x} \sqrt{b x+c x^2}}{3 c}-\frac{\left (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 c \sqrt{b x+c x^2}}+\frac{\left (2 (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 c \sqrt{b x+c x^2}}\\ &=\frac{2 e \sqrt{d+e x} \sqrt{b x+c x^2}}{3 c}+\frac{\left (2 (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 c \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{\left (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 c \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=\frac{2 e \sqrt{d+e x} \sqrt{b x+c x^2}}{3 c}+\frac{4 \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 c^{3/2} \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{2 \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 c^{3/2} \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 1.02004, size = 246, normalized size = 1.02 \[ \frac{2 i c x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (2 b^2 e^2-5 b c d e+3 c^2 d^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )-4 i b c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (b e-2 c d) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )-2 b (b+c x) (d+e x) (2 b e-c (4 d+e x))}{3 b c^2 \sqrt{x (b+c x)} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.278, size = 460, normalized size = 1.9 \begin{align*}{\frac{2}{3\,x \left ( ce{x}^{2}+bxe+cdx+bd \right ){c}^{3}}\sqrt{ex+d}\sqrt{x \left ( cx+b \right ) } \left ({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-\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}+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}-6\,\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+4\,\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{{\left (e x + d\right )}^{\frac{3}{2}}}{\sqrt{c x^{2} + b x}}\,{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{{\left (e x + d\right )}^{\frac{3}{2}}}{\sqrt{c x^{2} + b x}}, 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 )^{\frac{3}{2}}}{\sqrt{x \left (b + c 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 (e x + d\right )}^{\frac{3}{2}}}{\sqrt{c x^{2} + b x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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