Optimal. Leaf size=309 \[ -\frac{16 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{5 \sqrt{c} e^4 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (b^2 e^2-16 b c d e+16 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{5 \sqrt{c} e^4 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} (-7 b e+8 c d-6 c e x)}{5 e^3}-\frac{2 \left (b x+c x^2\right )^{3/2}}{e \sqrt{d+e x}} \]
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Rubi [A] time = 0.338442, antiderivative size = 309, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {732, 814, 843, 715, 112, 110, 117, 116} \[ \frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (b^2 e^2-16 b c d e+16 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{5 \sqrt{c} e^4 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} (-7 b e+8 c d-6 c e x)}{5 e^3}-\frac{16 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{5 \sqrt{c} e^4 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \left (b x+c x^2\right )^{3/2}}{e \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 732
Rule 814
Rule 843
Rule 715
Rule 112
Rule 110
Rule 117
Rule 116
Rubi steps
\begin{align*} \int \frac{\left (b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx &=-\frac{2 \left (b x+c x^2\right )^{3/2}}{e \sqrt{d+e x}}+\frac{3 \int \frac{(b+2 c x) \sqrt{b x+c x^2}}{\sqrt{d+e x}} \, dx}{e}\\ &=-\frac{2 \sqrt{d+e x} (8 c d-7 b e-6 c e x) \sqrt{b x+c x^2}}{5 e^3}-\frac{2 \left (b x+c x^2\right )^{3/2}}{e \sqrt{d+e x}}-\frac{2 \int \frac{-\frac{1}{2} b c d (8 c d-7 b e)-\frac{1}{2} c \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{5 c e^3}\\ &=-\frac{2 \sqrt{d+e x} (8 c d-7 b e-6 c e x) \sqrt{b x+c x^2}}{5 e^3}-\frac{2 \left (b x+c x^2\right )^{3/2}}{e \sqrt{d+e x}}-\frac{(8 d (c d-b e) (2 c d-b e)) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{5 e^4}+\frac{\left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{5 e^4}\\ &=-\frac{2 \sqrt{d+e x} (8 c d-7 b e-6 c e x) \sqrt{b x+c x^2}}{5 e^3}-\frac{2 \left (b x+c x^2\right )^{3/2}}{e \sqrt{d+e x}}-\frac{\left (8 d (c d-b e) (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}{5 e^4 \sqrt{b x+c x^2}}+\frac{\left (\left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{x} \sqrt{b+c x}} \, dx}{5 e^4 \sqrt{b x+c x^2}}\\ &=-\frac{2 \sqrt{d+e x} (8 c d-7 b e-6 c e x) \sqrt{b x+c x^2}}{5 e^3}-\frac{2 \left (b x+c x^2\right )^{3/2}}{e \sqrt{d+e x}}+\frac{\left (\left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) \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}{5 e^4 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{\left (8 d (c d-b e) (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}{5 e^4 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=-\frac{2 \sqrt{d+e x} (8 c d-7 b e-6 c e x) \sqrt{b x+c x^2}}{5 e^3}-\frac{2 \left (b x+c x^2\right )^{3/2}}{e \sqrt{d+e x}}+\frac{2 \sqrt{-b} \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) \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 )}{5 \sqrt{c} e^4 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{16 \sqrt{-b} d (c d-b e) (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 )}{5 \sqrt{c} e^4 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 1.68754, size = 340, normalized size = 1.1 \[ \frac{-2 i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (b^2 e^2-9 b c d e+8 c^2 d^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )+2 \left (b^2 c e \left (-16 d^2-8 d e x+3 e^2 x^2\right )+b^3 e^2 (d+e x)+b c^2 \left (-8 d^2 e x+16 d^3-11 d e^2 x^2+3 e^3 x^3\right )+c^3 x \left (8 d^2 e x+16 d^3-2 d e^2 x^2+e^3 x^3\right )\right )+2 i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (b^2 e^2-16 b c d e+16 c^2 d^2\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )}{5 c e^4 \sqrt{x (b+c x)} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.283, size = 685, normalized size = 2.2 \begin{align*} -{\frac{2}{5\,{c}^{2}x \left ( ce{x}^{2}+bxe+cdx+bd \right ){e}^{4}}\sqrt{x \left ( cx+b \right ) }\sqrt{ex+d} \left ( 8\,\sqrt{-{\frac{cx}{b}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) \sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}{b}^{3}cd{e}^{2}-24\,\sqrt{-{\frac{cx}{b}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) \sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}{b}^{2}{c}^{2}{d}^{2}e+16\,\sqrt{-{\frac{cx}{b}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) \sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}b{c}^{3}{d}^{3}+\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) \sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}{b}^{4}{e}^{3}-17\,\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) \sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}{b}^{3}cd{e}^{2}+32\,\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) \sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}{b}^{2}{c}^{2}{d}^{2}e-16\,\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) \sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}b{c}^{3}{d}^{3}-{x}^{4}{c}^{4}{e}^{3}-3\,{x}^{3}b{c}^{3}{e}^{3}+2\,{x}^{3}{c}^{4}d{e}^{2}-2\,{x}^{2}{b}^{2}{c}^{2}{e}^{3}-5\,{x}^{2}b{c}^{3}d{e}^{2}+8\,{x}^{2}{c}^{4}{d}^{2}e-7\,x{b}^{2}{c}^{2}d{e}^{2}+8\,xb{c}^{3}{d}^{2}e \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 (c x^{2} + b x\right )}^{\frac{3}{2}}}{{\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{{\left (c x^{2} + b x\right )}^{\frac{3}{2}} \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{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}{\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{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}}{{\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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