Optimal. Leaf size=478 \[ -\frac{4 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{3 (-b)^{3/2} d^2 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)^2}-\frac{4 e \sqrt{b x+c x^2} \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right )}{3 b^2 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac{2 e \sqrt{b x+c x^2} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right )}{3 b^2 d^3 \sqrt{d+e x} (c d-b e)^3}+\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{3/2} d^3 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^3}-\frac{2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (d+e x)^{3/2} (c d-b e)} \]
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Rubi [A] time = 0.58809, antiderivative size = 478, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {740, 834, 843, 715, 112, 110, 117, 116} \[ -\frac{4 e \sqrt{b x+c x^2} \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right )}{3 b^2 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac{2 e \sqrt{b x+c x^2} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right )}{3 b^2 d^3 \sqrt{d+e x} (c d-b e)^3}-\frac{4 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{3/2} d^2 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)^2}+\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{3/2} d^3 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^3}-\frac{2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (d+e x)^{3/2} (c d-b e)} \]
Antiderivative was successfully verified.
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Rule 740
Rule 834
Rule 843
Rule 715
Rule 112
Rule 110
Rule 117
Rule 116
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^{5/2} \left (b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x)^{3/2} \sqrt{b x+c x^2}}-\frac{2 \int \frac{\frac{1}{2} b e (3 c d-4 b e)+\frac{3}{2} c e (2 c d-b e) x}{(d+e x)^{5/2} \sqrt{b x+c x^2}} \, dx}{b^2 d (c d-b e)}\\ &=-\frac{2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x)^{3/2} \sqrt{b x+c x^2}}-\frac{4 e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt{b x+c x^2}}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}+\frac{4 \int \frac{-\frac{1}{4} b e \left (3 c^2 d^2-15 b c d e+8 b^2 e^2\right )-\frac{1}{2} c e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) x}{(d+e x)^{3/2} \sqrt{b x+c x^2}} \, dx}{3 b^2 d^2 (c d-b e)^2}\\ &=-\frac{2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x)^{3/2} \sqrt{b x+c x^2}}-\frac{4 e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt{b x+c x^2}}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac{2 e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) \sqrt{b x+c x^2}}{3 b^2 d^3 (c d-b e)^3 \sqrt{d+e x}}-\frac{8 \int \frac{-\frac{1}{8} b c d e \left (3 c^2 d^2+9 b c d e-4 b^2 e^2\right )-\frac{1}{8} c e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{3 b^2 d^3 (c d-b e)^3}\\ &=-\frac{2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x)^{3/2} \sqrt{b x+c x^2}}-\frac{4 e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt{b x+c x^2}}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac{2 e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) \sqrt{b x+c x^2}}{3 b^2 d^3 (c d-b e)^3 \sqrt{d+e x}}-\frac{\left (2 c \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{3 b^2 d^2 (c d-b e)^2}+\frac{\left (c (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{3 b^2 d^3 (c d-b e)^3}\\ &=-\frac{2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x)^{3/2} \sqrt{b x+c x^2}}-\frac{4 e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt{b x+c x^2}}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac{2 e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) \sqrt{b x+c x^2}}{3 b^2 d^3 (c d-b e)^3 \sqrt{d+e x}}-\frac{\left (2 c \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x} \sqrt{d+e x}} \, dx}{3 b^2 d^2 (c d-b e)^2 \sqrt{b x+c x^2}}+\frac{\left (c (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 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}{3 b^2 d^3 (c d-b e)^3 \sqrt{b x+c x^2}}\\ &=-\frac{2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x)^{3/2} \sqrt{b x+c x^2}}-\frac{4 e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt{b x+c x^2}}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac{2 e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) \sqrt{b x+c x^2}}{3 b^2 d^3 (c d-b e)^3 \sqrt{d+e x}}+\frac{\left (c (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 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}{3 b^2 d^3 (c d-b e)^3 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{\left (2 c \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \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 b^2 d^2 (c d-b e)^2 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=-\frac{2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x)^{3/2} \sqrt{b x+c x^2}}-\frac{4 e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt{b x+c x^2}}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac{2 e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) \sqrt{b x+c x^2}}{3 b^2 d^3 (c d-b e)^3 \sqrt{d+e x}}+\frac{2 \sqrt{c} (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 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 )}{3 (-b)^{3/2} d^3 (c d-b e)^3 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{4 \sqrt{c} \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \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 (-b)^{3/2} d^2 (c d-b e)^2 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 1.63144, size = 420, normalized size = 0.88 \[ -\frac{2 \left (b \left (b^2 d e^3 x (b+c x) (c d-b e)-5 b^2 e^3 x (b+c x) (d+e x) (b e-2 c d)+3 (b+c x) (d+e x)^2 (c d-b e)^3+3 c^4 d^3 x (d+e x)^2\right )-c \sqrt{\frac{b}{c}} (d+e x) \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (23 b^2 c d e^2-8 b^3 e^3-18 b c^2 d^2 e+3 c^3 d^3\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (19 b^2 c d e^2-8 b^3 e^3-9 b c^2 d^2 e+6 c^3 d^3\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (19 b^2 c d e^2-8 b^3 e^3-9 b c^2 d^2 e+6 c^3 d^3\right )\right )\right )}{3 b^3 d^3 \sqrt{x (b+c x)} (d+e x)^{3/2} (c d-b e)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.313, size = 1708, normalized size = 3.6 \begin{align*} \text{result too large to display} \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}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{\frac{5}{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^{2} e^{3} x^{7} + b^{2} d^{3} x^{2} +{\left (3 \, c^{2} d e^{2} + 2 \, b c e^{3}\right )} x^{6} +{\left (3 \, c^{2} d^{2} e + 6 \, b c d e^{2} + b^{2} e^{3}\right )} x^{5} +{\left (c^{2} d^{3} + 6 \, b c d^{2} e + 3 \, b^{2} d e^{2}\right )} x^{4} +{\left (2 \, b c d^{3} + 3 \, b^{2} d^{2} e\right )} x^{3}}, 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}{\left (x \left (b + c x\right )\right )^{\frac{3}{2}} \left (d + e x\right )^{\frac{5}{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}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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