Optimal. Leaf size=317 \[ -\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{3 d \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)}-\frac{4 e \sqrt{b x+c x^2} (2 c d-b e)}{3 d^2 \sqrt{d+e x} (c d-b e)^2}+\frac{4 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 d^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^2}-\frac{2 e \sqrt{b x+c x^2}}{3 d (d+e x)^{3/2} (c d-b e)} \]
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Rubi [A] time = 0.287765, antiderivative size = 317, 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 = {744, 834, 843, 715, 112, 110, 117, 116} \[ -\frac{4 e \sqrt{b x+c x^2} (2 c d-b e)}{3 d^2 \sqrt{d+e x} (c d-b e)^2}+\frac{4 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 d^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^2}-\frac{2 e \sqrt{b x+c x^2}}{3 d (d+e x)^{3/2} (c d-b e)}-\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 d \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)} \]
Antiderivative was successfully verified.
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Rule 744
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} \sqrt{b x+c x^2}} \, dx &=-\frac{2 e \sqrt{b x+c x^2}}{3 d (c d-b e) (d+e x)^{3/2}}-\frac{2 \int \frac{\frac{1}{2} (-3 c d+2 b e)+\frac{c e x}{2}}{(d+e x)^{3/2} \sqrt{b x+c x^2}} \, dx}{3 d (c d-b e)}\\ &=-\frac{2 e \sqrt{b x+c x^2}}{3 d (c d-b e) (d+e x)^{3/2}}-\frac{4 e (2 c d-b e) \sqrt{b x+c x^2}}{3 d^2 (c d-b e)^2 \sqrt{d+e x}}+\frac{4 \int \frac{\frac{1}{4} c d (3 c d-b e)+\frac{1}{2} c e (2 c d-b e) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{3 d^2 (c d-b e)^2}\\ &=-\frac{2 e \sqrt{b x+c x^2}}{3 d (c d-b e) (d+e x)^{3/2}}-\frac{4 e (2 c d-b e) \sqrt{b x+c x^2}}{3 d^2 (c d-b e)^2 \sqrt{d+e x}}-\frac{c \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{3 d (c d-b e)}+\frac{(2 c (2 c d-b e)) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{3 d^2 (c d-b e)^2}\\ &=-\frac{2 e \sqrt{b x+c x^2}}{3 d (c d-b e) (d+e x)^{3/2}}-\frac{4 e (2 c d-b e) \sqrt{b x+c x^2}}{3 d^2 (c d-b e)^2 \sqrt{d+e x}}-\frac{\left (c \sqrt{x} \sqrt{b+c x}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x} \sqrt{d+e x}} \, dx}{3 d (c d-b e) \sqrt{b x+c x^2}}+\frac{\left (2 c (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 d^2 (c d-b e)^2 \sqrt{b x+c x^2}}\\ &=-\frac{2 e \sqrt{b x+c x^2}}{3 d (c d-b e) (d+e x)^{3/2}}-\frac{4 e (2 c d-b e) \sqrt{b x+c x^2}}{3 d^2 (c d-b e)^2 \sqrt{d+e x}}+\frac{\left (2 c (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 d^2 (c d-b e)^2 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{\left (c \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 d (c d-b e) \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=-\frac{2 e \sqrt{b x+c x^2}}{3 d (c d-b e) (d+e x)^{3/2}}-\frac{4 e (2 c d-b e) \sqrt{b x+c x^2}}{3 d^2 (c d-b e)^2 \sqrt{d+e x}}+\frac{4 \sqrt{-b} \sqrt{c} (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 d^2 (c d-b e)^2 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{2 \sqrt{-b} \sqrt{c} \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 d (c d-b e) \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 1.18275, size = 290, normalized size = 0.91 \[ -\frac{2 \left (-b e x (b+c x) (b e (3 d+2 e x)-c d (5 d+4 e x))-c \sqrt{\frac{b}{c}} (d+e x) \left (i x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (2 b^2 e^2-5 b c d e+3 c^2 d^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )+2 i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (2 c d-b e) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )-2 \sqrt{\frac{b}{c}} (b+c x) (d+e x) (b e-2 c d)\right )\right )}{3 b d^2 \sqrt{x (b+c x)} (d+e x)^{3/2} (c d-b e)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.319, size = 897, normalized size = 2.8 \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}{\sqrt{c x^{2} + b x}{\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 e^{3} x^{5} + b d^{3} x +{\left (3 \, c d e^{2} + b e^{3}\right )} x^{4} + 3 \,{\left (c d^{2} e + b d e^{2}\right )} x^{3} +{\left (c d^{3} + 3 \, b d^{2} e\right )} x^{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{1}{\sqrt{x \left (b + c x\right )} \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}{\sqrt{c x^{2} + b x}{\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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