Optimal. Leaf size=301 \[ \frac{4 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{3 e^2 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 d e^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)}+\frac{2 \sqrt{b x+c x^2} (2 c d-b e)}{3 d e \sqrt{d+e x} (c d-b e)}-\frac{2 \sqrt{b x+c x^2}}{3 e (d+e x)^{3/2}} \]
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Rubi [A] time = 0.280329, antiderivative size = 301, 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, 834, 843, 715, 112, 110, 117, 116} \[ \frac{4 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 e^2 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 d e^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)}+\frac{2 \sqrt{b x+c x^2} (2 c d-b e)}{3 d e \sqrt{d+e x} (c d-b e)}-\frac{2 \sqrt{b x+c x^2}}{3 e (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 732
Rule 834
Rule 843
Rule 715
Rule 112
Rule 110
Rule 117
Rule 116
Rubi steps
\begin{align*} \int \frac{\sqrt{b x+c x^2}}{(d+e x)^{5/2}} \, dx &=-\frac{2 \sqrt{b x+c x^2}}{3 e (d+e x)^{3/2}}+\frac{\int \frac{b+2 c x}{(d+e x)^{3/2} \sqrt{b x+c x^2}} \, dx}{3 e}\\ &=-\frac{2 \sqrt{b x+c x^2}}{3 e (d+e x)^{3/2}}+\frac{2 (2 c d-b e) \sqrt{b x+c x^2}}{3 d e (c d-b e) \sqrt{d+e x}}-\frac{2 \int \frac{\frac{b c d}{2}+\frac{1}{2} c (2 c d-b e) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{3 d e (c d-b e)}\\ &=-\frac{2 \sqrt{b x+c x^2}}{3 e (d+e x)^{3/2}}+\frac{2 (2 c d-b e) \sqrt{b x+c x^2}}{3 d e (c d-b e) \sqrt{d+e x}}+\frac{(2 c) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{3 e^2}-\frac{(c (2 c d-b e)) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{3 d e^2 (c d-b e)}\\ &=-\frac{2 \sqrt{b x+c x^2}}{3 e (d+e x)^{3/2}}+\frac{2 (2 c d-b e) \sqrt{b x+c x^2}}{3 d e (c d-b e) \sqrt{d+e x}}+\frac{\left (2 c \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 (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 e^2 (c d-b e) \sqrt{b x+c x^2}}\\ &=-\frac{2 \sqrt{b x+c x^2}}{3 e (d+e x)^{3/2}}+\frac{2 (2 c d-b e) \sqrt{b x+c x^2}}{3 d e (c d-b e) \sqrt{d+e x}}-\frac{\left (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 e^2 (c d-b e) \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{\left (2 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 e^2 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=-\frac{2 \sqrt{b x+c x^2}}{3 e (d+e x)^{3/2}}+\frac{2 (2 c d-b e) \sqrt{b x+c x^2}}{3 d e (c d-b e) \sqrt{d+e x}}-\frac{2 \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 e^2 (c d-b e) \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{4 \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 e^2 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 1.14323, size = 265, normalized size = 0.88 \[ -\frac{2 \left (e x (b+c x) \left (b e^2 x-c d (d+2 e x)\right )+(d+e x) \left (i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (b e-c d) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )-i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (b e-2 c d) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+(b+c x) (d+e x) (2 c d-b e)\right )\right )}{3 d e^2 \sqrt{x (b+c x)} (d+e x)^{3/2} (c d-b e)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.293, size = 887, normalized size = 3. \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{\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}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{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{\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{\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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