Optimal. Leaf size=149 \[ \frac{e \sqrt{b x+c x^2} \left (15 b^2 e^2+10 c e x (2 c d-b e)-54 b c d e+64 c^2 d^2\right )}{24 c^3}+\frac{(2 c d-b e) \left (5 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{7/2}}+\frac{e \sqrt{b x+c x^2} (d+e x)^2}{3 c} \]
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Rubi [A] time = 0.149362, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {742, 779, 620, 206} \[ \frac{e \sqrt{b x+c x^2} \left (15 b^2 e^2+10 c e x (2 c d-b e)-54 b c d e+64 c^2 d^2\right )}{24 c^3}+\frac{(2 c d-b e) \left (5 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{7/2}}+\frac{e \sqrt{b x+c x^2} (d+e x)^2}{3 c} \]
Antiderivative was successfully verified.
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Rule 742
Rule 779
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{(d+e x)^3}{\sqrt{b x+c x^2}} \, dx &=\frac{e (d+e x)^2 \sqrt{b x+c x^2}}{3 c}+\frac{\int \frac{(d+e x) \left (\frac{1}{2} d (6 c d-b e)+\frac{5}{2} e (2 c d-b e) x\right )}{\sqrt{b x+c x^2}} \, dx}{3 c}\\ &=\frac{e (d+e x)^2 \sqrt{b x+c x^2}}{3 c}+\frac{e \left (64 c^2 d^2-54 b c d e+15 b^2 e^2+10 c e (2 c d-b e) x\right ) \sqrt{b x+c x^2}}{24 c^3}+\frac{\left ((2 c d-b e) \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right )\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{16 c^3}\\ &=\frac{e (d+e x)^2 \sqrt{b x+c x^2}}{3 c}+\frac{e \left (64 c^2 d^2-54 b c d e+15 b^2 e^2+10 c e (2 c d-b e) x\right ) \sqrt{b x+c x^2}}{24 c^3}+\frac{\left ((2 c d-b e) \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{8 c^3}\\ &=\frac{e (d+e x)^2 \sqrt{b x+c x^2}}{3 c}+\frac{e \left (64 c^2 d^2-54 b c d e+15 b^2 e^2+10 c e (2 c d-b e) x\right ) \sqrt{b x+c x^2}}{24 c^3}+\frac{(2 c d-b e) \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.465465, size = 152, normalized size = 1.02 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} e \left (15 b^2 e^2-2 b c e (27 d+5 e x)+4 c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+\frac{3 \left (18 b^2 c d e^2-5 b^3 e^3-24 b c^2 d^2 e+16 c^3 d^3\right ) \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{x} \sqrt{\frac{c x}{b}+1}}\right )}{24 c^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 265, normalized size = 1.8 \begin{align*}{\frac{{e}^{3}{x}^{2}}{3\,c}\sqrt{c{x}^{2}+bx}}-{\frac{5\,b{e}^{3}x}{12\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{5\,{e}^{3}{b}^{2}}{8\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{5\,{b}^{3}{e}^{3}}{16}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}}+{\frac{3\,d{e}^{2}x}{2\,c}\sqrt{c{x}^{2}+bx}}-{\frac{9\,bd{e}^{2}}{4\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{9\,{b}^{2}d{e}^{2}}{8}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}}+3\,{\frac{{d}^{2}e\sqrt{c{x}^{2}+bx}}{c}}-{\frac{3\,b{d}^{2}e}{2}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}}+{{d}^{3}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){\frac{1}{\sqrt{c}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03101, size = 666, normalized size = 4.47 \begin{align*} \left [-\frac{3 \,{\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 18 \, b^{2} c d e^{2} - 5 \, b^{3} e^{3}\right )} \sqrt{c} \log \left (2 \, c x + b - 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) - 2 \,{\left (8 \, c^{3} e^{3} x^{2} + 72 \, c^{3} d^{2} e - 54 \, b c^{2} d e^{2} + 15 \, b^{2} c e^{3} + 2 \,{\left (18 \, c^{3} d e^{2} - 5 \, b c^{2} e^{3}\right )} x\right )} \sqrt{c x^{2} + b x}}{48 \, c^{4}}, -\frac{3 \,{\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 18 \, b^{2} c d e^{2} - 5 \, b^{3} e^{3}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (8 \, c^{3} e^{3} x^{2} + 72 \, c^{3} d^{2} e - 54 \, b c^{2} d e^{2} + 15 \, b^{2} c e^{3} + 2 \,{\left (18 \, c^{3} d e^{2} - 5 \, b c^{2} e^{3}\right )} x\right )} \sqrt{c x^{2} + b x}}{24 \, c^{4}}\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 )^{3}}{\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 [A] time = 1.32395, size = 198, normalized size = 1.33 \begin{align*} \frac{1}{24} \, \sqrt{c x^{2} + b x}{\left (2 \, x{\left (\frac{4 \, x e^{3}}{c} + \frac{18 \, c^{2} d e^{2} - 5 \, b c e^{3}}{c^{3}}\right )} + \frac{3 \,{\left (24 \, c^{2} d^{2} e - 18 \, b c d e^{2} + 5 \, b^{2} e^{3}\right )}}{c^{3}}\right )} - \frac{{\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 18 \, b^{2} c d e^{2} - 5 \, b^{3} e^{3}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{16 \, c^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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