Optimal. Leaf size=71 \[ -\frac{2 x}{b^2 \sqrt{a x+b x^2}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{b^{5/2}}-\frac{2 x^3}{3 b \left (a x+b x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0300053, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {668, 652, 620, 206} \[ -\frac{2 x}{b^2 \sqrt{a x+b x^2}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{b^{5/2}}-\frac{2 x^3}{3 b \left (a x+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 668
Rule 652
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{x^4}{\left (a x+b x^2\right )^{5/2}} \, dx &=-\frac{2 x^3}{3 b \left (a x+b x^2\right )^{3/2}}+\frac{\int \frac{x^2}{\left (a x+b x^2\right )^{3/2}} \, dx}{b}\\ &=-\frac{2 x^3}{3 b \left (a x+b x^2\right )^{3/2}}-\frac{2 x}{b^2 \sqrt{a x+b x^2}}+\frac{\int \frac{1}{\sqrt{a x+b x^2}} \, dx}{b^2}\\ &=-\frac{2 x^3}{3 b \left (a x+b x^2\right )^{3/2}}-\frac{2 x}{b^2 \sqrt{a x+b x^2}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a x+b x^2}}\right )}{b^2}\\ &=-\frac{2 x^3}{3 b \left (a x+b x^2\right )^{3/2}}-\frac{2 x}{b^2 \sqrt{a x+b x^2}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.160041, size = 84, normalized size = 1.18 \[ \frac{x \left (6 \sqrt{a} \sqrt{x} (a+b x) \sqrt{\frac{b x}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )-2 \sqrt{b} x (3 a+4 b x)\right )}{3 b^{5/2} (x (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.044, size = 123, normalized size = 1.7 \begin{align*} -{\frac{{x}^{3}}{3\,b} \left ( b{x}^{2}+ax \right ) ^{-{\frac{3}{2}}}}+{\frac{a{x}^{2}}{2\,{b}^{2}} \left ( b{x}^{2}+ax \right ) ^{-{\frac{3}{2}}}}+{\frac{{a}^{2}x}{6\,{b}^{3}} \left ( b{x}^{2}+ax \right ) ^{-{\frac{3}{2}}}}-{\frac{7\,x}{3\,{b}^{2}}{\frac{1}{\sqrt{b{x}^{2}+ax}}}}-{\frac{a}{6\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{2}+ax}}}}+{\ln \left ({ \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{5}{2}}}} \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 = 2.02773, size = 432, normalized size = 6.08 \begin{align*} \left [\frac{3 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{b} \log \left (2 \, b x + a + 2 \, \sqrt{b x^{2} + a x} \sqrt{b}\right ) - 2 \,{\left (4 \, b^{2} x + 3 \, a b\right )} \sqrt{b x^{2} + a x}}{3 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}, -\frac{2 \,{\left (3 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{b x^{2} + a x} \sqrt{-b}}{b x}\right ) +{\left (4 \, b^{2} x + 3 \, a b\right )} \sqrt{b x^{2} + a x}\right )}}{3 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}\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{x^{4}}{\left (x \left (a + b x\right )\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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