Optimal. Leaf size=94 \[ -\frac{10 x^2}{3 b^2 \sqrt{a x+b x^2}}+\frac{5 \sqrt{a x+b x^2}}{b^3}-\frac{5 a \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{b^{7/2}}-\frac{2 x^4}{3 b \left (a x+b x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0414738, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {668, 640, 620, 206} \[ -\frac{10 x^2}{3 b^2 \sqrt{a x+b x^2}}+\frac{5 \sqrt{a x+b x^2}}{b^3}-\frac{5 a \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{b^{7/2}}-\frac{2 x^4}{3 b \left (a x+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 668
Rule 640
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{x^5}{\left (a x+b x^2\right )^{5/2}} \, dx &=-\frac{2 x^4}{3 b \left (a x+b x^2\right )^{3/2}}+\frac{5 \int \frac{x^3}{\left (a x+b x^2\right )^{3/2}} \, dx}{3 b}\\ &=-\frac{2 x^4}{3 b \left (a x+b x^2\right )^{3/2}}-\frac{10 x^2}{3 b^2 \sqrt{a x+b x^2}}+\frac{5 \int \frac{x}{\sqrt{a x+b x^2}} \, dx}{b^2}\\ &=-\frac{2 x^4}{3 b \left (a x+b x^2\right )^{3/2}}-\frac{10 x^2}{3 b^2 \sqrt{a x+b x^2}}+\frac{5 \sqrt{a x+b x^2}}{b^3}-\frac{(5 a) \int \frac{1}{\sqrt{a x+b x^2}} \, dx}{2 b^3}\\ &=-\frac{2 x^4}{3 b \left (a x+b x^2\right )^{3/2}}-\frac{10 x^2}{3 b^2 \sqrt{a x+b x^2}}+\frac{5 \sqrt{a x+b x^2}}{b^3}-\frac{(5 a) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a x+b x^2}}\right )}{b^3}\\ &=-\frac{2 x^4}{3 b \left (a x+b x^2\right )^{3/2}}-\frac{10 x^2}{3 b^2 \sqrt{a x+b x^2}}+\frac{5 \sqrt{a x+b x^2}}{b^3}-\frac{5 a \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{b^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0131385, size = 50, normalized size = 0.53 \[ \frac{2 x^4 \sqrt{\frac{b x}{a}+1} \, _2F_1\left (\frac{5}{2},\frac{7}{2};\frac{9}{2};-\frac{b x}{a}\right )}{7 a^2 \sqrt{x (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 149, normalized size = 1.6 \begin{align*}{\frac{{x}^{4}}{b} \left ( b{x}^{2}+ax \right ) ^{-{\frac{3}{2}}}}+{\frac{5\,a{x}^{3}}{6\,{b}^{2}} \left ( b{x}^{2}+ax \right ) ^{-{\frac{3}{2}}}}-{\frac{5\,{a}^{2}{x}^{2}}{4\,{b}^{3}} \left ( b{x}^{2}+ax \right ) ^{-{\frac{3}{2}}}}-{\frac{5\,x{a}^{3}}{12\,{b}^{4}} \left ( b{x}^{2}+ax \right ) ^{-{\frac{3}{2}}}}+{\frac{35\,ax}{6\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{2}+ax}}}}+{\frac{5\,{a}^{2}}{12\,{b}^{4}}{\frac{1}{\sqrt{b{x}^{2}+ax}}}}-{\frac{5\,a}{2}\ln \left ({ \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{7}{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 = 1.92178, size = 493, normalized size = 5.24 \begin{align*} \left [\frac{15 \,{\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )} \sqrt{b} \log \left (2 \, b x + a - 2 \, \sqrt{b x^{2} + a x} \sqrt{b}\right ) + 2 \,{\left (3 \, b^{3} x^{2} + 20 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt{b x^{2} + a x}}{6 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, \frac{15 \,{\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{b x^{2} + a x} \sqrt{-b}}{b x}\right ) +{\left (3 \, b^{3} x^{2} + 20 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt{b x^{2} + a x}}{3 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\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^{5}}{\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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