Optimal. Leaf size=139 \[ -\frac{5 a^5 (a+2 b x) \sqrt{a x+b x^2}}{1024 b^4}+\frac{5 a^3 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{384 b^3}+\frac{5 a^7 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{1024 b^{9/2}}-\frac{a (a+2 b x) \left (a x+b x^2\right )^{5/2}}{24 b^2}+\frac{\left (a x+b x^2\right )^{7/2}}{7 b} \]
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Rubi [A] time = 0.0504348, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {640, 612, 620, 206} \[ -\frac{5 a^5 (a+2 b x) \sqrt{a x+b x^2}}{1024 b^4}+\frac{5 a^3 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{384 b^3}+\frac{5 a^7 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{1024 b^{9/2}}-\frac{a (a+2 b x) \left (a x+b x^2\right )^{5/2}}{24 b^2}+\frac{\left (a x+b x^2\right )^{7/2}}{7 b} \]
Antiderivative was successfully verified.
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Rule 640
Rule 612
Rule 620
Rule 206
Rubi steps
\begin{align*} \int x \left (a x+b x^2\right )^{5/2} \, dx &=\frac{\left (a x+b x^2\right )^{7/2}}{7 b}-\frac{a \int \left (a x+b x^2\right )^{5/2} \, dx}{2 b}\\ &=-\frac{a (a+2 b x) \left (a x+b x^2\right )^{5/2}}{24 b^2}+\frac{\left (a x+b x^2\right )^{7/2}}{7 b}+\frac{\left (5 a^3\right ) \int \left (a x+b x^2\right )^{3/2} \, dx}{48 b^2}\\ &=\frac{5 a^3 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{384 b^3}-\frac{a (a+2 b x) \left (a x+b x^2\right )^{5/2}}{24 b^2}+\frac{\left (a x+b x^2\right )^{7/2}}{7 b}-\frac{\left (5 a^5\right ) \int \sqrt{a x+b x^2} \, dx}{256 b^3}\\ &=-\frac{5 a^5 (a+2 b x) \sqrt{a x+b x^2}}{1024 b^4}+\frac{5 a^3 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{384 b^3}-\frac{a (a+2 b x) \left (a x+b x^2\right )^{5/2}}{24 b^2}+\frac{\left (a x+b x^2\right )^{7/2}}{7 b}+\frac{\left (5 a^7\right ) \int \frac{1}{\sqrt{a x+b x^2}} \, dx}{2048 b^4}\\ &=-\frac{5 a^5 (a+2 b x) \sqrt{a x+b x^2}}{1024 b^4}+\frac{5 a^3 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{384 b^3}-\frac{a (a+2 b x) \left (a x+b x^2\right )^{5/2}}{24 b^2}+\frac{\left (a x+b x^2\right )^{7/2}}{7 b}+\frac{\left (5 a^7\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a x+b x^2}}\right )}{1024 b^4}\\ &=-\frac{5 a^5 (a+2 b x) \sqrt{a x+b x^2}}{1024 b^4}+\frac{5 a^3 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{384 b^3}-\frac{a (a+2 b x) \left (a x+b x^2\right )^{5/2}}{24 b^2}+\frac{\left (a x+b x^2\right )^{7/2}}{7 b}+\frac{5 a^7 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{1024 b^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.177274, size = 131, normalized size = 0.94 \[ \frac{\sqrt{x (a+b x)} \left (\sqrt{b} \left (-56 a^4 b^2 x^2+48 a^3 b^3 x^3+4736 a^2 b^4 x^4+70 a^5 b x-105 a^6+7424 a b^5 x^5+3072 b^6 x^6\right )+\frac{105 a^{13/2} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{x} \sqrt{\frac{b x}{a}+1}}\right )}{21504 b^{9/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 165, normalized size = 1.2 \begin{align*}{\frac{1}{7\,b} \left ( b{x}^{2}+ax \right ) ^{{\frac{7}{2}}}}-{\frac{ax}{12\,b} \left ( b{x}^{2}+ax \right ) ^{{\frac{5}{2}}}}-{\frac{{a}^{2}}{24\,{b}^{2}} \left ( b{x}^{2}+ax \right ) ^{{\frac{5}{2}}}}+{\frac{5\,x{a}^{3}}{192\,{b}^{2}} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{4}}{384\,{b}^{3}} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{a}^{5}x}{512\,{b}^{3}}\sqrt{b{x}^{2}+ax}}-{\frac{5\,{a}^{6}}{1024\,{b}^{4}}\sqrt{b{x}^{2}+ax}}+{\frac{5\,{a}^{7}}{2048}\ln \left ({ \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{9}{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 = 2.01484, size = 567, normalized size = 4.08 \begin{align*} \left [\frac{105 \, a^{7} \sqrt{b} \log \left (2 \, b x + a + 2 \, \sqrt{b x^{2} + a x} \sqrt{b}\right ) + 2 \,{\left (3072 \, b^{7} x^{6} + 7424 \, a b^{6} x^{5} + 4736 \, a^{2} b^{5} x^{4} + 48 \, a^{3} b^{4} x^{3} - 56 \, a^{4} b^{3} x^{2} + 70 \, a^{5} b^{2} x - 105 \, a^{6} b\right )} \sqrt{b x^{2} + a x}}{43008 \, b^{5}}, -\frac{105 \, a^{7} \sqrt{-b} \arctan \left (\frac{\sqrt{b x^{2} + a x} \sqrt{-b}}{b x}\right ) -{\left (3072 \, b^{7} x^{6} + 7424 \, a b^{6} x^{5} + 4736 \, a^{2} b^{5} x^{4} + 48 \, a^{3} b^{4} x^{3} - 56 \, a^{4} b^{3} x^{2} + 70 \, a^{5} b^{2} x - 105 \, a^{6} b\right )} \sqrt{b x^{2} + a x}}{21504 \, b^{5}}\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 x \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 [A] time = 1.29223, size = 162, normalized size = 1.17 \begin{align*} -\frac{5 \, a^{7} \log \left ({\left | -2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} \sqrt{b} - a \right |}\right )}{2048 \, b^{\frac{9}{2}}} - \frac{1}{21504} \, \sqrt{b x^{2} + a x}{\left (\frac{105 \, a^{6}}{b^{4}} - 2 \,{\left (\frac{35 \, a^{5}}{b^{3}} - 4 \,{\left (\frac{7 \, a^{4}}{b^{2}} - 2 \,{\left (\frac{3 \, a^{3}}{b} + 8 \,{\left (37 \, a^{2} + 2 \,{\left (12 \, b^{2} x + 29 \, a b\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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