Optimal. Leaf size=92 \[ \frac{15}{4} a^2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )-\frac{2 \left (a x+b x^2\right )^{5/2}}{x^3}+\frac{5 b \left (a x+b x^2\right )^{3/2}}{2 x}+\frac{15}{4} a b \sqrt{a x+b x^2} \]
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Rubi [A] time = 0.04064, antiderivative size = 92, 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 = {662, 664, 620, 206} \[ \frac{15}{4} a^2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )-\frac{2 \left (a x+b x^2\right )^{5/2}}{x^3}+\frac{5 b \left (a x+b x^2\right )^{3/2}}{2 x}+\frac{15}{4} a b \sqrt{a x+b x^2} \]
Antiderivative was successfully verified.
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Rule 662
Rule 664
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (a x+b x^2\right )^{5/2}}{x^4} \, dx &=-\frac{2 \left (a x+b x^2\right )^{5/2}}{x^3}+(5 b) \int \frac{\left (a x+b x^2\right )^{3/2}}{x^2} \, dx\\ &=\frac{5 b \left (a x+b x^2\right )^{3/2}}{2 x}-\frac{2 \left (a x+b x^2\right )^{5/2}}{x^3}+\frac{1}{4} (15 a b) \int \frac{\sqrt{a x+b x^2}}{x} \, dx\\ &=\frac{15}{4} a b \sqrt{a x+b x^2}+\frac{5 b \left (a x+b x^2\right )^{3/2}}{2 x}-\frac{2 \left (a x+b x^2\right )^{5/2}}{x^3}+\frac{1}{8} \left (15 a^2 b\right ) \int \frac{1}{\sqrt{a x+b x^2}} \, dx\\ &=\frac{15}{4} a b \sqrt{a x+b x^2}+\frac{5 b \left (a x+b x^2\right )^{3/2}}{2 x}-\frac{2 \left (a x+b x^2\right )^{5/2}}{x^3}+\frac{1}{4} \left (15 a^2 b\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a x+b x^2}}\right )\\ &=\frac{15}{4} a b \sqrt{a x+b x^2}+\frac{5 b \left (a x+b x^2\right )^{3/2}}{2 x}-\frac{2 \left (a x+b x^2\right )^{5/2}}{x^3}+\frac{15}{4} a^2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )\\ \end{align*}
Mathematica [C] time = 0.0124159, size = 48, normalized size = 0.52 \[ -\frac{2 a^2 \sqrt{x (a+b x)} \, _2F_1\left (-\frac{5}{2},-\frac{1}{2};\frac{1}{2};-\frac{b x}{a}\right )}{x \sqrt{\frac{b x}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.048, size = 185, normalized size = 2. \begin{align*} -2\,{\frac{ \left ( b{x}^{2}+ax \right ) ^{7/2}}{a{x}^{4}}}+12\,{\frac{b \left ( b{x}^{2}+ax \right ) ^{7/2}}{{a}^{2}{x}^{3}}}-32\,{\frac{{b}^{2} \left ( b{x}^{2}+ax \right ) ^{7/2}}{{x}^{2}{a}^{3}}}+32\,{\frac{{b}^{3} \left ( b{x}^{2}+ax \right ) ^{5/2}}{{a}^{3}}}+20\,{\frac{{b}^{3} \left ( b{x}^{2}+ax \right ) ^{3/2}x}{{a}^{2}}}+10\,{\frac{{b}^{2} \left ( b{x}^{2}+ax \right ) ^{3/2}}{a}}-{\frac{15\,{b}^{2}x}{2}\sqrt{b{x}^{2}+ax}}-{\frac{15\,ab}{4}\sqrt{b{x}^{2}+ax}}+{\frac{15\,{a}^{2}}{8}\sqrt{b}\ln \left ({ \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ) } \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 = 2.08301, size = 332, normalized size = 3.61 \begin{align*} \left [\frac{15 \, a^{2} \sqrt{b} x \log \left (2 \, b x + a + 2 \, \sqrt{b x^{2} + a x} \sqrt{b}\right ) + 2 \,{\left (2 \, b^{2} x^{2} + 9 \, a b x - 8 \, a^{2}\right )} \sqrt{b x^{2} + a x}}{8 \, x}, -\frac{15 \, a^{2} \sqrt{-b} x \arctan \left (\frac{\sqrt{b x^{2} + a x} \sqrt{-b}}{b x}\right ) -{\left (2 \, b^{2} x^{2} + 9 \, a b x - 8 \, a^{2}\right )} \sqrt{b x^{2} + a x}}{4 \, 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{\left (x \left (a + b x\right )\right )^{\frac{5}{2}}}{x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26531, size = 120, normalized size = 1.3 \begin{align*} -\frac{15}{8} \, a^{2} \sqrt{b} \log \left ({\left | -2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} \sqrt{b} - a \right |}\right ) + \frac{2 \, a^{3}}{\sqrt{b} x - \sqrt{b x^{2} + a x}} + \frac{1}{4} \,{\left (2 \, b^{2} x + 9 \, a b\right )} \sqrt{b x^{2} + a x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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