Optimal. Leaf size=115 \[ -\frac{5 b^2 \sqrt{a x^2+b x^3}}{8 a^3 x^2}+\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{8 a^{7/2}}+\frac{5 b \sqrt{a x^2+b x^3}}{12 a^2 x^3}-\frac{\sqrt{a x^2+b x^3}}{3 a x^4} \]
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Rubi [A] time = 0.135388, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2025, 2008, 206} \[ -\frac{5 b^2 \sqrt{a x^2+b x^3}}{8 a^3 x^2}+\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{8 a^{7/2}}+\frac{5 b \sqrt{a x^2+b x^3}}{12 a^2 x^3}-\frac{\sqrt{a x^2+b x^3}}{3 a x^4} \]
Antiderivative was successfully verified.
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Rule 2025
Rule 2008
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x^3 \sqrt{a x^2+b x^3}} \, dx &=-\frac{\sqrt{a x^2+b x^3}}{3 a x^4}-\frac{(5 b) \int \frac{1}{x^2 \sqrt{a x^2+b x^3}} \, dx}{6 a}\\ &=-\frac{\sqrt{a x^2+b x^3}}{3 a x^4}+\frac{5 b \sqrt{a x^2+b x^3}}{12 a^2 x^3}+\frac{\left (5 b^2\right ) \int \frac{1}{x \sqrt{a x^2+b x^3}} \, dx}{8 a^2}\\ &=-\frac{\sqrt{a x^2+b x^3}}{3 a x^4}+\frac{5 b \sqrt{a x^2+b x^3}}{12 a^2 x^3}-\frac{5 b^2 \sqrt{a x^2+b x^3}}{8 a^3 x^2}-\frac{\left (5 b^3\right ) \int \frac{1}{\sqrt{a x^2+b x^3}} \, dx}{16 a^3}\\ &=-\frac{\sqrt{a x^2+b x^3}}{3 a x^4}+\frac{5 b \sqrt{a x^2+b x^3}}{12 a^2 x^3}-\frac{5 b^2 \sqrt{a x^2+b x^3}}{8 a^3 x^2}+\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{x}{\sqrt{a x^2+b x^3}}\right )}{8 a^3}\\ &=-\frac{\sqrt{a x^2+b x^3}}{3 a x^4}+\frac{5 b \sqrt{a x^2+b x^3}}{12 a^2 x^3}-\frac{5 b^2 \sqrt{a x^2+b x^3}}{8 a^3 x^2}+\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{8 a^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0102403, size = 40, normalized size = 0.35 \[ \frac{2 b^3 \sqrt{x^2 (a+b x)} \, _2F_1\left (\frac{1}{2},4;\frac{3}{2};\frac{b x}{a}+1\right )}{a^4 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 95, normalized size = 0.8 \begin{align*} -{\frac{1}{24\,{x}^{2}}\sqrt{bx+a} \left ( 15\,{a}^{3/2}\sqrt{bx+a}{x}^{2}{b}^{2}-15\,{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ){x}^{3}a{b}^{3}-10\,{a}^{5/2}\sqrt{bx+a}xb+8\,{a}^{7/2}\sqrt{bx+a} \right ){\frac{1}{\sqrt{b{x}^{3}+a{x}^{2}}}}{a}^{-{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x^{3} + a x^{2}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.809012, size = 402, normalized size = 3.5 \begin{align*} \left [\frac{15 \, \sqrt{a} b^{3} x^{4} \log \left (\frac{b x^{2} + 2 \, a x + 2 \, \sqrt{b x^{3} + a x^{2}} \sqrt{a}}{x^{2}}\right ) - 2 \,{\left (15 \, a b^{2} x^{2} - 10 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt{b x^{3} + a x^{2}}}{48 \, a^{4} x^{4}}, -\frac{15 \, \sqrt{-a} b^{3} x^{4} \arctan \left (\frac{\sqrt{b x^{3} + a x^{2}} \sqrt{-a}}{a x}\right ) +{\left (15 \, a b^{2} x^{2} - 10 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt{b x^{3} + a x^{2}}}{24 \, a^{4} x^{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{1}{x^{3} \sqrt{x^{2} \left (a + b x\right )}}\, 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: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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