Optimal. Leaf size=138 \[ -\frac{35 b^2 \sqrt{a x^2+b x^3}}{8 a^4 x^2}+\frac{35 b^3 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{8 a^{9/2}}+\frac{35 b \sqrt{a x^2+b x^3}}{12 a^3 x^3}-\frac{7 \sqrt{a x^2+b x^3}}{3 a^2 x^4}+\frac{2}{a x^2 \sqrt{a x^2+b x^3}} \]
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Rubi [A] time = 0.186272, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2023, 2025, 2008, 206} \[ -\frac{35 b^2 \sqrt{a x^2+b x^3}}{8 a^4 x^2}+\frac{35 b^3 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{8 a^{9/2}}+\frac{35 b \sqrt{a x^2+b x^3}}{12 a^3 x^3}-\frac{7 \sqrt{a x^2+b x^3}}{3 a^2 x^4}+\frac{2}{a x^2 \sqrt{a x^2+b x^3}} \]
Antiderivative was successfully verified.
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Rule 2023
Rule 2025
Rule 2008
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x \left (a x^2+b x^3\right )^{3/2}} \, dx &=\frac{2}{a x^2 \sqrt{a x^2+b x^3}}+\frac{7 \int \frac{1}{x^3 \sqrt{a x^2+b x^3}} \, dx}{a}\\ &=\frac{2}{a x^2 \sqrt{a x^2+b x^3}}-\frac{7 \sqrt{a x^2+b x^3}}{3 a^2 x^4}-\frac{(35 b) \int \frac{1}{x^2 \sqrt{a x^2+b x^3}} \, dx}{6 a^2}\\ &=\frac{2}{a x^2 \sqrt{a x^2+b x^3}}-\frac{7 \sqrt{a x^2+b x^3}}{3 a^2 x^4}+\frac{35 b \sqrt{a x^2+b x^3}}{12 a^3 x^3}+\frac{\left (35 b^2\right ) \int \frac{1}{x \sqrt{a x^2+b x^3}} \, dx}{8 a^3}\\ &=\frac{2}{a x^2 \sqrt{a x^2+b x^3}}-\frac{7 \sqrt{a x^2+b x^3}}{3 a^2 x^4}+\frac{35 b \sqrt{a x^2+b x^3}}{12 a^3 x^3}-\frac{35 b^2 \sqrt{a x^2+b x^3}}{8 a^4 x^2}-\frac{\left (35 b^3\right ) \int \frac{1}{\sqrt{a x^2+b x^3}} \, dx}{16 a^4}\\ &=\frac{2}{a x^2 \sqrt{a x^2+b x^3}}-\frac{7 \sqrt{a x^2+b x^3}}{3 a^2 x^4}+\frac{35 b \sqrt{a x^2+b x^3}}{12 a^3 x^3}-\frac{35 b^2 \sqrt{a x^2+b x^3}}{8 a^4 x^2}+\frac{\left (35 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^4}\\ &=\frac{2}{a x^2 \sqrt{a x^2+b x^3}}-\frac{7 \sqrt{a x^2+b x^3}}{3 a^2 x^4}+\frac{35 b \sqrt{a x^2+b x^3}}{12 a^3 x^3}-\frac{35 b^2 \sqrt{a x^2+b x^3}}{8 a^4 x^2}+\frac{35 b^3 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{8 a^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.0087413, size = 38, normalized size = 0.28 \[ -\frac{2 b^3 x \, _2F_1\left (-\frac{1}{2},4;\frac{1}{2};\frac{b x}{a}+1\right )}{a^4 \sqrt{x^2 (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 86, normalized size = 0.6 \begin{align*} -{\frac{bx+a}{24} \left ( -105\,{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) \sqrt{bx+a}{x}^{3}{b}^{3}-14\,{a}^{5/2}xb+35\,{a}^{3/2}{x}^{2}{b}^{2}+105\,{b}^{3}{x}^{3}\sqrt{a}+8\,{a}^{7/2} \right ) \left ( b{x}^{3}+a{x}^{2} \right ) ^{-{\frac{3}{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}{{\left (b x^{3} + a x^{2}\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.816872, size = 524, normalized size = 3.8 \begin{align*} \left [\frac{105 \,{\left (b^{4} x^{5} + a b^{3} x^{4}\right )} \sqrt{a} \log \left (\frac{b x^{2} + 2 \, a x + 2 \, \sqrt{b x^{3} + a x^{2}} \sqrt{a}}{x^{2}}\right ) - 2 \,{\left (105 \, a b^{3} x^{3} + 35 \, a^{2} b^{2} x^{2} - 14 \, a^{3} b x + 8 \, a^{4}\right )} \sqrt{b x^{3} + a x^{2}}}{48 \,{\left (a^{5} b x^{5} + a^{6} x^{4}\right )}}, -\frac{105 \,{\left (b^{4} x^{5} + a b^{3} x^{4}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{b x^{3} + a x^{2}} \sqrt{-a}}{a x}\right ) +{\left (105 \, a b^{3} x^{3} + 35 \, a^{2} b^{2} x^{2} - 14 \, a^{3} b x + 8 \, a^{4}\right )} \sqrt{b x^{3} + a x^{2}}}{24 \,{\left (a^{5} b x^{5} + a^{6} x^{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{1}{x \left (x^{2} \left (a + b x\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{3} + a x^{2}\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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