Optimal. Leaf size=109 \[ \frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{8 a^{3/2}}-\frac{b^2 \sqrt{a x^2+b x^3}}{8 a x^2}-\frac{b \sqrt{a x^2+b x^3}}{4 x^3}-\frac{\left (a x^2+b x^3\right )^{3/2}}{3 x^6} \]
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Rubi [A] time = 0.134152, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2020, 2025, 2008, 206} \[ \frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{8 a^{3/2}}-\frac{b^2 \sqrt{a x^2+b x^3}}{8 a x^2}-\frac{b \sqrt{a x^2+b x^3}}{4 x^3}-\frac{\left (a x^2+b x^3\right )^{3/2}}{3 x^6} \]
Antiderivative was successfully verified.
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Rule 2020
Rule 2025
Rule 2008
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (a x^2+b x^3\right )^{3/2}}{x^7} \, dx &=-\frac{\left (a x^2+b x^3\right )^{3/2}}{3 x^6}+\frac{1}{2} b \int \frac{\sqrt{a x^2+b x^3}}{x^4} \, dx\\ &=-\frac{b \sqrt{a x^2+b x^3}}{4 x^3}-\frac{\left (a x^2+b x^3\right )^{3/2}}{3 x^6}+\frac{1}{8} b^2 \int \frac{1}{x \sqrt{a x^2+b x^3}} \, dx\\ &=-\frac{b \sqrt{a x^2+b x^3}}{4 x^3}-\frac{b^2 \sqrt{a x^2+b x^3}}{8 a x^2}-\frac{\left (a x^2+b x^3\right )^{3/2}}{3 x^6}-\frac{b^3 \int \frac{1}{\sqrt{a x^2+b x^3}} \, dx}{16 a}\\ &=-\frac{b \sqrt{a x^2+b x^3}}{4 x^3}-\frac{b^2 \sqrt{a x^2+b x^3}}{8 a x^2}-\frac{\left (a x^2+b x^3\right )^{3/2}}{3 x^6}+\frac{b^3 \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{x}{\sqrt{a x^2+b x^3}}\right )}{8 a}\\ &=-\frac{b \sqrt{a x^2+b x^3}}{4 x^3}-\frac{b^2 \sqrt{a x^2+b x^3}}{8 a x^2}-\frac{\left (a x^2+b x^3\right )^{3/2}}{3 x^6}+\frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{8 a^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0157798, size = 42, normalized size = 0.39 \[ \frac{2 b^3 \left (x^2 (a+b x)\right )^{5/2} \, _2F_1\left (\frac{5}{2},4;\frac{7}{2};\frac{b x}{a}+1\right )}{5 a^4 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 87, normalized size = 0.8 \begin{align*}{\frac{1}{24\,{x}^{6}} \left ( b{x}^{3}+a{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 3\,{a}^{7/2}\sqrt{bx+a}-8\,{a}^{5/2} \left ( bx+a \right ) ^{3/2}-3\,{a}^{3/2} \left ( bx+a \right ) ^{5/2}+3\,{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ){x}^{3}a{b}^{3} \right ){a}^{-{\frac{5}{2}}} \left ( bx+a \right ) ^{-{\frac{3}{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{{\left (b x^{3} + a x^{2}\right )}^{\frac{3}{2}}}{x^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.785699, size = 397, normalized size = 3.64 \begin{align*} \left [\frac{3 \, \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 (3 \, a b^{2} x^{2} + 14 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt{b x^{3} + a x^{2}}}{48 \, a^{2} x^{4}}, -\frac{3 \, \sqrt{-a} b^{3} x^{4} \arctan \left (\frac{\sqrt{b x^{3} + a x^{2}} \sqrt{-a}}{a x}\right ) +{\left (3 \, a b^{2} x^{2} + 14 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt{b x^{3} + a x^{2}}}{24 \, a^{2} 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{\left (x^{2} \left (a + b x\right )\right )^{\frac{3}{2}}}{x^{7}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21501, size = 124, normalized size = 1.14 \begin{align*} -\frac{\frac{3 \, b^{4} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) \mathrm{sgn}\left (x\right )}{\sqrt{-a} a} + \frac{3 \,{\left (b x + a\right )}^{\frac{5}{2}} b^{4} \mathrm{sgn}\left (x\right ) + 8 \,{\left (b x + a\right )}^{\frac{3}{2}} a b^{4} \mathrm{sgn}\left (x\right ) - 3 \, \sqrt{b x + a} a^{2} b^{4} \mathrm{sgn}\left (x\right )}{a b^{3} x^{3}}}{24 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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