Optimal. Leaf size=87 \[ \frac{15 b^2}{4 a^3 \sqrt{a+b x}}-\frac{15 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{7/2}}+\frac{5 b}{4 a^2 x \sqrt{a+b x}}-\frac{1}{2 a x^2 \sqrt{a+b x}} \]
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Rubi [A] time = 0.0231892, antiderivative size = 85, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {51, 63, 208} \[ -\frac{15 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{7/2}}-\frac{5 \sqrt{a+b x}}{2 a^2 x^2}+\frac{15 b \sqrt{a+b x}}{4 a^3 x}+\frac{2}{a x^2 \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x^3 (a+b x)^{3/2}} \, dx &=\frac{2}{a x^2 \sqrt{a+b x}}+\frac{5 \int \frac{1}{x^3 \sqrt{a+b x}} \, dx}{a}\\ &=\frac{2}{a x^2 \sqrt{a+b x}}-\frac{5 \sqrt{a+b x}}{2 a^2 x^2}-\frac{(15 b) \int \frac{1}{x^2 \sqrt{a+b x}} \, dx}{4 a^2}\\ &=\frac{2}{a x^2 \sqrt{a+b x}}-\frac{5 \sqrt{a+b x}}{2 a^2 x^2}+\frac{15 b \sqrt{a+b x}}{4 a^3 x}+\frac{\left (15 b^2\right ) \int \frac{1}{x \sqrt{a+b x}} \, dx}{8 a^3}\\ &=\frac{2}{a x^2 \sqrt{a+b x}}-\frac{5 \sqrt{a+b x}}{2 a^2 x^2}+\frac{15 b \sqrt{a+b x}}{4 a^3 x}+\frac{(15 b) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{4 a^3}\\ &=\frac{2}{a x^2 \sqrt{a+b x}}-\frac{5 \sqrt{a+b x}}{2 a^2 x^2}+\frac{15 b \sqrt{a+b x}}{4 a^3 x}-\frac{15 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0093782, size = 33, normalized size = 0.38 \[ \frac{2 b^2 \, _2F_1\left (-\frac{1}{2},3;\frac{1}{2};\frac{b x}{a}+1\right )}{a^3 \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 67, normalized size = 0.8 \begin{align*} 2\,{b}^{2} \left ({\frac{1}{{a}^{3}} \left ({\frac{1}{{b}^{2}{x}^{2}} \left ({\frac{7\, \left ( bx+a \right ) ^{3/2}}{8}}-{\frac{9\,a\sqrt{bx+a}}{8}} \right ) }-{\frac{15}{8\,\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) }+{\frac{1}{\sqrt{bx+a}{a}^{3}}} \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 = 1.57, size = 420, normalized size = 4.83 \begin{align*} \left [\frac{15 \,{\left (b^{3} x^{3} + a b^{2} x^{2}\right )} \sqrt{a} \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (15 \, a b^{2} x^{2} + 5 \, a^{2} b x - 2 \, a^{3}\right )} \sqrt{b x + a}}{8 \,{\left (a^{4} b x^{3} + a^{5} x^{2}\right )}}, \frac{15 \,{\left (b^{3} x^{3} + a b^{2} x^{2}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) +{\left (15 \, a b^{2} x^{2} + 5 \, a^{2} b x - 2 \, a^{3}\right )} \sqrt{b x + a}}{4 \,{\left (a^{4} b x^{3} + a^{5} x^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.50486, size = 107, normalized size = 1.23 \begin{align*} - \frac{1}{2 a \sqrt{b} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{5 \sqrt{b}}{4 a^{2} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{15 b^{\frac{3}{2}}}{4 a^{3} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} - \frac{15 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4 a^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17616, size = 108, normalized size = 1.24 \begin{align*} \frac{15 \, b^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{4 \, \sqrt{-a} a^{3}} + \frac{2 \, b^{2}}{\sqrt{b x + a} a^{3}} + \frac{7 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{2} - 9 \, \sqrt{b x + a} a b^{2}}{4 \, a^{3} b^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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