Optimal. Leaf size=97 \[ -\frac{35 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{9/2}}-\frac{7}{4 a^2 x^{3/2} (a-b x)}+\frac{35 b}{4 a^4 \sqrt{x}}+\frac{35}{12 a^3 x^{3/2}}-\frac{1}{2 a x^{3/2} (a-b x)^2} \]
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Rubi [A] time = 0.0303405, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {51, 63, 208} \[ -\frac{35 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{9/2}}-\frac{7}{4 a^2 x^{3/2} (a-b x)}+\frac{35 b}{4 a^4 \sqrt{x}}+\frac{35}{12 a^3 x^{3/2}}-\frac{1}{2 a x^{3/2} (a-b x)^2} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x^{5/2} (-a+b x)^3} \, dx &=-\frac{1}{2 a x^{3/2} (a-b x)^2}-\frac{7 \int \frac{1}{x^{5/2} (-a+b x)^2} \, dx}{4 a}\\ &=-\frac{1}{2 a x^{3/2} (a-b x)^2}-\frac{7}{4 a^2 x^{3/2} (a-b x)}+\frac{35 \int \frac{1}{x^{5/2} (-a+b x)} \, dx}{8 a^2}\\ &=\frac{35}{12 a^3 x^{3/2}}-\frac{1}{2 a x^{3/2} (a-b x)^2}-\frac{7}{4 a^2 x^{3/2} (a-b x)}+\frac{(35 b) \int \frac{1}{x^{3/2} (-a+b x)} \, dx}{8 a^3}\\ &=\frac{35}{12 a^3 x^{3/2}}+\frac{35 b}{4 a^4 \sqrt{x}}-\frac{1}{2 a x^{3/2} (a-b x)^2}-\frac{7}{4 a^2 x^{3/2} (a-b x)}+\frac{\left (35 b^2\right ) \int \frac{1}{\sqrt{x} (-a+b x)} \, dx}{8 a^4}\\ &=\frac{35}{12 a^3 x^{3/2}}+\frac{35 b}{4 a^4 \sqrt{x}}-\frac{1}{2 a x^{3/2} (a-b x)^2}-\frac{7}{4 a^2 x^{3/2} (a-b x)}+\frac{\left (35 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-a+b x^2} \, dx,x,\sqrt{x}\right )}{4 a^4}\\ &=\frac{35}{12 a^3 x^{3/2}}+\frac{35 b}{4 a^4 \sqrt{x}}-\frac{1}{2 a x^{3/2} (a-b x)^2}-\frac{7}{4 a^2 x^{3/2} (a-b x)}-\frac{35 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.005165, size = 26, normalized size = 0.27 \[ \frac{2 \, _2F_1\left (-\frac{3}{2},3;-\frac{1}{2};\frac{b x}{a}\right )}{3 a^3 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 69, normalized size = 0.7 \begin{align*}{\frac{2}{3\,{a}^{3}}{x}^{-{\frac{3}{2}}}}+6\,{\frac{b}{{a}^{4}\sqrt{x}}}+2\,{\frac{{b}^{2}}{{a}^{4}} \left ({\frac{1}{ \left ( bx-a \right ) ^{2}} \left ({\frac{11\,b{x}^{3/2}}{8}}-{\frac{13\,a\sqrt{x}}{8}} \right ) }-{\frac{35}{8\,\sqrt{ab}}{\it Artanh} \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \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 = 1.56587, size = 544, normalized size = 5.61 \begin{align*} \left [\frac{105 \,{\left (b^{3} x^{4} - 2 \, a b^{2} x^{3} + a^{2} b x^{2}\right )} \sqrt{\frac{b}{a}} \log \left (\frac{b x - 2 \, a \sqrt{x} \sqrt{\frac{b}{a}} + a}{b x - a}\right ) + 2 \,{\left (105 \, b^{3} x^{3} - 175 \, a b^{2} x^{2} + 56 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt{x}}{24 \,{\left (a^{4} b^{2} x^{4} - 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}, \frac{105 \,{\left (b^{3} x^{4} - 2 \, a b^{2} x^{3} + a^{2} b x^{2}\right )} \sqrt{-\frac{b}{a}} \arctan \left (\frac{a \sqrt{-\frac{b}{a}}}{b \sqrt{x}}\right ) +{\left (105 \, b^{3} x^{3} - 175 \, a b^{2} x^{2} + 56 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt{x}}{12 \,{\left (a^{4} b^{2} x^{4} - 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25547, size = 99, normalized size = 1.02 \begin{align*} \frac{35 \, b^{2} \arctan \left (\frac{b \sqrt{x}}{\sqrt{-a b}}\right )}{4 \, \sqrt{-a b} a^{4}} + \frac{2 \,{\left (9 \, b x + a\right )}}{3 \, a^{4} x^{\frac{3}{2}}} + \frac{11 \, b^{3} x^{\frac{3}{2}} - 13 \, a b^{2} \sqrt{x}}{4 \,{\left (b x - a\right )}^{2} a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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