Optimal. Leaf size=97 \[ -\frac{35 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{9/2}}+\frac{7 x^{5/2}}{4 b^2 (a-b x)}+\frac{35 a \sqrt{x}}{4 b^4}-\frac{x^{7/2}}{2 b (a-b x)^2}+\frac{35 x^{3/2}}{12 b^3} \]
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Rubi [A] time = 0.0301179, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {47, 50, 63, 208} \[ -\frac{35 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{9/2}}+\frac{7 x^{5/2}}{4 b^2 (a-b x)}+\frac{35 a \sqrt{x}}{4 b^4}-\frac{x^{7/2}}{2 b (a-b x)^2}+\frac{35 x^{3/2}}{12 b^3} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x^{7/2}}{(-a+b x)^3} \, dx &=-\frac{x^{7/2}}{2 b (a-b x)^2}+\frac{7 \int \frac{x^{5/2}}{(-a+b x)^2} \, dx}{4 b}\\ &=-\frac{x^{7/2}}{2 b (a-b x)^2}+\frac{7 x^{5/2}}{4 b^2 (a-b x)}+\frac{35 \int \frac{x^{3/2}}{-a+b x} \, dx}{8 b^2}\\ &=\frac{35 x^{3/2}}{12 b^3}-\frac{x^{7/2}}{2 b (a-b x)^2}+\frac{7 x^{5/2}}{4 b^2 (a-b x)}+\frac{(35 a) \int \frac{\sqrt{x}}{-a+b x} \, dx}{8 b^3}\\ &=\frac{35 a \sqrt{x}}{4 b^4}+\frac{35 x^{3/2}}{12 b^3}-\frac{x^{7/2}}{2 b (a-b x)^2}+\frac{7 x^{5/2}}{4 b^2 (a-b x)}+\frac{\left (35 a^2\right ) \int \frac{1}{\sqrt{x} (-a+b x)} \, dx}{8 b^4}\\ &=\frac{35 a \sqrt{x}}{4 b^4}+\frac{35 x^{3/2}}{12 b^3}-\frac{x^{7/2}}{2 b (a-b x)^2}+\frac{7 x^{5/2}}{4 b^2 (a-b x)}+\frac{\left (35 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{-a+b x^2} \, dx,x,\sqrt{x}\right )}{4 b^4}\\ &=\frac{35 a \sqrt{x}}{4 b^4}+\frac{35 x^{3/2}}{12 b^3}-\frac{x^{7/2}}{2 b (a-b x)^2}+\frac{7 x^{5/2}}{4 b^2 (a-b x)}-\frac{35 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.0052807, size = 26, normalized size = 0.27 \[ -\frac{2 x^{9/2} \, _2F_1\left (3,\frac{9}{2};\frac{11}{2};\frac{b x}{a}\right )}{9 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 70, normalized size = 0.7 \begin{align*} 2\,{\frac{1/3\,b{x}^{3/2}+3\,a\sqrt{x}}{{b}^{4}}}+2\,{\frac{{a}^{2}}{{b}^{4}} \left ({\frac{1}{ \left ( bx-a \right ) ^{2}} \left ( -{\frac{13\,b{x}^{3/2}}{8}}+{\frac{11\,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 = 1.5717, size = 509, normalized size = 5.25 \begin{align*} \left [\frac{105 \,{\left (a b^{2} x^{2} - 2 \, a^{2} b x + a^{3}\right )} \sqrt{\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{\frac{a}{b}} + a}{b x - a}\right ) + 2 \,{\left (8 \, b^{3} x^{3} + 56 \, a b^{2} x^{2} - 175 \, a^{2} b x + 105 \, a^{3}\right )} \sqrt{x}}{24 \,{\left (b^{6} x^{2} - 2 \, a b^{5} x + a^{2} b^{4}\right )}}, \frac{105 \,{\left (a b^{2} x^{2} - 2 \, a^{2} b x + a^{3}\right )} \sqrt{-\frac{a}{b}} \arctan \left (\frac{b \sqrt{x} \sqrt{-\frac{a}{b}}}{a}\right ) +{\left (8 \, b^{3} x^{3} + 56 \, a b^{2} x^{2} - 175 \, a^{2} b x + 105 \, a^{3}\right )} \sqrt{x}}{12 \,{\left (b^{6} x^{2} - 2 \, a b^{5} x + a^{2} b^{4}\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 = 1.25906, size = 109, normalized size = 1.12 \begin{align*} \frac{35 \, a^{2} \arctan \left (\frac{b \sqrt{x}}{\sqrt{-a b}}\right )}{4 \, \sqrt{-a b} b^{4}} - \frac{13 \, a^{2} b x^{\frac{3}{2}} - 11 \, a^{3} \sqrt{x}}{4 \,{\left (b x - a\right )}^{2} b^{4}} + \frac{2 \,{\left (b^{6} x^{\frac{3}{2}} + 9 \, a b^{5} \sqrt{x}\right )}}{3 \, b^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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