Optimal. Leaf size=68 \[ \frac{2 b^2}{a^3 \sqrt{x}}-\frac{2 b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{2 b}{3 a^2 x^{3/2}}+\frac{2}{5 a x^{5/2}} \]
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Rubi [A] time = 0.0224059, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {51, 63, 208} \[ \frac{2 b^2}{a^3 \sqrt{x}}-\frac{2 b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{2 b}{3 a^2 x^{3/2}}+\frac{2}{5 a x^{5/2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x^{7/2} (-a+b x)} \, dx &=\frac{2}{5 a x^{5/2}}+\frac{b \int \frac{1}{x^{5/2} (-a+b x)} \, dx}{a}\\ &=\frac{2}{5 a x^{5/2}}+\frac{2 b}{3 a^2 x^{3/2}}+\frac{b^2 \int \frac{1}{x^{3/2} (-a+b x)} \, dx}{a^2}\\ &=\frac{2}{5 a x^{5/2}}+\frac{2 b}{3 a^2 x^{3/2}}+\frac{2 b^2}{a^3 \sqrt{x}}+\frac{b^3 \int \frac{1}{\sqrt{x} (-a+b x)} \, dx}{a^3}\\ &=\frac{2}{5 a x^{5/2}}+\frac{2 b}{3 a^2 x^{3/2}}+\frac{2 b^2}{a^3 \sqrt{x}}+\frac{\left (2 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{-a+b x^2} \, dx,x,\sqrt{x}\right )}{a^3}\\ &=\frac{2}{5 a x^{5/2}}+\frac{2 b}{3 a^2 x^{3/2}}+\frac{2 b^2}{a^3 \sqrt{x}}-\frac{2 b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.005193, size = 26, normalized size = 0.38 \[ \frac{2 \, _2F_1\left (-\frac{5}{2},1;-\frac{3}{2};\frac{b x}{a}\right )}{5 a x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 54, normalized size = 0.8 \begin{align*}{\frac{2}{5\,a}{x}^{-{\frac{5}{2}}}}+{\frac{2\,b}{3\,{a}^{2}}{x}^{-{\frac{3}{2}}}}+2\,{\frac{{b}^{2}}{{a}^{3}\sqrt{x}}}-2\,{\frac{{b}^{3}}{{a}^{3}\sqrt{ab}}{\it Artanh} \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \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 = 1.54962, size = 336, normalized size = 4.94 \begin{align*} \left [\frac{15 \, b^{2} x^{3} \sqrt{\frac{b}{a}} \log \left (\frac{b x - 2 \, a \sqrt{x} \sqrt{\frac{b}{a}} + a}{b x - a}\right ) + 2 \,{\left (15 \, b^{2} x^{2} + 5 \, a b x + 3 \, a^{2}\right )} \sqrt{x}}{15 \, a^{3} x^{3}}, \frac{2 \,{\left (15 \, b^{2} x^{3} \sqrt{-\frac{b}{a}} \arctan \left (\frac{a \sqrt{-\frac{b}{a}}}{b \sqrt{x}}\right ) +{\left (15 \, b^{2} x^{2} + 5 \, a b x + 3 \, a^{2}\right )} \sqrt{x}\right )}}{15 \, a^{3} x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 76.1262, size = 131, normalized size = 1.93 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{7}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{7 b x^{\frac{7}{2}}} & \text{for}\: a = 0 \\\frac{2}{5 a x^{\frac{5}{2}}} & \text{for}\: b = 0 \\\frac{2}{5 a x^{\frac{5}{2}}} + \frac{2 b}{3 a^{2} x^{\frac{3}{2}}} + \frac{2 b^{2}}{a^{3} \sqrt{x}} + \frac{b^{2} \log{\left (- \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{a^{\frac{7}{2}} \sqrt{\frac{1}{b}}} - \frac{b^{2} \log{\left (\sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{a^{\frac{7}{2}} \sqrt{\frac{1}{b}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25937, size = 73, normalized size = 1.07 \begin{align*} \frac{2 \, b^{3} \arctan \left (\frac{b \sqrt{x}}{\sqrt{-a b}}\right )}{\sqrt{-a b} a^{3}} + \frac{2 \,{\left (15 \, b^{2} x^{2} + 5 \, a b x + 3 \, a^{2}\right )}}{15 \, a^{3} x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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