Optimal. Leaf size=69 \[ \frac{5 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{5 b}{a^3 \sqrt{x}}-\frac{5}{3 a^2 x^{3/2}}+\frac{1}{a x^{3/2} (a+b x)} \]
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Rubi [A] time = 0.0218529, antiderivative size = 69, normalized size of antiderivative = 1., 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, 205} \[ \frac{5 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{5 b}{a^3 \sqrt{x}}-\frac{5}{3 a^2 x^{3/2}}+\frac{1}{a x^{3/2} (a+b x)} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{x^{5/2} (a+b x)^2} \, dx &=\frac{1}{a x^{3/2} (a+b x)}+\frac{5 \int \frac{1}{x^{5/2} (a+b x)} \, dx}{2 a}\\ &=-\frac{5}{3 a^2 x^{3/2}}+\frac{1}{a x^{3/2} (a+b x)}-\frac{(5 b) \int \frac{1}{x^{3/2} (a+b x)} \, dx}{2 a^2}\\ &=-\frac{5}{3 a^2 x^{3/2}}+\frac{5 b}{a^3 \sqrt{x}}+\frac{1}{a x^{3/2} (a+b x)}+\frac{\left (5 b^2\right ) \int \frac{1}{\sqrt{x} (a+b x)} \, dx}{2 a^3}\\ &=-\frac{5}{3 a^2 x^{3/2}}+\frac{5 b}{a^3 \sqrt{x}}+\frac{1}{a x^{3/2} (a+b x)}+\frac{\left (5 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{a^3}\\ &=-\frac{5}{3 a^2 x^{3/2}}+\frac{5 b}{a^3 \sqrt{x}}+\frac{1}{a x^{3/2} (a+b x)}+\frac{5 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0049991, size = 27, normalized size = 0.39 \[ -\frac{2 \, _2F_1\left (-\frac{3}{2},2;-\frac{1}{2};-\frac{b x}{a}\right )}{3 a^2 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 60, normalized size = 0.9 \begin{align*} -{\frac{2}{3\,{a}^{2}}{x}^{-{\frac{3}{2}}}}+4\,{\frac{b}{{a}^{3}\sqrt{x}}}+{\frac{{b}^{2}}{{a}^{3} \left ( bx+a \right ) }\sqrt{x}}+5\,{\frac{{b}^{2}}{{a}^{3}\sqrt{ab}}\arctan \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 = 1.33587, size = 402, normalized size = 5.83 \begin{align*} \left [\frac{15 \,{\left (b^{2} x^{3} + a 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 (15 \, b^{2} x^{2} + 10 \, a b x - 2 \, a^{2}\right )} \sqrt{x}}{6 \,{\left (a^{3} b x^{3} + a^{4} x^{2}\right )}}, -\frac{15 \,{\left (b^{2} x^{3} + a b x^{2}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) -{\left (15 \, b^{2} x^{2} + 10 \, a b x - 2 \, a^{2}\right )} \sqrt{x}}{3 \,{\left (a^{3} b x^{3} + a^{4} x^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 116.113, size = 507, normalized size = 7.35 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{7}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{7 b^{2} x^{\frac{7}{2}}} & \text{for}\: a = 0 \\- \frac{2}{3 a^{2} x^{\frac{3}{2}}} & \text{for}\: b = 0 \\- \frac{4 i a^{\frac{5}{2}} \sqrt{\frac{1}{b}}}{6 i a^{\frac{9}{2}} x^{\frac{3}{2}} \sqrt{\frac{1}{b}} + 6 i a^{\frac{7}{2}} b x^{\frac{5}{2}} \sqrt{\frac{1}{b}}} + \frac{20 i a^{\frac{3}{2}} b x \sqrt{\frac{1}{b}}}{6 i a^{\frac{9}{2}} x^{\frac{3}{2}} \sqrt{\frac{1}{b}} + 6 i a^{\frac{7}{2}} b x^{\frac{5}{2}} \sqrt{\frac{1}{b}}} + \frac{30 i \sqrt{a} b^{2} x^{2} \sqrt{\frac{1}{b}}}{6 i a^{\frac{9}{2}} x^{\frac{3}{2}} \sqrt{\frac{1}{b}} + 6 i a^{\frac{7}{2}} b x^{\frac{5}{2}} \sqrt{\frac{1}{b}}} + \frac{15 a b x^{\frac{3}{2}} \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{6 i a^{\frac{9}{2}} x^{\frac{3}{2}} \sqrt{\frac{1}{b}} + 6 i a^{\frac{7}{2}} b x^{\frac{5}{2}} \sqrt{\frac{1}{b}}} - \frac{15 a b x^{\frac{3}{2}} \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{6 i a^{\frac{9}{2}} x^{\frac{3}{2}} \sqrt{\frac{1}{b}} + 6 i a^{\frac{7}{2}} b x^{\frac{5}{2}} \sqrt{\frac{1}{b}}} + \frac{15 b^{2} x^{\frac{5}{2}} \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{6 i a^{\frac{9}{2}} x^{\frac{3}{2}} \sqrt{\frac{1}{b}} + 6 i a^{\frac{7}{2}} b x^{\frac{5}{2}} \sqrt{\frac{1}{b}}} - \frac{15 b^{2} x^{\frac{5}{2}} \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{6 i a^{\frac{9}{2}} x^{\frac{3}{2}} \sqrt{\frac{1}{b}} + 6 i a^{\frac{7}{2}} b x^{\frac{5}{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 = 1.23766, size = 78, normalized size = 1.13 \begin{align*} \frac{5 \, b^{2} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{3}} + \frac{b^{2} \sqrt{x}}{{\left (b x + a\right )} a^{3}} + \frac{2 \,{\left (6 \, b x - a\right )}}{3 \, a^{3} x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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