Optimal. Leaf size=82 \[ \frac{5}{4 a^2 \sqrt{x} (a+b x)}-\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{7/2}}-\frac{15}{4 a^3 \sqrt{x}}+\frac{1}{2 a \sqrt{x} (a+b x)^2} \]
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Rubi [A] time = 0.0251132, antiderivative size = 82, 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} \[ \frac{5}{4 a^2 \sqrt{x} (a+b x)}-\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{7/2}}-\frac{15}{4 a^3 \sqrt{x}}+\frac{1}{2 a \sqrt{x} (a+b x)^2} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{x^{3/2} (a+b x)^3} \, dx &=\frac{1}{2 a \sqrt{x} (a+b x)^2}+\frac{5 \int \frac{1}{x^{3/2} (a+b x)^2} \, dx}{4 a}\\ &=\frac{1}{2 a \sqrt{x} (a+b x)^2}+\frac{5}{4 a^2 \sqrt{x} (a+b x)}+\frac{15 \int \frac{1}{x^{3/2} (a+b x)} \, dx}{8 a^2}\\ &=-\frac{15}{4 a^3 \sqrt{x}}+\frac{1}{2 a \sqrt{x} (a+b x)^2}+\frac{5}{4 a^2 \sqrt{x} (a+b x)}-\frac{(15 b) \int \frac{1}{\sqrt{x} (a+b x)} \, dx}{8 a^3}\\ &=-\frac{15}{4 a^3 \sqrt{x}}+\frac{1}{2 a \sqrt{x} (a+b x)^2}+\frac{5}{4 a^2 \sqrt{x} (a+b x)}-\frac{(15 b) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{4 a^3}\\ &=-\frac{15}{4 a^3 \sqrt{x}}+\frac{1}{2 a \sqrt{x} (a+b x)^2}+\frac{5}{4 a^2 \sqrt{x} (a+b x)}-\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0049071, size = 25, normalized size = 0.3 \[ -\frac{2 \, _2F_1\left (-\frac{1}{2},3;\frac{1}{2};-\frac{b x}{a}\right )}{a^3 \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 66, normalized size = 0.8 \begin{align*} -2\,{\frac{1}{{a}^{3}\sqrt{x}}}-{\frac{7\,{b}^{2}}{4\,{a}^{3} \left ( bx+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{9\,b}{4\,{a}^{2} \left ( bx+a \right ) ^{2}}\sqrt{x}}-{\frac{15\,b}{4\,{a}^{3}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \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 = 1.44803, size = 466, normalized size = 5.68 \begin{align*} \left [\frac{15 \,{\left (b^{2} x^{3} + 2 \, a b x^{2} + a^{2} x\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} + 25 \, a b x + 8 \, a^{2}\right )} \sqrt{x}}{8 \,{\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}}, \frac{15 \,{\left (b^{2} x^{3} + 2 \, a b x^{2} + a^{2} x\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) -{\left (15 \, b^{2} x^{2} + 25 \, a b x + 8 \, a^{2}\right )} \sqrt{x}}{4 \,{\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 155.787, size = 865, normalized size = 10.55 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22685, size = 80, normalized size = 0.98 \begin{align*} -\frac{15 \, b \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} a^{3}} - \frac{2}{a^{3} \sqrt{x}} - \frac{7 \, b^{2} x^{\frac{3}{2}} + 9 \, a b \sqrt{x}}{4 \,{\left (b x + a\right )}^{2} a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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