Optimal. Leaf size=204 \[ -\frac{a^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{7/4}}+\frac{a^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{7/4}}+\frac{a^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} b^{7/4}}-\frac{a^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} b^{7/4}}+\frac{2 x^{3/2}}{3 b} \]
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Rubi [A] time = 0.152454, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533, Rules used = {321, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac{a^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{7/4}}+\frac{a^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{7/4}}+\frac{a^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} b^{7/4}}-\frac{a^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} b^{7/4}}+\frac{2 x^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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Rule 321
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{x^{5/2}}{a+b x^2} \, dx &=\frac{2 x^{3/2}}{3 b}-\frac{a \int \frac{\sqrt{x}}{a+b x^2} \, dx}{b}\\ &=\frac{2 x^{3/2}}{3 b}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{b}\\ &=\frac{2 x^{3/2}}{3 b}+\frac{a \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{b^{3/2}}-\frac{a \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{b^{3/2}}\\ &=\frac{2 x^{3/2}}{3 b}-\frac{a \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{x}\right )}{2 b^2}-\frac{a \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{x}\right )}{2 b^2}-\frac{a^{3/4} \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt{x}\right )}{2 \sqrt{2} b^{7/4}}-\frac{a^{3/4} \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt{x}\right )}{2 \sqrt{2} b^{7/4}}\\ &=\frac{2 x^{3/2}}{3 b}-\frac{a^{3/4} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} b^{7/4}}+\frac{a^{3/4} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} b^{7/4}}-\frac{a^{3/4} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} b^{7/4}}+\frac{a^{3/4} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} b^{7/4}}\\ &=\frac{2 x^{3/2}}{3 b}+\frac{a^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} b^{7/4}}-\frac{a^{3/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} b^{7/4}}-\frac{a^{3/4} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} b^{7/4}}+\frac{a^{3/4} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} b^{7/4}}\\ \end{align*}
Mathematica [A] time = 0.0253083, size = 78, normalized size = 0.38 \[ \frac{(-a)^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{-a}}\right )}{b^{7/4}}-\frac{(-a)^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{-a}}\right )}{b^{7/4}}+\frac{2 x^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 143, normalized size = 0.7 \begin{align*}{\frac{2}{3\,b}{x}^{{\frac{3}{2}}}}-{\frac{a\sqrt{2}}{4\,{b}^{2}}\ln \left ({ \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{a\sqrt{2}}{2\,{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{a\sqrt{2}}{2\,{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \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 = 1.41165, size = 377, normalized size = 1.85 \begin{align*} \frac{12 \, b \left (-\frac{a^{3}}{b^{7}}\right )^{\frac{1}{4}} \arctan \left (-\frac{a^{2} b^{2} \sqrt{x} \left (-\frac{a^{3}}{b^{7}}\right )^{\frac{1}{4}} - \sqrt{-a^{3} b^{3} \sqrt{-\frac{a^{3}}{b^{7}}} + a^{4} x} b^{2} \left (-\frac{a^{3}}{b^{7}}\right )^{\frac{1}{4}}}{a^{3}}\right ) - 3 \, b \left (-\frac{a^{3}}{b^{7}}\right )^{\frac{1}{4}} \log \left (b^{5} \left (-\frac{a^{3}}{b^{7}}\right )^{\frac{3}{4}} + a^{2} \sqrt{x}\right ) + 3 \, b \left (-\frac{a^{3}}{b^{7}}\right )^{\frac{1}{4}} \log \left (-b^{5} \left (-\frac{a^{3}}{b^{7}}\right )^{\frac{3}{4}} + a^{2} \sqrt{x}\right ) + 4 \, x^{\frac{3}{2}}}{6 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 25.5627, size = 180, normalized size = 0.88 \begin{align*} \begin{cases} \tilde{\infty } x^{\frac{3}{2}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{2 x^{\frac{7}{2}}}{7 a} & \text{for}\: b = 0 \\\frac{2 x^{\frac{3}{2}}}{3 b} & \text{for}\: a = 0 \\\frac{\left (-1\right )^{\frac{3}{4}} a^{\frac{3}{4}} \log{\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + \sqrt{x} \right )}}{2 b^{8} \left (\frac{1}{b}\right )^{\frac{25}{4}}} - \frac{\left (-1\right )^{\frac{3}{4}} a^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{3}{4}} \sqrt{x}}{\sqrt [4]{a} \sqrt [4]{\frac{1}{b}}} \right )}}{b^{8} \left (\frac{1}{b}\right )^{\frac{25}{4}}} - \frac{\left (-1\right )^{\frac{3}{4}} a^{\frac{3}{4}} \log{\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + \sqrt{x} \right )}}{2 b^{15} \left (\frac{1}{b}\right )^{\frac{53}{4}}} + \frac{2 x^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.24192, size = 240, normalized size = 1.18 \begin{align*} \frac{2 \, x^{\frac{3}{2}}}{3 \, b} - \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, b^{4}} - \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, b^{4}} + \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{3}{4}} \log \left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, b^{4}} - \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{3}{4}} \log \left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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