Optimal. Leaf size=255 \[ \frac{2 \sqrt{x} (A b-a B)}{b^2}+\frac{\sqrt [4]{a} (A b-a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{9/4}}-\frac{\sqrt [4]{a} (A b-a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{9/4}}+\frac{\sqrt [4]{a} (A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} b^{9/4}}-\frac{\sqrt [4]{a} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} b^{9/4}}+\frac{2 B x^{5/2}}{5 b} \]
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Rubi [A] time = 0.202483, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {459, 321, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{2 \sqrt{x} (A b-a B)}{b^2}+\frac{\sqrt [4]{a} (A b-a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{9/4}}-\frac{\sqrt [4]{a} (A b-a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{9/4}}+\frac{\sqrt [4]{a} (A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} b^{9/4}}-\frac{\sqrt [4]{a} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} b^{9/4}}+\frac{2 B x^{5/2}}{5 b} \]
Antiderivative was successfully verified.
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Rule 459
Rule 321
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{x^{3/2} \left (A+B x^2\right )}{a+b x^2} \, dx &=\frac{2 B x^{5/2}}{5 b}-\frac{\left (2 \left (-\frac{5 A b}{2}+\frac{5 a B}{2}\right )\right ) \int \frac{x^{3/2}}{a+b x^2} \, dx}{5 b}\\ &=\frac{2 (A b-a B) \sqrt{x}}{b^2}+\frac{2 B x^{5/2}}{5 b}-\frac{(a (A b-a B)) \int \frac{1}{\sqrt{x} \left (a+b x^2\right )} \, dx}{b^2}\\ &=\frac{2 (A b-a B) \sqrt{x}}{b^2}+\frac{2 B x^{5/2}}{5 b}-\frac{(2 a (A b-a B)) \operatorname{Subst}\left (\int \frac{1}{a+b x^4} \, dx,x,\sqrt{x}\right )}{b^2}\\ &=\frac{2 (A b-a B) \sqrt{x}}{b^2}+\frac{2 B x^{5/2}}{5 b}-\frac{\left (\sqrt{a} (A b-a B)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{b^2}-\frac{\left (\sqrt{a} (A b-a B)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{b^2}\\ &=\frac{2 (A b-a B) \sqrt{x}}{b^2}+\frac{2 B x^{5/2}}{5 b}-\frac{\left (\sqrt{a} (A b-a B)\right ) \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^{5/2}}-\frac{\left (\sqrt{a} (A b-a B)\right ) \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^{5/2}}+\frac{\left (\sqrt [4]{a} (A b-a B)\right ) \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^{9/4}}+\frac{\left (\sqrt [4]{a} (A b-a B)\right ) \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^{9/4}}\\ &=\frac{2 (A b-a B) \sqrt{x}}{b^2}+\frac{2 B x^{5/2}}{5 b}+\frac{\sqrt [4]{a} (A b-a B) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} b^{9/4}}-\frac{\sqrt [4]{a} (A b-a B) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} b^{9/4}}-\frac{\left (\sqrt [4]{a} (A b-a B)\right ) \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^{9/4}}+\frac{\left (\sqrt [4]{a} (A b-a B)\right ) \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^{9/4}}\\ &=\frac{2 (A b-a B) \sqrt{x}}{b^2}+\frac{2 B x^{5/2}}{5 b}+\frac{\sqrt [4]{a} (A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} b^{9/4}}-\frac{\sqrt [4]{a} (A b-a B) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} b^{9/4}}+\frac{\sqrt [4]{a} (A b-a B) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} b^{9/4}}-\frac{\sqrt [4]{a} (A b-a B) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} b^{9/4}}\\ \end{align*}
Mathematica [A] time = 0.205342, size = 208, normalized size = 0.82 \[ \frac{40 \sqrt{x} (A b-a B)+\frac{5 \sqrt{2} \sqrt [4]{a} (A b-a B) \left (\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )\right )}{\sqrt [4]{b}}+\frac{10 \sqrt{2} \sqrt [4]{a} (A b-a B) \left (\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )-\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )\right )}{\sqrt [4]{b}}+8 b B x^{5/2}}{20 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 299, normalized size = 1.2 \begin{align*}{\frac{2\,B}{5\,b}{x}^{{\frac{5}{2}}}}+2\,{\frac{A\sqrt{x}}{b}}-2\,{\frac{Ba\sqrt{x}}{{b}^{2}}}-{\frac{\sqrt{2}A}{2\,b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }-{\frac{\sqrt{2}A}{2\,b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }-{\frac{\sqrt{2}A}{4\,b}\sqrt [4]{{\frac{a}{b}}}\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{\sqrt{2}Ba}{2\,{b}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{\sqrt{2}Ba}{2\,{b}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{\sqrt{2}Ba}{4\,{b}^{2}}\sqrt [4]{{\frac{a}{b}}}\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 ) } \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 [B] time = 1.0013, size = 1365, normalized size = 5.35 \begin{align*} -\frac{20 \, b^{2} \left (-\frac{B^{4} a^{5} - 4 \, A B^{3} a^{4} b + 6 \, A^{2} B^{2} a^{3} b^{2} - 4 \, A^{3} B a^{2} b^{3} + A^{4} a b^{4}}{b^{9}}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{b^{4} \sqrt{-\frac{B^{4} a^{5} - 4 \, A B^{3} a^{4} b + 6 \, A^{2} B^{2} a^{3} b^{2} - 4 \, A^{3} B a^{2} b^{3} + A^{4} a b^{4}}{b^{9}}} +{\left (B^{2} a^{2} - 2 \, A B a b + A^{2} b^{2}\right )} x} b^{7} \left (-\frac{B^{4} a^{5} - 4 \, A B^{3} a^{4} b + 6 \, A^{2} B^{2} a^{3} b^{2} - 4 \, A^{3} B a^{2} b^{3} + A^{4} a b^{4}}{b^{9}}\right )^{\frac{3}{4}} +{\left (B a b^{7} - A b^{8}\right )} \sqrt{x} \left (-\frac{B^{4} a^{5} - 4 \, A B^{3} a^{4} b + 6 \, A^{2} B^{2} a^{3} b^{2} - 4 \, A^{3} B a^{2} b^{3} + A^{4} a b^{4}}{b^{9}}\right )^{\frac{3}{4}}}{B^{4} a^{5} - 4 \, A B^{3} a^{4} b + 6 \, A^{2} B^{2} a^{3} b^{2} - 4 \, A^{3} B a^{2} b^{3} + A^{4} a b^{4}}\right ) + 5 \, b^{2} \left (-\frac{B^{4} a^{5} - 4 \, A B^{3} a^{4} b + 6 \, A^{2} B^{2} a^{3} b^{2} - 4 \, A^{3} B a^{2} b^{3} + A^{4} a b^{4}}{b^{9}}\right )^{\frac{1}{4}} \log \left (b^{2} \left (-\frac{B^{4} a^{5} - 4 \, A B^{3} a^{4} b + 6 \, A^{2} B^{2} a^{3} b^{2} - 4 \, A^{3} B a^{2} b^{3} + A^{4} a b^{4}}{b^{9}}\right )^{\frac{1}{4}} -{\left (B a - A b\right )} \sqrt{x}\right ) - 5 \, b^{2} \left (-\frac{B^{4} a^{5} - 4 \, A B^{3} a^{4} b + 6 \, A^{2} B^{2} a^{3} b^{2} - 4 \, A^{3} B a^{2} b^{3} + A^{4} a b^{4}}{b^{9}}\right )^{\frac{1}{4}} \log \left (-b^{2} \left (-\frac{B^{4} a^{5} - 4 \, A B^{3} a^{4} b + 6 \, A^{2} B^{2} a^{3} b^{2} - 4 \, A^{3} B a^{2} b^{3} + A^{4} a b^{4}}{b^{9}}\right )^{\frac{1}{4}} -{\left (B a - A b\right )} \sqrt{x}\right ) - 4 \,{\left (B b x^{2} - 5 \, B a + 5 \, A b\right )} \sqrt{x}}{10 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 34.8597, size = 382, normalized size = 1.5 \begin{align*} \begin{cases} \tilde{\infty } \left (2 A \sqrt{x} + \frac{2 B x^{\frac{5}{2}}}{5}\right ) & \text{for}\: a = 0 \wedge b = 0 \\\frac{\frac{2 A x^{\frac{5}{2}}}{5} + \frac{2 B x^{\frac{9}{2}}}{9}}{a} & \text{for}\: b = 0 \\\frac{2 A \sqrt{x} + \frac{2 B x^{\frac{5}{2}}}{5}}{b} & \text{for}\: a = 0 \\\frac{\sqrt [4]{-1} A \sqrt [4]{a} b \left (\frac{1}{b}\right )^{\frac{9}{4}} \log{\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + \sqrt{x} \right )}}{2} - \frac{\sqrt [4]{-1} A \sqrt [4]{a} b \left (\frac{1}{b}\right )^{\frac{9}{4}} \log{\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + \sqrt{x} \right )}}{2} + \sqrt [4]{-1} A \sqrt [4]{a} b \left (\frac{1}{b}\right )^{\frac{9}{4}} \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{3}{4}} \sqrt{x}}{\sqrt [4]{a} \sqrt [4]{\frac{1}{b}}} \right )} + \frac{2 A \sqrt{x}}{b} - \frac{\sqrt [4]{-1} B a^{\frac{5}{4}} \left (\frac{1}{b}\right )^{\frac{9}{4}} \log{\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + \sqrt{x} \right )}}{2} + \frac{\sqrt [4]{-1} B a^{\frac{5}{4}} \left (\frac{1}{b}\right )^{\frac{9}{4}} \log{\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + \sqrt{x} \right )}}{2} - \sqrt [4]{-1} B a^{\frac{5}{4}} \left (\frac{1}{b}\right )^{\frac{9}{4}} \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{3}{4}} \sqrt{x}}{\sqrt [4]{a} \sqrt [4]{\frac{1}{b}}} \right )} - \frac{2 B a \sqrt{x}}{b^{2}} + \frac{2 B x^{\frac{5}{2}}}{5 b} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17794, size = 355, normalized size = 1.39 \begin{align*} \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a - \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \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^{3}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a - \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \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^{3}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a - \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \log \left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, b^{3}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a - \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \log \left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, b^{3}} + \frac{2 \,{\left (B b^{4} x^{\frac{5}{2}} - 5 \, B a b^{3} \sqrt{x} + 5 \, A b^{4} \sqrt{x}\right )}}{5 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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