Optimal. Leaf size=289 \[ -\frac{7 A b-3 a B}{6 a^2 b x^{3/2}}+\frac{(7 A b-3 a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4} \sqrt [4]{b}}-\frac{(7 A b-3 a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4} \sqrt [4]{b}}+\frac{(7 A b-3 a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{11/4} \sqrt [4]{b}}-\frac{(7 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{11/4} \sqrt [4]{b}}+\frac{A b-a B}{2 a b x^{3/2} \left (a+b x^2\right )} \]
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Rubi [A] time = 0.213974, antiderivative size = 289, 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 = {457, 325, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{7 A b-3 a B}{6 a^2 b x^{3/2}}+\frac{(7 A b-3 a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4} \sqrt [4]{b}}-\frac{(7 A b-3 a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4} \sqrt [4]{b}}+\frac{(7 A b-3 a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{11/4} \sqrt [4]{b}}-\frac{(7 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{11/4} \sqrt [4]{b}}+\frac{A b-a B}{2 a b x^{3/2} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 457
Rule 325
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{A+B x^2}{x^{5/2} \left (a+b x^2\right )^2} \, dx &=\frac{A b-a B}{2 a b x^{3/2} \left (a+b x^2\right )}+\frac{\left (\frac{7 A b}{2}-\frac{3 a B}{2}\right ) \int \frac{1}{x^{5/2} \left (a+b x^2\right )} \, dx}{2 a b}\\ &=-\frac{7 A b-3 a B}{6 a^2 b x^{3/2}}+\frac{A b-a B}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac{(7 A b-3 a B) \int \frac{1}{\sqrt{x} \left (a+b x^2\right )} \, dx}{4 a^2}\\ &=-\frac{7 A b-3 a B}{6 a^2 b x^{3/2}}+\frac{A b-a B}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac{(7 A b-3 a B) \operatorname{Subst}\left (\int \frac{1}{a+b x^4} \, dx,x,\sqrt{x}\right )}{2 a^2}\\ &=-\frac{7 A b-3 a B}{6 a^2 b x^{3/2}}+\frac{A b-a B}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac{(7 A b-3 a B) \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{4 a^{5/2}}-\frac{(7 A b-3 a B) \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{4 a^{5/2}}\\ &=-\frac{7 A b-3 a B}{6 a^2 b x^{3/2}}+\frac{A b-a B}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac{(7 A b-3 a B) \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 )}{8 a^{5/2} \sqrt{b}}-\frac{(7 A b-3 a B) \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 )}{8 a^{5/2} \sqrt{b}}+\frac{(7 A b-3 a B) \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 )}{8 \sqrt{2} a^{11/4} \sqrt [4]{b}}+\frac{(7 A b-3 a B) \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 )}{8 \sqrt{2} a^{11/4} \sqrt [4]{b}}\\ &=-\frac{7 A b-3 a B}{6 a^2 b x^{3/2}}+\frac{A b-a B}{2 a b x^{3/2} \left (a+b x^2\right )}+\frac{(7 A b-3 a B) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4} \sqrt [4]{b}}-\frac{(7 A b-3 a B) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4} \sqrt [4]{b}}-\frac{(7 A b-3 a B) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{11/4} \sqrt [4]{b}}+\frac{(7 A b-3 a B) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{11/4} \sqrt [4]{b}}\\ &=-\frac{7 A b-3 a B}{6 a^2 b x^{3/2}}+\frac{A b-a B}{2 a b x^{3/2} \left (a+b x^2\right )}+\frac{(7 A b-3 a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{11/4} \sqrt [4]{b}}-\frac{(7 A b-3 a B) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{11/4} \sqrt [4]{b}}+\frac{(7 A b-3 a B) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4} \sqrt [4]{b}}-\frac{(7 A b-3 a B) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4} \sqrt [4]{b}}\\ \end{align*}
Mathematica [A] time = 0.401288, size = 355, normalized size = 1.23 \[ \frac{-\frac{24 a^{3/4} A b \sqrt{x}}{a+b x^2}-\frac{32 a^{3/4} A}{x^{3/2}}+\frac{24 a^{7/4} B \sqrt{x}}{a+b x^2}+21 \sqrt{2} A b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-21 \sqrt{2} A b^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+\frac{6 \sqrt{2} (7 A b-3 a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{b}}-\frac{6 \sqrt{2} (7 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{b}}-\frac{9 \sqrt{2} a B \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{b}}+\frac{9 \sqrt{2} a B \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{b}}}{48 a^{11/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 317, normalized size = 1.1 \begin{align*} -{\frac{2\,A}{3\,{a}^{2}}{x}^{-{\frac{3}{2}}}}-{\frac{Ab}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }\sqrt{x}}+{\frac{B}{2\,a \left ( b{x}^{2}+a \right ) }\sqrt{x}}-{\frac{7\,\sqrt{2}Ab}{8\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }-{\frac{7\,\sqrt{2}Ab}{8\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }-{\frac{7\,\sqrt{2}Ab}{16\,{a}^{3}}\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{3\,\sqrt{2}B}{8\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{3\,\sqrt{2}B}{8\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{3\,\sqrt{2}B}{16\,{a}^{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 = 0.89974, size = 1700, normalized size = 5.88 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18449, size = 382, normalized size = 1.32 \begin{align*} \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{1}{4}} B a - 7 \, \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 )}{8 \, a^{3} b} + \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{1}{4}} B a - 7 \, \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 )}{8 \, a^{3} b} + \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{1}{4}} B a - 7 \, \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 )}{16 \, a^{3} b} - \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{1}{4}} B a - 7 \, \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 )}{16 \, a^{3} b} + \frac{B a \sqrt{x} - A b \sqrt{x}}{2 \,{\left (b x^{2} + a\right )} a^{2}} - \frac{2 \, A}{3 \, a^{2} x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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