Optimal. Leaf size=322 \[ -\frac{5 (9 A b-a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{13/4} b^{3/4}}+\frac{5 (9 A b-a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{13/4} b^{3/4}}+\frac{5 (9 A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{13/4} b^{3/4}}-\frac{5 (9 A b-a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{13/4} b^{3/4}}+\frac{9 A b-a B}{16 a^2 b \sqrt{x} \left (a+b x^2\right )}-\frac{5 (9 A b-a B)}{16 a^3 b \sqrt{x}}+\frac{A b-a B}{4 a b \sqrt{x} \left (a+b x^2\right )^2} \]
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Rubi [A] time = 0.233806, antiderivative size = 322, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {457, 290, 325, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac{5 (9 A b-a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{13/4} b^{3/4}}+\frac{5 (9 A b-a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{13/4} b^{3/4}}+\frac{5 (9 A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{13/4} b^{3/4}}-\frac{5 (9 A b-a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{13/4} b^{3/4}}+\frac{9 A b-a B}{16 a^2 b \sqrt{x} \left (a+b x^2\right )}-\frac{5 (9 A b-a B)}{16 a^3 b \sqrt{x}}+\frac{A b-a B}{4 a b \sqrt{x} \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 457
Rule 290
Rule 325
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{A+B x^2}{x^{3/2} \left (a+b x^2\right )^3} \, dx &=\frac{A b-a B}{4 a b \sqrt{x} \left (a+b x^2\right )^2}+\frac{\left (\frac{9 A b}{2}-\frac{a B}{2}\right ) \int \frac{1}{x^{3/2} \left (a+b x^2\right )^2} \, dx}{4 a b}\\ &=\frac{A b-a B}{4 a b \sqrt{x} \left (a+b x^2\right )^2}+\frac{9 A b-a B}{16 a^2 b \sqrt{x} \left (a+b x^2\right )}+\frac{(5 (9 A b-a B)) \int \frac{1}{x^{3/2} \left (a+b x^2\right )} \, dx}{32 a^2 b}\\ &=-\frac{5 (9 A b-a B)}{16 a^3 b \sqrt{x}}+\frac{A b-a B}{4 a b \sqrt{x} \left (a+b x^2\right )^2}+\frac{9 A b-a B}{16 a^2 b \sqrt{x} \left (a+b x^2\right )}-\frac{(5 (9 A b-a B)) \int \frac{\sqrt{x}}{a+b x^2} \, dx}{32 a^3}\\ &=-\frac{5 (9 A b-a B)}{16 a^3 b \sqrt{x}}+\frac{A b-a B}{4 a b \sqrt{x} \left (a+b x^2\right )^2}+\frac{9 A b-a B}{16 a^2 b \sqrt{x} \left (a+b x^2\right )}-\frac{(5 (9 A b-a B)) \operatorname{Subst}\left (\int \frac{x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{16 a^3}\\ &=-\frac{5 (9 A b-a B)}{16 a^3 b \sqrt{x}}+\frac{A b-a B}{4 a b \sqrt{x} \left (a+b x^2\right )^2}+\frac{9 A b-a B}{16 a^2 b \sqrt{x} \left (a+b x^2\right )}+\frac{(5 (9 A b-a B)) \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{32 a^3 \sqrt{b}}-\frac{(5 (9 A b-a B)) \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{32 a^3 \sqrt{b}}\\ &=-\frac{5 (9 A b-a B)}{16 a^3 b \sqrt{x}}+\frac{A b-a B}{4 a b \sqrt{x} \left (a+b x^2\right )^2}+\frac{9 A b-a B}{16 a^2 b \sqrt{x} \left (a+b x^2\right )}-\frac{(5 (9 A b-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 )}{64 a^3 b}-\frac{(5 (9 A b-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 )}{64 a^3 b}-\frac{(5 (9 A b-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 )}{64 \sqrt{2} a^{13/4} b^{3/4}}-\frac{(5 (9 A b-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 )}{64 \sqrt{2} a^{13/4} b^{3/4}}\\ &=-\frac{5 (9 A b-a B)}{16 a^3 b \sqrt{x}}+\frac{A b-a B}{4 a b \sqrt{x} \left (a+b x^2\right )^2}+\frac{9 A b-a B}{16 a^2 b \sqrt{x} \left (a+b x^2\right )}-\frac{5 (9 A b-a B) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{64 \sqrt{2} a^{13/4} b^{3/4}}+\frac{5 (9 A b-a B) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{64 \sqrt{2} a^{13/4} b^{3/4}}-\frac{(5 (9 A b-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 )}{32 \sqrt{2} a^{13/4} b^{3/4}}+\frac{(5 (9 A b-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 )}{32 \sqrt{2} a^{13/4} b^{3/4}}\\ &=-\frac{5 (9 A b-a B)}{16 a^3 b \sqrt{x}}+\frac{A b-a B}{4 a b \sqrt{x} \left (a+b x^2\right )^2}+\frac{9 A b-a B}{16 a^2 b \sqrt{x} \left (a+b x^2\right )}+\frac{5 (9 A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{13/4} b^{3/4}}-\frac{5 (9 A b-a B) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{13/4} b^{3/4}}-\frac{5 (9 A b-a B) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{64 \sqrt{2} a^{13/4} b^{3/4}}+\frac{5 (9 A b-a B) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{64 \sqrt{2} a^{13/4} b^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.194449, size = 147, normalized size = 0.46 \[ \frac{2 x^{3/2} (a B-A b) \, _2F_1\left (\frac{3}{4},3;\frac{7}{4};-\frac{b x^2}{a}\right )}{3 a^4}-\frac{2 A b x^{3/2} \, _2F_1\left (\frac{3}{4},2;\frac{7}{4};-\frac{b x^2}{a}\right )}{3 a^4}-\frac{2 A}{a^3 \sqrt{x}}+\frac{A \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{-a}}\right )}{(-a)^{13/4}}+\frac{a A \sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{-a}}\right )}{(-a)^{17/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 363, normalized size = 1.1 \begin{align*} -2\,{\frac{A}{{a}^{3}\sqrt{x}}}-{\frac{13\,A{b}^{2}}{16\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}{x}^{{\frac{7}{2}}}}+{\frac{5\,Bb}{16\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}{x}^{{\frac{7}{2}}}}-{\frac{17\,Ab}{16\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{9\,B}{16\,a \left ( b{x}^{2}+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{45\,\sqrt{2}A}{128\,{a}^{3}}\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{45\,\sqrt{2}A}{64\,{a}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{45\,\sqrt{2}A}{64\,{a}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{5\,\sqrt{2}B}{128\,{a}^{2}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{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{5\,\sqrt{2}B}{64\,{a}^{2}b}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{5\,\sqrt{2}B}{64\,{a}^{2}b}\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 [B] time = 1.06905, size = 2244, normalized size = 6.97 \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 = 1.22196, size = 405, normalized size = 1.26 \begin{align*} -\frac{2 \, A}{a^{3} \sqrt{x}} + \frac{5 \, B a b x^{\frac{7}{2}} - 13 \, A b^{2} x^{\frac{7}{2}} + 9 \, B a^{2} x^{\frac{3}{2}} - 17 \, A a b x^{\frac{3}{2}}}{16 \,{\left (b x^{2} + a\right )}^{2} a^{3}} + \frac{5 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac{3}{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 )}{64 \, a^{4} b^{3}} + \frac{5 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac{3}{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 )}{64 \, a^{4} b^{3}} - \frac{5 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac{3}{4}} A b\right )} \log \left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{128 \, a^{4} b^{3}} + \frac{5 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac{3}{4}} A b\right )} \log \left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{128 \, a^{4} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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