Optimal. Leaf size=343 \[ \frac{13 A b-5 a B}{16 a^2 b x^{5/2} \left (a+b x^2\right )}-\frac{9 (13 A b-5 a B)}{80 a^3 b x^{5/2}}+\frac{9 (13 A b-5 a B)}{16 a^4 \sqrt{x}}+\frac{9 \sqrt [4]{b} (13 A b-5 a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{17/4}}-\frac{9 \sqrt [4]{b} (13 A b-5 a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{17/4}}-\frac{9 \sqrt [4]{b} (13 A b-5 a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{17/4}}+\frac{9 \sqrt [4]{b} (13 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{17/4}}+\frac{A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2} \]
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Rubi [A] time = 0.26158, antiderivative size = 343, normalized size of antiderivative = 1., number of steps used = 14, 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} \[ \frac{13 A b-5 a B}{16 a^2 b x^{5/2} \left (a+b x^2\right )}-\frac{9 (13 A b-5 a B)}{80 a^3 b x^{5/2}}+\frac{9 (13 A b-5 a B)}{16 a^4 \sqrt{x}}+\frac{9 \sqrt [4]{b} (13 A b-5 a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{17/4}}-\frac{9 \sqrt [4]{b} (13 A b-5 a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{17/4}}-\frac{9 \sqrt [4]{b} (13 A b-5 a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{17/4}}+\frac{9 \sqrt [4]{b} (13 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{17/4}}+\frac{A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2} \]
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^{7/2} \left (a+b x^2\right )^3} \, dx &=\frac{A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}+\frac{\left (\frac{13 A b}{2}-\frac{5 a B}{2}\right ) \int \frac{1}{x^{7/2} \left (a+b x^2\right )^2} \, dx}{4 a b}\\ &=\frac{A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}+\frac{13 A b-5 a B}{16 a^2 b x^{5/2} \left (a+b x^2\right )}+\frac{(9 (13 A b-5 a B)) \int \frac{1}{x^{7/2} \left (a+b x^2\right )} \, dx}{32 a^2 b}\\ &=-\frac{9 (13 A b-5 a B)}{80 a^3 b x^{5/2}}+\frac{A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}+\frac{13 A b-5 a B}{16 a^2 b x^{5/2} \left (a+b x^2\right )}-\frac{(9 (13 A b-5 a B)) \int \frac{1}{x^{3/2} \left (a+b x^2\right )} \, dx}{32 a^3}\\ &=-\frac{9 (13 A b-5 a B)}{80 a^3 b x^{5/2}}+\frac{9 (13 A b-5 a B)}{16 a^4 \sqrt{x}}+\frac{A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}+\frac{13 A b-5 a B}{16 a^2 b x^{5/2} \left (a+b x^2\right )}+\frac{(9 b (13 A b-5 a B)) \int \frac{\sqrt{x}}{a+b x^2} \, dx}{32 a^4}\\ &=-\frac{9 (13 A b-5 a B)}{80 a^3 b x^{5/2}}+\frac{9 (13 A b-5 a B)}{16 a^4 \sqrt{x}}+\frac{A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}+\frac{13 A b-5 a B}{16 a^2 b x^{5/2} \left (a+b x^2\right )}+\frac{(9 b (13 A b-5 a B)) \operatorname{Subst}\left (\int \frac{x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{16 a^4}\\ &=-\frac{9 (13 A b-5 a B)}{80 a^3 b x^{5/2}}+\frac{9 (13 A b-5 a B)}{16 a^4 \sqrt{x}}+\frac{A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}+\frac{13 A b-5 a B}{16 a^2 b x^{5/2} \left (a+b x^2\right )}-\frac{\left (9 \sqrt{b} (13 A b-5 a B)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{32 a^4}+\frac{\left (9 \sqrt{b} (13 A b-5 a B)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{32 a^4}\\ &=-\frac{9 (13 A b-5 a B)}{80 a^3 b x^{5/2}}+\frac{9 (13 A b-5 a B)}{16 a^4 \sqrt{x}}+\frac{A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}+\frac{13 A b-5 a B}{16 a^2 b x^{5/2} \left (a+b x^2\right )}+\frac{(9 (13 A b-5 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^4}+\frac{(9 (13 A b-5 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^4}+\frac{\left (9 \sqrt [4]{b} (13 A b-5 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 )}{64 \sqrt{2} a^{17/4}}+\frac{\left (9 \sqrt [4]{b} (13 A b-5 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 )}{64 \sqrt{2} a^{17/4}}\\ &=-\frac{9 (13 A b-5 a B)}{80 a^3 b x^{5/2}}+\frac{9 (13 A b-5 a B)}{16 a^4 \sqrt{x}}+\frac{A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}+\frac{13 A b-5 a B}{16 a^2 b x^{5/2} \left (a+b x^2\right )}+\frac{9 \sqrt [4]{b} (13 A b-5 a B) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{64 \sqrt{2} a^{17/4}}-\frac{9 \sqrt [4]{b} (13 A b-5 a B) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{64 \sqrt{2} a^{17/4}}+\frac{\left (9 \sqrt [4]{b} (13 A b-5 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 )}{32 \sqrt{2} a^{17/4}}-\frac{\left (9 \sqrt [4]{b} (13 A b-5 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 )}{32 \sqrt{2} a^{17/4}}\\ &=-\frac{9 (13 A b-5 a B)}{80 a^3 b x^{5/2}}+\frac{9 (13 A b-5 a B)}{16 a^4 \sqrt{x}}+\frac{A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}+\frac{13 A b-5 a B}{16 a^2 b x^{5/2} \left (a+b x^2\right )}-\frac{9 \sqrt [4]{b} (13 A b-5 a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{17/4}}+\frac{9 \sqrt [4]{b} (13 A b-5 a B) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{17/4}}+\frac{9 \sqrt [4]{b} (13 A b-5 a B) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{64 \sqrt{2} a^{17/4}}-\frac{9 \sqrt [4]{b} (13 A b-5 a B) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{64 \sqrt{2} a^{17/4}}\\ \end{align*}
Mathematica [C] time = 0.473284, size = 189, normalized size = 0.55 \[ -\frac{2 b x^{3/2} (a B-2 A b) \, _2F_1\left (\frac{3}{4},2;\frac{7}{4};-\frac{b x^2}{a}\right )}{3 a^5}+\frac{2 b x^{3/2} (A b-a B) \, _2F_1\left (\frac{3}{4},3;\frac{7}{4};-\frac{b x^2}{a}\right )}{3 a^5}+\frac{6 A b-2 a B}{a^4 \sqrt{x}}-\frac{2 A}{5 a^3 x^{5/2}}+\frac{\sqrt [4]{b} (3 A b-a B) \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{-a}}\right )}{(-a)^{17/4}}+\frac{\sqrt [4]{b} (a B-3 A b) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{-a}}\right )}{(-a)^{17/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 381, normalized size = 1.1 \begin{align*} -{\frac{2\,A}{5\,{a}^{3}}{x}^{-{\frac{5}{2}}}}+6\,{\frac{Ab}{{a}^{4}\sqrt{x}}}-2\,{\frac{B}{{a}^{3}\sqrt{x}}}+{\frac{21\,{b}^{3}A}{16\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{2}}{x}^{{\frac{7}{2}}}}-{\frac{13\,{b}^{2}B}{16\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}{x}^{{\frac{7}{2}}}}+{\frac{25\,A{b}^{2}}{16\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{17\,Bb}{16\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{117\,b\sqrt{2}A}{128\,{a}^{4}}\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{117\,b\sqrt{2}A}{64\,{a}^{4}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{117\,b\sqrt{2}A}{64\,{a}^{4}}\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}B}{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}B}{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}B}{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}}}}}} \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 = 0.883611, size = 2507, normalized size = 7.31 \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 = 1.21917, size = 440, normalized size = 1.28 \begin{align*} -\frac{9 \, \sqrt{2}{\left (5 \, \left (a b^{3}\right )^{\frac{3}{4}} B a - 13 \, \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^{5} b^{2}} - \frac{9 \, \sqrt{2}{\left (5 \, \left (a b^{3}\right )^{\frac{3}{4}} B a - 13 \, \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^{5} b^{2}} + \frac{9 \, \sqrt{2}{\left (5 \, \left (a b^{3}\right )^{\frac{3}{4}} B a - 13 \, \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^{5} b^{2}} - \frac{9 \, \sqrt{2}{\left (5 \, \left (a b^{3}\right )^{\frac{3}{4}} B a - 13 \, \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^{5} b^{2}} - \frac{13 \, B a b^{2} x^{\frac{7}{2}} - 21 \, A b^{3} x^{\frac{7}{2}} + 17 \, B a^{2} b x^{\frac{3}{2}} - 25 \, A a b^{2} x^{\frac{3}{2}}}{16 \,{\left (b x^{2} + a\right )}^{2} a^{4}} - \frac{2 \,{\left (5 \, B a x^{2} - 15 \, A b x^{2} + A a\right )}}{5 \, a^{4} x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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