Optimal. Leaf size=215 \[ \frac{b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4}}-\frac{b^{5/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4}}-\frac{b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{9/4}}+\frac{b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{9/4}}+\frac{2 b}{a^2 \sqrt{x}}-\frac{2}{5 a x^{5/2}} \]
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Rubi [A] time = 0.175071, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533, Rules used = {325, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac{b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4}}-\frac{b^{5/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4}}-\frac{b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{9/4}}+\frac{b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{9/4}}+\frac{2 b}{a^2 \sqrt{x}}-\frac{2}{5 a x^{5/2}} \]
Antiderivative was successfully verified.
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Rule 325
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{x^{7/2} \left (a+b x^2\right )} \, dx &=-\frac{2}{5 a x^{5/2}}-\frac{b \int \frac{1}{x^{3/2} \left (a+b x^2\right )} \, dx}{a}\\ &=-\frac{2}{5 a x^{5/2}}+\frac{2 b}{a^2 \sqrt{x}}+\frac{b^2 \int \frac{\sqrt{x}}{a+b x^2} \, dx}{a^2}\\ &=-\frac{2}{5 a x^{5/2}}+\frac{2 b}{a^2 \sqrt{x}}+\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{a^2}\\ &=-\frac{2}{5 a x^{5/2}}+\frac{2 b}{a^2 \sqrt{x}}-\frac{b^{3/2} \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{a^2}+\frac{b^{3/2} \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{a^2}\\ &=-\frac{2}{5 a x^{5/2}}+\frac{2 b}{a^2 \sqrt{x}}+\frac{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 )}{2 a^2}+\frac{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 )}{2 a^2}+\frac{b^{5/4} \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} a^{9/4}}+\frac{b^{5/4} \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} a^{9/4}}\\ &=-\frac{2}{5 a x^{5/2}}+\frac{2 b}{a^2 \sqrt{x}}+\frac{b^{5/4} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4}}-\frac{b^{5/4} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4}}+\frac{b^{5/4} \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} a^{9/4}}-\frac{b^{5/4} \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} a^{9/4}}\\ &=-\frac{2}{5 a x^{5/2}}+\frac{2 b}{a^2 \sqrt{x}}-\frac{b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{9/4}}+\frac{b^{5/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{9/4}}+\frac{b^{5/4} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4}}-\frac{b^{5/4} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4}}\\ \end{align*}
Mathematica [C] time = 0.0065314, size = 29, normalized size = 0.13 \[ -\frac{2 \, _2F_1\left (-\frac{5}{4},1;-\frac{1}{4};-\frac{b x^2}{a}\right )}{5 a x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 152, normalized size = 0.7 \begin{align*}{\frac{b\sqrt{2}}{4\,{a}^{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{b\sqrt{2}}{2\,{a}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{b\sqrt{2}}{2\,{a}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{2}{5\,a}{x}^{-{\frac{5}{2}}}}+2\,{\frac{b}{{a}^{2}\sqrt{x}}} \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 = 1.434, size = 433, normalized size = 2.01 \begin{align*} -\frac{20 \, a^{2} x^{3} \left (-\frac{b^{5}}{a^{9}}\right )^{\frac{1}{4}} \arctan \left (-\frac{a^{2} b^{4} \sqrt{x} \left (-\frac{b^{5}}{a^{9}}\right )^{\frac{1}{4}} - \sqrt{-a^{5} b^{5} \sqrt{-\frac{b^{5}}{a^{9}}} + b^{8} x} a^{2} \left (-\frac{b^{5}}{a^{9}}\right )^{\frac{1}{4}}}{b^{5}}\right ) - 5 \, a^{2} x^{3} \left (-\frac{b^{5}}{a^{9}}\right )^{\frac{1}{4}} \log \left (a^{7} \left (-\frac{b^{5}}{a^{9}}\right )^{\frac{3}{4}} + b^{4} \sqrt{x}\right ) + 5 \, a^{2} x^{3} \left (-\frac{b^{5}}{a^{9}}\right )^{\frac{1}{4}} \log \left (-a^{7} \left (-\frac{b^{5}}{a^{9}}\right )^{\frac{3}{4}} + b^{4} \sqrt{x}\right ) - 4 \,{\left (5 \, b x^{2} - a\right )} \sqrt{x}}{10 \, a^{2} x^{3}} \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 = 3.07966, size = 270, normalized size = 1.26 \begin{align*} \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 \, a^{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 \, a^{3} b} - \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 \, a^{3} b} + \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 \, a^{3} b} + \frac{2 \,{\left (5 \, b x^{2} - a\right )}}{5 \, a^{2} x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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