Optimal. Leaf size=298 \[ \frac{5 \sqrt [4]{a} d^{7/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} b^{9/4}}-\frac{5 \sqrt [4]{a} d^{7/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} b^{9/4}}+\frac{5 \sqrt [4]{a} d^{7/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} b^{9/4}}-\frac{5 \sqrt [4]{a} d^{7/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{4 \sqrt{2} b^{9/4}}-\frac{d (d x)^{5/2}}{2 b \left (a+b x^2\right )}+\frac{5 d^3 \sqrt{d x}}{2 b^2} \]
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Rubi [A] time = 0.294855, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {28, 288, 321, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{5 \sqrt [4]{a} d^{7/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} b^{9/4}}-\frac{5 \sqrt [4]{a} d^{7/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} b^{9/4}}+\frac{5 \sqrt [4]{a} d^{7/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} b^{9/4}}-\frac{5 \sqrt [4]{a} d^{7/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{4 \sqrt{2} b^{9/4}}-\frac{d (d x)^{5/2}}{2 b \left (a+b x^2\right )}+\frac{5 d^3 \sqrt{d x}}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 28
Rule 288
Rule 321
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{(d x)^{7/2}}{a^2+2 a b x^2+b^2 x^4} \, dx &=b^2 \int \frac{(d x)^{7/2}}{\left (a b+b^2 x^2\right )^2} \, dx\\ &=-\frac{d (d x)^{5/2}}{2 b \left (a+b x^2\right )}+\frac{1}{4} \left (5 d^2\right ) \int \frac{(d x)^{3/2}}{a b+b^2 x^2} \, dx\\ &=\frac{5 d^3 \sqrt{d x}}{2 b^2}-\frac{d (d x)^{5/2}}{2 b \left (a+b x^2\right )}-\frac{\left (5 a d^4\right ) \int \frac{1}{\sqrt{d x} \left (a b+b^2 x^2\right )} \, dx}{4 b}\\ &=\frac{5 d^3 \sqrt{d x}}{2 b^2}-\frac{d (d x)^{5/2}}{2 b \left (a+b x^2\right )}-\frac{\left (5 a d^3\right ) \operatorname{Subst}\left (\int \frac{1}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{2 b}\\ &=\frac{5 d^3 \sqrt{d x}}{2 b^2}-\frac{d (d x)^{5/2}}{2 b \left (a+b x^2\right )}-\frac{\left (5 \sqrt{a} d^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a} d-\sqrt{b} x^2}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{4 b}-\frac{\left (5 \sqrt{a} d^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a} d+\sqrt{b} x^2}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{4 b}\\ &=\frac{5 d^3 \sqrt{d x}}{2 b^2}-\frac{d (d x)^{5/2}}{2 b \left (a+b x^2\right )}+\frac{\left (5 \sqrt [4]{a} d^{7/2}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d}}{\sqrt [4]{b}}+2 x}{-\frac{\sqrt{a} d}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt{d x}\right )}{8 \sqrt{2} b^{9/4}}+\frac{\left (5 \sqrt [4]{a} d^{7/2}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d}}{\sqrt [4]{b}}-2 x}{-\frac{\sqrt{a} d}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt{d x}\right )}{8 \sqrt{2} b^{9/4}}-\frac{\left (5 \sqrt{a} d^4\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a} d}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{d x}\right )}{8 b^{5/2}}-\frac{\left (5 \sqrt{a} d^4\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a} d}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{d x}\right )}{8 b^{5/2}}\\ &=\frac{5 d^3 \sqrt{d x}}{2 b^2}-\frac{d (d x)^{5/2}}{2 b \left (a+b x^2\right )}+\frac{5 \sqrt [4]{a} d^{7/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{8 \sqrt{2} b^{9/4}}-\frac{5 \sqrt [4]{a} d^{7/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{8 \sqrt{2} b^{9/4}}-\frac{\left (5 \sqrt [4]{a} d^{7/2}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} b^{9/4}}+\frac{\left (5 \sqrt [4]{a} d^{7/2}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} b^{9/4}}\\ &=\frac{5 d^3 \sqrt{d x}}{2 b^2}-\frac{d (d x)^{5/2}}{2 b \left (a+b x^2\right )}+\frac{5 \sqrt [4]{a} d^{7/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} b^{9/4}}-\frac{5 \sqrt [4]{a} d^{7/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} b^{9/4}}+\frac{5 \sqrt [4]{a} d^{7/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{8 \sqrt{2} b^{9/4}}-\frac{5 \sqrt [4]{a} d^{7/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{8 \sqrt{2} b^{9/4}}\\ \end{align*}
Mathematica [A] time = 0.154868, size = 244, normalized size = 0.82 \[ \frac{d^3 \sqrt{d x} \left (\frac{32 b^{5/4} x^2}{a+b x^2}+\frac{40 a \sqrt [4]{b}}{a+b x^2}+\frac{5 \sqrt{2} \sqrt [4]{a} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt{x}}-\frac{5 \sqrt{2} \sqrt [4]{a} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt{x}}+\frac{10 \sqrt{2} \sqrt [4]{a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{x}}-\frac{10 \sqrt{2} \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{x}}\right )}{16 b^{9/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 223, normalized size = 0.8 \begin{align*} 2\,{\frac{{d}^{3}\sqrt{dx}}{{b}^{2}}}+{\frac{{d}^{5}a}{2\,{b}^{2} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) }\sqrt{dx}}-{\frac{5\,{d}^{3}\sqrt{2}}{16\,{b}^{2}}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\ln \left ({ \left ( dx+\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) \left ( dx-\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) ^{-1}} \right ) }-{\frac{5\,{d}^{3}\sqrt{2}}{8\,{b}^{2}}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ) }-{\frac{5\,{d}^{3}\sqrt{2}}{8\,{b}^{2}}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-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 [A] time = 1.38641, size = 549, normalized size = 1.84 \begin{align*} -\frac{20 \, \left (-\frac{a d^{14}}{b^{9}}\right )^{\frac{1}{4}}{\left (b^{3} x^{2} + a b^{2}\right )} \arctan \left (-\frac{\left (-\frac{a d^{14}}{b^{9}}\right )^{\frac{3}{4}} \sqrt{d x} b^{7} d^{3} - \sqrt{d^{7} x + \sqrt{-\frac{a d^{14}}{b^{9}}} b^{4}} \left (-\frac{a d^{14}}{b^{9}}\right )^{\frac{3}{4}} b^{7}}{a d^{14}}\right ) + 5 \, \left (-\frac{a d^{14}}{b^{9}}\right )^{\frac{1}{4}}{\left (b^{3} x^{2} + a b^{2}\right )} \log \left (5 \, \sqrt{d x} d^{3} + 5 \, \left (-\frac{a d^{14}}{b^{9}}\right )^{\frac{1}{4}} b^{2}\right ) - 5 \, \left (-\frac{a d^{14}}{b^{9}}\right )^{\frac{1}{4}}{\left (b^{3} x^{2} + a b^{2}\right )} \log \left (5 \, \sqrt{d x} d^{3} - 5 \, \left (-\frac{a d^{14}}{b^{9}}\right )^{\frac{1}{4}} b^{2}\right ) - 4 \,{\left (4 \, b d^{3} x^{2} + 5 \, a d^{3}\right )} \sqrt{d x}}{8 \,{\left (b^{3} x^{2} + a b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d x\right )^{\frac{7}{2}}}{\left (a + b x^{2}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15487, size = 362, normalized size = 1.21 \begin{align*} \frac{1}{16} \,{\left (\frac{8 \, \sqrt{d x} a d^{3}}{{\left (b d^{2} x^{2} + a d^{2}\right )} b^{2}} - \frac{10 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{b^{3}} - \frac{10 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{b^{3}} - \frac{5 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \log \left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{b^{3}} + \frac{5 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \log \left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{b^{3}} + \frac{32 \, \sqrt{d x} d}{b^{2}}\right )} d^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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