Optimal. Leaf size=318 \[ \frac{9 b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} a^{13/4} d^{7/2}}-\frac{9 b^{5/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} a^{13/4} d^{7/2}}-\frac{9 b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} a^{13/4} d^{7/2}}+\frac{9 b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{4 \sqrt{2} a^{13/4} d^{7/2}}+\frac{9 b}{2 a^3 d^3 \sqrt{d x}}-\frac{9}{10 a^2 d (d x)^{5/2}}+\frac{1}{2 a d (d x)^{5/2} \left (a+b x^2\right )} \]
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Rubi [A] time = 0.342626, antiderivative size = 318, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {28, 290, 325, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac{9 b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} a^{13/4} d^{7/2}}-\frac{9 b^{5/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} a^{13/4} d^{7/2}}-\frac{9 b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} a^{13/4} d^{7/2}}+\frac{9 b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{4 \sqrt{2} a^{13/4} d^{7/2}}+\frac{9 b}{2 a^3 d^3 \sqrt{d x}}-\frac{9}{10 a^2 d (d x)^{5/2}}+\frac{1}{2 a d (d x)^{5/2} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 28
Rule 290
Rule 325
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx &=b^2 \int \frac{1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^2} \, dx\\ &=\frac{1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}+\frac{(9 b) \int \frac{1}{(d x)^{7/2} \left (a b+b^2 x^2\right )} \, dx}{4 a}\\ &=-\frac{9}{10 a^2 d (d x)^{5/2}}+\frac{1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}-\frac{\left (9 b^2\right ) \int \frac{1}{(d x)^{3/2} \left (a b+b^2 x^2\right )} \, dx}{4 a^2 d^2}\\ &=-\frac{9}{10 a^2 d (d x)^{5/2}}+\frac{9 b}{2 a^3 d^3 \sqrt{d x}}+\frac{1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}+\frac{\left (9 b^3\right ) \int \frac{\sqrt{d x}}{a b+b^2 x^2} \, dx}{4 a^3 d^4}\\ &=-\frac{9}{10 a^2 d (d x)^{5/2}}+\frac{9 b}{2 a^3 d^3 \sqrt{d x}}+\frac{1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}+\frac{\left (9 b^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{2 a^3 d^5}\\ &=-\frac{9}{10 a^2 d (d x)^{5/2}}+\frac{9 b}{2 a^3 d^3 \sqrt{d x}}+\frac{1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}-\frac{\left (9 b^{5/2}\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 a^3 d^5}+\frac{\left (9 b^{5/2}\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 a^3 d^5}\\ &=-\frac{9}{10 a^2 d (d x)^{5/2}}+\frac{9 b}{2 a^3 d^3 \sqrt{d x}}+\frac{1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}+\frac{\left (9 b^{5/4}\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} a^{13/4} d^{7/2}}+\frac{\left (9 b^{5/4}\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} a^{13/4} d^{7/2}}+\frac{(9 b) \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 a^3 d^3}+\frac{(9 b) \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 a^3 d^3}\\ &=-\frac{9}{10 a^2 d (d x)^{5/2}}+\frac{9 b}{2 a^3 d^3 \sqrt{d x}}+\frac{1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}+\frac{9 b^{5/4} \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} a^{13/4} d^{7/2}}-\frac{9 b^{5/4} \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} a^{13/4} d^{7/2}}+\frac{\left (9 b^{5/4}\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} a^{13/4} d^{7/2}}-\frac{\left (9 b^{5/4}\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} a^{13/4} d^{7/2}}\\ &=-\frac{9}{10 a^2 d (d x)^{5/2}}+\frac{9 b}{2 a^3 d^3 \sqrt{d x}}+\frac{1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}-\frac{9 b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} a^{13/4} d^{7/2}}+\frac{9 b^{5/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} a^{13/4} d^{7/2}}+\frac{9 b^{5/4} \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} a^{13/4} d^{7/2}}-\frac{9 b^{5/4} \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} a^{13/4} d^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0112613, size = 37, normalized size = 0.12 \[ -\frac{2 \sqrt{d x} \, _2F_1\left (-\frac{5}{4},2;-\frac{1}{4};-\frac{b x^2}{a}\right )}{5 a^2 d^4 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.063, size = 242, normalized size = 0.8 \begin{align*} -{\frac{2}{5\,{a}^{2}d} \left ( dx \right ) ^{-{\frac{5}{2}}}}+4\,{\frac{b}{{a}^{3}{d}^{3}\sqrt{dx}}}+{\frac{{b}^{2}}{2\,{a}^{3}{d}^{3} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) } \left ( dx \right ) ^{{\frac{3}{2}}}}+{\frac{9\,b\sqrt{2}}{16\,{a}^{3}{d}^{3}}\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{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+{\frac{9\,b\sqrt{2}}{8\,{a}^{3}{d}^{3}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+{\frac{9\,b\sqrt{2}}{8\,{a}^{3}{d}^{3}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{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 [A] time = 1.3864, size = 767, normalized size = 2.41 \begin{align*} -\frac{180 \,{\left (a^{3} b d^{4} x^{5} + a^{4} d^{4} x^{3}\right )} \left (-\frac{b^{5}}{a^{13} d^{14}}\right )^{\frac{1}{4}} \arctan \left (-\frac{729 \, \sqrt{d x} a^{3} b^{4} d^{3} \left (-\frac{b^{5}}{a^{13} d^{14}}\right )^{\frac{1}{4}} - \sqrt{-531441 \, a^{7} b^{5} d^{8} \sqrt{-\frac{b^{5}}{a^{13} d^{14}}} + 531441 \, b^{8} d x} a^{3} d^{3} \left (-\frac{b^{5}}{a^{13} d^{14}}\right )^{\frac{1}{4}}}{729 \, b^{5}}\right ) - 45 \,{\left (a^{3} b d^{4} x^{5} + a^{4} d^{4} x^{3}\right )} \left (-\frac{b^{5}}{a^{13} d^{14}}\right )^{\frac{1}{4}} \log \left (729 \, a^{10} d^{11} \left (-\frac{b^{5}}{a^{13} d^{14}}\right )^{\frac{3}{4}} + 729 \, \sqrt{d x} b^{4}\right ) + 45 \,{\left (a^{3} b d^{4} x^{5} + a^{4} d^{4} x^{3}\right )} \left (-\frac{b^{5}}{a^{13} d^{14}}\right )^{\frac{1}{4}} \log \left (-729 \, a^{10} d^{11} \left (-\frac{b^{5}}{a^{13} d^{14}}\right )^{\frac{3}{4}} + 729 \, \sqrt{d x} b^{4}\right ) - 4 \,{\left (45 \, b^{2} x^{4} + 36 \, a b x^{2} - 4 \, a^{2}\right )} \sqrt{d x}}{40 \,{\left (a^{3} b d^{4} x^{5} + a^{4} d^{4} x^{3}\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{1}{\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 = 1.25903, size = 414, normalized size = 1.3 \begin{align*} \frac{\sqrt{d x} b^{2} x}{2 \,{\left (b d^{2} x^{2} + a d^{2}\right )} a^{3} d^{2}} + \frac{9 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \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 )}{8 \, a^{4} b d^{5}} + \frac{9 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \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 )}{8 \, a^{4} b d^{5}} - \frac{9 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \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 )}{16 \, a^{4} b d^{5}} + \frac{9 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \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 )}{16 \, a^{4} b d^{5}} + \frac{2 \,{\left (10 \, b d^{2} x^{2} - a d^{2}\right )}}{5 \, \sqrt{d x} a^{3} d^{5} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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