Optimal. Leaf size=298 \[ -\frac{7 a^{3/4} d^{9/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} b^{11/4}}+\frac{7 a^{3/4} d^{9/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} b^{11/4}}+\frac{7 a^{3/4} d^{9/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} b^{11/4}}-\frac{7 a^{3/4} d^{9/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{4 \sqrt{2} b^{11/4}}-\frac{d (d x)^{7/2}}{2 b \left (a+b x^2\right )}+\frac{7 d^3 (d x)^{3/2}}{6 b^2} \]
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Rubi [A] time = 0.299031, 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, 297, 1162, 617, 204, 1165, 628} \[ -\frac{7 a^{3/4} d^{9/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} b^{11/4}}+\frac{7 a^{3/4} d^{9/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} b^{11/4}}+\frac{7 a^{3/4} d^{9/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} b^{11/4}}-\frac{7 a^{3/4} d^{9/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{4 \sqrt{2} b^{11/4}}-\frac{d (d x)^{7/2}}{2 b \left (a+b x^2\right )}+\frac{7 d^3 (d x)^{3/2}}{6 b^2} \]
Antiderivative was successfully verified.
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Rule 28
Rule 288
Rule 321
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{(d x)^{9/2}}{a^2+2 a b x^2+b^2 x^4} \, dx &=b^2 \int \frac{(d x)^{9/2}}{\left (a b+b^2 x^2\right )^2} \, dx\\ &=-\frac{d (d x)^{7/2}}{2 b \left (a+b x^2\right )}+\frac{1}{4} \left (7 d^2\right ) \int \frac{(d x)^{5/2}}{a b+b^2 x^2} \, dx\\ &=\frac{7 d^3 (d x)^{3/2}}{6 b^2}-\frac{d (d x)^{7/2}}{2 b \left (a+b x^2\right )}-\frac{\left (7 a d^4\right ) \int \frac{\sqrt{d x}}{a b+b^2 x^2} \, dx}{4 b}\\ &=\frac{7 d^3 (d x)^{3/2}}{6 b^2}-\frac{d (d x)^{7/2}}{2 b \left (a+b x^2\right )}-\frac{\left (7 a d^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 b}\\ &=\frac{7 d^3 (d x)^{3/2}}{6 b^2}-\frac{d (d x)^{7/2}}{2 b \left (a+b x^2\right )}+\frac{\left (7 a d^3\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^{3/2}}-\frac{\left (7 a d^3\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^{3/2}}\\ &=\frac{7 d^3 (d x)^{3/2}}{6 b^2}-\frac{d (d x)^{7/2}}{2 b \left (a+b x^2\right )}-\frac{\left (7 a^{3/4} d^{9/2}\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^{11/4}}-\frac{\left (7 a^{3/4} d^{9/2}\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^{11/4}}-\frac{\left (7 a d^5\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^3}-\frac{\left (7 a d^5\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^3}\\ &=\frac{7 d^3 (d x)^{3/2}}{6 b^2}-\frac{d (d x)^{7/2}}{2 b \left (a+b x^2\right )}-\frac{7 a^{3/4} d^{9/2} \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^{11/4}}+\frac{7 a^{3/4} d^{9/2} \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^{11/4}}-\frac{\left (7 a^{3/4} d^{9/2}\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^{11/4}}+\frac{\left (7 a^{3/4} d^{9/2}\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^{11/4}}\\ &=\frac{7 d^3 (d x)^{3/2}}{6 b^2}-\frac{d (d x)^{7/2}}{2 b \left (a+b x^2\right )}+\frac{7 a^{3/4} d^{9/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} b^{11/4}}-\frac{7 a^{3/4} d^{9/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} b^{11/4}}-\frac{7 a^{3/4} d^{9/2} \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^{11/4}}+\frac{7 a^{3/4} d^{9/2} \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^{11/4}}\\ \end{align*}
Mathematica [C] time = 0.0183977, size = 63, normalized size = 0.21 \[ -\frac{2 d^4 x \sqrt{d x} \left (7 \left (a+b x^2\right ) \, _2F_1\left (\frac{3}{4},2;\frac{7}{4};-\frac{b x^2}{a}\right )-7 a-b x^2\right )}{3 b^2 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 226, normalized size = 0.8 \begin{align*}{\frac{2\,{d}^{3}}{3\,{b}^{2}} \left ( dx \right ) ^{{\frac{3}{2}}}}+{\frac{{d}^{5}a}{2\,{b}^{2} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) } \left ( dx \right ) ^{{\frac{3}{2}}}}-{\frac{7\,{d}^{5}a\sqrt{2}}{16\,{b}^{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{7\,{d}^{5}a\sqrt{2}}{8\,{b}^{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{7\,{d}^{5}a\sqrt{2}}{8\,{b}^{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 = 1.40596, size = 640, normalized size = 2.15 \begin{align*} \frac{84 \, \left (-\frac{a^{3} d^{18}}{b^{11}}\right )^{\frac{1}{4}}{\left (b^{3} x^{2} + a b^{2}\right )} \arctan \left (-\frac{\left (-\frac{a^{3} d^{18}}{b^{11}}\right )^{\frac{1}{4}} \sqrt{d x} a^{2} b^{3} d^{13} - \sqrt{a^{4} d^{27} x - \sqrt{-\frac{a^{3} d^{18}}{b^{11}}} a^{3} b^{5} d^{18}} \left (-\frac{a^{3} d^{18}}{b^{11}}\right )^{\frac{1}{4}} b^{3}}{a^{3} d^{18}}\right ) - 21 \, \left (-\frac{a^{3} d^{18}}{b^{11}}\right )^{\frac{1}{4}}{\left (b^{3} x^{2} + a b^{2}\right )} \log \left (343 \, \sqrt{d x} a^{2} d^{13} + 343 \, \left (-\frac{a^{3} d^{18}}{b^{11}}\right )^{\frac{3}{4}} b^{8}\right ) + 21 \, \left (-\frac{a^{3} d^{18}}{b^{11}}\right )^{\frac{1}{4}}{\left (b^{3} x^{2} + a b^{2}\right )} \log \left (343 \, \sqrt{d x} a^{2} d^{13} - 343 \, \left (-\frac{a^{3} d^{18}}{b^{11}}\right )^{\frac{3}{4}} b^{8}\right ) + 4 \,{\left (4 \, b d^{4} x^{3} + 7 \, a d^{4} x\right )} \sqrt{d x}}{24 \,{\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{9}{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 = 1.14137, size = 359, normalized size = 1.2 \begin{align*} \frac{1}{48} \,{\left (\frac{24 \, \sqrt{d x} a d^{3} x}{{\left (b d^{2} x^{2} + a d^{2}\right )} b^{2}} + \frac{32 \, \sqrt{d x} d x}{b^{2}} - \frac{42 \, \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 )}{b^{5}} - \frac{42 \, \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 )}{b^{5}} + \frac{21 \, \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 )}{b^{5}} - \frac{21 \, \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 )}{b^{5}}\right )} d^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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