Optimal. Leaf size=300 \[ \frac{7 b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} a^{11/4} d^{5/2}}-\frac{7 b^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} a^{11/4} d^{5/2}}+\frac{7 b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} a^{11/4} d^{5/2}}-\frac{7 b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{4 \sqrt{2} a^{11/4} d^{5/2}}-\frac{7}{6 a^2 d (d x)^{3/2}}+\frac{1}{2 a d (d x)^{3/2} \left (a+b x^2\right )} \]
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Rubi [A] time = 0.299124, antiderivative size = 300, 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, 290, 325, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{7 b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} a^{11/4} d^{5/2}}-\frac{7 b^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} a^{11/4} d^{5/2}}+\frac{7 b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} a^{11/4} d^{5/2}}-\frac{7 b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{4 \sqrt{2} a^{11/4} d^{5/2}}-\frac{7}{6 a^2 d (d x)^{3/2}}+\frac{1}{2 a d (d x)^{3/2} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 28
Rule 290
Rule 325
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx &=b^2 \int \frac{1}{(d x)^{5/2} \left (a b+b^2 x^2\right )^2} \, dx\\ &=\frac{1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}+\frac{(7 b) \int \frac{1}{(d x)^{5/2} \left (a b+b^2 x^2\right )} \, dx}{4 a}\\ &=-\frac{7}{6 a^2 d (d x)^{3/2}}+\frac{1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}-\frac{\left (7 b^2\right ) \int \frac{1}{\sqrt{d x} \left (a b+b^2 x^2\right )} \, dx}{4 a^2 d^2}\\ &=-\frac{7}{6 a^2 d (d x)^{3/2}}+\frac{1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}-\frac{\left (7 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{2 a^2 d^3}\\ &=-\frac{7}{6 a^2 d (d x)^{3/2}}+\frac{1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}-\frac{\left (7 b^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^{5/2} d^4}-\frac{\left (7 b^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^{5/2} d^4}\\ &=-\frac{7}{6 a^2 d (d x)^{3/2}}+\frac{1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}+\frac{\left (7 b^{3/4}\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^{11/4} d^{5/2}}+\frac{\left (7 b^{3/4}\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^{11/4} d^{5/2}}-\frac{\left (7 \sqrt{b}\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 a^{5/2} d^2}-\frac{\left (7 \sqrt{b}\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 a^{5/2} d^2}\\ &=-\frac{7}{6 a^2 d (d x)^{3/2}}+\frac{1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}+\frac{7 b^{3/4} \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^{11/4} d^{5/2}}-\frac{7 b^{3/4} \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^{11/4} d^{5/2}}-\frac{\left (7 b^{3/4}\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^{11/4} d^{5/2}}+\frac{\left (7 b^{3/4}\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^{11/4} d^{5/2}}\\ &=-\frac{7}{6 a^2 d (d x)^{3/2}}+\frac{1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}+\frac{7 b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} a^{11/4} d^{5/2}}-\frac{7 b^{3/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} a^{11/4} d^{5/2}}+\frac{7 b^{3/4} \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^{11/4} d^{5/2}}-\frac{7 b^{3/4} \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^{11/4} d^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0122955, size = 32, normalized size = 0.11 \[ -\frac{2 x \, _2F_1\left (-\frac{3}{4},2;\frac{1}{4};-\frac{b x^2}{a}\right )}{3 a^2 (d x)^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.064, size = 226, normalized size = 0.8 \begin{align*} -{\frac{2}{3\,{a}^{2}d} \left ( dx \right ) ^{-{\frac{3}{2}}}}-{\frac{b}{2\,{a}^{2}d \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) }\sqrt{dx}}-{\frac{7\,b\sqrt{2}}{16\,{a}^{3}{d}^{3}}\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{7\,b\sqrt{2}}{8\,{a}^{3}{d}^{3}}\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{7\,b\sqrt{2}}{8\,{a}^{3}{d}^{3}}\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 = 1.4402, size = 678, normalized size = 2.26 \begin{align*} -\frac{84 \,{\left (a^{2} b d^{3} x^{4} + a^{3} d^{3} x^{2}\right )} \left (-\frac{b^{3}}{a^{11} d^{10}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{d x} a^{8} b d^{7} \left (-\frac{b^{3}}{a^{11} d^{10}}\right )^{\frac{3}{4}} - \sqrt{a^{6} d^{6} \sqrt{-\frac{b^{3}}{a^{11} d^{10}}} + b^{2} d x} a^{8} d^{7} \left (-\frac{b^{3}}{a^{11} d^{10}}\right )^{\frac{3}{4}}}{b^{3}}\right ) + 21 \,{\left (a^{2} b d^{3} x^{4} + a^{3} d^{3} x^{2}\right )} \left (-\frac{b^{3}}{a^{11} d^{10}}\right )^{\frac{1}{4}} \log \left (7 \, a^{3} d^{3} \left (-\frac{b^{3}}{a^{11} d^{10}}\right )^{\frac{1}{4}} + 7 \, \sqrt{d x} b\right ) - 21 \,{\left (a^{2} b d^{3} x^{4} + a^{3} d^{3} x^{2}\right )} \left (-\frac{b^{3}}{a^{11} d^{10}}\right )^{\frac{1}{4}} \log \left (-7 \, a^{3} d^{3} \left (-\frac{b^{3}}{a^{11} d^{10}}\right )^{\frac{1}{4}} + 7 \, \sqrt{d x} b\right ) + 4 \,{\left (7 \, b x^{2} + 4 \, a\right )} \sqrt{d x}}{24 \,{\left (a^{2} b d^{3} x^{4} + a^{3} d^{3} x^{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{1}{\left (d x\right )^{\frac{5}{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 = 1.26881, size = 373, normalized size = 1.24 \begin{align*} -\frac{\sqrt{d x} b}{2 \,{\left (b d^{2} x^{2} + a d^{2}\right )} a^{2} d} - \frac{7 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{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^{3} d^{3}} - \frac{7 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{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^{3} d^{3}} - \frac{7 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{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^{3} d^{3}} + \frac{7 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{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^{3} d^{3}} - \frac{2}{3 \, \sqrt{d x} a^{2} d^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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