Optimal. Leaf size=352 \[ -\frac{195 \sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{17/4} d^{3/2}}+\frac{195 \sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{17/4} d^{3/2}}+\frac{195 \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} a^{17/4} d^{3/2}}-\frac{195 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} a^{17/4} d^{3/2}}+\frac{39}{64 a^3 d \sqrt{d x} \left (a+b x^2\right )}+\frac{13}{48 a^2 d \sqrt{d x} \left (a+b x^2\right )^2}-\frac{195}{64 a^4 d \sqrt{d x}}+\frac{1}{6 a d \sqrt{d x} \left (a+b x^2\right )^3} \]
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Rubi [A] time = 0.402453, antiderivative size = 352, normalized size of antiderivative = 1., number of steps used = 15, 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} \[ -\frac{195 \sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{17/4} d^{3/2}}+\frac{195 \sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{17/4} d^{3/2}}+\frac{195 \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} a^{17/4} d^{3/2}}-\frac{195 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} a^{17/4} d^{3/2}}+\frac{39}{64 a^3 d \sqrt{d x} \left (a+b x^2\right )}+\frac{13}{48 a^2 d \sqrt{d x} \left (a+b x^2\right )^2}-\frac{195}{64 a^4 d \sqrt{d x}}+\frac{1}{6 a d \sqrt{d x} \left (a+b x^2\right )^3} \]
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)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac{1}{(d x)^{3/2} \left (a b+b^2 x^2\right )^4} \, dx\\ &=\frac{1}{6 a d \sqrt{d x} \left (a+b x^2\right )^3}+\frac{\left (13 b^3\right ) \int \frac{1}{(d x)^{3/2} \left (a b+b^2 x^2\right )^3} \, dx}{12 a}\\ &=\frac{1}{6 a d \sqrt{d x} \left (a+b x^2\right )^3}+\frac{13}{48 a^2 d \sqrt{d x} \left (a+b x^2\right )^2}+\frac{\left (39 b^2\right ) \int \frac{1}{(d x)^{3/2} \left (a b+b^2 x^2\right )^2} \, dx}{32 a^2}\\ &=\frac{1}{6 a d \sqrt{d x} \left (a+b x^2\right )^3}+\frac{13}{48 a^2 d \sqrt{d x} \left (a+b x^2\right )^2}+\frac{39}{64 a^3 d \sqrt{d x} \left (a+b x^2\right )}+\frac{(195 b) \int \frac{1}{(d x)^{3/2} \left (a b+b^2 x^2\right )} \, dx}{128 a^3}\\ &=-\frac{195}{64 a^4 d \sqrt{d x}}+\frac{1}{6 a d \sqrt{d x} \left (a+b x^2\right )^3}+\frac{13}{48 a^2 d \sqrt{d x} \left (a+b x^2\right )^2}+\frac{39}{64 a^3 d \sqrt{d x} \left (a+b x^2\right )}-\frac{\left (195 b^2\right ) \int \frac{\sqrt{d x}}{a b+b^2 x^2} \, dx}{128 a^4 d^2}\\ &=-\frac{195}{64 a^4 d \sqrt{d x}}+\frac{1}{6 a d \sqrt{d x} \left (a+b x^2\right )^3}+\frac{13}{48 a^2 d \sqrt{d x} \left (a+b x^2\right )^2}+\frac{39}{64 a^3 d \sqrt{d x} \left (a+b x^2\right )}-\frac{\left (195 b^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{64 a^4 d^3}\\ &=-\frac{195}{64 a^4 d \sqrt{d x}}+\frac{1}{6 a d \sqrt{d x} \left (a+b x^2\right )^3}+\frac{13}{48 a^2 d \sqrt{d x} \left (a+b x^2\right )^2}+\frac{39}{64 a^3 d \sqrt{d x} \left (a+b x^2\right )}+\frac{\left (195 b^{3/2}\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 )}{128 a^4 d^3}-\frac{\left (195 b^{3/2}\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 )}{128 a^4 d^3}\\ &=-\frac{195}{64 a^4 d \sqrt{d x}}+\frac{1}{6 a d \sqrt{d x} \left (a+b x^2\right )^3}+\frac{13}{48 a^2 d \sqrt{d x} \left (a+b x^2\right )^2}+\frac{39}{64 a^3 d \sqrt{d x} \left (a+b x^2\right )}-\frac{\left (195 \sqrt [4]{b}\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 )}{256 \sqrt{2} a^{17/4} d^{3/2}}-\frac{\left (195 \sqrt [4]{b}\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 )}{256 \sqrt{2} a^{17/4} d^{3/2}}-\frac{195 \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 )}{256 a^4 d}-\frac{195 \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 )}{256 a^4 d}\\ &=-\frac{195}{64 a^4 d \sqrt{d x}}+\frac{1}{6 a d \sqrt{d x} \left (a+b x^2\right )^3}+\frac{13}{48 a^2 d \sqrt{d x} \left (a+b x^2\right )^2}+\frac{39}{64 a^3 d \sqrt{d x} \left (a+b x^2\right )}-\frac{195 \sqrt [4]{b} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{256 \sqrt{2} a^{17/4} d^{3/2}}+\frac{195 \sqrt [4]{b} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{256 \sqrt{2} a^{17/4} d^{3/2}}-\frac{\left (195 \sqrt [4]{b}\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 )}{128 \sqrt{2} a^{17/4} d^{3/2}}+\frac{\left (195 \sqrt [4]{b}\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 )}{128 \sqrt{2} a^{17/4} d^{3/2}}\\ &=-\frac{195}{64 a^4 d \sqrt{d x}}+\frac{1}{6 a d \sqrt{d x} \left (a+b x^2\right )^3}+\frac{13}{48 a^2 d \sqrt{d x} \left (a+b x^2\right )^2}+\frac{39}{64 a^3 d \sqrt{d x} \left (a+b x^2\right )}+\frac{195 \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} a^{17/4} d^{3/2}}-\frac{195 \sqrt [4]{b} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} a^{17/4} d^{3/2}}-\frac{195 \sqrt [4]{b} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{256 \sqrt{2} a^{17/4} d^{3/2}}+\frac{195 \sqrt [4]{b} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{256 \sqrt{2} a^{17/4} d^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0118885, size = 30, normalized size = 0.09 \[ -\frac{2 x \, _2F_1\left (-\frac{1}{4},4;\frac{3}{4};-\frac{b x^2}{a}\right )}{a^4 (d x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.07, size = 285, normalized size = 0.8 \begin{align*} -2\,{\frac{1}{{a}^{4}d\sqrt{dx}}}-{\frac{67\,{b}^{3}}{64\,{a}^{4}d \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}} \left ( dx \right ) ^{{\frac{11}{2}}}}-{\frac{81\,{b}^{2}d}{32\,{a}^{3} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}} \left ( dx \right ) ^{{\frac{7}{2}}}}-{\frac{317\,{d}^{3}b}{192\,{a}^{2} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}} \left ( dx \right ) ^{{\frac{3}{2}}}}-{\frac{195\,\sqrt{2}}{512\,{a}^{4}d}\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{195\,\sqrt{2}}{256\,{a}^{4}d}\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{195\,\sqrt{2}}{256\,{a}^{4}d}\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 = 1.5407, size = 1008, normalized size = 2.86 \begin{align*} \frac{2340 \,{\left (a^{4} b^{3} d^{2} x^{7} + 3 \, a^{5} b^{2} d^{2} x^{5} + 3 \, a^{6} b d^{2} x^{3} + a^{7} d^{2} x\right )} \left (-\frac{b}{a^{17} d^{6}}\right )^{\frac{1}{4}} \arctan \left (-\frac{7414875 \, \sqrt{d x} a^{4} b d \left (-\frac{b}{a^{17} d^{6}}\right )^{\frac{1}{4}} - \sqrt{-54980371265625 \, a^{9} b d^{4} \sqrt{-\frac{b}{a^{17} d^{6}}} + 54980371265625 \, b^{2} d x} a^{4} d \left (-\frac{b}{a^{17} d^{6}}\right )^{\frac{1}{4}}}{7414875 \, b}\right ) - 585 \,{\left (a^{4} b^{3} d^{2} x^{7} + 3 \, a^{5} b^{2} d^{2} x^{5} + 3 \, a^{6} b d^{2} x^{3} + a^{7} d^{2} x\right )} \left (-\frac{b}{a^{17} d^{6}}\right )^{\frac{1}{4}} \log \left (7414875 \, a^{13} d^{5} \left (-\frac{b}{a^{17} d^{6}}\right )^{\frac{3}{4}} + 7414875 \, \sqrt{d x} b\right ) + 585 \,{\left (a^{4} b^{3} d^{2} x^{7} + 3 \, a^{5} b^{2} d^{2} x^{5} + 3 \, a^{6} b d^{2} x^{3} + a^{7} d^{2} x\right )} \left (-\frac{b}{a^{17} d^{6}}\right )^{\frac{1}{4}} \log \left (-7414875 \, a^{13} d^{5} \left (-\frac{b}{a^{17} d^{6}}\right )^{\frac{3}{4}} + 7414875 \, \sqrt{d x} b\right ) - 4 \,{\left (585 \, b^{3} x^{6} + 1638 \, a b^{2} x^{4} + 1469 \, a^{2} b x^{2} + 384 \, a^{3}\right )} \sqrt{d x}}{768 \,{\left (a^{4} b^{3} d^{2} x^{7} + 3 \, a^{5} b^{2} d^{2} x^{5} + 3 \, a^{6} b d^{2} x^{3} + a^{7} d^{2} x\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{3}{2}} \left (a + b x^{2}\right )^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22216, size = 441, normalized size = 1.25 \begin{align*} -\frac{\frac{3072}{\sqrt{d x} a^{4}} + \frac{8 \,{\left (201 \, \sqrt{d x} b^{3} d^{5} x^{5} + 486 \, \sqrt{d x} a b^{2} d^{5} x^{3} + 317 \, \sqrt{d x} a^{2} b d^{5} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{3} a^{4}} + \frac{1170 \, \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 )}{a^{5} b^{2} d^{2}} + \frac{1170 \, \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 )}{a^{5} b^{2} d^{2}} - \frac{585 \, \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 )}{a^{5} b^{2} d^{2}} + \frac{585 \, \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 )}{a^{5} b^{2} d^{2}}}{1536 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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