Optimal. Leaf size=506 \[ -\frac{45 \sqrt [4]{b} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{13/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 \sqrt [4]{b} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{13/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 \sqrt [4]{b} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} a^{13/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \sqrt [4]{b} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{32 \sqrt{2} a^{13/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \left (a+b x^2\right )}{16 a^3 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{9}{16 a^2 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )} \]
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Rubi [A] time = 0.384298, antiderivative size = 506, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1112, 290, 325, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac{45 \sqrt [4]{b} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{13/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 \sqrt [4]{b} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{13/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 \sqrt [4]{b} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} a^{13/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \sqrt [4]{b} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{32 \sqrt{2} a^{13/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \left (a+b x^2\right )}{16 a^3 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{9}{16 a^2 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 1112
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 )^{3/2}} \, dx &=\frac{\left (b^2 \left (a b+b^2 x^2\right )\right ) \int \frac{1}{(d x)^{3/2} \left (a b+b^2 x^2\right )^3} \, dx}{\sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{1}{4 a d \sqrt{d x} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (9 b \left (a b+b^2 x^2\right )\right ) \int \frac{1}{(d x)^{3/2} \left (a b+b^2 x^2\right )^2} \, dx}{8 a \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{9}{16 a^2 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a d \sqrt{d x} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (45 \left (a b+b^2 x^2\right )\right ) \int \frac{1}{(d x)^{3/2} \left (a b+b^2 x^2\right )} \, dx}{32 a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{9}{16 a^2 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a d \sqrt{d x} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \left (a+b x^2\right )}{16 a^3 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (45 b \left (a b+b^2 x^2\right )\right ) \int \frac{\sqrt{d x}}{a b+b^2 x^2} \, dx}{32 a^3 d^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{9}{16 a^2 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a d \sqrt{d x} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \left (a+b x^2\right )}{16 a^3 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (45 b \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \frac{x^2}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{16 a^3 d^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{9}{16 a^2 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a d \sqrt{d x} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \left (a+b x^2\right )}{16 a^3 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (45 \sqrt{b} \left (a b+b^2 x^2\right )\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 )}{32 a^3 d^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (45 \sqrt{b} \left (a b+b^2 x^2\right )\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 )}{32 a^3 d^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{9}{16 a^2 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a d \sqrt{d x} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \left (a+b x^2\right )}{16 a^3 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (45 \left (a b+b^2 x^2\right )\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 )}{64 \sqrt{2} a^{13/4} b^{3/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (45 \left (a b+b^2 x^2\right )\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 )}{64 \sqrt{2} a^{13/4} b^{3/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (45 \left (a b+b^2 x^2\right )\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 )}{64 a^3 b d \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (45 \left (a b+b^2 x^2\right )\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 )}{64 a^3 b d \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{9}{16 a^2 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a d \sqrt{d x} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \left (a+b x^2\right )}{16 a^3 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \sqrt [4]{b} \left (a+b x^2\right ) \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{64 \sqrt{2} a^{13/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 \sqrt [4]{b} \left (a+b x^2\right ) \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{64 \sqrt{2} a^{13/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (45 \left (a b+b^2 x^2\right )\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 )}{32 \sqrt{2} a^{13/4} b^{3/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (45 \left (a b+b^2 x^2\right )\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 )}{32 \sqrt{2} a^{13/4} b^{3/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{9}{16 a^2 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a d \sqrt{d x} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \left (a+b x^2\right )}{16 a^3 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 \sqrt [4]{b} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} a^{13/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \sqrt [4]{b} \left (a+b x^2\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} a^{13/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \sqrt [4]{b} \left (a+b x^2\right ) \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{64 \sqrt{2} a^{13/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 \sqrt [4]{b} \left (a+b x^2\right ) \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{64 \sqrt{2} a^{13/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [C] time = 0.0151114, size = 52, normalized size = 0.1 \[ -\frac{2 x \left (a+b x^2\right )^3 \, _2F_1\left (-\frac{1}{4},3;\frac{3}{4};-\frac{b x^2}{a}\right )}{a^3 (d x)^{3/2} \left (\left (a+b x^2\right )^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.237, size = 645, normalized size = 1.3 \begin{align*} \text{result too large to display} \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 = 1.77116, size = 837, normalized size = 1.65 \begin{align*} \frac{180 \,{\left (a^{3} b^{2} d^{2} x^{5} + 2 \, a^{4} b d^{2} x^{3} + a^{5} d^{2} x\right )} \left (-\frac{b}{a^{13} d^{6}}\right )^{\frac{1}{4}} \arctan \left (-\frac{91125 \, \sqrt{d x} a^{3} b d \left (-\frac{b}{a^{13} d^{6}}\right )^{\frac{1}{4}} - \sqrt{-8303765625 \, a^{7} b d^{4} \sqrt{-\frac{b}{a^{13} d^{6}}} + 8303765625 \, b^{2} d x} a^{3} d \left (-\frac{b}{a^{13} d^{6}}\right )^{\frac{1}{4}}}{91125 \, b}\right ) - 45 \,{\left (a^{3} b^{2} d^{2} x^{5} + 2 \, a^{4} b d^{2} x^{3} + a^{5} d^{2} x\right )} \left (-\frac{b}{a^{13} d^{6}}\right )^{\frac{1}{4}} \log \left (91125 \, a^{10} d^{5} \left (-\frac{b}{a^{13} d^{6}}\right )^{\frac{3}{4}} + 91125 \, \sqrt{d x} b\right ) + 45 \,{\left (a^{3} b^{2} d^{2} x^{5} + 2 \, a^{4} b d^{2} x^{3} + a^{5} d^{2} x\right )} \left (-\frac{b}{a^{13} d^{6}}\right )^{\frac{1}{4}} \log \left (-91125 \, a^{10} d^{5} \left (-\frac{b}{a^{13} d^{6}}\right )^{\frac{3}{4}} + 91125 \, \sqrt{d x} b\right ) - 4 \,{\left (45 \, b^{2} x^{4} + 81 \, a b x^{2} + 32 \, a^{2}\right )} \sqrt{d x}}{64 \,{\left (a^{3} b^{2} d^{2} x^{5} + 2 \, a^{4} b d^{2} x^{3} + a^{5} 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 (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.40361, size = 554, normalized size = 1.09 \begin{align*} -\frac{\frac{256}{\sqrt{d x} a^{3} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{8 \,{\left (13 \, \sqrt{d x} b^{2} d^{3} x^{3} + 17 \, \sqrt{d x} a b d^{3} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{2} a^{3} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{90 \, \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^{4} b^{2} d^{2} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{90 \, \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^{4} b^{2} d^{2} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{45 \, \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^{4} b^{2} d^{2} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{45 \, \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^{4} b^{2} d^{2} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )}}{128 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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