Optimal. Leaf size=459 \[ -\frac{3 d^{3/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{7/4} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3 d^{3/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{7/4} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 d^{3/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} a^{7/4} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3 d^{3/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{32 \sqrt{2} a^{7/4} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{d \sqrt{d x}}{16 a b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d \sqrt{d x}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.326663, antiderivative size = 459, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1112, 288, 290, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{3 d^{3/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{7/4} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3 d^{3/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{7/4} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 d^{3/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} a^{7/4} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3 d^{3/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{32 \sqrt{2} a^{7/4} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{d \sqrt{d x}}{16 a b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d \sqrt{d x}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1112
Rule 288
Rule 290
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{(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{(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{d \sqrt{d x}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (d^2 \left (a b+b^2 x^2\right )\right ) \int \frac{1}{\sqrt{d x} \left (a b+b^2 x^2\right )^2} \, dx}{8 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{d \sqrt{d x}}{16 a b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d \sqrt{d x}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (3 d^2 \left (a b+b^2 x^2\right )\right ) \int \frac{1}{\sqrt{d x} \left (a b+b^2 x^2\right )} \, dx}{32 a b \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{d \sqrt{d x}}{16 a b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d \sqrt{d x}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (3 d \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{16 a b \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{d \sqrt{d x}}{16 a b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d \sqrt{d x}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (3 \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^{3/2} b \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (3 \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^{3/2} b \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{d \sqrt{d x}}{16 a b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d \sqrt{d x}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (3 d^{3/2} \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^{7/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (3 d^{3/2} \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^{7/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (3 d^2 \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^{3/2} b^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (3 d^2 \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^{3/2} b^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{d \sqrt{d x}}{16 a b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d \sqrt{d x}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 d^{3/2} \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^{7/4} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3 d^{3/2} \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^{7/4} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (3 d^{3/2} \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^{7/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (3 d^{3/2} \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^{7/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{d \sqrt{d x}}{16 a b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d \sqrt{d x}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 d^{3/2} \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^{7/4} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3 d^{3/2} \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^{7/4} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 d^{3/2} \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^{7/4} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3 d^{3/2} \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^{7/4} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [A] time = 0.216866, size = 272, normalized size = 0.59 \[ \frac{(d x)^{3/2} \left (a+b x^2\right ) \left (8 a^{3/4} \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )-32 a^{7/4} \sqrt [4]{b} \sqrt{x}-3 \sqrt{2} \left (a+b x^2\right )^2 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+3 \sqrt{2} \left (a+b x^2\right )^2 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-6 \sqrt{2} \left (a+b x^2\right )^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+6 \sqrt{2} \left (a+b x^2\right )^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )\right )}{128 a^{7/4} b^{5/4} x^{3/2} \left (\left (a+b x^2\right )^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.237, size = 668, normalized size = 1.5 \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 = 1.92385, size = 679, normalized size = 1.48 \begin{align*} \frac{12 \,{\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )} \left (-\frac{d^{6}}{a^{7} b^{5}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{d x} a^{5} b^{4} d \left (-\frac{d^{6}}{a^{7} b^{5}}\right )^{\frac{3}{4}} - \sqrt{a^{4} b^{2} \sqrt{-\frac{d^{6}}{a^{7} b^{5}}} + d^{3} x} a^{5} b^{4} \left (-\frac{d^{6}}{a^{7} b^{5}}\right )^{\frac{3}{4}}}{d^{6}}\right ) + 3 \,{\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )} \left (-\frac{d^{6}}{a^{7} b^{5}}\right )^{\frac{1}{4}} \log \left (3 \, a^{2} b \left (-\frac{d^{6}}{a^{7} b^{5}}\right )^{\frac{1}{4}} + 3 \, \sqrt{d x} d\right ) - 3 \,{\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )} \left (-\frac{d^{6}}{a^{7} b^{5}}\right )^{\frac{1}{4}} \log \left (-3 \, a^{2} b \left (-\frac{d^{6}}{a^{7} b^{5}}\right )^{\frac{1}{4}} + 3 \, \sqrt{d x} d\right ) + 4 \,{\left (b d x^{2} - 3 \, a d\right )} \sqrt{d x}}{64 \,{\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\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{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 = 1.39991, size = 497, normalized size = 1.08 \begin{align*} \frac{3 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \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 )}{64 \, a^{2} b^{2} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{3 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \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 )}{64 \, a^{2} b^{2} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{3 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \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 )}{128 \, a^{2} b^{2} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{3 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \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 )}{128 \, a^{2} b^{2} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{\sqrt{d x} b d^{5} x^{2} - 3 \, \sqrt{d x} a d^{5}}{16 \,{\left (b d^{2} x^{2} + a d^{2}\right )}^{2} a b \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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