Optimal. Leaf size=412 \[ -\frac{\sqrt [4]{b} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{2 \sqrt{2} a^{5/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\sqrt [4]{b} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{2 \sqrt{2} a^{5/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\sqrt [4]{b} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{\sqrt{2} a^{5/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\sqrt [4]{b} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{\sqrt{2} a^{5/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{2 \left (a+b x^2\right )}{a d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.278618, antiderivative size = 412, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1112, 325, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac{\sqrt [4]{b} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{2 \sqrt{2} a^{5/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\sqrt [4]{b} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{2 \sqrt{2} a^{5/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\sqrt [4]{b} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{\sqrt{2} a^{5/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\sqrt [4]{b} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{\sqrt{2} a^{5/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{2 \left (a+b x^2\right )}{a d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1112
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} \sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx &=\frac{\left (a b+b^2 x^2\right ) \int \frac{1}{(d x)^{3/2} \left (a b+b^2 x^2\right )} \, dx}{\sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{2 \left (a+b x^2\right )}{a d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (b \left (a b+b^2 x^2\right )\right ) \int \frac{\sqrt{d x}}{a b+b^2 x^2} \, dx}{a d^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{2 \left (a+b x^2\right )}{a d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (2 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 )}{a d^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{2 \left (a+b x^2\right )}{a d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (\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 )}{a d^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (\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 )}{a d^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{2 \left (a+b x^2\right )}{a d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (a b+b^2 x^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 )}{2 \sqrt{2} a^{5/4} b^{3/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (a b+b^2 x^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 )}{2 \sqrt{2} a^{5/4} b^{3/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (a b+b^2 x^2\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 )}{2 a b d \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (a b+b^2 x^2\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 )}{2 a b d \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{2 \left (a+b x^2\right )}{a d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\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 )}{2 \sqrt{2} a^{5/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\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 )}{2 \sqrt{2} a^{5/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (a b+b^2 x^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 )}{\sqrt{2} a^{5/4} b^{3/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a b+b^2 x^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 )}{\sqrt{2} a^{5/4} b^{3/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{2 \left (a+b x^2\right )}{a d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\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 )}{\sqrt{2} a^{5/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\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 )}{\sqrt{2} a^{5/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\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 )}{2 \sqrt{2} a^{5/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\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 )}{2 \sqrt{2} a^{5/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [C] time = 0.0127298, size = 50, normalized size = 0.12 \[ -\frac{2 x \left (a+b x^2\right ) \, _2F_1\left (-\frac{1}{4},1;\frac{3}{4};-\frac{b x^2}{a}\right )}{a (d x)^{3/2} \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.227, size = 224, normalized size = 0.5 \begin{align*} -{\frac{b{x}^{2}+a}{4\,ad} \left ( \sqrt{2}\sqrt{dx}\ln \left ( -{ \left ( \sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}-dx-\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 ) +2\,\sqrt{2}\sqrt{dx}\arctan \left ({ \left ( \sqrt{2}\sqrt{dx}+\sqrt [4]{{\frac{a{d}^{2}}{b}}} \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}} \right ) +2\,\sqrt{2}\sqrt{dx}\arctan \left ({ \left ( \sqrt{2}\sqrt{dx}-\sqrt [4]{{\frac{a{d}^{2}}{b}}} \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}} \right ) +8\,\sqrt [4]{{\frac{a{d}^{2}}{b}}} \right ){\frac{1}{\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}}}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}{\frac{1}{\sqrt{dx}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{{\left (b x^{2} + a\right )}^{2}} \left (d x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61482, size = 464, normalized size = 1.13 \begin{align*} \frac{4 \, a d^{2} x \left (-\frac{b}{a^{5} d^{6}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{d x} a b d \left (-\frac{b}{a^{5} d^{6}}\right )^{\frac{1}{4}} - \sqrt{-a^{3} b d^{4} \sqrt{-\frac{b}{a^{5} d^{6}}} + b^{2} d x} a d \left (-\frac{b}{a^{5} d^{6}}\right )^{\frac{1}{4}}}{b}\right ) - a d^{2} x \left (-\frac{b}{a^{5} d^{6}}\right )^{\frac{1}{4}} \log \left (a^{4} d^{5} \left (-\frac{b}{a^{5} d^{6}}\right )^{\frac{3}{4}} + \sqrt{d x} b\right ) + a d^{2} x \left (-\frac{b}{a^{5} d^{6}}\right )^{\frac{1}{4}} \log \left (-a^{4} d^{5} \left (-\frac{b}{a^{5} d^{6}}\right )^{\frac{3}{4}} + \sqrt{d x} b\right ) - 4 \, \sqrt{d x}}{2 \, a d^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16243, size = 356, normalized size = 0.86 \begin{align*} -\frac{{\left (\frac{8}{\sqrt{d x} a} + \frac{2 \, \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^{2} b^{2} d^{2}} + \frac{2 \, \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^{2} b^{2} d^{2}} - \frac{\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^{2} b^{2} d^{2}} + \frac{\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^{2} b^{2} d^{2}}\right )} \mathrm{sgn}\left (b x^{2} + a\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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