Optimal. Leaf size=410 \[ \frac{\sqrt [4]{a} d^{3/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{2 \sqrt{2} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\sqrt [4]{a} d^{3/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{2 \sqrt{2} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\sqrt [4]{a} d^{3/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{\sqrt{2} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\sqrt [4]{a} d^{3/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{\sqrt{2} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{2 d \sqrt{d x} \left (a+b x^2\right )}{b \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.27544, antiderivative size = 410, 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, 321, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{\sqrt [4]{a} d^{3/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{2 \sqrt{2} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\sqrt [4]{a} d^{3/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{2 \sqrt{2} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\sqrt [4]{a} d^{3/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{\sqrt{2} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\sqrt [4]{a} d^{3/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{\sqrt{2} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{2 d \sqrt{d x} \left (a+b x^2\right )}{b \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1112
Rule 321
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{(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{(d x)^{3/2}}{a b+b^2 x^2} \, dx}{\sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{2 d \sqrt{d x} \left (a+b x^2\right )}{b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (a 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}{b \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{2 d \sqrt{d x} \left (a+b x^2\right )}{b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (2 a 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 )}{b \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{2 d \sqrt{d x} \left (a+b x^2\right )}{b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (\sqrt{a} \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 )}{b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (\sqrt{a} \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 )}{b \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{2 d \sqrt{d x} \left (a+b x^2\right )}{b \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (\sqrt [4]{a} 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 )}{2 \sqrt{2} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (\sqrt [4]{a} 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 )}{2 \sqrt{2} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (\sqrt{a} 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 )}{2 b^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (\sqrt{a} 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 )}{2 b^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{2 d \sqrt{d x} \left (a+b x^2\right )}{b \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\sqrt [4]{a} 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 )}{2 \sqrt{2} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\sqrt [4]{a} 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 )}{2 \sqrt{2} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (\sqrt [4]{a} 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 )}{\sqrt{2} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (\sqrt [4]{a} 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 )}{\sqrt{2} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{2 d \sqrt{d x} \left (a+b x^2\right )}{b \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\sqrt [4]{a} 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 )}{\sqrt{2} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\sqrt [4]{a} 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 )}{\sqrt{2} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\sqrt [4]{a} 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 )}{2 \sqrt{2} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\sqrt [4]{a} 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 )}{2 \sqrt{2} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [A] time = 0.0732777, size = 221, normalized size = 0.54 \[ \frac{(d x)^{3/2} \left (a+b x^2\right ) \left (\sqrt{2} \sqrt [4]{a} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-\sqrt{2} \sqrt [4]{a} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+2 \sqrt{2} \sqrt [4]{a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )-2 \sqrt{2} \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )+8 \sqrt [4]{b} \sqrt{x}\right )}{4 b^{5/4} x^{3/2} \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.227, size = 214, normalized size = 0.5 \begin{align*} -{\frac{ \left ( b{x}^{2}+a \right ) d}{4\,b} \left ( \sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{2}\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 ) +2\,\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{2}\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 [4]{{\frac{a{d}^{2}}{b}}}\sqrt{2}\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{dx} \right ){\frac{1}{\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d x\right )^{\frac{3}{2}}}{\sqrt{{\left (b x^{2} + a\right )}^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.58735, size = 390, normalized size = 0.95 \begin{align*} -\frac{4 \, \left (-\frac{a d^{6}}{b^{5}}\right )^{\frac{1}{4}} b \arctan \left (-\frac{\left (-\frac{a d^{6}}{b^{5}}\right )^{\frac{3}{4}} \sqrt{d x} b^{4} d - \sqrt{d^{3} x + \sqrt{-\frac{a d^{6}}{b^{5}}} b^{2}} \left (-\frac{a d^{6}}{b^{5}}\right )^{\frac{3}{4}} b^{4}}{a d^{6}}\right ) + \left (-\frac{a d^{6}}{b^{5}}\right )^{\frac{1}{4}} b \log \left (\sqrt{d x} d + \left (-\frac{a d^{6}}{b^{5}}\right )^{\frac{1}{4}} b\right ) - \left (-\frac{a d^{6}}{b^{5}}\right )^{\frac{1}{4}} b \log \left (\sqrt{d x} d - \left (-\frac{a d^{6}}{b^{5}}\right )^{\frac{1}{4}} b\right ) - 4 \, \sqrt{d x} d}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.27476, size = 327, normalized size = 0.8 \begin{align*} -\frac{1}{4} \,{\left (\frac{2 \, \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 )}{b^{2}} + \frac{2 \, \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 )}{b^{2}} + \frac{\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 )}{b^{2}} - \frac{\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 )}{b^{2}} - \frac{8 \, \sqrt{d x} d}{b}\right )} \mathrm{sgn}\left (b x^{2} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]