Optimal. Leaf size=332 \[ -\frac{\sqrt{x} \left (7 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{6 c^2 d \left (c+d x^2\right )}-\frac{2 a^2}{3 c x^{3/2} \left (c+d x^2\right )}-\frac{(b c-a d) (7 a d+b c) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{11/4} d^{5/4}}+\frac{(b c-a d) (7 a d+b c) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{11/4} d^{5/4}}-\frac{(b c-a d) (7 a d+b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{11/4} d^{5/4}}+\frac{(b c-a d) (7 a d+b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{11/4} d^{5/4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.343347, antiderivative size = 332, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {462, 457, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{\sqrt{x} \left (7 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{6 c^2 d \left (c+d x^2\right )}-\frac{2 a^2}{3 c x^{3/2} \left (c+d x^2\right )}-\frac{(b c-a d) (7 a d+b c) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{11/4} d^{5/4}}+\frac{(b c-a d) (7 a d+b c) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{11/4} d^{5/4}}-\frac{(b c-a d) (7 a d+b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{11/4} d^{5/4}}+\frac{(b c-a d) (7 a d+b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{11/4} d^{5/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 462
Rule 457
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )^2} \, dx &=-\frac{2 a^2}{3 c x^{3/2} \left (c+d x^2\right )}+\frac{2 \int \frac{\frac{1}{2} a (6 b c-7 a d)+\frac{3}{2} b^2 c x^2}{\sqrt{x} \left (c+d x^2\right )^2} \, dx}{3 c}\\ &=-\frac{2 a^2}{3 c x^{3/2} \left (c+d x^2\right )}-\frac{\left (3 b^2 c^2-6 a b c d+7 a^2 d^2\right ) \sqrt{x}}{6 c^2 d \left (c+d x^2\right )}+\frac{((b c-a d) (b c+7 a d)) \int \frac{1}{\sqrt{x} \left (c+d x^2\right )} \, dx}{4 c^2 d}\\ &=-\frac{2 a^2}{3 c x^{3/2} \left (c+d x^2\right )}-\frac{\left (3 b^2 c^2-6 a b c d+7 a^2 d^2\right ) \sqrt{x}}{6 c^2 d \left (c+d x^2\right )}+\frac{((b c-a d) (b c+7 a d)) \operatorname{Subst}\left (\int \frac{1}{c+d x^4} \, dx,x,\sqrt{x}\right )}{2 c^2 d}\\ &=-\frac{2 a^2}{3 c x^{3/2} \left (c+d x^2\right )}-\frac{\left (3 b^2 c^2-6 a b c d+7 a^2 d^2\right ) \sqrt{x}}{6 c^2 d \left (c+d x^2\right )}+\frac{((b c-a d) (b c+7 a d)) \operatorname{Subst}\left (\int \frac{\sqrt{c}-\sqrt{d} x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{4 c^{5/2} d}+\frac{((b c-a d) (b c+7 a d)) \operatorname{Subst}\left (\int \frac{\sqrt{c}+\sqrt{d} x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{4 c^{5/2} d}\\ &=-\frac{2 a^2}{3 c x^{3/2} \left (c+d x^2\right )}-\frac{\left (3 b^2 c^2-6 a b c d+7 a^2 d^2\right ) \sqrt{x}}{6 c^2 d \left (c+d x^2\right )}+\frac{((b c-a d) (b c+7 a d)) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{c}}{\sqrt{d}}-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt{x}\right )}{8 c^{5/2} d^{3/2}}+\frac{((b c-a d) (b c+7 a d)) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{c}}{\sqrt{d}}+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt{x}\right )}{8 c^{5/2} d^{3/2}}-\frac{((b c-a d) (b c+7 a d)) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac{\sqrt{c}}{\sqrt{d}}-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt{x}\right )}{8 \sqrt{2} c^{11/4} d^{5/4}}-\frac{((b c-a d) (b c+7 a d)) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac{\sqrt{c}}{\sqrt{d}}+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt{x}\right )}{8 \sqrt{2} c^{11/4} d^{5/4}}\\ &=-\frac{2 a^2}{3 c x^{3/2} \left (c+d x^2\right )}-\frac{\left (3 b^2 c^2-6 a b c d+7 a^2 d^2\right ) \sqrt{x}}{6 c^2 d \left (c+d x^2\right )}-\frac{(b c-a d) (b c+7 a d) \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{8 \sqrt{2} c^{11/4} d^{5/4}}+\frac{(b c-a d) (b c+7 a d) \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{8 \sqrt{2} c^{11/4} d^{5/4}}+\frac{((b c-a d) (b c+7 a d)) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{11/4} d^{5/4}}-\frac{((b c-a d) (b c+7 a d)) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{11/4} d^{5/4}}\\ &=-\frac{2 a^2}{3 c x^{3/2} \left (c+d x^2\right )}-\frac{\left (3 b^2 c^2-6 a b c d+7 a^2 d^2\right ) \sqrt{x}}{6 c^2 d \left (c+d x^2\right )}-\frac{(b c-a d) (b c+7 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{11/4} d^{5/4}}+\frac{(b c-a d) (b c+7 a d) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{11/4} d^{5/4}}-\frac{(b c-a d) (b c+7 a d) \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{8 \sqrt{2} c^{11/4} d^{5/4}}+\frac{(b c-a d) (b c+7 a d) \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{8 \sqrt{2} c^{11/4} d^{5/4}}\\ \end{align*}
Mathematica [A] time = 0.184479, size = 315, normalized size = 0.95 \[ \frac{-\frac{3 \sqrt{2} \left (-7 a^2 d^2+6 a b c d+b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{d^{5/4}}+\frac{3 \sqrt{2} \left (-7 a^2 d^2+6 a b c d+b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{d^{5/4}}-\frac{6 \sqrt{2} \left (-7 a^2 d^2+6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{d^{5/4}}+\frac{6 \sqrt{2} \left (-7 a^2 d^2+6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{d^{5/4}}-\frac{32 a^2 c^{3/4}}{x^{3/2}}-\frac{24 c^{3/4} \sqrt{x} (b c-a d)^2}{d \left (c+d x^2\right )}}{48 c^{11/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.017, size = 498, normalized size = 1.5 \begin{align*} -{\frac{{a}^{2}d}{2\,{c}^{2} \left ( d{x}^{2}+c \right ) }\sqrt{x}}+{\frac{ab}{c \left ( d{x}^{2}+c \right ) }\sqrt{x}}-{\frac{{b}^{2}}{2\,d \left ( d{x}^{2}+c \right ) }\sqrt{x}}-{\frac{7\,d\sqrt{2}{a}^{2}}{8\,{c}^{3}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }+{\frac{3\,\sqrt{2}ab}{4\,{c}^{2}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }+{\frac{\sqrt{2}{b}^{2}}{8\,cd}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }-{\frac{7\,d\sqrt{2}{a}^{2}}{8\,{c}^{3}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{3\,\sqrt{2}ab}{4\,{c}^{2}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{\sqrt{2}{b}^{2}}{8\,cd}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }-{\frac{7\,d\sqrt{2}{a}^{2}}{16\,{c}^{3}}\sqrt [4]{{\frac{c}{d}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ( x-\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }+{\frac{3\,\sqrt{2}ab}{8\,{c}^{2}}\sqrt [4]{{\frac{c}{d}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ( x-\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}{b}^{2}}{16\,cd}\sqrt [4]{{\frac{c}{d}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ( x-\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }-{\frac{2\,{a}^{2}}{3\,{c}^{2}}{x}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 0.967523, size = 2965, normalized size = 8.93 \begin{align*} \text{result too large to display} \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 = 1.20827, size = 518, normalized size = 1.56 \begin{align*} -\frac{2 \, a^{2}}{3 \, c^{2} x^{\frac{3}{2}}} + \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d - 7 \, \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{8 \, c^{3} d^{2}} + \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d - 7 \, \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{8 \, c^{3} d^{2}} + \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d - 7 \, \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \log \left (\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{16 \, c^{3} d^{2}} - \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d - 7 \, \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{16 \, c^{3} d^{2}} - \frac{b^{2} c^{2} \sqrt{x} - 2 \, a b c d \sqrt{x} + a^{2} d^{2} \sqrt{x}}{2 \,{\left (d x^{2} + c\right )} c^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]