Optimal. Leaf size=230 \[ -\frac{7 b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{11/4}}+\frac{7 b^{3/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{11/4}}+\frac{7 b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} c^{11/4}}-\frac{7 b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} c^{11/4}}-\frac{x^{7/2}}{2 c \left (b+c x^2\right )}+\frac{7 x^{3/2}}{6 c^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.182154, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526, Rules used = {1584, 288, 321, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac{7 b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{11/4}}+\frac{7 b^{3/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{11/4}}+\frac{7 b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} c^{11/4}}-\frac{7 b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} c^{11/4}}-\frac{x^{7/2}}{2 c \left (b+c x^2\right )}+\frac{7 x^{3/2}}{6 c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1584
Rule 288
Rule 321
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{x^{17/2}}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac{x^{9/2}}{\left (b+c x^2\right )^2} \, dx\\ &=-\frac{x^{7/2}}{2 c \left (b+c x^2\right )}+\frac{7 \int \frac{x^{5/2}}{b+c x^2} \, dx}{4 c}\\ &=\frac{7 x^{3/2}}{6 c^2}-\frac{x^{7/2}}{2 c \left (b+c x^2\right )}-\frac{(7 b) \int \frac{\sqrt{x}}{b+c x^2} \, dx}{4 c^2}\\ &=\frac{7 x^{3/2}}{6 c^2}-\frac{x^{7/2}}{2 c \left (b+c x^2\right )}-\frac{(7 b) \operatorname{Subst}\left (\int \frac{x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{2 c^2}\\ &=\frac{7 x^{3/2}}{6 c^2}-\frac{x^{7/2}}{2 c \left (b+c x^2\right )}+\frac{(7 b) \operatorname{Subst}\left (\int \frac{\sqrt{b}-\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{4 c^{5/2}}-\frac{(7 b) \operatorname{Subst}\left (\int \frac{\sqrt{b}+\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{4 c^{5/2}}\\ &=\frac{7 x^{3/2}}{6 c^2}-\frac{x^{7/2}}{2 c \left (b+c x^2\right )}-\frac{(7 b) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{b}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{x}\right )}{8 c^3}-\frac{(7 b) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{b}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{x}\right )}{8 c^3}-\frac{\left (7 b^{3/4}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{b}}{\sqrt [4]{c}}+2 x}{-\frac{\sqrt{b}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt{x}\right )}{8 \sqrt{2} c^{11/4}}-\frac{\left (7 b^{3/4}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{b}}{\sqrt [4]{c}}-2 x}{-\frac{\sqrt{b}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt{x}\right )}{8 \sqrt{2} c^{11/4}}\\ &=\frac{7 x^{3/2}}{6 c^2}-\frac{x^{7/2}}{2 c \left (b+c x^2\right )}-\frac{7 b^{3/4} \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} c^{11/4}}+\frac{7 b^{3/4} \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} c^{11/4}}-\frac{\left (7 b^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} c^{11/4}}+\frac{\left (7 b^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} c^{11/4}}\\ &=\frac{7 x^{3/2}}{6 c^2}-\frac{x^{7/2}}{2 c \left (b+c x^2\right )}+\frac{7 b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} c^{11/4}}-\frac{7 b^{3/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} c^{11/4}}-\frac{7 b^{3/4} \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} c^{11/4}}+\frac{7 b^{3/4} \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} c^{11/4}}\\ \end{align*}
Mathematica [C] time = 0.0167356, size = 57, normalized size = 0.25 \[ -\frac{2 x^{3/2} \left (7 \left (b+c x^2\right ) \, _2F_1\left (\frac{3}{4},2;\frac{7}{4};-\frac{c x^2}{b}\right )-7 b-c x^2\right )}{3 c^2 \left (b+c x^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.061, size = 161, normalized size = 0.7 \begin{align*}{\frac{2}{3\,{c}^{2}}{x}^{{\frac{3}{2}}}}+{\frac{b}{2\,{c}^{2} \left ( c{x}^{2}+b \right ) }{x}^{{\frac{3}{2}}}}-{\frac{7\,b\sqrt{2}}{16\,{c}^{3}}\ln \left ({ \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{7\,b\sqrt{2}}{8\,{c}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{7\,b\sqrt{2}}{8\,{c}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}} \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 [A] time = 1.47583, size = 556, normalized size = 2.42 \begin{align*} \frac{84 \,{\left (c^{3} x^{2} + b c^{2}\right )} \left (-\frac{b^{3}}{c^{11}}\right )^{\frac{1}{4}} \arctan \left (-\frac{343 \, b^{2} c^{3} \sqrt{x} \left (-\frac{b^{3}}{c^{11}}\right )^{\frac{1}{4}} - \sqrt{-117649 \, b^{3} c^{5} \sqrt{-\frac{b^{3}}{c^{11}}} + 117649 \, b^{4} x} c^{3} \left (-\frac{b^{3}}{c^{11}}\right )^{\frac{1}{4}}}{343 \, b^{3}}\right ) - 21 \,{\left (c^{3} x^{2} + b c^{2}\right )} \left (-\frac{b^{3}}{c^{11}}\right )^{\frac{1}{4}} \log \left (343 \, c^{8} \left (-\frac{b^{3}}{c^{11}}\right )^{\frac{3}{4}} + 343 \, b^{2} \sqrt{x}\right ) + 21 \,{\left (c^{3} x^{2} + b c^{2}\right )} \left (-\frac{b^{3}}{c^{11}}\right )^{\frac{1}{4}} \log \left (-343 \, c^{8} \left (-\frac{b^{3}}{c^{11}}\right )^{\frac{3}{4}} + 343 \, b^{2} \sqrt{x}\right ) + 4 \,{\left (4 \, c x^{3} + 7 \, b x\right )} \sqrt{x}}{24 \,{\left (c^{3} x^{2} + b c^{2}\right )}} \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 = 1.19619, size = 265, normalized size = 1.15 \begin{align*} \frac{b x^{\frac{3}{2}}}{2 \,{\left (c x^{2} + b\right )} c^{2}} + \frac{2 \, x^{\frac{3}{2}}}{3 \, c^{2}} - \frac{7 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, c^{5}} - \frac{7 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, c^{5}} + \frac{7 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \log \left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, c^{5}} - \frac{7 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \log \left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]