Optimal. Leaf size=255 \[ -\frac{2 \sqrt{x} (b B-A c)}{c^2}-\frac{\sqrt [4]{b} (b B-A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} c^{9/4}}+\frac{\sqrt [4]{b} (b B-A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} c^{9/4}}-\frac{\sqrt [4]{b} (b B-A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} c^{9/4}}+\frac{\sqrt [4]{b} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} c^{9/4}}+\frac{2 B x^{5/2}}{5 c} \]
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Rubi [A] time = 0.209264, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {1584, 459, 321, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{2 \sqrt{x} (b B-A c)}{c^2}-\frac{\sqrt [4]{b} (b B-A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} c^{9/4}}+\frac{\sqrt [4]{b} (b B-A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} c^{9/4}}-\frac{\sqrt [4]{b} (b B-A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} c^{9/4}}+\frac{\sqrt [4]{b} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} c^{9/4}}+\frac{2 B x^{5/2}}{5 c} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 459
Rule 321
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{x^{7/2} \left (A+B x^2\right )}{b x^2+c x^4} \, dx &=\int \frac{x^{3/2} \left (A+B x^2\right )}{b+c x^2} \, dx\\ &=\frac{2 B x^{5/2}}{5 c}-\frac{\left (2 \left (\frac{5 b B}{2}-\frac{5 A c}{2}\right )\right ) \int \frac{x^{3/2}}{b+c x^2} \, dx}{5 c}\\ &=-\frac{2 (b B-A c) \sqrt{x}}{c^2}+\frac{2 B x^{5/2}}{5 c}+\frac{(b (b B-A c)) \int \frac{1}{\sqrt{x} \left (b+c x^2\right )} \, dx}{c^2}\\ &=-\frac{2 (b B-A c) \sqrt{x}}{c^2}+\frac{2 B x^{5/2}}{5 c}+\frac{(2 b (b B-A c)) \operatorname{Subst}\left (\int \frac{1}{b+c x^4} \, dx,x,\sqrt{x}\right )}{c^2}\\ &=-\frac{2 (b B-A c) \sqrt{x}}{c^2}+\frac{2 B x^{5/2}}{5 c}+\frac{\left (\sqrt{b} (b B-A c)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b}-\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{c^2}+\frac{\left (\sqrt{b} (b B-A c)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b}+\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{c^2}\\ &=-\frac{2 (b B-A c) \sqrt{x}}{c^2}+\frac{2 B x^{5/2}}{5 c}+\frac{\left (\sqrt{b} (b B-A c)\right ) \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 )}{2 c^{5/2}}+\frac{\left (\sqrt{b} (b B-A c)\right ) \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 )}{2 c^{5/2}}-\frac{\left (\sqrt [4]{b} (b B-A c)\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 )}{2 \sqrt{2} c^{9/4}}-\frac{\left (\sqrt [4]{b} (b B-A c)\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 )}{2 \sqrt{2} c^{9/4}}\\ &=-\frac{2 (b B-A c) \sqrt{x}}{c^2}+\frac{2 B x^{5/2}}{5 c}-\frac{\sqrt [4]{b} (b B-A c) \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} c^{9/4}}+\frac{\sqrt [4]{b} (b B-A c) \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} c^{9/4}}+\frac{\left (\sqrt [4]{b} (b B-A c)\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 )}{\sqrt{2} c^{9/4}}-\frac{\left (\sqrt [4]{b} (b B-A c)\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 )}{\sqrt{2} c^{9/4}}\\ &=-\frac{2 (b B-A c) \sqrt{x}}{c^2}+\frac{2 B x^{5/2}}{5 c}-\frac{\sqrt [4]{b} (b B-A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} c^{9/4}}+\frac{\sqrt [4]{b} (b B-A c) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} c^{9/4}}-\frac{\sqrt [4]{b} (b B-A c) \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} c^{9/4}}+\frac{\sqrt [4]{b} (b B-A c) \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} c^{9/4}}\\ \end{align*}
Mathematica [A] time = 0.201955, size = 208, normalized size = 0.82 \[ \frac{-40 \sqrt{x} (b B-A c)+\frac{5 \sqrt{2} \sqrt [4]{b} (A c-b B) \left (\log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-\log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )\right )}{\sqrt [4]{c}}-\frac{10 \sqrt{2} \sqrt [4]{b} (b B-A c) \left (\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )-\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )\right )}{\sqrt [4]{c}}+8 B c x^{5/2}}{20 c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 299, normalized size = 1.2 \begin{align*}{\frac{2\,B}{5\,c}{x}^{{\frac{5}{2}}}}+2\,{\frac{A\sqrt{x}}{c}}-2\,{\frac{Bb\sqrt{x}}{{c}^{2}}}-{\frac{\sqrt{2}A}{2\,c}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }-{\frac{\sqrt{2}A}{2\,c}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }-{\frac{\sqrt{2}A}{4\,c}\sqrt [4]{{\frac{b}{c}}}\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{\sqrt{2}Bb}{2\,{c}^{2}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }+{\frac{\sqrt{2}Bb}{2\,{c}^{2}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }+{\frac{\sqrt{2}Bb}{4\,{c}^{2}}\sqrt [4]{{\frac{b}{c}}}\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 ) } \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 [B] time = 2.07589, size = 1365, normalized size = 5.35 \begin{align*} -\frac{20 \, c^{2} \left (-\frac{B^{4} b^{5} - 4 \, A B^{3} b^{4} c + 6 \, A^{2} B^{2} b^{3} c^{2} - 4 \, A^{3} B b^{2} c^{3} + A^{4} b c^{4}}{c^{9}}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{c^{4} \sqrt{-\frac{B^{4} b^{5} - 4 \, A B^{3} b^{4} c + 6 \, A^{2} B^{2} b^{3} c^{2} - 4 \, A^{3} B b^{2} c^{3} + A^{4} b c^{4}}{c^{9}}} +{\left (B^{2} b^{2} - 2 \, A B b c + A^{2} c^{2}\right )} x} c^{7} \left (-\frac{B^{4} b^{5} - 4 \, A B^{3} b^{4} c + 6 \, A^{2} B^{2} b^{3} c^{2} - 4 \, A^{3} B b^{2} c^{3} + A^{4} b c^{4}}{c^{9}}\right )^{\frac{3}{4}} +{\left (B b c^{7} - A c^{8}\right )} \sqrt{x} \left (-\frac{B^{4} b^{5} - 4 \, A B^{3} b^{4} c + 6 \, A^{2} B^{2} b^{3} c^{2} - 4 \, A^{3} B b^{2} c^{3} + A^{4} b c^{4}}{c^{9}}\right )^{\frac{3}{4}}}{B^{4} b^{5} - 4 \, A B^{3} b^{4} c + 6 \, A^{2} B^{2} b^{3} c^{2} - 4 \, A^{3} B b^{2} c^{3} + A^{4} b c^{4}}\right ) + 5 \, c^{2} \left (-\frac{B^{4} b^{5} - 4 \, A B^{3} b^{4} c + 6 \, A^{2} B^{2} b^{3} c^{2} - 4 \, A^{3} B b^{2} c^{3} + A^{4} b c^{4}}{c^{9}}\right )^{\frac{1}{4}} \log \left (c^{2} \left (-\frac{B^{4} b^{5} - 4 \, A B^{3} b^{4} c + 6 \, A^{2} B^{2} b^{3} c^{2} - 4 \, A^{3} B b^{2} c^{3} + A^{4} b c^{4}}{c^{9}}\right )^{\frac{1}{4}} -{\left (B b - A c\right )} \sqrt{x}\right ) - 5 \, c^{2} \left (-\frac{B^{4} b^{5} - 4 \, A B^{3} b^{4} c + 6 \, A^{2} B^{2} b^{3} c^{2} - 4 \, A^{3} B b^{2} c^{3} + A^{4} b c^{4}}{c^{9}}\right )^{\frac{1}{4}} \log \left (-c^{2} \left (-\frac{B^{4} b^{5} - 4 \, A B^{3} b^{4} c + 6 \, A^{2} B^{2} b^{3} c^{2} - 4 \, A^{3} B b^{2} c^{3} + A^{4} b c^{4}}{c^{9}}\right )^{\frac{1}{4}} -{\left (B b - A c\right )} \sqrt{x}\right ) - 4 \,{\left (B c x^{2} - 5 \, B b + 5 \, A c\right )} \sqrt{x}}{10 \, c^{2}} \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 = 1.12993, size = 355, normalized size = 1.39 \begin{align*} \frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{1}{4}} B b - \left (b c^{3}\right )^{\frac{1}{4}} A c\right )} \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 )}{2 \, c^{3}} + \frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{1}{4}} B b - \left (b c^{3}\right )^{\frac{1}{4}} A c\right )} \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 )}{2 \, c^{3}} + \frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{1}{4}} B b - \left (b c^{3}\right )^{\frac{1}{4}} A c\right )} \log \left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, c^{3}} - \frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{1}{4}} B b - \left (b c^{3}\right )^{\frac{1}{4}} A c\right )} \log \left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, c^{3}} + \frac{2 \,{\left (B c^{4} x^{\frac{5}{2}} - 5 \, B b c^{3} \sqrt{x} + 5 \, A c^{4} \sqrt{x}\right )}}{5 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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