Optimal. Leaf size=251 \[ \frac{77 c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{15/4}}-\frac{77 c^{3/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{15/4}}+\frac{77 c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{15/4}}-\frac{77 c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{15/4}}-\frac{77}{48 b^3 x^{3/2}}+\frac{11}{16 b^2 x^{3/2} \left (b+c x^2\right )}+\frac{1}{4 b x^{3/2} \left (b+c x^2\right )^2} \]
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Rubi [A] time = 0.430924, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526 \[ \frac{77 c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{15/4}}-\frac{77 c^{3/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{15/4}}+\frac{77 c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{15/4}}-\frac{77 c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{15/4}}-\frac{77}{48 b^3 x^{3/2}}+\frac{11}{16 b^2 x^{3/2} \left (b+c x^2\right )}+\frac{1}{4 b x^{3/2} \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
[In] Int[x^(7/2)/(b*x^2 + c*x^4)^3,x]
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Rubi in Sympy [A] time = 81.4421, size = 238, normalized size = 0.95 \[ \frac{1}{4 b x^{\frac{3}{2}} \left (b + c x^{2}\right )^{2}} + \frac{11}{16 b^{2} x^{\frac{3}{2}} \left (b + c x^{2}\right )} - \frac{77}{48 b^{3} x^{\frac{3}{2}}} + \frac{77 \sqrt{2} c^{\frac{3}{4}} \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 b^{\frac{15}{4}}} - \frac{77 \sqrt{2} c^{\frac{3}{4}} \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 b^{\frac{15}{4}}} + \frac{77 \sqrt{2} c^{\frac{3}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 b^{\frac{15}{4}}} - \frac{77 \sqrt{2} c^{\frac{3}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 b^{\frac{15}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(7/2)/(c*x**4+b*x**2)**3,x)
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Mathematica [A] time = 0.209879, size = 234, normalized size = 0.93 \[ \frac{-\frac{96 b^{7/4} c \sqrt{x}}{\left (b+c x^2\right )^2}-\frac{360 b^{3/4} c \sqrt{x}}{b+c x^2}-\frac{256 b^{3/4}}{x^{3/2}}+231 \sqrt{2} c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-231 \sqrt{2} c^{3/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+462 \sqrt{2} c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )-462 \sqrt{2} c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{384 b^{15/4}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(7/2)/(b*x^2 + c*x^4)^3,x]
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Maple [A] time = 0.025, size = 181, normalized size = 0.7 \[ -{\frac{15\,{c}^{2}}{16\,{b}^{3} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{5}{2}}}}-{\frac{19\,c}{16\,{b}^{2} \left ( c{x}^{2}+b \right ) ^{2}}\sqrt{x}}-{\frac{77\,c\sqrt{2}}{128\,{b}^{4}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }-{\frac{77\,c\sqrt{2}}{64\,{b}^{4}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }-{\frac{77\,c\sqrt{2}}{64\,{b}^{4}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }-{\frac{2}{3\,{b}^{3}}{x}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(7/2)/(c*x^4+b*x^2)^3,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(7/2)/(c*x^4 + b*x^2)^3,x, algorithm="maxima")
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Fricas [A] time = 0.290843, size = 356, normalized size = 1.42 \[ -\frac{308 \, c^{2} x^{4} + 484 \, b c x^{2} - 924 \,{\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )} \sqrt{x} \left (-\frac{c^{3}}{b^{15}}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{4} \left (-\frac{c^{3}}{b^{15}}\right )^{\frac{1}{4}}}{c \sqrt{x} + \sqrt{b^{8} \sqrt{-\frac{c^{3}}{b^{15}}} + c^{2} x}}\right ) + 231 \,{\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )} \sqrt{x} \left (-\frac{c^{3}}{b^{15}}\right )^{\frac{1}{4}} \log \left (77 \, b^{4} \left (-\frac{c^{3}}{b^{15}}\right )^{\frac{1}{4}} + 77 \, c \sqrt{x}\right ) - 231 \,{\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )} \sqrt{x} \left (-\frac{c^{3}}{b^{15}}\right )^{\frac{1}{4}} \log \left (-77 \, b^{4} \left (-\frac{c^{3}}{b^{15}}\right )^{\frac{1}{4}} + 77 \, c \sqrt{x}\right ) + 128 \, b^{2}}{192 \,{\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )} \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(7/2)/(c*x^4 + b*x^2)^3,x, algorithm="fricas")
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(7/2)/(c*x**4+b*x**2)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.280037, size = 281, normalized size = 1.12 \[ -\frac{77 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b^{4}} - \frac{77 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b^{4}} - \frac{77 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{4}} + \frac{77 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{4}} - \frac{15 \, c^{2} x^{\frac{5}{2}} + 19 \, b c \sqrt{x}}{16 \,{\left (c x^{2} + b\right )}^{2} b^{3}} - \frac{2}{3 \, b^{3} x^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(7/2)/(c*x^4 + b*x^2)^3,x, algorithm="giac")
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