Optimal. Leaf size=291 \[ -\frac{21 b^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{10 c^{11/4} \sqrt{b x^2+c x^4}}+\frac{21 b^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 c^{11/4} \sqrt{b x^2+c x^4}}-\frac{21 b x^{3/2} \left (b+c x^2\right )}{5 c^{5/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}+\frac{7 \sqrt{x} \sqrt{b x^2+c x^4}}{5 c^2}-\frac{x^{9/2}}{c \sqrt{b x^2+c x^4}} \]
[Out]
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Rubi [A] time = 0.624546, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{21 b^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{10 c^{11/4} \sqrt{b x^2+c x^4}}+\frac{21 b^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 c^{11/4} \sqrt{b x^2+c x^4}}-\frac{21 b x^{3/2} \left (b+c x^2\right )}{5 c^{5/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}+\frac{7 \sqrt{x} \sqrt{b x^2+c x^4}}{5 c^2}-\frac{x^{9/2}}{c \sqrt{b x^2+c x^4}} \]
Antiderivative was successfully verified.
[In] Int[x^(15/2)/(b*x^2 + c*x^4)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 59.285, size = 272, normalized size = 0.93 \[ \frac{21 b^{\frac{5}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \sqrt{b x^{2} + c x^{4}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{5 c^{\frac{11}{4}} x \left (b + c x^{2}\right )} - \frac{21 b^{\frac{5}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{10 c^{\frac{11}{4}} x \left (b + c x^{2}\right )} - \frac{21 b \sqrt{b x^{2} + c x^{4}}}{5 c^{\frac{5}{2}} \sqrt{x} \left (\sqrt{b} + \sqrt{c} x\right )} - \frac{x^{\frac{9}{2}}}{c \sqrt{b x^{2} + c x^{4}}} + \frac{7 \sqrt{x} \sqrt{b x^{2} + c x^{4}}}{5 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(15/2)/(c*x**4+b*x**2)**(3/2),x)
[Out]
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Mathematica [C] time = 0.209121, size = 179, normalized size = 0.62 \[ \frac{x^{3/2} \left (21 b^{3/2} \sqrt{\frac{c x^2}{b}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}}\right )\right |-1\right )-21 b^{3/2} \sqrt{\frac{c x^2}{b}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}}\right )\right |-1\right )+\sqrt{c} x \sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}} \left (7 b+2 c x^2\right )\right )}{5 c^{5/2} \sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}} \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(15/2)/(b*x^2 + c*x^4)^(3/2),x]
[Out]
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Maple [A] time = 0.046, size = 213, normalized size = 0.7 \[ -{\frac{c{x}^{2}+b}{10\,{c}^{3}}{x}^{{\frac{5}{2}}} \left ( 42\,{b}^{2}\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) -21\,{b}^{2}\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) -4\,{c}^{2}{x}^{4}-14\,bc{x}^{2} \right ) \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(15/2)/(c*x^4+b*x^2)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{15}{2}}}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(15/2)/(c*x^4 + b*x^2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{\frac{11}{2}}}{\sqrt{c x^{4} + b x^{2}}{\left (c x^{2} + b\right )}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(15/2)/(c*x^4 + b*x^2)^(3/2),x, algorithm="fricas")
[Out]
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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**(15/2)/(c*x**4+b*x**2)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{15}{2}}}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(15/2)/(c*x^4 + b*x^2)^(3/2),x, algorithm="giac")
[Out]