Optimal. Leaf size=149 \[ \frac{5 c^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{21 b^{9/4} \sqrt{b x^2+c x^4}}+\frac{10 c \sqrt{b x^2+c x^4}}{21 b^2 x^{5/2}}-\frac{2 \sqrt{b x^2+c x^4}}{7 b x^{9/2}} \]
[Out]
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Rubi [A] time = 0.366148, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{5 c^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{21 b^{9/4} \sqrt{b x^2+c x^4}}+\frac{10 c \sqrt{b x^2+c x^4}}{21 b^2 x^{5/2}}-\frac{2 \sqrt{b x^2+c x^4}}{7 b x^{9/2}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^(7/2)*Sqrt[b*x^2 + c*x^4]),x]
[Out]
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Rubi in Sympy [A] time = 34.905, size = 143, normalized size = 0.96 \[ - \frac{2 \sqrt{b x^{2} + c x^{4}}}{7 b x^{\frac{9}{2}}} + \frac{10 c \sqrt{b x^{2} + c x^{4}}}{21 b^{2} x^{\frac{5}{2}}} + \frac{5 c^{\frac{7}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{21 b^{\frac{9}{4}} x \left (b + c x^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**(7/2)/(c*x**4+b*x**2)**(1/2),x)
[Out]
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Mathematica [C] time = 0.161682, size = 144, normalized size = 0.97 \[ \frac{x \left (\frac{10 c}{21 b^2 x^{3/2}}-\frac{2}{7 b x^{7/2}}\right ) \left (b+c x^2\right )}{\sqrt{x^2 \left (b+c x^2\right )}}+\frac{10 i c^2 x^2 \sqrt{\frac{b}{c x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )}{21 b^2 \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^(7/2)*Sqrt[b*x^2 + c*x^4]),x]
[Out]
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Maple [A] time = 0.022, size = 134, normalized size = 0.9 \[{\frac{1}{21\,{b}^{2}} \left ( 5\,\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}\sqrt{2}{x}^{3}c+10\,{c}^{2}{x}^{4}+4\,bc{x}^{2}-6\,{b}^{2} \right ){x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^(7/2)/(c*x^4+b*x^2)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{4} + b x^{2}} x^{\frac{7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*x^4 + b*x^2)*x^(7/2)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{c x^{4} + b x^{2}} x^{\frac{7}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*x^4 + b*x^2)*x^(7/2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{\frac{7}{2}} \sqrt{x^{2} \left (b + c x^{2}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**(7/2)/(c*x**4+b*x**2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{4} + b x^{2}} x^{\frac{7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*x^4 + b*x^2)*x^(7/2)),x, algorithm="giac")
[Out]