Optimal. Leaf size=293 \[ -\frac{2 b^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 c^{7/4} \sqrt{b x^2+c x^4}}+\frac{4 b^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 c^{7/4} \sqrt{b x^2+c x^4}}-\frac{4 b^2 x^{3/2} \left (b+c x^2\right )}{15 c^{3/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}+\frac{4 b \sqrt{x} \sqrt{b x^2+c x^4}}{45 c}+\frac{2}{9} x^{5/2} \sqrt{b x^2+c x^4} \]
[Out]
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Rubi [A] time = 0.621729, antiderivative size = 293, 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 \[ -\frac{2 b^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 c^{7/4} \sqrt{b x^2+c x^4}}+\frac{4 b^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 c^{7/4} \sqrt{b x^2+c x^4}}-\frac{4 b^2 x^{3/2} \left (b+c x^2\right )}{15 c^{3/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}+\frac{4 b \sqrt{x} \sqrt{b x^2+c x^4}}{45 c}+\frac{2}{9} x^{5/2} \sqrt{b x^2+c x^4} \]
Antiderivative was successfully verified.
[In] Int[x^(3/2)*Sqrt[b*x^2 + c*x^4],x]
[Out]
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Rubi in Sympy [A] time = 56.5116, size = 275, normalized size = 0.94 \[ \frac{4 b^{\frac{9}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \sqrt{b x^{2} + c x^{4}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{15 c^{\frac{7}{4}} x \left (b + c x^{2}\right )} - \frac{2 b^{\frac{9}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{15 c^{\frac{7}{4}} x \left (b + c x^{2}\right )} - \frac{4 b^{2} \sqrt{b x^{2} + c x^{4}}}{15 c^{\frac{3}{2}} \sqrt{x} \left (\sqrt{b} + \sqrt{c} x\right )} + \frac{4 b \sqrt{x} \sqrt{b x^{2} + c x^{4}}}{45 c} + \frac{2 x^{\frac{5}{2}} \sqrt{b x^{2} + c x^{4}}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(3/2)*(c*x**4+b*x**2)**(1/2),x)
[Out]
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Mathematica [C] time = 0.314554, size = 190, normalized size = 0.65 \[ \frac{2 x^{3/2} \left (6 b^{5/2} \sqrt{\frac{c x^2}{b}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}}\right )\right |-1\right )-6 b^{5/2} \sqrt{\frac{c x^2}{b}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}}\right )\right |-1\right )+\sqrt{c} x \sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}} \left (2 b^2+7 b c x^2+5 c^2 x^4\right )\right )}{45 c^{3/2} \sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}} \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(3/2)*Sqrt[b*x^2 + c*x^4],x]
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Maple [A] time = 0.033, size = 226, normalized size = 0.8 \[ -{\frac{2}{ \left ( 45\,c{x}^{2}+45\,b \right ){c}^{2}}\sqrt{c{x}^{4}+b{x}^{2}} \left ( -5\,{c}^{3}{x}^{6}+6\,{b}^{3}\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) -3\,{b}^{3}\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) -7\,b{c}^{2}{x}^{4}-2\,{b}^{2}c{x}^{2} \right ){x}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(3/2)*(c*x^4+b*x^2)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c x^{4} + b x^{2}} x^{\frac{3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^4 + b*x^2)*x^(3/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{c x^{4} + b x^{2}} x^{\frac{3}{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^4 + b*x^2)*x^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x^{\frac{3}{2}} \sqrt{x^{2} \left (b + c x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(3/2)*(c*x**4+b*x**2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c x^{4} + b x^{2}} x^{\frac{3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^4 + b*x^2)*x^(3/2),x, algorithm="giac")
[Out]