Optimal. Leaf size=146 \[ -\frac{2 c^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{21 b^{5/4} \sqrt{b x^2+c x^4}}-\frac{4 c \sqrt{b x^2+c x^4}}{21 b x^{5/2}}-\frac{2 \sqrt{b x^2+c x^4}}{7 x^{9/2}} \]
[Out]
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Rubi [A] time = 0.359456, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{2 c^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{21 b^{5/4} \sqrt{b x^2+c x^4}}-\frac{4 c \sqrt{b x^2+c x^4}}{21 b x^{5/2}}-\frac{2 \sqrt{b x^2+c x^4}}{7 x^{9/2}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[b*x^2 + c*x^4]/x^(11/2),x]
[Out]
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Rubi in Sympy [A] time = 32.5988, size = 141, normalized size = 0.97 \[ - \frac{2 \sqrt{b x^{2} + c x^{4}}}{7 x^{\frac{9}{2}}} - \frac{4 c \sqrt{b x^{2} + c x^{4}}}{21 b x^{\frac{5}{2}}} - \frac{2 c^{\frac{7}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{21 b^{\frac{5}{4}} x \left (b + c x^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**4+b*x**2)**(1/2)/x**(11/2),x)
[Out]
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Mathematica [C] time = 0.429399, size = 122, normalized size = 0.84 \[ \frac{1}{21} \sqrt{x^2 \left (b+c x^2\right )} \left (-\frac{2 \left (3 b+2 c x^2\right )}{b x^{9/2}}-\frac{4 i c^2 \sqrt{\frac{b}{c x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )}{b \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \left (b+c x^2\right )}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[b*x^2 + c*x^4]/x^(11/2),x]
[Out]
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Maple [A] time = 0.039, size = 142, normalized size = 1. \[ -{\frac{2}{ \left ( 21\,c{x}^{2}+21\,b \right ) b}\sqrt{c{x}^{4}+b{x}^{2}} \left ( \sqrt{{1 \left ( cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}}\sqrt{{1 \left ( -cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}}\sqrt{-{cx{\frac{1}{\sqrt{-bc}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-bc}\sqrt{2}{x}^{3}c+2\,{c}^{2}{x}^{4}+5\,bc{x}^{2}+3\,{b}^{2} \right ){x}^{-{\frac{9}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^4+b*x^2)^(1/2)/x^(11/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{4} + b x^{2}}}{x^{\frac{11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^4 + b*x^2)/x^(11/2),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x^{4} + b x^{2}}}{x^{\frac{11}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^4 + b*x^2)/x^(11/2),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((c*x**4+b*x**2)**(1/2)/x**(11/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{4} + b x^{2}}}{x^{\frac{11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^4 + b*x^2)/x^(11/2),x, algorithm="giac")
[Out]