Optimal. Leaf size=63 \[ -\frac{2 \sqrt{c x} \sqrt [4]{\frac{a}{b x^2}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} \sqrt{b} \sqrt [4]{a+b x^2}} \]
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Rubi [A] time = 0.0742066, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{2 \sqrt{c x} \sqrt [4]{\frac{a}{b x^2}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} \sqrt{b} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c*x]/(a + b*x^2)^(5/4),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\sqrt{c x} \sqrt [4]{\frac{a}{b x^{2}} + 1} \int ^{\frac{1}{x}} \frac{1}{\sqrt [4]{\frac{a x^{2}}{b} + 1}}\, dx}{b \sqrt [4]{a + b x^{2}}} - \frac{2 \sqrt{c x}}{b x \sqrt [4]{a + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x)**(1/2)/(b*x**2+a)**(5/4),x)
[Out]
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Mathematica [C] time = 0.0503388, size = 63, normalized size = 1. \[ -\frac{2 x \sqrt{c x} \left (2 \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )-3\right )}{3 a \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[c*x]/(a + b*x^2)^(5/4),x]
[Out]
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Maple [F] time = 0.039, size = 0, normalized size = 0. \[ \int{1\sqrt{cx} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x)^(1/2)/(b*x^2+a)^(5/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x}}{{\left (b x^{2} + a\right )}^{\frac{5}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x)/(b*x^2 + a)^(5/4),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x}}{{\left (b x^{2} + a\right )}^{\frac{5}{4}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x)/(b*x^2 + a)^(5/4),x, algorithm="fricas")
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Sympy [A] time = 14.1861, size = 44, normalized size = 0.7 \[ \frac{\sqrt{c} x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{5}{4}} \Gamma \left (\frac{7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x)**(1/2)/(b*x**2+a)**(5/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x}}{{\left (b x^{2} + a\right )}^{\frac{5}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x)/(b*x^2 + a)^(5/4),x, algorithm="giac")
[Out]