Optimal. Leaf size=97 \[ \frac{\left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{\sqrt [4]{a} \sqrt [4]{b} \sqrt{c} \sqrt{a+b x^2}} \]
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Rubi [A] time = 0.156603, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{\left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{\sqrt [4]{a} \sqrt [4]{b} \sqrt{c} \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[c*x]*Sqrt[a + b*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 12.7659, size = 88, normalized size = 0.91 \[ \frac{\sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{\sqrt [4]{a} \sqrt [4]{b} \sqrt{c} \sqrt{a + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(c*x)**(1/2)/(b*x**2+a)**(1/2),x)
[Out]
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Mathematica [C] time = 0.049535, size = 90, normalized size = 0.93 \[ \frac{2 i x^{3/2} \sqrt{\frac{a}{b x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \sqrt{c x} \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[c*x]*Sqrt[a + b*x^2]),x]
[Out]
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Maple [A] time = 0.022, size = 104, normalized size = 1.1 \[{\frac{\sqrt{2}}{b}\sqrt{-ab}\sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{{1 \left ( -bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{cx}}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(c*x)^(1/2)/(b*x^2+a)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{2} + a} \sqrt{c x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x^2 + a)*sqrt(c*x)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{b x^{2} + a} \sqrt{c x}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x^2 + a)*sqrt(c*x)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.45906, size = 44, normalized size = 0.45 \[ \frac{\sqrt{x} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{a} \sqrt{c} \Gamma \left (\frac{5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(c*x)**(1/2)/(b*x**2+a)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{2} + a} \sqrt{c x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x^2 + a)*sqrt(c*x)),x, algorithm="giac")
[Out]