Optimal. Leaf size=236 \[ \frac{\sqrt [4]{a} \sqrt{c} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{b^{3/4} \sqrt{a+b x^2}}-\frac{2 \sqrt [4]{a} \sqrt{c} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{b^{3/4} \sqrt{a+b x^2}}+\frac{2 \sqrt{c x} \sqrt{a+b x^2}}{\sqrt{b} \left (\sqrt{a}+\sqrt{b} x\right )} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.434289, antiderivative size = 236, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{\sqrt [4]{a} \sqrt{c} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{b^{3/4} \sqrt{a+b x^2}}-\frac{2 \sqrt [4]{a} \sqrt{c} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{b^{3/4} \sqrt{a+b x^2}}+\frac{2 \sqrt{c x} \sqrt{a+b x^2}}{\sqrt{b} \left (\sqrt{a}+\sqrt{b} x\right )} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c*x]/Sqrt[a + b*x^2],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 39.2971, size = 216, normalized size = 0.92 \[ - \frac{2 \sqrt [4]{a} \sqrt{c} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{b^{\frac{3}{4}} \sqrt{a + b x^{2}}} + \frac{\sqrt [4]{a} \sqrt{c} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{b^{\frac{3}{4}} \sqrt{a + b x^{2}}} + \frac{2 \sqrt{c x} \sqrt{a + b x^{2}}}{\sqrt{b} \left (\sqrt{a} + \sqrt{b} x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x)**(1/2)/(b*x**2+a)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.0941198, size = 111, normalized size = 0.47 \[ \frac{2 i x \sqrt{c x} \sqrt{\frac{b x^2}{a}+1} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )\right )}{\left (\frac{i \sqrt{b} x}{\sqrt{a}}\right )^{3/2} \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[c*x]/Sqrt[a + b*x^2],x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.015, size = 132, normalized size = 0.6 \[{\frac{a\sqrt{2}}{bx}\sqrt{cx}\sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{{1 \left ( -bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}} \left ( 2\,{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) -{\it EllipticF} \left ( \sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x)^(1/2)/(b*x^2+a)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x}}{\sqrt{b x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x)/sqrt(b*x^2 + a),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x}}{\sqrt{b x^{2} + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x)/sqrt(b*x^2 + a),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 2.60509, size = 44, normalized size = 0.19 \[ \frac{\sqrt{c} x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{a} \Gamma \left (\frac{7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x)**(1/2)/(b*x**2+a)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x}}{\sqrt{b x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x)/sqrt(b*x^2 + a),x, algorithm="giac")
[Out]