Optimal. Leaf size=51 \[ \frac{2 \sqrt{d+e x} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} \sqrt{x}\right )|-\frac{2 e}{3 d}\right )}{\sqrt{3} \sqrt{\frac{e x}{d}+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0122931, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {112, 110} \[ \frac{2 \sqrt{d+e x} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} \sqrt{x}\right )|-\frac{2 e}{3 d}\right )}{\sqrt{3} \sqrt{\frac{e x}{d}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 112
Rule 110
Rubi steps
\begin{align*} \int \frac{\sqrt{d+e x}}{\sqrt{2-3 x} \sqrt{x}} \, dx &=\frac{\left (\sqrt{1-\frac{3 x}{2}} \sqrt{d+e x}\right ) \int \frac{\sqrt{1+\frac{e x}{d}}}{\sqrt{1-\frac{3 x}{2}} \sqrt{x}} \, dx}{\sqrt{2-3 x} \sqrt{1+\frac{e x}{d}}}\\ &=\frac{2 \sqrt{d+e x} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} \sqrt{x}\right )|-\frac{2 e}{3 d}\right )}{\sqrt{3} \sqrt{1+\frac{e x}{d}}}\\ \end{align*}
Mathematica [B] time = 0.782408, size = 125, normalized size = 2.45 \[ \frac{2 \sqrt{x} \left (\frac{3 (d+e x)}{\sqrt{2-3 x}}-\frac{(3 d+2 e) \sqrt{\frac{d+e x}{e (3 x-2)}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{3 d}{e}+2}}{\sqrt{2-3 x}}\right )|\frac{2 e}{3 d+2 e}\right )}{\sqrt{\frac{x}{3 x-2}} \sqrt{\frac{3 d}{e}+2}}\right )}{3 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.015, size = 212, normalized size = 4.2 \begin{align*} -{\frac{2\,d}{3\,e \left ( 3\,e{x}^{2}+3\,dx-2\,ex-2\,d \right ) }\sqrt{ex+d}\sqrt{2-3\,x}\sqrt{{\frac{ex+d}{d}}}\sqrt{-{\frac{ \left ( -2+3\,x \right ) e}{3\,d+2\,e}}}\sqrt{-{\frac{ex}{d}}} \left ( 3\,d{\it EllipticF} \left ( \sqrt{{\frac{ex+d}{d}}},\sqrt{3}\sqrt{{\frac{d}{3\,d+2\,e}}} \right ) +2\,{\it EllipticF} \left ( \sqrt{{\frac{ex+d}{d}}},\sqrt{3}\sqrt{{\frac{d}{3\,d+2\,e}}} \right ) e-3\,{\it EllipticE} \left ( \sqrt{{\frac{ex+d}{d}}},\sqrt{3}\sqrt{{\frac{d}{3\,d+2\,e}}} \right ) d-2\,{\it EllipticE} \left ( \sqrt{{\frac{ex+d}{d}}},\sqrt{3}\sqrt{{\frac{d}{3\,d+2\,e}}} \right ) e \right ){\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e x + d}}{\sqrt{x} \sqrt{-3 \, x + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{e x + d} \sqrt{x} \sqrt{-3 \, x + 2}}{3 \, x^{2} - 2 \, x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]