Optimal. Leaf size=103 \[ -\frac{\sqrt{\frac{6}{11}} \sqrt{5-2 x} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right ),\frac{1}{3}\right )}{5 \sqrt{2 x-5}}-\frac{3 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{5 \sqrt{11} \sqrt{2 x-5}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0957722, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {175, 121, 119, 168, 538, 537} \[ -\frac{\sqrt{\frac{6}{11}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{5 \sqrt{2 x-5}}-\frac{3 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{5 \sqrt{11} \sqrt{2 x-5}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 175
Rule 121
Rule 119
Rule 168
Rule 538
Rule 537
Rubi steps
\begin{align*} \int \frac{\sqrt{2-3 x}}{\sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)} \, dx &=-\left (\frac{3}{5} \int \frac{1}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx\right )+\frac{31}{5} \int \frac{1}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)} \, dx\\ &=-\left (\frac{62}{5} \operatorname{Subst}\left (\int \frac{1}{\left (31-5 x^2\right ) \sqrt{\frac{11}{3}-\frac{4 x^2}{3}} \sqrt{-\frac{11}{3}-\frac{2 x^2}{3}}} \, dx,x,\sqrt{2-3 x}\right )\right )-\frac{\left (3 \sqrt{\frac{2}{11}} \sqrt{5-2 x}\right ) \int \frac{1}{\sqrt{2-3 x} \sqrt{\frac{10}{11}-\frac{4 x}{11}} \sqrt{1+4 x}} \, dx}{5 \sqrt{-5+2 x}}\\ &=-\frac{\sqrt{\frac{6}{11}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{1+4 x}\right )|\frac{1}{3}\right )}{5 \sqrt{-5+2 x}}-\frac{\left (62 \sqrt{\frac{3}{11}} \sqrt{5-2 x}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (31-5 x^2\right ) \sqrt{\frac{11}{3}-\frac{4 x^2}{3}} \sqrt{1+\frac{2 x^2}{11}}} \, dx,x,\sqrt{2-3 x}\right )}{5 \sqrt{-5+2 x}}\\ &=-\frac{\sqrt{\frac{6}{11}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{1+4 x}\right )|\frac{1}{3}\right )}{5 \sqrt{-5+2 x}}-\frac{3 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{5 \sqrt{11} \sqrt{-5+2 x}}\\ \end{align*}
Mathematica [A] time = 0.400946, size = 70, normalized size = 0.68 \[ \frac{3 \sqrt{5-2 x} \left (\text{EllipticF}\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right ),-\frac{1}{2}\right )+\Pi \left (\frac{55}{124};-\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )\right )}{5 \sqrt{22 x-55}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.015, size = 56, normalized size = 0.5 \begin{align*}{\frac{3\,\sqrt{11}}{55} \left ({\it EllipticF} \left ({\frac{2}{11}\sqrt{22-33\,x}},{\frac{i}{2}}\sqrt{2} \right ) -{\it EllipticPi} \left ({\frac{2}{11}\sqrt{22-33\,x}},{\frac{55}{124}},{\frac{i}{2}}\sqrt{2} \right ) \right ) \sqrt{5-2\,x}{\frac{1}{\sqrt{2\,x-5}}}} \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{-3 \, x + 2}}{{\left (5 \, x + 7\right )} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5}}\,{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{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}{40 \, x^{3} - 34 \, x^{2} - 151 \, x - 35}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{2 - 3 x}}{\sqrt{2 x - 5} \sqrt{4 x + 1} \left (5 x + 7\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-3 \, x + 2}}{{\left (5 \, x + 7\right )} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]