3.48 \(\int \frac{\sqrt{2-3 x} \sqrt{1+4 x}}{\sqrt{-5+2 x} (7+5 x)} \, dx\)

Optimal. Leaf size=151 \[ -\frac{41 \sqrt{\frac{2}{33}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{25 \sqrt{2 x-5}}+\frac{2 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{5 \sqrt{5-2 x}}+\frac{69 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{25 \sqrt{11} \sqrt{2 x-5}} \]

[Out]

(2*Sqrt[11]*Sqrt[-5 + 2*x]*EllipticE[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/
(5*Sqrt[5 - 2*x]) - (41*Sqrt[2/33]*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqr
t[1 + 4*x]], 1/3])/(25*Sqrt[-5 + 2*x]) + (69*Sqrt[5 - 2*x]*EllipticPi[55/124, Ar
cSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/(25*Sqrt[11]*Sqrt[-5 + 2*x])

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Rubi [A]  time = 0.639643, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.257 \[ -\frac{41 \sqrt{\frac{2}{33}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{25 \sqrt{2 x-5}}+\frac{2 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{5 \sqrt{5-2 x}}+\frac{69 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{25 \sqrt{11} \sqrt{2 x-5}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[2 - 3*x]*Sqrt[1 + 4*x])/(Sqrt[-5 + 2*x]*(7 + 5*x)),x]

[Out]

(2*Sqrt[11]*Sqrt[-5 + 2*x]*EllipticE[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/
(5*Sqrt[5 - 2*x]) - (41*Sqrt[2/33]*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqr
t[1 + 4*x]], 1/3])/(25*Sqrt[-5 + 2*x]) + (69*Sqrt[5 - 2*x]*EllipticPi[55/124, Ar
cSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/(25*Sqrt[11]*Sqrt[-5 + 2*x])

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Rubi in Sympy [A]  time = 39.3961, size = 214, normalized size = 1.42 \[ \frac{2 \sqrt{11} \sqrt{\frac{12 x}{11} + \frac{3}{11}} \sqrt{2 x - 5} E\left (\operatorname{asin}{\left (\frac{2 \sqrt{11} \sqrt{- 3 x + 2}}{11} \right )}\middle | - \frac{1}{2}\right )}{5 \sqrt{- \frac{6 x}{11} + \frac{15}{11}} \sqrt{4 x + 1}} - \frac{41 \sqrt{11} \sqrt{- \frac{12 x}{11} + \frac{8}{11}} \sqrt{- \frac{4 x}{11} + \frac{10}{11}} F\left (\operatorname{asin}{\left (\frac{\sqrt{11} \sqrt{4 x + 1}}{11} \right )}\middle | 3\right )}{50 \sqrt{- 3 x + 2} \sqrt{2 x - 5}} + \frac{713 \sqrt{22} i \sqrt{\frac{4 x}{11} + \frac{1}{11}} \sqrt{\frac{6 x}{11} - \frac{4}{11}} \Pi \left (\frac{55}{78}; i \operatorname{asinh}{\left (\frac{\sqrt{22} \sqrt{2 x - 5}}{11} \right )}\middle | \frac{3}{2}\right )}{975 \sqrt{- 3 x + 2} \sqrt{4 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2-3*x)**(1/2)*(1+4*x)**(1/2)/(7+5*x)/(-5+2*x)**(1/2),x)

[Out]

2*sqrt(11)*sqrt(12*x/11 + 3/11)*sqrt(2*x - 5)*elliptic_e(asin(2*sqrt(11)*sqrt(-3
*x + 2)/11), -1/2)/(5*sqrt(-6*x/11 + 15/11)*sqrt(4*x + 1)) - 41*sqrt(11)*sqrt(-1
2*x/11 + 8/11)*sqrt(-4*x/11 + 10/11)*elliptic_f(asin(sqrt(11)*sqrt(4*x + 1)/11),
3)/(50*sqrt(-3*x + 2)*sqrt(2*x - 5)) + 713*sqrt(22)*I*sqrt(4*x/11 + 1/11)*sqrt(
6*x/11 - 4/11)*elliptic_pi(55/78, I*asinh(sqrt(22)*sqrt(2*x - 5)/11), 3/2)/(975*
sqrt(-3*x + 2)*sqrt(4*x + 1))

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Mathematica [A]  time = 0.207531, size = 97, normalized size = 0.64 \[ \frac{\sqrt{5-2 x} \left (41 F\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )-110 E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )-69 \Pi \left (\frac{55}{124};-\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )\right )}{25 \sqrt{22 x-55}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[2 - 3*x]*Sqrt[1 + 4*x])/(Sqrt[-5 + 2*x]*(7 + 5*x)),x]

[Out]

(Sqrt[5 - 2*x]*(-110*EllipticE[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2] + 41*El
lipticF[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2] - 69*EllipticPi[55/124, -ArcSi
n[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2]))/(25*Sqrt[-55 + 22*x])

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Maple [A]  time = 0.02, size = 85, normalized size = 0.6 \[{\frac{\sqrt{11}}{275} \left ( 41\,{\it EllipticF} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ) -110\,{\it EllipticE} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ) +69\,{\it EllipticPi} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},{\frac{55}{124}},i/2\sqrt{2} \right ) \right ) \sqrt{5-2\,x}{\frac{1}{\sqrt{-5+2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2-3*x)^(1/2)*(1+4*x)^(1/2)/(7+5*x)/(-5+2*x)^(1/2),x)

[Out]

1/275*(41*EllipticF(2/11*(2-3*x)^(1/2)*11^(1/2),1/2*I*2^(1/2))-110*EllipticE(2/1
1*(2-3*x)^(1/2)*11^(1/2),1/2*I*2^(1/2))+69*EllipticPi(2/11*(2-3*x)^(1/2)*11^(1/2
),55/124,1/2*I*2^(1/2)))*(5-2*x)^(1/2)*11^(1/2)/(-5+2*x)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{4 \, x + 1} \sqrt{-3 \, x + 2}}{{\left (5 \, x + 7\right )} \sqrt{2 \, x - 5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(4*x + 1)*sqrt(-3*x + 2)/((5*x + 7)*sqrt(2*x - 5)),x, algorithm="maxima")

[Out]

integrate(sqrt(4*x + 1)*sqrt(-3*x + 2)/((5*x + 7)*sqrt(2*x - 5)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{4 \, x + 1} \sqrt{-3 \, x + 2}}{{\left (5 \, x + 7\right )} \sqrt{2 \, x - 5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(4*x + 1)*sqrt(-3*x + 2)/((5*x + 7)*sqrt(2*x - 5)),x, algorithm="fricas")

[Out]

integral(sqrt(4*x + 1)*sqrt(-3*x + 2)/((5*x + 7)*sqrt(2*x - 5)), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2-3*x)**(1/2)*(1+4*x)**(1/2)/(7+5*x)/(-5+2*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{4 \, x + 1} \sqrt{-3 \, x + 2}}{{\left (5 \, x + 7\right )} \sqrt{2 \, x - 5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(4*x + 1)*sqrt(-3*x + 2)/((5*x + 7)*sqrt(2*x - 5)),x, algorithm="giac")

[Out]

integrate(sqrt(4*x + 1)*sqrt(-3*x + 2)/((5*x + 7)*sqrt(2*x - 5)), x)