3.2959 \(\int \frac{1}{(1-2 x)^{5/2} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=125 \[ \frac{544 \sqrt{3 x+2} \sqrt{5 x+3}}{17787 \sqrt{1-2 x}}+\frac{4 \sqrt{3 x+2} \sqrt{5 x+3}}{231 (1-2 x)^{3/2}}-\frac{202 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{539 \sqrt{33}}+\frac{272 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{539 \sqrt{33}} \]

[Out]

(4*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/(231*(1 - 2*x)^(3/2)) + (544*Sqrt[2 + 3*x]*Sqrt[
3 + 5*x])/(17787*Sqrt[1 - 2*x]) + (272*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]]
, 35/33])/(539*Sqrt[33]) - (202*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33
])/(539*Sqrt[33])

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Rubi [A]  time = 0.269793, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{544 \sqrt{3 x+2} \sqrt{5 x+3}}{17787 \sqrt{1-2 x}}+\frac{4 \sqrt{3 x+2} \sqrt{5 x+3}}{231 (1-2 x)^{3/2}}-\frac{202 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{539 \sqrt{33}}+\frac{272 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{539 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/(231*(1 - 2*x)^(3/2)) + (544*Sqrt[2 + 3*x]*Sqrt[
3 + 5*x])/(17787*Sqrt[1 - 2*x]) + (272*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]]
, 35/33])/(539*Sqrt[33]) - (202*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33
])/(539*Sqrt[33])

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Rubi in Sympy [A]  time = 23.6227, size = 114, normalized size = 0.91 \[ \frac{272 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{17787} - \frac{202 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{17787} + \frac{544 \sqrt{3 x + 2} \sqrt{5 x + 3}}{17787 \sqrt{- 2 x + 1}} + \frac{4 \sqrt{3 x + 2} \sqrt{5 x + 3}}{231 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

272*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/17787 - 202*sqrt
(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/17787 + 544*sqrt(3*x + 2
)*sqrt(5*x + 3)/(17787*sqrt(-2*x + 1)) + 4*sqrt(3*x + 2)*sqrt(5*x + 3)/(231*(-2*
x + 1)**(3/2))

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Mathematica [A]  time = 0.19446, size = 115, normalized size = 0.92 \[ -\frac{4 \sqrt{3 x+2} \sqrt{5 x+3} (272 x-213)+3605 \sqrt{2-4 x} (2 x-1) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-272 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{17787 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(4*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-213 + 272*x) - 272*Sqrt[2 - 4*x]*(-1 + 2*x)*El
lipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 3605*Sqrt[2 - 4*x]*(-1 + 2*x)
*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(17787*(1 - 2*x)^(3/2))

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Maple [C]  time = 0.033, size = 276, normalized size = 2.2 \[ -{\frac{1}{17787\, \left ( -1+2\,x \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( 7210\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-544\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-3605\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +272\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +16320\,{x}^{3}+7892\,{x}^{2}-9660\,x-5112 \right ) \sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/17787*(7210*2^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1
/2)*3^(1/2)*2^(1/2))*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-544*2^(1/2)*Ell
ipticE(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))*x*(3+
5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-3605*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*
(1-2*x)^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/
2)*2^(1/2))+272*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/11
*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))+16320*x^3+7892*x
^2-9660*x-5112)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)/(-1+2*x)^2/(15*x^2+19*
x+6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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