3.2958 \(\int \frac{\sqrt{2+3 x}}{(1-2 x)^{5/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=125 \[ \frac{74 \sqrt{3 x+2} \sqrt{5 x+3}}{2541 \sqrt{1-2 x}}+\frac{2 \sqrt{3 x+2} \sqrt{5 x+3}}{33 (1-2 x)^{3/2}}-\frac{2 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{77 \sqrt{33}}+\frac{37 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{77 \sqrt{33}} \]

[Out]

(2*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/(33*(1 - 2*x)^(3/2)) + (74*Sqrt[2 + 3*x]*Sqrt[3
+ 5*x])/(2541*Sqrt[1 - 2*x]) + (37*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35
/33])/(77*Sqrt[33]) - (2*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(77*
Sqrt[33])

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Rubi [A]  time = 0.264263, 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 \[ \frac{74 \sqrt{3 x+2} \sqrt{5 x+3}}{2541 \sqrt{1-2 x}}+\frac{2 \sqrt{3 x+2} \sqrt{5 x+3}}{33 (1-2 x)^{3/2}}-\frac{2 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{77 \sqrt{33}}+\frac{37 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{77 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/(33*(1 - 2*x)^(3/2)) + (74*Sqrt[2 + 3*x]*Sqrt[3
+ 5*x])/(2541*Sqrt[1 - 2*x]) + (37*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35
/33])/(77*Sqrt[33]) - (2*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(77*
Sqrt[33])

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Rubi in Sympy [A]  time = 23.7677, size = 114, normalized size = 0.91 \[ \frac{37 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{2541} - \frac{2 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{2695} + \frac{74 \sqrt{3 x + 2} \sqrt{5 x + 3}}{2541 \sqrt{- 2 x + 1}} + \frac{2 \sqrt{3 x + 2} \sqrt{5 x + 3}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

37*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/2541 - 2*sqrt(35)
*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/2695 + 74*sqrt(3*x + 2)*sqr
t(5*x + 3)/(2541*sqrt(-2*x + 1)) + 2*sqrt(3*x + 2)*sqrt(5*x + 3)/(33*(-2*x + 1)*
*(3/2))

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Mathematica [A]  time = 0.203294, size = 115, normalized size = 0.92 \[ -\frac{4 \sqrt{3 x+2} \sqrt{5 x+3} (37 x-57)+70 \sqrt{2-4 x} (2 x-1) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-37 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{2541 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(4*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-57 + 37*x) - 37*Sqrt[2 - 4*x]*(-1 + 2*x)*Ellip
ticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 70*Sqrt[2 - 4*x]*(-1 + 2*x)*Elli
pticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(2541*(1 - 2*x)^(3/2))

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Maple [C]  time = 0.03, size = 276, normalized size = 2.2 \[ -{\frac{1}{2541\, \left ( -1+2\,x \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( 140\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-74\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-70\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +37\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +2220\,{x}^{3}-608\,{x}^{2}-3444\,x-1368 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}\sqrt{2+3\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2541*(140*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)-74*2^(1/2)*Ellipt
icE(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)-70*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))+37*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))+2220*x^3-608*x^2-3444*
x-1368)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2+3*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. \[ \int \frac{\sqrt{3 \, x + 2}}{\sqrt{5 \, x + 3}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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