3.2933 \(\int \frac{\sqrt{3+5 x}}{(1-2 x)^{5/2} \sqrt{2+3 x}} \, dx\)

Optimal. Leaf size=125 \[ \frac{62 \sqrt{3 x+2} \sqrt{5 x+3}}{1617 \sqrt{1-2 x}}+\frac{2 \sqrt{3 x+2} \sqrt{5 x+3}}{21 (1-2 x)^{3/2}}+\frac{4 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{49 \sqrt{33}}+\frac{31 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{49 \sqrt{33}} \]

[Out]

(2*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)) + (62*Sqrt[2 + 3*x]*Sqrt[3
+ 5*x])/(1617*Sqrt[1 - 2*x]) + (31*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35
/33])/(49*Sqrt[33]) + (4*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(49*
Sqrt[33])

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Rubi [A]  time = 0.262843, 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 \[ \frac{62 \sqrt{3 x+2} \sqrt{5 x+3}}{1617 \sqrt{1-2 x}}+\frac{2 \sqrt{3 x+2} \sqrt{5 x+3}}{21 (1-2 x)^{3/2}}+\frac{4 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{49 \sqrt{33}}+\frac{31 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{49 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)) + (62*Sqrt[2 + 3*x]*Sqrt[3
+ 5*x])/(1617*Sqrt[1 - 2*x]) + (31*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35
/33])/(49*Sqrt[33]) + (4*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(49*
Sqrt[33])

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Rubi in Sympy [A]  time = 24.0679, size = 114, normalized size = 0.91 \[ \frac{31 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1617} + \frac{4 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1617} + \frac{62 \sqrt{3 x + 2} \sqrt{5 x + 3}}{1617 \sqrt{- 2 x + 1}} + \frac{2 \sqrt{3 x + 2} \sqrt{5 x + 3}}{21 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

31*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/1617 + 4*sqrt(33)
*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/1617 + 62*sqrt(3*x + 2)*sqrt
(5*x + 3)/(1617*sqrt(-2*x + 1)) + 2*sqrt(3*x + 2)*sqrt(5*x + 3)/(21*(-2*x + 1)**
(3/2))

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Mathematica [A]  time = 0.304002, size = 115, normalized size = 0.92 \[ \frac{4 \sqrt{3 x+2} \sqrt{5 x+3} (54-31 x)+35 \sqrt{2-4 x} (2 x-1) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+31 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{1617 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(4*(54 - 31*x)*Sqrt[2 + 3*x]*Sqrt[3 + 5*x] + 31*Sqrt[2 - 4*x]*(-1 + 2*x)*Ellipti
cE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 35*Sqrt[2 - 4*x]*(-1 + 2*x)*Ellipt
icF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(1617*(1 - 2*x)^(3/2))

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Maple [C]  time = 0.029, size = 276, normalized size = 2.2 \[{\frac{1}{1617\, \left ( -1+2\,x \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( 70\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+62\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-35\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -31\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -1860\,{x}^{3}+884\,{x}^{2}+3360\,x+1296 \right ) \sqrt{2+3\,x}\sqrt{1-2\,x}\sqrt{3+5\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1617*(70*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)+62*2^(1/2)*Elliptic
E(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)-35*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))-31*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))-1860*x^3+884*x^2+3360*x+
1296)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*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. \[ \int \frac{\sqrt{5 \, x + 3}}{\sqrt{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(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. \[{\rm integral}\left (\frac{\sqrt{5 \, x + 3}}{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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