3.2879 \(\int \frac{\sqrt{3+5 x}}{(1-2 x)^{3/2} (2+3 x)^{3/2}} \, dx\)

Optimal. Leaf size=125 \[ -\frac{12 \sqrt{1-2 x} \sqrt{5 x+3}}{49 \sqrt{3 x+2}}+\frac{2 \sqrt{5 x+3}}{7 \sqrt{1-2 x} \sqrt{3 x+2}}-\frac{62 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{49 \sqrt{33}}+\frac{4}{49} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(2*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]) - (12*Sqrt[1 - 2*x]*Sqrt[3 + 5
*x])/(49*Sqrt[2 + 3*x]) + (4*Sqrt[33]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]],
35/33])/49 - (62*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(49*Sqrt[33
])

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Rubi [A]  time = 0.266459, 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 \[ -\frac{12 \sqrt{1-2 x} \sqrt{5 x+3}}{49 \sqrt{3 x+2}}+\frac{2 \sqrt{5 x+3}}{7 \sqrt{1-2 x} \sqrt{3 x+2}}-\frac{62 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{49 \sqrt{33}}+\frac{4}{49} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]) - (12*Sqrt[1 - 2*x]*Sqrt[3 + 5
*x])/(49*Sqrt[2 + 3*x]) + (4*Sqrt[33]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]],
35/33])/49 - (62*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(49*Sqrt[33
])

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Rubi in Sympy [A]  time = 24.2113, size = 114, normalized size = 0.91 \[ - \frac{12 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{49 \sqrt{3 x + 2}} + \frac{4 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{49} - \frac{62 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{1715} + \frac{2 \sqrt{5 x + 3}}{7 \sqrt{- 2 x + 1} \sqrt{3 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-12*sqrt(-2*x + 1)*sqrt(5*x + 3)/(49*sqrt(3*x + 2)) + 4*sqrt(33)*elliptic_e(asin
(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/49 - 62*sqrt(35)*elliptic_f(asin(sqrt(55)*sq
rt(-2*x + 1)/11), 33/35)/1715 + 2*sqrt(5*x + 3)/(7*sqrt(-2*x + 1)*sqrt(3*x + 2))

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Mathematica [A]  time = 0.182159, size = 122, normalized size = 0.98 \[ \frac{2 \sqrt{3 x+2} \sqrt{5 x+3} (12 x+1)+35 \sqrt{2-4 x} (3 x+2) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-4 \sqrt{2-4 x} (3 x+2) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{49 \sqrt{1-2 x} (3 x+2)} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(1 + 12*x) - 4*Sqrt[2 - 4*x]*(2 + 3*x)*EllipticE[
ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 35*Sqrt[2 - 4*x]*(2 + 3*x)*EllipticF[
ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(49*Sqrt[1 - 2*x]*(2 + 3*x))

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Maple [C]  time = 0.027, size = 159, normalized size = 1.3 \[ -{\frac{1}{1470\,{x}^{3}+1127\,{x}^{2}-343\,x-294}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 35\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -4\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +120\,{x}^{2}+82\,x+6 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/49*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2+3*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))-4*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(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))+120*x^2+82
*x+6)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{\sqrt{5 \, x + 3}}{{\left (6 \, x^{2} + x - 2\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-sqrt(5*x + 3)/((6*x^2 + x - 2)*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)**(3/2)/(2+3*x)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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