3.2686 \(\int \frac{(1-2 x)^{3/2} \sqrt{3+5 x}}{(2+3 x)^{3/2}} \, dx\)

Optimal. Leaf size=129 \[ -\frac{2 \sqrt{5 x+3} (1-2 x)^{3/2}}{3 \sqrt{3 x+2}}-\frac{16}{27} \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{214}{135} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{494}{135} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(-2*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(3*Sqrt[2 + 3*x]) - (16*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x]*Sqrt[3 + 5*x])/27 + (494*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2
*x]], 35/33])/135 - (214*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 3
5/33])/135

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Rubi [A]  time = 0.255645, antiderivative size = 129, 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 \[ -\frac{2 \sqrt{5 x+3} (1-2 x)^{3/2}}{3 \sqrt{3 x+2}}-\frac{16}{27} \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{214}{135} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{494}{135} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(3*Sqrt[2 + 3*x]) - (16*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x]*Sqrt[3 + 5*x])/27 + (494*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2
*x]], 35/33])/135 - (214*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 3
5/33])/135

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Rubi in Sympy [A]  time = 24.3155, size = 114, normalized size = 0.88 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{3 \sqrt{3 x + 2}} - \frac{16 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{27} + \frac{494 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{405} - \frac{214 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{405} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(3*sqrt(3*x + 2)) - 16*sqrt(-2*x + 1)*sqrt(3*
x + 2)*sqrt(5*x + 3)/27 + 494*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7
), 35/33)/405 - 214*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/
405

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Mathematica [A]  time = 0.249964, size = 97, normalized size = 0.75 \[ \frac{1}{405} \left (-\frac{30 \sqrt{1-2 x} \sqrt{5 x+3} (6 x+25)}{\sqrt{3 x+2}}+4025 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-494 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-30*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(25 + 6*x))/Sqrt[2 + 3*x] - 494*Sqrt[2]*Ellipt
icE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 4025*Sqrt[2]*EllipticF[ArcSin[Sqr
t[2/11]*Sqrt[3 + 5*x]], -33/2])/405

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Maple [C]  time = 0.024, size = 164, normalized size = 1.3 \[ -{\frac{1}{12150\,{x}^{3}+9315\,{x}^{2}-2835\,x-2430}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 4025\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -494\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +1800\,{x}^{3}+7680\,{x}^{2}+210\,x-2250 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/405*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(4025*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))-494*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(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))+1800*
x^3+7680*x^2+210*x-2250)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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