3.2876 \(\int \frac{(2+3 x)^{3/2} \sqrt{3+5 x}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=122 \[ \frac{\sqrt{5 x+3} (3 x+2)^{3/2}}{\sqrt{1-2 x}}+2 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}+\frac{23 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5 \sqrt{33}}+\frac{139}{10} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x] + ((2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/Sq
rt[1 - 2*x] + (139*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])
/10 + (23*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(5*Sqrt[33])

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Rubi [A]  time = 0.24398, antiderivative size = 122, 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 \[ \frac{\sqrt{5 x+3} (3 x+2)^{3/2}}{\sqrt{1-2 x}}+2 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}+\frac{23 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5 \sqrt{33}}+\frac{139}{10} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x] + ((2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/Sq
rt[1 - 2*x] + (139*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])
/10 + (23*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(5*Sqrt[33])

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Rubi in Sympy [A]  time = 24.2159, size = 109, normalized size = 0.89 \[ 2 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3} + \frac{139 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{30} + \frac{23 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{165} + \frac{\left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{\sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3) + 139*sqrt(33)*elliptic_e(asin(sqrt
(21)*sqrt(-2*x + 1)/7), 35/33)/30 + 23*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2
*x + 1)/7), 35/33)/165 + (3*x + 2)**(3/2)*sqrt(5*x + 3)/sqrt(-2*x + 1)

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Mathematica [A]  time = 0.21864, size = 103, normalized size = 0.84 \[ \frac{-30 \sqrt{3 x+2} \sqrt{5 x+3} (x-4)+70 \sqrt{2-4 x} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-139 \sqrt{2-4 x} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{30 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(-30*(-4 + x)*Sqrt[2 + 3*x]*Sqrt[3 + 5*x] - 139*Sqrt[2 - 4*x]*EllipticE[ArcSin[S
qrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 70*Sqrt[2 - 4*x]*EllipticF[ArcSin[Sqrt[2/11]*
Sqrt[3 + 5*x]], -33/2])/(30*Sqrt[1 - 2*x])

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Maple [C]  time = 0.023, size = 164, normalized size = 1.3 \[ -{\frac{1}{900\,{x}^{3}+690\,{x}^{2}-210\,x-180}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 70\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -139\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -450\,{x}^{3}+1230\,{x}^{2}+2100\,x+720 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/30*(2+3*x)^(1/2)*(3+5*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))-139*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(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))-450*x^3+
1230*x^2+2100*x+720)/(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. \[{\rm integral}\left (-\frac{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{3}{2}}}{{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}}, x\right ) \]

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