3.2930 \(\int \frac{(2+3 x)^{5/2} \sqrt{3+5 x}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=156 \[ \frac{\sqrt{5 x+3} (3 x+2)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{100 \sqrt{5 x+3} (3 x+2)^{3/2}}{33 \sqrt{1-2 x}}-\frac{133}{22} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}-\frac{139 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{10 \sqrt{33}}-\frac{4621 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{10 \sqrt{33}} \]

[Out]

(-133*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/22 - (100*(2 + 3*x)^(3/2)*Sqrt[
3 + 5*x])/(33*Sqrt[1 - 2*x]) + ((2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/(3*(1 - 2*x)^(3/2
)) - (4621*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(10*Sqrt[33]) - (1
39*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(10*Sqrt[33])

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Rubi [A]  time = 0.332967, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{\sqrt{5 x+3} (3 x+2)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{100 \sqrt{5 x+3} (3 x+2)^{3/2}}{33 \sqrt{1-2 x}}-\frac{133}{22} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}-\frac{139 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{10 \sqrt{33}}-\frac{4621 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{10 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-133*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/22 - (100*(2 + 3*x)^(3/2)*Sqrt[
3 + 5*x])/(33*Sqrt[1 - 2*x]) + ((2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/(3*(1 - 2*x)^(3/2
)) - (4621*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(10*Sqrt[33]) - (1
39*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(10*Sqrt[33])

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Rubi in Sympy [A]  time = 31.723, size = 141, normalized size = 0.9 \[ - \frac{133 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{22} - \frac{4621 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{330} - \frac{139 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{330} - \frac{100 \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{33 \sqrt{- 2 x + 1}} + \frac{\left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{3 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-133*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/22 - 4621*sqrt(33)*elliptic_e(as
in(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/330 - 139*sqrt(33)*elliptic_f(asin(sqrt(21
)*sqrt(-2*x + 1)/7), 35/33)/330 - 100*(3*x + 2)**(3/2)*sqrt(5*x + 3)/(33*sqrt(-2
*x + 1)) + (3*x + 2)**(5/2)*sqrt(5*x + 3)/(3*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.312573, size = 120, normalized size = 0.77 \[ -\frac{10 \sqrt{3 x+2} \sqrt{5 x+3} \left (198 x^2-2060 x+711\right )-4655 \sqrt{2-4 x} (2 x-1) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+9242 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{660 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(10*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(711 - 2060*x + 198*x^2) + 9242*Sqrt[2 - 4*x]*(
-1 + 2*x)*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 4655*Sqrt[2 - 4*x
]*(-1 + 2*x)*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(660*(1 - 2*x)^
(3/2))

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Maple [C]  time = 0.028, size = 286, normalized size = 1.8 \[{\frac{1}{ \left ( 19800\,{x}^{3}+15180\,{x}^{2}-4620\,x-3960 \right ) \left ( -1+2\,x \right ) } \left ( 9310\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-18484\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-4655\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +9242\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -29700\,{x}^{4}+271380\,{x}^{3}+272870\,{x}^{2}-11490\,x-42660 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/660*(9310*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)-18484*2^(1/2)*Elli
pticE(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)-4655*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))+9242*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))-29700*x^4+271380
*x^3+272870*x^2-11490*x-42660)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)/(30*x^3
+23*x^2-7*x-6)/(-1+2*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((9*x^2 + 12*x + 4)*sqrt(5*x + 3)*sqrt(3*x + 2)/((4*x^2 - 4*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)**(5/2)*(3+5*x)**(1/2)/(1-2*x)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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