3.2875 \(\int \frac{(2+3 x)^{5/2} \sqrt{3+5 x}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=155 \[ \frac{\sqrt{5 x+3} (3 x+2)^{5/2}}{\sqrt{1-2 x}}+\frac{9}{5} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}+\frac{419}{50} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}+\frac{4817 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{250 \sqrt{33}}+\frac{7279}{125} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(419*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/50 + (9*Sqrt[1 - 2*x]*(2 + 3*x)^
(3/2)*Sqrt[3 + 5*x])/5 + ((2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/Sqrt[1 - 2*x] + (7279*S
qrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/125 + (4817*Ellipti
cF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(250*Sqrt[33])

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Rubi [A]  time = 0.316622, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{\sqrt{5 x+3} (3 x+2)^{5/2}}{\sqrt{1-2 x}}+\frac{9}{5} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}+\frac{419}{50} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}+\frac{4817 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{250 \sqrt{33}}+\frac{7279}{125} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(419*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/50 + (9*Sqrt[1 - 2*x]*(2 + 3*x)^
(3/2)*Sqrt[3 + 5*x])/5 + ((2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/Sqrt[1 - 2*x] + (7279*S
qrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/125 + (4817*Ellipti
cF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(250*Sqrt[33])

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Rubi in Sympy [A]  time = 31.5326, size = 139, normalized size = 0.9 \[ \frac{9 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{5} + \frac{419 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{50} + \frac{7279 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{375} + \frac{4817 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{8750} + \frac{\left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{\sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/5 + 419*sqrt(-2*x + 1)*sqrt(3*x
+ 2)*sqrt(5*x + 3)/50 + 7279*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7)
, 35/33)/375 + 4817*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)
/8750 + (3*x + 2)**(5/2)*sqrt(5*x + 3)/sqrt(-2*x + 1)

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Mathematica [A]  time = 0.257679, size = 110, normalized size = 0.71 \[ \frac{-30 \sqrt{3 x+2} \sqrt{5 x+3} \left (90 x^2+328 x-799\right )+14665 \sqrt{2-4 x} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-29116 \sqrt{2-4 x} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{1500 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(-30*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-799 + 328*x + 90*x^2) - 29116*Sqrt[2 - 4*x]*E
llipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 14665*Sqrt[2 - 4*x]*Elliptic
F[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(1500*Sqrt[1 - 2*x])

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Maple [C]  time = 0.024, size = 169, normalized size = 1.1 \[ -{\frac{1}{45000\,{x}^{3}+34500\,{x}^{2}-10500\,x-9000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 14665\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -29116\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -40500\,{x}^{4}-198900\,{x}^{3}+156390\,{x}^{2}+396390\,x+143820 \right ) } \]

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)^(3/2),x)

[Out]

-1/1500*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(14665*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*1
1^(1/2)*3^(1/2)*2^(1/2))-29116*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))-4
0500*x^4-198900*x^3+156390*x^2+396390*x+143820)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(9*x^2 + 12*x + 4)*sqrt(5*x + 3)*sqrt(3*x + 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)**(5/2)*(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. \[ \int \frac{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{5}{2}}}{{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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