3.2894 \(\int \frac{\sqrt{2+3 x} (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=155 \[ \frac{\sqrt{3 x+2} (5 x+3)^{5/2}}{\sqrt{1-2 x}}+3 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}+\frac{397}{18} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}+\frac{397}{90} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{6599}{45} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(397*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/18 + 3*Sqrt[1 - 2*x]*Sqrt[2 + 3*
x]*(3 + 5*x)^(3/2) + (Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/Sqrt[1 - 2*x] + (6599*Sqrt[
11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/45 + (397*Sqrt[11/3]*El
lipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/90

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Rubi [A]  time = 0.315608, 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 \[ \frac{\sqrt{3 x+2} (5 x+3)^{5/2}}{\sqrt{1-2 x}}+3 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}+\frac{397}{18} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}+\frac{397}{90} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{6599}{45} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(397*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/18 + 3*Sqrt[1 - 2*x]*Sqrt[2 + 3*
x]*(3 + 5*x)^(3/2) + (Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/Sqrt[1 - 2*x] + (6599*Sqrt[
11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/45 + (397*Sqrt[11/3]*El
lipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/90

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Rubi in Sympy [A]  time = 31.697, size = 138, normalized size = 0.89 \[ 3 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}} + \frac{397 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{18} + \frac{6599 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{135} + \frac{397 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{270} + \frac{\sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{5}{2}}}{\sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*sqrt(-2*x + 1)*sqrt(3*x + 2)*(5*x + 3)**(3/2) + 397*sqrt(-2*x + 1)*sqrt(3*x +
2)*sqrt(5*x + 3)/18 + 6599*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7),
35/33)/135 + 397*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/270
+ sqrt(3*x + 2)*(5*x + 3)**(5/2)/sqrt(-2*x + 1)

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Mathematica [A]  time = 0.257352, size = 110, normalized size = 0.71 \[ \frac{-30 \sqrt{3 x+2} \sqrt{5 x+3} \left (90 x^2+308 x-721\right )+13295 \sqrt{2-4 x} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-26396 \sqrt{2-4 x} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{540 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(-30*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-721 + 308*x + 90*x^2) - 26396*Sqrt[2 - 4*x]*E
llipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 13295*Sqrt[2 - 4*x]*Elliptic
F[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(540*Sqrt[1 - 2*x])

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Maple [C]  time = 0.023, size = 169, normalized size = 1.1 \[ -{\frac{1}{16200\,{x}^{3}+12420\,{x}^{2}-3780\,x-3240}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 13295\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -26396\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -40500\,{x}^{4}-189900\,{x}^{3}+132690\,{x}^{2}+355530\,x+129780 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/540*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(13295*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))-26396*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))-40
500*x^4-189900*x^3+132690*x^2+355530*x+129780)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}{{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(25*x^2 + 30*x + 9)*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((3+5*x)**(5/2)*(2+3*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. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{3 \, x + 2}}{{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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