3.2843 \(\int \frac{(2+3 x)^{5/2}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=127 \[ -\frac{3}{25} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}-\frac{74}{125} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}-\frac{857 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{625 \sqrt{33}}-\frac{5161 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1250} \]

[Out]

(-74*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/125 - (3*Sqrt[1 - 2*x]*(2 + 3*x)
^(3/2)*Sqrt[3 + 5*x])/25 - (5161*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 -
2*x]], 35/33])/1250 - (857*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(6
25*Sqrt[33])

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Rubi [A]  time = 0.263266, antiderivative size = 127, 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 \[ -\frac{3}{25} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}-\frac{74}{125} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}-\frac{857 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{625 \sqrt{33}}-\frac{5161 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1250} \]

Antiderivative was successfully verified.

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

[Out]

(-74*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/125 - (3*Sqrt[1 - 2*x]*(2 + 3*x)
^(3/2)*Sqrt[3 + 5*x])/25 - (5161*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 -
2*x]], 35/33])/1250 - (857*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(6
25*Sqrt[33])

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Rubi in Sympy [A]  time = 25.5849, size = 116, normalized size = 0.91 \[ - \frac{3 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{25} - \frac{74 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{125} - \frac{5161 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{3750} - \frac{857 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{20625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/25 - 74*sqrt(-2*x + 1)*sqrt(3*x
+ 2)*sqrt(5*x + 3)/125 - 5161*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/
7), 35/33)/3750 - 857*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33
)/20625

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Mathematica [A]  time = 0.267221, size = 95, normalized size = 0.75 \[ \frac{5161 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (3 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} (45 x+104)+518 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{1875 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(5161*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 5*(3*Sqrt[2 - 4*x]*Sq
rt[2 + 3*x]*Sqrt[3 + 5*x]*(104 + 45*x) + 518*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3
+ 5*x]], -33/2]))/(1875*Sqrt[2])

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Maple [C]  time = 0.023, size = 169, normalized size = 1.3 \[{\frac{1}{112500\,{x}^{3}+86250\,{x}^{2}-26250\,x-22500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 2590\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -5161\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -40500\,{x}^{4}-124650\,{x}^{3}-62310\,{x}^{2}+29940\,x+18720 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3750*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2590*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))-5161*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*El
lipticE(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))-4050
0*x^4-124650*x^3-62310*x^2+29940*x+18720)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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