3.2960 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=156 \[ \frac{5594 \sqrt{1-2 x} \sqrt{5 x+3}}{41503 \sqrt{3 x+2}}+\frac{808 \sqrt{5 x+3}}{17787 \sqrt{1-2 x} \sqrt{3 x+2}}+\frac{4 \sqrt{5 x+3}}{231 (1-2 x)^{3/2} \sqrt{3 x+2}}-\frac{1196 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3773 \sqrt{33}}-\frac{5594 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3773 \sqrt{33}} \]

[Out]

(4*Sqrt[3 + 5*x])/(231*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]) + (808*Sqrt[3 + 5*x])/(177
87*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]) + (5594*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(41503*Sqrt
[2 + 3*x]) - (5594*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(3773*Sqrt
[33]) - (1196*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(3773*Sqrt[33])

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Rubi [A]  time = 0.349597, antiderivative size = 156, 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 \[ \frac{5594 \sqrt{1-2 x} \sqrt{5 x+3}}{41503 \sqrt{3 x+2}}+\frac{808 \sqrt{5 x+3}}{17787 \sqrt{1-2 x} \sqrt{3 x+2}}+\frac{4 \sqrt{5 x+3}}{231 (1-2 x)^{3/2} \sqrt{3 x+2}}-\frac{1196 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3773 \sqrt{33}}-\frac{5594 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3773 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[3 + 5*x])/(231*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]) + (808*Sqrt[3 + 5*x])/(177
87*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]) + (5594*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(41503*Sqrt
[2 + 3*x]) - (5594*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(3773*Sqrt
[33]) - (1196*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(3773*Sqrt[33])

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Rubi in Sympy [A]  time = 30.4918, size = 143, normalized size = 0.92 \[ - \frac{5594 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{124509} - \frac{1196 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{124509} - \frac{11188 \sqrt{3 x + 2} \sqrt{5 x + 3}}{124509 \sqrt{- 2 x + 1}} + \frac{194 \sqrt{5 x + 3}}{539 \sqrt{- 2 x + 1} \sqrt{3 x + 2}} + \frac{4 \sqrt{5 x + 3}}{231 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5594*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/124509 - 1196*
sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/124509 - 11188*sqrt(
3*x + 2)*sqrt(5*x + 3)/(124509*sqrt(-2*x + 1)) + 194*sqrt(5*x + 3)/(539*sqrt(-2*
x + 1)*sqrt(3*x + 2)) + 4*sqrt(5*x + 3)/(231*(-2*x + 1)**(3/2)*sqrt(3*x + 2))

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Mathematica [A]  time = 0.203302, size = 98, normalized size = 0.63 \[ \frac{2 \left (\frac{\sqrt{5 x+3} \left (33564 x^2-39220 x+12297\right )}{(1-2 x)^{3/2} \sqrt{3 x+2}}+\sqrt{2} \left (7070 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+2797 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{124509} \]

Antiderivative was successfully verified.

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

[Out]

(2*((Sqrt[3 + 5*x]*(12297 - 39220*x + 33564*x^2))/((1 - 2*x)^(3/2)*Sqrt[2 + 3*x]
) + Sqrt[2]*(2797*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 7070*Elli
pticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/124509

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Maple [C]  time = 0.038, size = 276, normalized size = 1.8 \[ -{\frac{2}{ \left ( 1867635\,{x}^{2}+2365671\,x+747054 \right ) \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 14140\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+5594\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-7070\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -2797\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -167820\,{x}^{3}+95408\,{x}^{2}+56175\,x-36891 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/124509*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(14140*2^(1/2)*EllipticF(1/1
1*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)+5594*2^(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))*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1
/2)-7070*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))-2797*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))-167820*x^3+95408*x^2+56175*x-36891)/(15*x^2+19*x+6
)/(-1+2*x)^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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