3.2968 \(\int \frac{\sqrt{2+3 x}}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=156 \[ -\frac{2470 \sqrt{1-2 x} \sqrt{3 x+2}}{27951 \sqrt{5 x+3}}+\frac{214 \sqrt{3 x+2}}{2541 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{2 \sqrt{3 x+2}}{33 (1-2 x)^{3/2} \sqrt{5 x+3}}-\frac{214 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{847 \sqrt{33}}+\frac{494 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{847 \sqrt{33}} \]

[Out]

(2*Sqrt[2 + 3*x])/(33*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) + (214*Sqrt[2 + 3*x])/(2541
*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (2470*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(27951*Sqrt[3
+ 5*x]) + (494*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(847*Sqrt[33]
) - (214*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(847*Sqrt[33])

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Rubi [A]  time = 0.343505, 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 \[ -\frac{2470 \sqrt{1-2 x} \sqrt{3 x+2}}{27951 \sqrt{5 x+3}}+\frac{214 \sqrt{3 x+2}}{2541 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{2 \sqrt{3 x+2}}{33 (1-2 x)^{3/2} \sqrt{5 x+3}}-\frac{214 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{847 \sqrt{33}}+\frac{494 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{847 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[2 + 3*x])/(33*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) + (214*Sqrt[2 + 3*x])/(2541
*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (2470*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(27951*Sqrt[3
+ 5*x]) + (494*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(847*Sqrt[33]
) - (214*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(847*Sqrt[33])

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Rubi in Sympy [A]  time = 31.4802, size = 143, normalized size = 0.92 \[ \frac{494 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{27951} - \frac{214 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{29645} + \frac{988 \sqrt{3 x + 2} \sqrt{5 x + 3}}{27951 \sqrt{- 2 x + 1}} - \frac{40 \sqrt{3 x + 2}}{363 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} + \frac{2 \sqrt{3 x + 2}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

494*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/27951 - 214*sqrt
(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/29645 + 988*sqrt(3*x +
2)*sqrt(5*x + 3)/(27951*sqrt(-2*x + 1)) - 40*sqrt(3*x + 2)/(363*sqrt(-2*x + 1)*s
qrt(5*x + 3)) + 2*sqrt(3*x + 2)/(33*(-2*x + 1)**(3/2)*sqrt(5*x + 3))

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Mathematica [A]  time = 0.190041, size = 99, normalized size = 0.63 \[ \frac{\sqrt{2} \left (4025 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-494 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )-\frac{2 \sqrt{3 x+2} \left (4940 x^2-2586 x-789\right )}{(1-2 x)^{3/2} \sqrt{5 x+3}}}{27951} \]

Antiderivative was successfully verified.

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

[Out]

((-2*Sqrt[2 + 3*x]*(-789 - 2586*x + 4940*x^2))/((1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) +
Sqrt[2]*(-494*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 4025*Ellipti
cF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/27951

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Maple [C]  time = 0.035, size = 276, normalized size = 1.8 \[ -{\frac{1}{ \left ( 419265\,{x}^{2}+531069\,x+167706 \right ) \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 8050\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-988\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-4025\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +494\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +29640\,{x}^{3}+4244\,{x}^{2}-15078\,x-3156 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/27951*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(8050*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)-988*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)
-4025*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))+494*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))+29640*x^3+4244*x^2-15078*x-3156)/(15*x^2+19*x+6)/(-1+2
*x)^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{3 \, x + 2}}{{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(3*x + 2)/((20*x^3 - 8*x^2 - 7*x + 3)*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)**(1/2)/(1-2*x)**(5/2)/(3+5*x)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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