3.2923 \(\int \frac{\sqrt{2+3 x}}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=156 \[ -\frac{490 \sqrt{1-2 x} \sqrt{3 x+2}}{3993 \sqrt{5 x+3}}-\frac{40 \sqrt{1-2 x} \sqrt{3 x+2}}{363 (5 x+3)^{3/2}}+\frac{2 \sqrt{3 x+2}}{11 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{16 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{121 \sqrt{33}}+\frac{98 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{121 \sqrt{33}} \]

[Out]

(2*Sqrt[2 + 3*x])/(11*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (40*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x])/(363*(3 + 5*x)^(3/2)) - (490*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(3993*Sqrt[3 +
5*x]) + (98*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(121*Sqrt[33]) -
(16*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(121*Sqrt[33])

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Rubi [A]  time = 0.34258, 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 \[ -\frac{490 \sqrt{1-2 x} \sqrt{3 x+2}}{3993 \sqrt{5 x+3}}-\frac{40 \sqrt{1-2 x} \sqrt{3 x+2}}{363 (5 x+3)^{3/2}}+\frac{2 \sqrt{3 x+2}}{11 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{16 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{121 \sqrt{33}}+\frac{98 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{121 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[2 + 3*x])/(11*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (40*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x])/(363*(3 + 5*x)^(3/2)) - (490*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(3993*Sqrt[3 +
5*x]) + (98*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(121*Sqrt[33]) -
(16*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(121*Sqrt[33])

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Rubi in Sympy [A]  time = 31.483, size = 143, normalized size = 0.92 \[ - \frac{490 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{3993 \sqrt{5 x + 3}} - \frac{40 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{363 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{98 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{3993} - \frac{16 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{4235} + \frac{2 \sqrt{3 x + 2}}{11 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-490*sqrt(-2*x + 1)*sqrt(3*x + 2)/(3993*sqrt(5*x + 3)) - 40*sqrt(-2*x + 1)*sqrt(
3*x + 2)/(363*(5*x + 3)**(3/2)) + 98*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x
+ 1)/7), 35/33)/3993 - 16*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11),
33/35)/4235 + 2*sqrt(3*x + 2)/(11*sqrt(-2*x + 1)*(5*x + 3)**(3/2))

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Mathematica [A]  time = 0.341213, size = 96, normalized size = 0.62 \[ \frac{\sqrt{2} \left (\frac{\sqrt{6 x+4} \left (2450 x^2+685 x-592\right )}{\sqrt{1-2 x} (5 x+3)^{3/2}}+362 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-98 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{3993} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[2]*((Sqrt[4 + 6*x]*(-592 + 685*x + 2450*x^2))/(Sqrt[1 - 2*x]*(3 + 5*x)^(3/
2)) - 98*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 362*EllipticF[ArcS
in[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/3993

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Maple [C]  time = 0.033, size = 267, normalized size = 1.7 \[ -{\frac{2}{23958\,{x}^{2}+3993\,x-7986}\sqrt{2+3\,x}\sqrt{1-2\,x} \left ( 905\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-245\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+543\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -147\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +7350\,{x}^{3}+6955\,{x}^{2}-406\,x-1184 \right ) \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3993*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(905*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)-245*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)+543*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))-147*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))+7350*x^3+6955*x^2-406*x-1184)/(3+5*x)^(3/2)/(6*x^2+x-2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{\sqrt{3 \, x + 2}}{{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-sqrt(3*x + 2)/((50*x^3 + 35*x^2 - 12*x - 9)*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)**(3/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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