3.2888 \(\int \frac{(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^{5/2}} \, dx\)

Optimal. Leaf size=160 \[ -\frac{458 \sqrt{1-2 x} \sqrt{5 x+3}}{1029 \sqrt{3 x+2}}-\frac{97 \sqrt{1-2 x} \sqrt{5 x+3}}{147 (3 x+2)^{3/2}}+\frac{11 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^{3/2}}-\frac{178 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1029}+\frac{458 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1029} \]

[Out]

(11*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)) - (97*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(147*(2 + 3*x)^(3/2)) - (458*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1029*Sqrt[2 +
3*x]) + (458*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1029
- (178*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1029

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Rubi [A]  time = 0.343574, antiderivative size = 160, 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 \[ -\frac{458 \sqrt{1-2 x} \sqrt{5 x+3}}{1029 \sqrt{3 x+2}}-\frac{97 \sqrt{1-2 x} \sqrt{5 x+3}}{147 (3 x+2)^{3/2}}+\frac{11 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^{3/2}}-\frac{178 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1029}+\frac{458 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1029} \]

Antiderivative was successfully verified.

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

[Out]

(11*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)) - (97*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(147*(2 + 3*x)^(3/2)) - (458*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1029*Sqrt[2 +
3*x]) + (458*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1029
- (178*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1029

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Rubi in Sympy [A]  time = 31.4374, size = 143, normalized size = 0.89 \[ - \frac{458 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1029 \sqrt{3 x + 2}} - \frac{97 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{147 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{458 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{3087} - \frac{178 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{3087} + \frac{11 \sqrt{5 x + 3}}{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-458*sqrt(-2*x + 1)*sqrt(5*x + 3)/(1029*sqrt(3*x + 2)) - 97*sqrt(-2*x + 1)*sqrt(
5*x + 3)/(147*(3*x + 2)**(3/2)) + 458*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*
x + 1)/7), 35/33)/3087 - 178*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7)
, 35/33)/3087 + 11*sqrt(5*x + 3)/(7*sqrt(-2*x + 1)*(3*x + 2)**(3/2))

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Mathematica [A]  time = 0.354939, size = 97, normalized size = 0.61 \[ \frac{\sqrt{2} \left (\frac{3 \sqrt{10 x+6} \left (1374 x^2+908 x+11\right )}{\sqrt{1-2 x} (3 x+2)^{3/2}}+3395 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-458 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{3087} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[2]*((3*Sqrt[6 + 10*x]*(11 + 908*x + 1374*x^2))/(Sqrt[1 - 2*x]*(2 + 3*x)^(3
/2)) - 458*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 3395*EllipticF[A
rcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/3087

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Maple [C]  time = 0.033, size = 267, normalized size = 1.7 \[ -{\frac{1}{30870\,{x}^{2}+3087\,x-9261}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 10185\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-1374\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+6790\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -916\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +41220\,{x}^{3}+51972\,{x}^{2}+16674\,x+198 \right ) \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3087*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(10185*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)-1374*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)+6790*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))-916*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))+41220*x^3+51972*x^2+16674*x+198)/(2+3*x)^(3/2)/(10*x^2+x-3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(5*x + 3)^(3/2)/((18*x^3 + 15*x^2 - 4*x - 4)*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((3+5*x)**(3/2)/(1-2*x)**(3/2)/(2+3*x)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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