3.2907 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=160 \[ \frac{5256 \sqrt{1-2 x} \sqrt{5 x+3}}{3773 \sqrt{3 x+2}}+\frac{54 \sqrt{1-2 x} \sqrt{5 x+3}}{539 (3 x+2)^{3/2}}+\frac{4 \sqrt{5 x+3}}{77 \sqrt{1-2 x} (3 x+2)^{3/2}}-\frac{68}{343} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{1752}{343} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(4*Sqrt[3 + 5*x])/(77*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)) + (54*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(539*(2 + 3*x)^(3/2)) + (5256*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3773*Sqrt[2
+ 3*x]) - (1752*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/34
3 - (68*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/343

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Rubi [A]  time = 0.340875, 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 \[ \frac{5256 \sqrt{1-2 x} \sqrt{5 x+3}}{3773 \sqrt{3 x+2}}+\frac{54 \sqrt{1-2 x} \sqrt{5 x+3}}{539 (3 x+2)^{3/2}}+\frac{4 \sqrt{5 x+3}}{77 \sqrt{1-2 x} (3 x+2)^{3/2}}-\frac{68}{343} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{1752}{343} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[3 + 5*x])/(77*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)) + (54*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(539*(2 + 3*x)^(3/2)) + (5256*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3773*Sqrt[2
+ 3*x]) - (1752*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/34
3 - (68*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/343

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Rubi in Sympy [A]  time = 30.7215, size = 143, normalized size = 0.89 \[ \frac{5256 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{3773 \sqrt{3 x + 2}} + \frac{54 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{539 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{1752 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{3773} - \frac{204 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{12005} + \frac{4 \sqrt{5 x + 3}}{77 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5256*sqrt(-2*x + 1)*sqrt(5*x + 3)/(3773*sqrt(3*x + 2)) + 54*sqrt(-2*x + 1)*sqrt(
5*x + 3)/(539*(3*x + 2)**(3/2)) - 1752*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2
*x + 1)/7), 35/33)/3773 - 204*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/1
1), 33/35)/12005 + 4*sqrt(5*x + 3)/(77*sqrt(-2*x + 1)*(3*x + 2)**(3/2))

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Mathematica [A]  time = 0.22698, size = 99, normalized size = 0.62 \[ \frac{2 \left (\frac{\sqrt{5 x+3} \left (-15768 x^2-3006 x+5543\right )}{\sqrt{1-2 x} (3 x+2)^{3/2}}+3 \sqrt{2} \left (292 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-105 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{3773} \]

Antiderivative was successfully verified.

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

[Out]

(2*((Sqrt[3 + 5*x]*(5543 - 3006*x - 15768*x^2))/(Sqrt[1 - 2*x]*(2 + 3*x)^(3/2))
+ 3*Sqrt[2]*(292*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 105*Ellipt
icF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/3773

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Maple [C]  time = 0.036, size = 267, normalized size = 1.7 \[{\frac{2}{37730\,{x}^{2}+3773\,x-11319}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 945\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-2628\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+630\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -1752\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +78840\,{x}^{3}+62334\,{x}^{2}-18697\,x-16629 \right ) \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3773*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(945*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)-2628*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)+630*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))-1752*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))+78840*x^3+62334*x^2-18697*x-16629)/(2+3*x)^(3/2)/(10*x^2+x-3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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