3.2856 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=160 \[ -\frac{31940 \sqrt{1-2 x} \sqrt{3 x+2}}{539 \sqrt{5 x+3}}+\frac{288 \sqrt{1-2 x}}{49 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{2 \sqrt{1-2 x}}{7 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{192}{49} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{6388}{49} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(2*Sqrt[1 - 2*x])/(7*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + (288*Sqrt[1 - 2*x])/(49*Sq
rt[2 + 3*x]*Sqrt[3 + 5*x]) - (31940*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(539*Sqrt[3 + 5
*x]) + (6388*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/49 +
(192*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/49

_______________________________________________________________________________________

Rubi [A]  time = 0.35179, 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 \[ -\frac{31940 \sqrt{1-2 x} \sqrt{3 x+2}}{539 \sqrt{5 x+3}}+\frac{288 \sqrt{1-2 x}}{49 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{2 \sqrt{1-2 x}}{7 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{192}{49} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{6388}{49} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 - 2*x])/(7*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + (288*Sqrt[1 - 2*x])/(49*Sq
rt[2 + 3*x]*Sqrt[3 + 5*x]) - (31940*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(539*Sqrt[3 + 5
*x]) + (6388*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/49 +
(192*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/49

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 32.7397, size = 143, normalized size = 0.89 \[ - \frac{31940 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{539 \sqrt{5 x + 3}} + \frac{288 \sqrt{- 2 x + 1}}{49 \sqrt{3 x + 2} \sqrt{5 x + 3}} + \frac{2 \sqrt{- 2 x + 1}}{7 \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}} + \frac{6388 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{539} + \frac{576 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{1715} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-31940*sqrt(-2*x + 1)*sqrt(3*x + 2)/(539*sqrt(5*x + 3)) + 288*sqrt(-2*x + 1)/(49
*sqrt(3*x + 2)*sqrt(5*x + 3)) + 2*sqrt(-2*x + 1)/(7*(3*x + 2)**(3/2)*sqrt(5*x +
3)) + 6388*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/539 + 576
*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/1715

_______________________________________________________________________________________

Mathematica [A]  time = 0.196633, size = 100, normalized size = 0.62 \[ \frac{2}{539} \left (-\frac{\sqrt{1-2 x} \left (143730 x^2+186888 x+60635\right )}{(3 x+2)^{3/2} \sqrt{5 x+3}}-2 \sqrt{2} \left (1597 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-805 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*(-((Sqrt[1 - 2*x]*(60635 + 186888*x + 143730*x^2))/((2 + 3*x)^(3/2)*Sqrt[3 +
5*x])) - 2*Sqrt[2]*(1597*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 80
5*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/539

_______________________________________________________________________________________

Maple [C]  time = 0.034, size = 267, normalized size = 1.7 \[ -{\frac{2}{5390\,{x}^{2}+539\,x-1617}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 4830\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-9582\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+3220\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -6388\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +287460\,{x}^{3}+230046\,{x}^{2}-65618\,x-60635 \right ) \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/539*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(4830*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)-9582*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)+3220*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))-6388*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))+287460*x^3+230046*x^2-65618*x-60635)/(2+3*x)^(3/2)/(10*x^2+x-3)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/((45*x^3 + 87*x^2 + 56*x + 12)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x
+ 1)), x)

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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