3.2917 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=222 \[ -\frac{46585232 \sqrt{1-2 x} \sqrt{3 x+2}}{290521 \sqrt{5 x+3}}+\frac{2101332 \sqrt{1-2 x}}{132055 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{14928 \sqrt{1-2 x}}{18865 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{138 \sqrt{1-2 x}}{2695 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{4}{77 \sqrt{1-2 x} (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{1400888 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{132055}+\frac{46585232 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{132055} \]

[Out]

4/(77*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) + (138*Sqrt[1 - 2*x])/(2695*(
2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) + (14928*Sqrt[1 - 2*x])/(18865*(2 + 3*x)^(3/2)*Sqr
t[3 + 5*x]) + (2101332*Sqrt[1 - 2*x])/(132055*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) - (46
585232*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(290521*Sqrt[3 + 5*x]) + (46585232*Sqrt[3/11
]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/132055 + (1400888*Sqrt[3/11
]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/132055

_______________________________________________________________________________________

Rubi [A]  time = 0.523662, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{46585232 \sqrt{1-2 x} \sqrt{3 x+2}}{290521 \sqrt{5 x+3}}+\frac{2101332 \sqrt{1-2 x}}{132055 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{14928 \sqrt{1-2 x}}{18865 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{138 \sqrt{1-2 x}}{2695 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{4}{77 \sqrt{1-2 x} (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{1400888 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{132055}+\frac{46585232 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{132055} \]

Antiderivative was successfully verified.

[In]  Int[1/((1 - 2*x)^(3/2)*(2 + 3*x)^(7/2)*(3 + 5*x)^(3/2)),x]

[Out]

4/(77*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) + (138*Sqrt[1 - 2*x])/(2695*(
2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) + (14928*Sqrt[1 - 2*x])/(18865*(2 + 3*x)^(3/2)*Sqr
t[3 + 5*x]) + (2101332*Sqrt[1 - 2*x])/(132055*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) - (46
585232*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(290521*Sqrt[3 + 5*x]) + (46585232*Sqrt[3/11
]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/132055 + (1400888*Sqrt[3/11
]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/132055

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 45.9111, size = 201, normalized size = 0.91 \[ - \frac{46585232 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{290521 \sqrt{5 x + 3}} + \frac{2101332 \sqrt{- 2 x + 1}}{132055 \sqrt{3 x + 2} \sqrt{5 x + 3}} + \frac{14928 \sqrt{- 2 x + 1}}{18865 \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}} + \frac{138 \sqrt{- 2 x + 1}}{2695 \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}} + \frac{46585232 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1452605} + \frac{1400888 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1452605} + \frac{4}{77 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1-2*x)**(3/2)/(2+3*x)**(7/2)/(3+5*x)**(3/2),x)

[Out]

-46585232*sqrt(-2*x + 1)*sqrt(3*x + 2)/(290521*sqrt(5*x + 3)) + 2101332*sqrt(-2*
x + 1)/(132055*sqrt(3*x + 2)*sqrt(5*x + 3)) + 14928*sqrt(-2*x + 1)/(18865*(3*x +
2)**(3/2)*sqrt(5*x + 3)) + 138*sqrt(-2*x + 1)/(2695*(3*x + 2)**(5/2)*sqrt(5*x +
3)) + 46585232*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/1452
605 + 1400888*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/145260
5 + 4/(77*sqrt(-2*x + 1)*(3*x + 2)**(5/2)*sqrt(5*x + 3))

_______________________________________________________________________________________

Mathematica [A]  time = 0.268343, size = 109, normalized size = 0.49 \[ \frac{2 \left (\frac{6289006320 x^4+9225477612 x^3+1919527182 x^2-2283681406 x-884250959}{\sqrt{1-2 x} (3 x+2)^{5/2} \sqrt{5 x+3}}-2 \sqrt{2} \left (11646308 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5867645 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{1452605} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((1 - 2*x)^(3/2)*(2 + 3*x)^(7/2)*(3 + 5*x)^(3/2)),x]

[Out]

(2*((-884250959 - 2283681406*x + 1919527182*x^2 + 9225477612*x^3 + 6289006320*x^
4)/(Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) - 2*Sqrt[2]*(11646308*EllipticE
[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 5867645*EllipticF[ArcSin[Sqrt[2/11]*
Sqrt[3 + 5*x]], -33/2])))/1452605

_______________________________________________________________________________________

Maple [C]  time = 0.041, size = 386, normalized size = 1.7 \[{\frac{2}{14526050\,{x}^{2}+1452605\,x-4357815}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 209633544\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-105617610\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-140823480\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -46941160\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -6289006320\,{x}^{4}-9225477612\,{x}^{3}-1919527182\,{x}^{2}[phone]\,x+884250959 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-2*x)^(3/2)/(2+3*x)^(7/2)/(3+5*x)^(3/2),x)

[Out]

2/1452605*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(209633544*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^2*(3+5*x)^(1/2)*(2+3*x)
^(1/2)*(1-2*x)^(1/2)-105617610*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^2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/
2)[phone]*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)-140823480*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)[phone]*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))-46941160*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*E
llipticF(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))-628
9006320*x^4-9225477612*x^3-1919527182*x^2+2283681406*x+884250959)/(2+3*x)^(5/2)/
(10*x^2+x-3)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((5*x + 3)^(3/2)*(3*x + 2)^(7/2)*(-2*x + 1)^(3/2)),x, algorithm="maxima")

[Out]

integrate(1/((5*x + 3)^(3/2)*(3*x + 2)^(7/2)*(-2*x + 1)^(3/2)), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{1}{{\left (270 \, x^{5} + 567 \, x^{4} + 333 \, x^{3} - 46 \, x^{2} - 100 \, x - 24\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((5*x + 3)^(3/2)*(3*x + 2)^(7/2)*(-2*x + 1)^(3/2)),x, algorithm="fricas")

[Out]

integral(-1/((270*x^5 + 567*x^4 + 333*x^3 - 46*x^2 - 100*x - 24)*sqrt(5*x + 3)*s
qrt(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/(1-2*x)**(3/2)/(2+3*x)**(7/2)/(3+5*x)**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((5*x + 3)^(3/2)*(3*x + 2)^(7/2)*(-2*x + 1)^(3/2)),x, algorithm="giac")

[Out]

integrate(1/((5*x + 3)^(3/2)*(3*x + 2)^(7/2)*(-2*x + 1)^(3/2)), x)