3.2971 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=218 \[ -\frac{113693540 \sqrt{1-2 x} \sqrt{3 x+2}}{9587193 \sqrt{5 x+3}}+\frac{336536 \sqrt{1-2 x}}{290521 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{694 \sqrt{1-2 x}}{41503 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{1352}{17787 \sqrt{1-2 x} (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{4}{231 (1-2 x)^{3/2} (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{673072 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{290521 \sqrt{33}}+\frac{22738708 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{290521 \sqrt{33}} \]

[Out]

4/(231*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + 1352/(17787*Sqrt[1 - 2*x
]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + (694*Sqrt[1 - 2*x])/(41503*(2 + 3*x)^(3/2)*Sq
rt[3 + 5*x]) + (336536*Sqrt[1 - 2*x])/(290521*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) - (11
3693540*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(9587193*Sqrt[3 + 5*x]) + (22738708*Ellipti
cE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(290521*Sqrt[33]) + (673072*Elliptic
F[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(290521*Sqrt[33])

_______________________________________________________________________________________

Rubi [A]  time = 0.53128, antiderivative size = 218, 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 \[ -\frac{113693540 \sqrt{1-2 x} \sqrt{3 x+2}}{9587193 \sqrt{5 x+3}}+\frac{336536 \sqrt{1-2 x}}{290521 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{694 \sqrt{1-2 x}}{41503 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{1352}{17787 \sqrt{1-2 x} (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{4}{231 (1-2 x)^{3/2} (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{673072 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{290521 \sqrt{33}}+\frac{22738708 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{290521 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

4/(231*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + 1352/(17787*Sqrt[1 - 2*x
]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + (694*Sqrt[1 - 2*x])/(41503*(2 + 3*x)^(3/2)*Sq
rt[3 + 5*x]) + (336536*Sqrt[1 - 2*x])/(290521*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) - (11
3693540*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(9587193*Sqrt[3 + 5*x]) + (22738708*Ellipti
cE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(290521*Sqrt[33]) + (673072*Elliptic
F[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(290521*Sqrt[33])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 45.2212, size = 201, normalized size = 0.92 \[ \frac{22738708 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{9587193} + \frac{673072 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{9587193} + \frac{45477416 \sqrt{3 x + 2} \sqrt{5 x + 3}}{9587193 \sqrt{- 2 x + 1}} - \frac{3344540 \sqrt{3 x + 2}}{124509 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} + \frac{10156}{3773 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}} + \frac{62}{539 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}} + \frac{4}{231 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

22738708*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/9587193 + 6
73072*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/9587193 + 4547
7416*sqrt(3*x + 2)*sqrt(5*x + 3)/(9587193*sqrt(-2*x + 1)) - 3344540*sqrt(3*x + 2
)/(124509*sqrt(-2*x + 1)*sqrt(5*x + 3)) + 10156/(3773*sqrt(-2*x + 1)*sqrt(3*x +
2)*sqrt(5*x + 3)) + 62/(539*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3)) + 4/(
231*(-2*x + 1)**(3/2)*(3*x + 2)**(3/2)*sqrt(5*x + 3))

_______________________________________________________________________________________

Mathematica [A]  time = 0.348684, size = 109, normalized size = 0.5 \[ \frac{\frac{-4092967440 x^4-1231054224 x^3+2571169924 x^2+397147008 x-431507730}{(1-2 x)^{3/2} (3 x+2)^{3/2} \sqrt{5 x+3}}-4 \sqrt{2} \left (5684677 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-2908255 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{9587193} \]

Antiderivative was successfully verified.

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

[Out]

((-431507730 + 397147008*x + 2571169924*x^2 - 1231054224*x^3 - 4092967440*x^4)/(
(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) - 4*Sqrt[2]*(5684677*EllipticE[Ar
cSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 2908255*EllipticF[ArcSin[Sqrt[2/11]*Sqr
t[3 + 5*x]], -33/2]))/9587193

_______________________________________________________________________________________

Maple [C]  time = 0.039, size = 383, normalized size = 1.8 \[ -{\frac{2}{9587193\, \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 34899060\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-68216124\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+5816510\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-11369354\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-11633020\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{4}[phone]\,{x}^{3}-1285584962\,{x}^{2}-198573504\,x+215753865 \right ) \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9587193*(1-2*x)^(1/2)*(34899060*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)-68216124*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)+5816510*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)-11369354*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)-11633020*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)
)[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))[phone]*x^4+615527
112*x^3-1285584962*x^2-198573504*x+215753865)/(2+3*x)^(3/2)/(-1+2*x)^2/(3+5*x)^(
1/2)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/((180*x^5 + 168*x^4 - 79*x^3 - 89*x^2 + 8*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/(1-2*x)**(5/2)/(2+3*x)**(5/2)/(3+5*x)**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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