3.2795 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=191 \[ \frac{14 (1-2 x)^{3/2}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac{36968 \sqrt{3 x+2} \sqrt{1-2 x}}{9 \sqrt{5 x+3}}-\frac{6116 \sqrt{3 x+2} \sqrt{1-2 x}}{9 (5 x+3)^{3/2}}+\frac{308 \sqrt{1-2 x}}{3 \sqrt{3 x+2} (5 x+3)^{3/2}}-\frac{1112}{15} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{36968}{15} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(14*(1 - 2*x)^(3/2))/(9*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (308*Sqrt[1 - 2*x])/(
3*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (6116*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(9*(3 + 5*
x)^(3/2)) + (36968*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(9*Sqrt[3 + 5*x]) - (36968*Sqrt[
11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/15 - (1112*Sqrt[11/3]*E
llipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/15

_______________________________________________________________________________________

Rubi [A]  time = 0.422756, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{14 (1-2 x)^{3/2}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac{36968 \sqrt{3 x+2} \sqrt{1-2 x}}{9 \sqrt{5 x+3}}-\frac{6116 \sqrt{3 x+2} \sqrt{1-2 x}}{9 (5 x+3)^{3/2}}+\frac{308 \sqrt{1-2 x}}{3 \sqrt{3 x+2} (5 x+3)^{3/2}}-\frac{1112}{15} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{36968}{15} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(14*(1 - 2*x)^(3/2))/(9*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (308*Sqrt[1 - 2*x])/(
3*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (6116*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(9*(3 + 5*
x)^(3/2)) + (36968*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(9*Sqrt[3 + 5*x]) - (36968*Sqrt[
11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/15 - (1112*Sqrt[11/3]*E
llipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/15

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 39.4909, size = 172, normalized size = 0.9 \[ \frac{14 \left (- 2 x + 1\right )^{\frac{3}{2}}}{9 \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{36968 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{9 \sqrt{5 x + 3}} - \frac{6116 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{9 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{308 \sqrt{- 2 x + 1}}{3 \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{36968 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{45} - \frac{1112 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{45} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

14*(-2*x + 1)**(3/2)/(9*(3*x + 2)**(3/2)*(5*x + 3)**(3/2)) + 36968*sqrt(-2*x + 1
)*sqrt(3*x + 2)/(9*sqrt(5*x + 3)) - 6116*sqrt(-2*x + 1)*sqrt(3*x + 2)/(9*(5*x +
3)**(3/2)) + 308*sqrt(-2*x + 1)/(3*sqrt(3*x + 2)*(5*x + 3)**(3/2)) - 36968*sqrt(
33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/45 - 1112*sqrt(33)*ellipt
ic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/45

_______________________________________________________________________________________

Mathematica [A]  time = 0.24604, size = 105, normalized size = 0.55 \[ \frac{2 \sqrt{1-2 x} \left (277260 x^3+526862 x^2+333260 x+70169\right )}{3 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac{4}{45} \sqrt{2} \left (9242 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-4655 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 - 2*x]*(70169 + 333260*x + 526862*x^2 + 277260*x^3))/(3*(2 + 3*x)^(3/2
)*(3 + 5*x)^(3/2)) + (4*Sqrt[2]*(9242*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]]
, -33/2] - 4655*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/45

_______________________________________________________________________________________

Maple [C]  time = 0.033, size = 383, normalized size = 2. \[{\frac{2}{-45+90\,x}\sqrt{1-2\,x} \left ( 139650\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-277260\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+176890\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-351196\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+55860\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -110904\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +8317800\,{x}^{4}[phone]\,{x}^{3}+2094870\,{x}^{2}-2893830\,x-1052535 \right ) \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45*(1-2*x)^(1/2)*(139650*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)-2
77260*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)+176890*2^(1/2)*Ellipti
cF(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)-351196*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)+55860*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))-110904*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))+8317800*x^4+11646960*x^3+2094870*x^2-28
93830*x-1052535)/(2+3*x)^(3/2)/(3+5*x)^(3/2)/(-1+2*x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{-2 \, x + 1}}{{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((4*x^2 - 4*x + 1)*sqrt(-2*x + 1)/((225*x^4 + 570*x^3 + 541*x^2 + 228*x
+ 36)*sqrt(5*x + 3)*sqrt(3*x + 2)), 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*x)**(5/2)/(2+3*x)**(5/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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