3.2786 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=191 \[ \frac{14 (1-2 x)^{3/2}}{15 (3 x+2)^{5/2} \sqrt{5 x+3}}-\frac{105584 \sqrt{3 x+2} \sqrt{1-2 x}}{27 \sqrt{5 x+3}}+\frac{17468 \sqrt{1-2 x}}{45 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{2716 \sqrt{1-2 x}}{135 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{3176}{45} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{105584}{45} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(14*(1 - 2*x)^(3/2))/(15*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) + (2716*Sqrt[1 - 2*x])/(
135*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + (17468*Sqrt[1 - 2*x])/(45*Sqrt[2 + 3*x]*Sqr
t[3 + 5*x]) - (105584*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(27*Sqrt[3 + 5*x]) + (105584*
Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/45 + (3176*Sqrt[11
/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/45

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Rubi [A]  time = 0.426338, 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 \[ \frac{14 (1-2 x)^{3/2}}{15 (3 x+2)^{5/2} \sqrt{5 x+3}}-\frac{105584 \sqrt{3 x+2} \sqrt{1-2 x}}{27 \sqrt{5 x+3}}+\frac{17468 \sqrt{1-2 x}}{45 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{2716 \sqrt{1-2 x}}{135 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{3176}{45} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{105584}{45} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(14*(1 - 2*x)^(3/2))/(15*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) + (2716*Sqrt[1 - 2*x])/(
135*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + (17468*Sqrt[1 - 2*x])/(45*Sqrt[2 + 3*x]*Sqr
t[3 + 5*x]) - (105584*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(27*Sqrt[3 + 5*x]) + (105584*
Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/45 + (3176*Sqrt[11
/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/45

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Rubi in Sympy [A]  time = 40.6927, size = 172, normalized size = 0.9 \[ \frac{14 \left (- 2 x + 1\right )^{\frac{3}{2}}}{15 \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}} - \frac{105584 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{27 \sqrt{5 x + 3}} + \frac{17468 \sqrt{- 2 x + 1}}{45 \sqrt{3 x + 2} \sqrt{5 x + 3}} + \frac{2716 \sqrt{- 2 x + 1}}{135 \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}} + \frac{105584 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{135} + \frac{3176 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{135} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

14*(-2*x + 1)**(3/2)/(15*(3*x + 2)**(5/2)*sqrt(5*x + 3)) - 105584*sqrt(-2*x + 1)
*sqrt(3*x + 2)/(27*sqrt(5*x + 3)) + 17468*sqrt(-2*x + 1)/(45*sqrt(3*x + 2)*sqrt(
5*x + 3)) + 2716*sqrt(-2*x + 1)/(135*(3*x + 2)**(3/2)*sqrt(5*x + 3)) + 105584*sq
rt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/135 + 3176*sqrt(33)*el
liptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/135

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Mathematica [A]  time = 0.327842, size = 105, normalized size = 0.55 \[ \frac{2}{135} \left (-\frac{3 \sqrt{1-2 x} \left (2375640 x^3+4672674 x^2+3061396 x+668031\right )}{(3 x+2)^{5/2} \sqrt{5 x+3}}-2 \sqrt{2} \left (26396 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-13295 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*((-3*Sqrt[1 - 2*x]*(668031 + 3061396*x + 4672674*x^2 + 2375640*x^3))/((2 + 3*
x)^(5/2)*Sqrt[3 + 5*x]) - 2*Sqrt[2]*(26396*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]], -33/2] - 13295*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/135

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Maple [C]  time = 0.035, size = 386, normalized size = 2. \[ -{\frac{2}{1350\,{x}^{2}+135\,x-405}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 239310\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-475128\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+319080\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-633504\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+106360\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -211168\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{4}[phone]\,{x}^{3}+4350354\,{x}^{2}-5176002\,x-2004093 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/135*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(239310*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)-475128*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)+319080
*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)-633504*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)+106360*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
))-211168*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+20909124*
x^3+4350354*x^2-5176002*x-2004093)/(2+3*x)^(5/2)/(10*x^2+x-3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{-2 \, x + 1}}{{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((4*x^2 - 4*x + 1)*sqrt(-2*x + 1)/((135*x^4 + 351*x^3 + 342*x^2 + 148*x
+ 24)*sqrt(5*x + 3)*sqrt(3*x + 2)), 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((1-2*x)**(5/2)/(2+3*x)**(7/2)/(3+5*x)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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