3.2797 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^{9/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=251 \[ \frac{2 (1-2 x)^{3/2}}{3 (3 x+2)^{7/2} (5 x+3)^{3/2}}+\frac{11171040 \sqrt{3 x+2} \sqrt{1-2 x}}{49 \sqrt{5 x+3}}-\frac{5544440 \sqrt{3 x+2} \sqrt{1-2 x}}{147 (5 x+3)^{3/2}}+\frac{2488904 \sqrt{1-2 x}}{441 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{11924 \sqrt{1-2 x}}{63 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac{44 \sqrt{1-2 x}}{3 (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac{201616}{49} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{2234208}{49} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(2*(1 - 2*x)^(3/2))/(3*(2 + 3*x)^(7/2)*(3 + 5*x)^(3/2)) + (44*Sqrt[1 - 2*x])/(3*
(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (11924*Sqrt[1 - 2*x])/(63*(2 + 3*x)^(3/2)*(3
+ 5*x)^(3/2)) + (2488904*Sqrt[1 - 2*x])/(441*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (5
544440*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(147*(3 + 5*x)^(3/2)) + (11171040*Sqrt[1 - 2
*x]*Sqrt[2 + 3*x])/(49*Sqrt[3 + 5*x]) - (2234208*Sqrt[33]*EllipticE[ArcSin[Sqrt[
3/7]*Sqrt[1 - 2*x]], 35/33])/49 - (201616*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*
Sqrt[1 - 2*x]], 35/33])/49

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Rubi [A]  time = 0.609184, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{2 (1-2 x)^{3/2}}{3 (3 x+2)^{7/2} (5 x+3)^{3/2}}+\frac{11171040 \sqrt{3 x+2} \sqrt{1-2 x}}{49 \sqrt{5 x+3}}-\frac{5544440 \sqrt{3 x+2} \sqrt{1-2 x}}{147 (5 x+3)^{3/2}}+\frac{2488904 \sqrt{1-2 x}}{441 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{11924 \sqrt{1-2 x}}{63 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac{44 \sqrt{1-2 x}}{3 (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac{201616}{49} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{2234208}{49} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 - 2*x)^(5/2)/((2 + 3*x)^(9/2)*(3 + 5*x)^(5/2)),x]

[Out]

(2*(1 - 2*x)^(3/2))/(3*(2 + 3*x)^(7/2)*(3 + 5*x)^(3/2)) + (44*Sqrt[1 - 2*x])/(3*
(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (11924*Sqrt[1 - 2*x])/(63*(2 + 3*x)^(3/2)*(3
+ 5*x)^(3/2)) + (2488904*Sqrt[1 - 2*x])/(441*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (5
544440*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(147*(3 + 5*x)^(3/2)) + (11171040*Sqrt[1 - 2
*x]*Sqrt[2 + 3*x])/(49*Sqrt[3 + 5*x]) - (2234208*Sqrt[33]*EllipticE[ArcSin[Sqrt[
3/7]*Sqrt[1 - 2*x]], 35/33])/49 - (201616*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*
Sqrt[1 - 2*x]], 35/33])/49

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Rubi in Sympy [A]  time = 55.9894, size = 230, normalized size = 0.92 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{3 \left (3 x + 2\right )^{\frac{7}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{11171040 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{49 \sqrt{5 x + 3}} - \frac{5544440 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{147 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{2488904 \sqrt{- 2 x + 1}}{441 \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{11924 \sqrt{- 2 x + 1}}{63 \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{44 \sqrt{- 2 x + 1}}{3 \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{2234208 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{49} - \frac{2217776 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{1715} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)/(2+3*x)**(9/2)/(3+5*x)**(5/2),x)

[Out]

2*(-2*x + 1)**(3/2)/(3*(3*x + 2)**(7/2)*(5*x + 3)**(3/2)) + 11171040*sqrt(-2*x +
1)*sqrt(3*x + 2)/(49*sqrt(5*x + 3)) - 5544440*sqrt(-2*x + 1)*sqrt(3*x + 2)/(147
*(5*x + 3)**(3/2)) + 2488904*sqrt(-2*x + 1)/(441*sqrt(3*x + 2)*(5*x + 3)**(3/2))
+ 11924*sqrt(-2*x + 1)/(63*(3*x + 2)**(3/2)*(5*x + 3)**(3/2)) + 44*sqrt(-2*x +
1)/(3*(3*x + 2)**(5/2)*(5*x + 3)**(3/2)) - 2234208*sqrt(33)*elliptic_e(asin(sqrt
(21)*sqrt(-2*x + 1)/7), 35/33)/49 - 2217776*sqrt(35)*elliptic_f(asin(sqrt(55)*sq
rt(-2*x + 1)/11), 33/35)/1715

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Mathematica [A]  time = 0.403471, size = 114, normalized size = 0.45 \[ \frac{2}{147} \left (\frac{\sqrt{1-2 x} \left (6786406800 x^5+21944379060 x^4+28367736228 x^3+18325125498 x^2+5915384456 x+763335749\right )}{(3 x+2)^{7/2} (5 x+3)^{3/2}}+12 \sqrt{2} \left (279276 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-140665 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^(9/2)*(3 + 5*x)^(5/2)),x]

[Out]

(2*((Sqrt[1 - 2*x]*(763335749 + 5915384456*x + 18325125498*x^2 + 28367736228*x^3
+ 21944379060*x^4 + 6786406800*x^5))/((2 + 3*x)^(7/2)*(3 + 5*x)^(3/2)) + 12*Sqr
t[2]*(279276*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 140665*Ellipti
cF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/147

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Maple [C]  time = 0.036, size = 621, normalized size = 2.5 \[ -{\frac{2}{-147+294\,x}\sqrt{1-2\,x} \left ( 452427120\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{4}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-227877300\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{4}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}-592480980\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-577289160\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-249821040\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-13572813600\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -40511520\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -37102351320\,{x}^{5}-34791093396\,{x}^{4}-8282514768\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+763335749 \right ) \left ( 2+3\,x \right ) ^{-{\frac{7}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)/(2+3*x)^(9/2)/(3+5*x)^(5/2),x)

[Out]

-2/147*(1-2*x)^(1/2)*(452427120*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^4*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1
/2)-227877300*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^4*(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*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)-592480980*2^(1/2)*EllipticF(1/1
1*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*(1-2*x)^(1/
2)*(3+5*x)^(1/2)*(2+3*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^2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*
(1-2*x)^(1/2)-577289160*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)+4959
94176*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)-249821040*2^(1/2)*Ellipt
icF(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)-13572813600*x^6+80431488*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))-40511520*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))-37102351320*x^5-34791093396*x^4-8282514768*x^3+6494356586*x^2+4388712958*x
[phone])/(2+3*x)^(7/2)/(3+5*x)^(3/2)/(-1+2*x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{9}{2}}}\,{d x} \]

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)^(9/2)),x, algorithm="maxima")

[Out]

integrate((-2*x + 1)^(5/2)/((5*x + 3)^(5/2)*(3*x + 2)^(9/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{-2 \, x + 1}}{{\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}, x\right ) \]

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)^(9/2)),x, algorithm="fricas")

[Out]

integral((4*x^2 - 4*x + 1)*sqrt(-2*x + 1)/((2025*x^6 + 7830*x^5 + 12609*x^4 + 10
824*x^3 + 5224*x^2 + 1344*x + 144)*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)**(9/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{9}{2}}}\,{d x} \]

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)^(9/2)),x, algorithm="giac")

[Out]

integrate((-2*x + 1)^(5/2)/((5*x + 3)^(5/2)*(3*x + 2)^(9/2)), x)