3.2728 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=222 \[ -\frac{10312712 \sqrt{1-2 x} \sqrt{3 x+2}}{1029 \sqrt{5 x+3}}+\frac{1706144 \sqrt{1-2 x}}{1715 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{12276 \sqrt{1-2 x}}{245 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{176 \sqrt{1-2 x}}{35 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{2 \sqrt{1-2 x}}{3 (3 x+2)^{7/2} \sqrt{5 x+3}}+\frac{310208 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1715}+\frac{10312712 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1715} \]

[Out]

(2*Sqrt[1 - 2*x])/(3*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x]) + (176*Sqrt[1 - 2*x])/(35*(2
+ 3*x)^(5/2)*Sqrt[3 + 5*x]) + (12276*Sqrt[1 - 2*x])/(245*(2 + 3*x)^(3/2)*Sqrt[3
+ 5*x]) + (1706144*Sqrt[1 - 2*x])/(1715*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) - (1031271
2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(1029*Sqrt[3 + 5*x]) + (10312712*Sqrt[11/3]*Ellip
ticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1715 + (310208*Sqrt[11/3]*Elliptic
F[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1715

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Rubi [A]  time = 0.524784, 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 \[ -\frac{10312712 \sqrt{1-2 x} \sqrt{3 x+2}}{1029 \sqrt{5 x+3}}+\frac{1706144 \sqrt{1-2 x}}{1715 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{12276 \sqrt{1-2 x}}{245 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{176 \sqrt{1-2 x}}{35 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{2 \sqrt{1-2 x}}{3 (3 x+2)^{7/2} \sqrt{5 x+3}}+\frac{310208 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1715}+\frac{10312712 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1715} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 - 2*x])/(3*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x]) + (176*Sqrt[1 - 2*x])/(35*(2
+ 3*x)^(5/2)*Sqrt[3 + 5*x]) + (12276*Sqrt[1 - 2*x])/(245*(2 + 3*x)^(3/2)*Sqrt[3
+ 5*x]) + (1706144*Sqrt[1 - 2*x])/(1715*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) - (1031271
2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(1029*Sqrt[3 + 5*x]) + (10312712*Sqrt[11/3]*Ellip
ticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1715 + (310208*Sqrt[11/3]*Elliptic
F[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1715

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Rubi in Sympy [A]  time = 45.6682, size = 201, normalized size = 0.91 \[ - \frac{10312712 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1715 \sqrt{3 x + 2}} - \frac{148408 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{245 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{3188 \sqrt{- 2 x + 1}}{21 \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}} + \frac{176 \sqrt{- 2 x + 1}}{35 \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}} + \frac{2 \sqrt{- 2 x + 1}}{3 \left (3 x + 2\right )^{\frac{7}{2}} \sqrt{5 x + 3}} + \frac{10312712 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{5145} + \frac{3412288 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{60025} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-10312712*sqrt(-2*x + 1)*sqrt(5*x + 3)/(1715*sqrt(3*x + 2)) - 148408*sqrt(-2*x +
1)*sqrt(5*x + 3)/(245*(3*x + 2)**(3/2)) - 3188*sqrt(-2*x + 1)/(21*(3*x + 2)**(3
/2)*sqrt(5*x + 3)) + 176*sqrt(-2*x + 1)/(35*(3*x + 2)**(5/2)*sqrt(5*x + 3)) + 2*
sqrt(-2*x + 1)/(3*(3*x + 2)**(7/2)*sqrt(5*x + 3)) + 10312712*sqrt(33)*elliptic_e
(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/5145 + 3412288*sqrt(35)*elliptic_f(asin
(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/60025

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Mathematica [A]  time = 0.274668, size = 110, normalized size = 0.5 \[ \frac{2 \left (-\frac{3 \sqrt{1-2 x} \left (696108060 x^4+1833255216 x^3+1809835578 x^2+793777840 x+130497191\right )}{(3 x+2)^{7/2} \sqrt{5 x+3}}-4 \sqrt{2} \left (1289089 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-649285 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{5145} \]

Antiderivative was successfully verified.

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

[Out]

(2*((-3*Sqrt[1 - 2*x]*(130497191 + 793777840*x + 1809835578*x^2 + 1833255216*x^3
+ 696108060*x^4))/((2 + 3*x)^(7/2)*Sqrt[3 + 5*x]) - 4*Sqrt[2]*(1289089*Elliptic
E[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 649285*EllipticF[ArcSin[Sqrt[2/11]*
Sqrt[3 + 5*x]], -33/2])))/5145

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Maple [C]  time = 0.037, size = 505, normalized size = 2.3 \[{\frac{2}{51450\,{x}^{2}+5145\,x-15435}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 139221612\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}-70122780\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-140245560\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-93497040\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -20777120\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -4176648360\,{x}^{5}-8911207116\,{x}^{4}-5359247820\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+391491573 \right ) \left ( 2+3\,x \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5145*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(139221612*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)-70122780*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*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)+2
78443224*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)-140245560*2^(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))*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)-93497040*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))-20777120*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))-4176648360*x^5-8911207116*x^4-5359247820*x^3
[phone]*x^2+1598350374*x+391491573)/(2+3*x)^(7/2)/(10*x^2+x-3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(3/2)/((5*x + 3)^(3/2)*(3*x + 2)^(9/2)),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(3/2)/((5*x + 3)^(3/2)*(3*x + 2)^(9/2)),x, algorithm="fricas")

[Out]

integral((-2*x + 1)^(3/2)/((405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 4
8)*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)**(3/2)/(2+3*x)**(9/2)/(3+5*x)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(3/2)/((5*x + 3)^(3/2)*(3*x + 2)^(9/2)),x, algorithm="giac")

[Out]

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