3.2699 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{11/2}} \, dx\)

Optimal. Leaf size=222 \[ \frac{74 \sqrt{1-2 x} (5 x+3)^{3/2}}{189 (3 x+2)^{7/2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}+\frac{19885156 \sqrt{1-2 x} \sqrt{5 x+3}}{2917215 \sqrt{3 x+2}}+\frac{280904 \sqrt{1-2 x} \sqrt{5 x+3}}{416745 (3 x+2)^{3/2}}-\frac{8252 \sqrt{1-2 x} \sqrt{5 x+3}}{19845 (3 x+2)^{5/2}}-\frac{609304 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2917215}-\frac{19885156 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2917215} \]

[Out]

(-8252*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(19845*(2 + 3*x)^(5/2)) + (280904*Sqrt[1 - 2
*x]*Sqrt[3 + 5*x])/(416745*(2 + 3*x)^(3/2)) + (19885156*Sqrt[1 - 2*x]*Sqrt[3 + 5
*x])/(2917215*Sqrt[2 + 3*x]) - (2*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(27*(2 + 3*x)
^(9/2)) + (74*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(189*(2 + 3*x)^(7/2)) - (19885156*S
qrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2917215 - (609304*S
qrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2917215

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Rubi [A]  time = 0.493621, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{74 \sqrt{1-2 x} (5 x+3)^{3/2}}{189 (3 x+2)^{7/2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}+\frac{19885156 \sqrt{1-2 x} \sqrt{5 x+3}}{2917215 \sqrt{3 x+2}}+\frac{280904 \sqrt{1-2 x} \sqrt{5 x+3}}{416745 (3 x+2)^{3/2}}-\frac{8252 \sqrt{1-2 x} \sqrt{5 x+3}}{19845 (3 x+2)^{5/2}}-\frac{609304 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2917215}-\frac{19885156 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2917215} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^(11/2),x]

[Out]

(-8252*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(19845*(2 + 3*x)^(5/2)) + (280904*Sqrt[1 - 2
*x]*Sqrt[3 + 5*x])/(416745*(2 + 3*x)^(3/2)) + (19885156*Sqrt[1 - 2*x]*Sqrt[3 + 5
*x])/(2917215*Sqrt[2 + 3*x]) - (2*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(27*(2 + 3*x)
^(9/2)) + (74*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(189*(2 + 3*x)^(7/2)) - (19885156*S
qrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2917215 - (609304*S
qrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2917215

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Rubi in Sympy [A]  time = 45.5029, size = 201, normalized size = 0.91 \[ - \frac{74 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{1323 \left (3 x + 2\right )^{\frac{7}{2}}} - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{27 \left (3 x + 2\right )^{\frac{9}{2}}} + \frac{19885156 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2917215 \sqrt{3 x + 2}} + \frac{280904 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{416745 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{3958 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{19845 \left (3 x + 2\right )^{\frac{5}{2}}} - \frac{19885156 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{8751645} - \frac{6702344 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{102102525} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(3/2)*(3+5*x)**(3/2)/(2+3*x)**(11/2),x)

[Out]

-74*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(1323*(3*x + 2)**(7/2)) - 2*(-2*x + 1)**(3/2
)*(5*x + 3)**(3/2)/(27*(3*x + 2)**(9/2)) + 19885156*sqrt(-2*x + 1)*sqrt(5*x + 3)
/(2917215*sqrt(3*x + 2)) + 280904*sqrt(-2*x + 1)*sqrt(5*x + 3)/(416745*(3*x + 2)
**(3/2)) + 3958*sqrt(-2*x + 1)*sqrt(5*x + 3)/(19845*(3*x + 2)**(5/2)) - 19885156
*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/8751645 - 6702344*s
qrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/102102525

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Mathematica [A]  time = 0.362568, size = 111, normalized size = 0.5 \[ \frac{4 \left (\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (805348818 x^4+2174142276 x^3+2204875881 x^2+993561978 x+167622907\right )}{2 (3 x+2)^{9/2}}+\sqrt{2} \left (4971289 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-2457910 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{8751645} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^(11/2),x]

[Out]

(4*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(167622907 + 993561978*x + 2204875881*x^2 + 2
174142276*x^3 + 805348818*x^4))/(2*(2 + 3*x)^(9/2)) + Sqrt[2]*(4971289*EllipticE
[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 2457910*EllipticF[ArcSin[Sqrt[2/11]*
Sqrt[3 + 5*x]], -33/2])))/8751645

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Maple [C]  time = 0.03, size = 624, normalized size = 2.8 \[{\frac{2}{87516450\,{x}^{2}+8751645\,x-26254935} \left ( 398181420\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{4}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-805348818\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{4}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}-2147596848\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-2147596848\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-954487488\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -159081248\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}-11834509785\,{x}^{2}-8439189081\,x-1508606163 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^(11/2),x)

[Out]

2/8751645*(398181420*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)-8053488
18*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)[phone]*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*(1-2
*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)-2147596848*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)[phone]*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)-2147596848*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)[phone]*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)-954487488*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)[phone]*x^6+78653120*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))-159081248*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))+67
640314734*x^5+65420563896*x^4+16854206499*x^3-11834509785*x^2-8439189081*x-15086
06163)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(9/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)/(3*x + 2)^(11/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)/(3*x + 2)^(11/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (10 \, x^{2} + x - 3\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)/(3*x + 2)^(11/2),x, algorithm="fricas")

[Out]

integral(-(10*x^2 + x - 3)*sqrt(5*x + 3)*sqrt(-2*x + 1)/((243*x^5 + 810*x^4 + 10
80*x^3 + 720*x^2 + 240*x + 32)*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)*(3+5*x)**(3/2)/(2+3*x)**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)/(3*x + 2)^(11/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)/(3*x + 2)^(11/2), x)