3.2720 \(\int \frac{(1-2 x)^{3/2} (2+3 x)^{7/2}}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=222 \[ -\frac{8}{45} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{7/2}-\frac{2 (1-2 x)^{3/2} (3 x+2)^{7/2}}{5 \sqrt{5 x+3}}+\frac{958 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}}{1575}+\frac{5153 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{39375}-\frac{12601 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{196875}-\frac{31288 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{984375}-\frac{1473539 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1968750} \]

[Out]

(-2*(1 - 2*x)^(3/2)*(2 + 3*x)^(7/2))/(5*Sqrt[3 + 5*x]) - (12601*Sqrt[1 - 2*x]*Sq
rt[2 + 3*x]*Sqrt[3 + 5*x])/196875 + (5153*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 +
5*x])/39375 + (958*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/1575 - (8*Sqrt[
1 - 2*x]*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x])/45 - (1473539*Sqrt[11/3]*EllipticE[ArcSi
n[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1968750 - (31288*Sqrt[11/3]*EllipticF[ArcSin
[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/984375

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Rubi [A]  time = 0.484417, 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 \[ -\frac{8}{45} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{7/2}-\frac{2 (1-2 x)^{3/2} (3 x+2)^{7/2}}{5 \sqrt{5 x+3}}+\frac{958 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}}{1575}+\frac{5153 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{39375}-\frac{12601 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{196875}-\frac{31288 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{984375}-\frac{1473539 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1968750} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(3/2)*(2 + 3*x)^(7/2))/(5*Sqrt[3 + 5*x]) - (12601*Sqrt[1 - 2*x]*Sq
rt[2 + 3*x]*Sqrt[3 + 5*x])/196875 + (5153*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 +
5*x])/39375 + (958*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/1575 - (8*Sqrt[
1 - 2*x]*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x])/45 - (1473539*Sqrt[11/3]*EllipticE[ArcSi
n[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1968750 - (31288*Sqrt[11/3]*EllipticF[ArcSin
[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/984375

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Rubi in Sympy [A]  time = 47.4585, size = 201, normalized size = 0.91 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{7}{2}}}{5 \sqrt{5 x + 3}} - \frac{8 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{7}{2}} \sqrt{5 x + 3}}{45} + \frac{958 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{1575} + \frac{5153 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{39375} - \frac{12601 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{196875} - \frac{1473539 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{5906250} - \frac{31288 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{2953125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-2*x + 1)**(3/2)*(3*x + 2)**(7/2)/(5*sqrt(5*x + 3)) - 8*sqrt(-2*x + 1)*(3*x
+ 2)**(7/2)*sqrt(5*x + 3)/45 + 958*sqrt(-2*x + 1)*(3*x + 2)**(5/2)*sqrt(5*x + 3)
/1575 + 5153*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/39375 - 12601*sqrt(-2
*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/196875 - 1473539*sqrt(33)*elliptic_e(asin(sq
rt(21)*sqrt(-2*x + 1)/7), 35/33)/5906250 - 31288*sqrt(33)*elliptic_f(asin(sqrt(2
1)*sqrt(-2*x + 1)/7), 35/33)/2953125

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Mathematica [A]  time = 0.233998, size = 132, normalized size = 0.59 \[ \frac{1473539 \sqrt{2} (5 x+3) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (6 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (472500 x^4+517500 x^3-252225 x^2-377530 x-83787\right )+88207 \sqrt{2} (5 x+3) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{5906250 (5 x+3)} \]

Antiderivative was successfully verified.

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

[Out]

(1473539*Sqrt[2]*(3 + 5*x)*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] -
5*(6*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-83787 - 377530*x - 252225*x^2 +
517500*x^3 + 472500*x^4) + 88207*Sqrt[2]*(3 + 5*x)*EllipticF[ArcSin[Sqrt[2/11]*
Sqrt[3 + 5*x]], -33/2]))/(5906250*(3 + 5*x))

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Maple [C]  time = 0.026, size = 179, normalized size = 0.8 \[{\frac{1}{177187500\,{x}^{3}[phone]\,{x}^{2}-41343750\,x-35437500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -85050000\,{x}^{6}+441035\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -1473539\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -107325000\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-20138190\,x-5027220 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5906250*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(-85050000*x^6+441035*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))-1473539*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))-107325000*x^5+58225500*x^4+106572150*x^3+11274060*x^2-20138190
*x-5027220)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(5*
x + 3)^(3/2), 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)**(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. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{7}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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