3.2792 \(\int \frac{(1-2 x)^{5/2} \sqrt{2+3 x}}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=160 \[ -\frac{2 \sqrt{3 x+2} (1-2 x)^{5/2}}{15 (5 x+3)^{3/2}}-\frac{46 \sqrt{3 x+2} (1-2 x)^{3/2}}{75 \sqrt{5 x+3}}-\frac{76}{375} \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{992 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1875}+\frac{338 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1875} \]

[Out]

(-2*(1 - 2*x)^(5/2)*Sqrt[2 + 3*x])/(15*(3 + 5*x)^(3/2)) - (46*(1 - 2*x)^(3/2)*Sq
rt[2 + 3*x])/(75*Sqrt[3 + 5*x]) - (76*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])
/375 + (338*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1875 +
(992*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1875

_______________________________________________________________________________________

Rubi [A]  time = 0.333128, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{2 \sqrt{3 x+2} (1-2 x)^{5/2}}{15 (5 x+3)^{3/2}}-\frac{46 \sqrt{3 x+2} (1-2 x)^{3/2}}{75 \sqrt{5 x+3}}-\frac{76}{375} \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{992 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1875}+\frac{338 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1875} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(5/2)*Sqrt[2 + 3*x])/(15*(3 + 5*x)^(3/2)) - (46*(1 - 2*x)^(3/2)*Sq
rt[2 + 3*x])/(75*Sqrt[3 + 5*x]) - (76*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])
/375 + (338*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1875 +
(992*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1875

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 33.1744, size = 143, normalized size = 0.89 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{3 x + 2}}{15 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{46 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2}}{75 \sqrt{5 x + 3}} - \frac{76 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{375} + \frac{338 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{5625} + \frac{992 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{5625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-2*x + 1)**(5/2)*sqrt(3*x + 2)/(15*(5*x + 3)**(3/2)) - 46*(-2*x + 1)**(3/2)*
sqrt(3*x + 2)/(75*sqrt(5*x + 3)) - 76*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)
/375 + 338*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/5625 + 99
2*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/5625

_______________________________________________________________________________________

Mathematica [A]  time = 0.408513, size = 102, normalized size = 0.64 \[ \frac{2 \left (\frac{15 \sqrt{1-2 x} \sqrt{3 x+2} \left (100 x^2-925 x-712\right )}{(5 x+3)^{3/2}}-8015 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-169 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{5625} \]

Antiderivative was successfully verified.

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

[Out]

(2*((15*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(-712 - 925*x + 100*x^2))/(3 + 5*x)^(3/2) -
169*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 8015*Sqrt[2]*El
lipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/5625

_______________________________________________________________________________________

Maple [C]  time = 0.027, size = 272, normalized size = 1.7 \[{\frac{2}{33750\,{x}^{2}+5625\,x-11250} \left ( 40075\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+845\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+24045\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +507\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +9000\,{x}^{4}-81750\,{x}^{3}-80955\,{x}^{2}+17070\,x+21360 \right ) \sqrt{2+3\,x}\sqrt{1-2\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5625*(40075*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)+845*2^(1/2)*Elli
pticE(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)+24045*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))+507*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))+9000*x^4-81750*x
^3-80955*x^2+17070*x+21360)*(2+3*x)^(1/2)*(1-2*x)^(1/2)/(6*x^2+x-2)/(3+5*x)^(3/2
)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((4*x^2 - 4*x + 1)*sqrt(3*x + 2)*sqrt(-2*x + 1)/((25*x^2 + 30*x + 9)*sqr
t(5*x + 3)), x)

_______________________________________________________________________________________

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)**(1/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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