3.2696 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{5/2}} \, dx\)

Optimal. Leaf size=160 \[ \frac{74 \sqrt{1-2 x} (5 x+3)^{3/2}}{9 \sqrt{3 x+2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^{3/2}}-\frac{1150}{81} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{230}{81} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{592}{81} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(-1150*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/81 - (2*(1 - 2*x)^(3/2)*(3 + 5
*x)^(3/2))/(9*(2 + 3*x)^(3/2)) + (74*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(9*Sqrt[2 +
3*x]) + (592*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/81 -
(230*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/81

_______________________________________________________________________________________

Rubi [A]  time = 0.328648, 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 \[ \frac{74 \sqrt{1-2 x} (5 x+3)^{3/2}}{9 \sqrt{3 x+2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^{3/2}}-\frac{1150}{81} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{230}{81} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{592}{81} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-1150*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/81 - (2*(1 - 2*x)^(3/2)*(3 + 5
*x)^(3/2))/(9*(2 + 3*x)^(3/2)) + (74*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(9*Sqrt[2 +
3*x]) + (592*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/81 -
(230*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/81

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 31.1654, size = 143, normalized size = 0.89 \[ - \frac{74 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{63 \sqrt{3 x + 2}} - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{9 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{724 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{567} + \frac{592 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{243} - \frac{506 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{567} \]

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)**(5/2),x)

[Out]

-74*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(63*sqrt(3*x + 2)) - 2*(-2*x + 1)**(3/2)*(5*
x + 3)**(3/2)/(9*(3*x + 2)**(3/2)) - 724*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x +
3)/567 + 592*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/243 -
506*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/567

_______________________________________________________________________________________

Mathematica [A]  time = 0.325282, size = 102, normalized size = 0.64 \[ \frac{1}{243} \left (-\frac{6 \sqrt{1-2 x} \sqrt{5 x+3} \left (90 x^2+564 x+329\right )}{(3 x+2)^{3/2}}+4387 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-592 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-6*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(329 + 564*x + 90*x^2))/(2 + 3*x)^(3/2) - 592*S
qrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 4387*Sqrt[2]*Ellipti
cF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/243

_______________________________________________________________________________________

Maple [C]  time = 0.028, size = 272, normalized size = 1.7 \[ -{\frac{1}{2430\,{x}^{2}+243\,x-729} \left ( 13161\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-1776\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+8774\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -1184\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +5400\,{x}^{4}+34380\,{x}^{3}+21504\,{x}^{2}-8178\,x-5922 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}} \]

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)^(5/2),x)

[Out]

-1/243*(13161*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)-1776*2^(1/2)*Ell
ipticE(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)+8774*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))-1184*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(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))+5400*x^4+34380*
x^3+21504*x^2-8178*x-5922)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(3/2
)

_______________________________________________________________________________________

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

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)^(5/2),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (10 \, x^{2} + x - 3\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt{3 \, x + 2}}, x\right ) \]

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)^(5/2),x, algorithm="fricas")

[Out]

integral(-(10*x^2 + x - 3)*sqrt(5*x + 3)*sqrt(-2*x + 1)/((9*x^2 + 12*x + 4)*sqrt
(3*x + 2)), 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)**(3/2)*(3+5*x)**(3/2)/(2+3*x)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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

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)^(5/2),x, algorithm="giac")

[Out]

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