3.2901 \(\int \frac{(2+3 x)^{7/2}}{(1-2 x)^{3/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=160 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^{5/2}}{11 \sqrt{1-2 x}}+\frac{312}{275} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}+\frac{14517 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{2750}+\frac{5057 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1250}+\frac{168123 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1250} \]

[Out]

(14517*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/2750 + (312*Sqrt[1 - 2*x]*(2 +
3*x)^(3/2)*Sqrt[3 + 5*x])/275 + (7*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/(11*Sqrt[1 -
2*x]) + (168123*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/12
50 + (5057*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1250

_______________________________________________________________________________________

Rubi [A]  time = 0.335423, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^{5/2}}{11 \sqrt{1-2 x}}+\frac{312}{275} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}+\frac{14517 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{2750}+\frac{5057 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1250}+\frac{168123 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1250} \]

Antiderivative was successfully verified.

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

[Out]

(14517*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/2750 + (312*Sqrt[1 - 2*x]*(2 +
3*x)^(3/2)*Sqrt[3 + 5*x])/275 + (7*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/(11*Sqrt[1 -
2*x]) + (168123*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/12
50 + (5057*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1250

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 31.9538, size = 143, normalized size = 0.89 \[ \frac{312 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{275} + \frac{14517 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{2750} + \frac{168123 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{13750} + \frac{15171 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{43750} + \frac{7 \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{11 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

312*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/275 + 14517*sqrt(-2*x + 1)*sqr
t(3*x + 2)*sqrt(5*x + 3)/2750 + 168123*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2
*x + 1)/7), 35/33)/13750 + 15171*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1
)/11), 33/35)/43750 + 7*(3*x + 2)**(5/2)*sqrt(5*x + 3)/(11*sqrt(-2*x + 1))

_______________________________________________________________________________________

Mathematica [A]  time = 0.188605, size = 110, normalized size = 0.69 \[ \frac{-10 \sqrt{3 x+2} \sqrt{5 x+3} \left (2970 x^2+11154 x-27757\right )+169365 \sqrt{2-4 x} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-336246 \sqrt{2-4 x} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{27500 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-27757 + 11154*x + 2970*x^2) - 336246*Sqrt[2 -
4*x]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 169365*Sqrt[2 - 4*x]*
EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(27500*Sqrt[1 - 2*x])

_______________________________________________________________________________________

Maple [C]  time = 0.025, size = 169, normalized size = 1.1 \[ -{\frac{1}{825000\,{x}^{3}+632500\,{x}^{2}-192500\,x-165000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 169365\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -336246\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -445500\,{x}^{4}-2237400\,{x}^{3}+1866090\,{x}^{2}+4604590\,x+1665420 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/27500*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(169365*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))-336246*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)
)-445500*x^4-2237400*x^3+1866090*x^2+4604590*x+1665420)/(30*x^3+23*x^2-7*x-6)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt{3 \, x + 2}}{\sqrt{5 \, x + 3}{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(27*x^3 + 54*x^2 + 36*x + 8)*sqrt(3*x + 2)/(sqrt(5*x + 3)*(2*x - 1)*sq
rt(-2*x + 1)), 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((2+3*x)**(7/2)/(1-2*x)**(3/2)/(3+5*x)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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