3.2673 \(\int \frac{\sqrt{1-2 x} (2+3 x)^{9/2}}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=218 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^{9/2}}{15 (5 x+3)^{3/2}}-\frac{118 \sqrt{1-2 x} (3 x+2)^{7/2}}{165 \sqrt{5 x+3}}+\frac{958 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}}{1925}+\frac{5153 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{48125}-\frac{12601 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{240625}-\frac{31288 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{109375 \sqrt{33}}-\frac{1473539 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{218750 \sqrt{33}} \]

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^(9/2))/(15*(3 + 5*x)^(3/2)) - (118*Sqrt[1 - 2*x]*(2
+ 3*x)^(7/2))/(165*Sqrt[3 + 5*x]) - (12601*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 +
5*x])/240625 + (5153*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/48125 + (958*S
qrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/1925 - (1473539*EllipticE[ArcSin[Sqr
t[3/7]*Sqrt[1 - 2*x]], 35/33])/(218750*Sqrt[33]) - (31288*EllipticF[ArcSin[Sqrt[
3/7]*Sqrt[1 - 2*x]], 35/33])/(109375*Sqrt[33])

_______________________________________________________________________________________

Rubi [A]  time = 0.495995, antiderivative size = 218, 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 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^{9/2}}{15 (5 x+3)^{3/2}}-\frac{118 \sqrt{1-2 x} (3 x+2)^{7/2}}{165 \sqrt{5 x+3}}+\frac{958 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}}{1925}+\frac{5153 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{48125}-\frac{12601 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{240625}-\frac{31288 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{109375 \sqrt{33}}-\frac{1473539 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{218750 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[1 - 2*x]*(2 + 3*x)^(9/2))/(3 + 5*x)^(5/2),x]

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^(9/2))/(15*(3 + 5*x)^(3/2)) - (118*Sqrt[1 - 2*x]*(2
+ 3*x)^(7/2))/(165*Sqrt[3 + 5*x]) - (12601*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 +
5*x])/240625 + (5153*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/48125 + (958*S
qrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/1925 - (1473539*EllipticE[ArcSin[Sqr
t[3/7]*Sqrt[1 - 2*x]], 35/33])/(218750*Sqrt[33]) - (31288*EllipticF[ArcSin[Sqrt[
3/7]*Sqrt[1 - 2*x]], 35/33])/(109375*Sqrt[33])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 47.1336, size = 201, normalized size = 0.92 \[ - \frac{2 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{9}{2}}}{15 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{118 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{7}{2}}}{165 \sqrt{5 x + 3}} + \frac{958 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{1925} + \frac{5153 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{48125} - \frac{12601 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{240625} - \frac{1473539 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{7218750} - \frac{31288 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{3828125} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**(9/2)*(1-2*x)**(1/2)/(3+5*x)**(5/2),x)

[Out]

-2*sqrt(-2*x + 1)*(3*x + 2)**(9/2)/(15*(5*x + 3)**(3/2)) - 118*sqrt(-2*x + 1)*(3
*x + 2)**(7/2)/(165*sqrt(5*x + 3)) + 958*sqrt(-2*x + 1)*(3*x + 2)**(5/2)*sqrt(5*
x + 3)/1925 + 5153*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/48125 - 12601*s
qrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/240625 - 1473539*sqrt(33)*elliptic_e(a
sin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/7218750 - 31288*sqrt(35)*elliptic_f(asin(
sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/3828125

_______________________________________________________________________________________

Mathematica [A]  time = 0.459184, size = 112, normalized size = 0.51 \[ \frac{\frac{10 \sqrt{1-2 x} \sqrt{3 x+2} \left (3341250 x^4+8575875 x^3+6882975 x^2+1854575 x+54083\right )}{(5 x+3)^{3/2}}-441035 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+1473539 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{7218750} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[1 - 2*x]*(2 + 3*x)^(9/2))/(3 + 5*x)^(5/2),x]

[Out]

((10*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(54083 + 1854575*x + 6882975*x^2 + 8575875*x^3
+ 3341250*x^4))/(3 + 5*x)^(3/2) + 1473539*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sq
rt[3 + 5*x]], -33/2] - 441035*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]]
, -33/2])/7218750

_______________________________________________________________________________________

Maple [C]  time = 0.052, size = 282, normalized size = 1.3 \[{\frac{1}{43312500\,{x}^{2}+7218750\,x-14437500} \left ( 2205175\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-7367695\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,{x}^{6}+1323105\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -4420617\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}+8586750\,{x}^{3}-115868770\,{x}^{2}-36550670\,x-1081660 \right ) \sqrt{1-2\,x}\sqrt{2+3\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^(9/2)*(1-2*x)^(1/2)/(3+5*x)^(5/2),x)

[Out]

1/7218750*(2205175*2^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*1
1^(1/2)*3^(1/2)*2^(1/2))*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-7367695*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+1323105*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))-4420617*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))[phone]*x^5+431912250*x^4+8586750*x^3-115868770*x^2-36550670*x-10
81660)*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(6*x^2+x-2)/(3+5*x)^(3/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{9}{2}} \sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^(9/2)*sqrt(-2*x + 1)/(5*x + 3)^(5/2),x, algorithm="maxima")

[Out]

integrate((3*x + 2)^(9/2)*sqrt(-2*x + 1)/(5*x + 3)^(5/2), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^(9/2)*sqrt(-2*x + 1)/(5*x + 3)^(5/2),x, algorithm="fricas")

[Out]

integral((81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(
(25*x^2 + 30*x + 9)*sqrt(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((2+3*x)**(9/2)*(1-2*x)**(1/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{9}{2}} \sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^(9/2)*sqrt(-2*x + 1)/(5*x + 3)^(5/2),x, algorithm="giac")

[Out]

integrate((3*x + 2)^(9/2)*sqrt(-2*x + 1)/(5*x + 3)^(5/2), x)