3.2814 \(\int \frac{(2+3 x)^{5/2} (3+5 x)^{5/2}}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=218 \[ -\frac{1}{11} \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}-\frac{34}{99} \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}-\frac{1053}{770} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{329683 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{34650}-\frac{43624697 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{623700}-\frac{43624697 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{283500 \sqrt{33}}-\frac{725140729 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{141750 \sqrt{33}} \]

[Out]

(-43624697*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/623700 - (329683*Sqrt[1 -
2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/34650 - (1053*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3
+ 5*x)^(5/2))/770 - (34*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/99 - (Sq
rt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/11 - (725140729*EllipticE[ArcSin[Sq
rt[3/7]*Sqrt[1 - 2*x]], 35/33])/(141750*Sqrt[33]) - (43624697*EllipticF[ArcSin[S
qrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(283500*Sqrt[33])

_______________________________________________________________________________________

Rubi [A]  time = 0.495028, antiderivative size = 218, 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 \[ -\frac{1}{11} \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}-\frac{34}{99} \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}-\frac{1053}{770} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{329683 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{34650}-\frac{43624697 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{623700}-\frac{43624697 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{283500 \sqrt{33}}-\frac{725140729 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{141750 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-43624697*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/623700 - (329683*Sqrt[1 -
2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/34650 - (1053*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3
+ 5*x)^(5/2))/770 - (34*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/99 - (Sq
rt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/11 - (725140729*EllipticE[ArcSin[Sq
rt[3/7]*Sqrt[1 - 2*x]], 35/33])/(141750*Sqrt[33]) - (43624697*EllipticF[ArcSin[S
qrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(283500*Sqrt[33])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 49.1627, size = 201, normalized size = 0.92 \[ - \frac{\sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{11} - \frac{170 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{297} - \frac{9001 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{4158} - \frac{156944 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{10395} - \frac{41741369 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{623700} - \frac{725140729 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{4677750} - \frac{43624697 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{9355500} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-2*x + 1)*(3*x + 2)**(5/2)*(5*x + 3)**(5/2)/11 - 170*sqrt(-2*x + 1)*(3*x +
2)**(5/2)*(5*x + 3)**(3/2)/297 - 9001*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*(5*x + 3)
**(3/2)/4158 - 156944*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/10395 - 4174
1369*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/623700 - 725140729*sqrt(33)*elli
ptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/4677750 - 43624697*sqrt(33)*ellip
tic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/9355500

_______________________________________________________________________________________

Mathematica [A]  time = 0.375741, size = 110, normalized size = 0.5 \[ \frac{2900562916 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (3 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (12757500 x^4+48384000 x^3+81985950 x^2+86822370 x+75000749\right )[phone] F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{9355500 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(2900562916*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 5*(3*Sqrt[2 - 4
*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(75000749 + 86822370*x + 81985950*x^2 + 48384000
*x^3 + 12757500*x^4) + 292189583*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33
/2]))/(9355500*Sqrt[2])

_______________________________________________________________________________________

Maple [C]  time = 0.018, size = 184, normalized size = 0.8 \[{\frac{1}{561330000\,{x}^{3}[phone]\,{x}^{2}-130977000\,x-112266000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -11481750000\,{x}^{7}-52348275000\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -2900562916\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -104493240000\,{x}^{5}-122253448500\,{x}^{4}-101481939900\,{x}^{3}-18760348110\,{x}^{2}[phone]\,x+13500134820 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18711000*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(-11481750000*x^7-523482750
00*x^6+1460947915*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))-2900562916*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))-104493240000*x^5-122253448500*x^4-1
01481939900*x^3-18760348110*x^2+31378183890*x+13500134820)/(30*x^3+23*x^2-7*x-6)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}{\sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*sqrt(5*x + 3)*sqrt(3*x + 2)/
sqrt(-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)**(5/2)*(3+5*x)**(5/2)/(1-2*x)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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