3.2805 \(\int \frac{(2+3 x)^{5/2} (3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=191 \[ -\frac{1}{9} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}-\frac{137}{315} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}-\frac{9547 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{5250}-\frac{663409 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{47250}-\frac{663409 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{236250}-\frac{44109377 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{472500} \]

[Out]

(-663409*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/47250 - (9547*Sqrt[1 - 2*x]*
Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/5250 - (137*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*
x)^(3/2))/315 - (Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2))/9 - (44109377*Sq
rt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/472500 - (663409*Sqr
t[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/236250

_______________________________________________________________________________________

Rubi [A]  time = 0.403233, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{1}{9} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}-\frac{137}{315} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}-\frac{9547 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{5250}-\frac{663409 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{47250}-\frac{663409 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{236250}-\frac{44109377 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{472500} \]

Antiderivative was successfully verified.

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

[Out]

(-663409*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/47250 - (9547*Sqrt[1 - 2*x]*
Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/5250 - (137*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*
x)^(3/2))/315 - (Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2))/9 - (44109377*Sq
rt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/472500 - (663409*Sqr
t[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/236250

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 40.9908, size = 172, normalized size = 0.9 \[ - \frac{\sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{9} - \frac{137 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{189} - \frac{27271 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{9450} - \frac{317384 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{23625} - \frac{44109377 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1417500} - \frac{663409 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{708750} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-2*x + 1)*(3*x + 2)**(5/2)*(5*x + 3)**(3/2)/9 - 137*sqrt(-2*x + 1)*(3*x +
2)**(5/2)*sqrt(5*x + 3)/189 - 27271*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3
)/9450 - 317384*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/23625 - 44109377*sqrt
(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/1417500 - 663409*sqrt(33
)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/708750

_______________________________________________________________________________________

Mathematica [A]  time = 0.348435, size = 105, normalized size = 0.55 \[ \frac{44109377 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (3 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (236250 x^3+765000 x^2+1114065 x+1107478\right )+4443376 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{708750 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(44109377*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]*(1107478 + 1114065*x + 765000*x^2 + 236250*x^3) +
4443376*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/(708750*Sqrt[2])

_______________________________________________________________________________________

Maple [C]  time = 0.019, size = 179, normalized size = 0.9 \[{\frac{1}{42525000\,{x}^{3}[phone]\,{x}^{2}-9922500\,x-8505000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -212625000\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -44109377\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -851512500\,{x}^{5}-1480896000\,{x}^{4}-1562260050\,{x}^{3}-392506170\,{x}^{2}[phone]\,x+199346040 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1417500*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(-212625000*x^6+22216880*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))-44109377*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))-851512500*x^5-1480896000*x^4-1562260050*x^3-392506170*x^2+
433102080*x+199346040)/(30*x^3+23*x^2-7*x-6)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}{\sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((45*x^3 + 87*x^2 + 56*x + 12)*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)**(3/2)/(1-2*x)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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