3.2953 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{9/2}} \, dx\)

Optimal. Leaf size=253 \[ \frac{11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{7/2}}-\frac{98642 \sqrt{1-2 x} \sqrt{5 x+3}}{823543 \sqrt{3 x+2}}-\frac{33778 \sqrt{1-2 x} \sqrt{5 x+3}}{117649 (3 x+2)^{3/2}}-\frac{11433 \sqrt{1-2 x} \sqrt{5 x+3}}{16807 (3 x+2)^{5/2}}-\frac{4545 \sqrt{1-2 x} \sqrt{5 x+3}}{2401 (3 x+2)^{7/2}}+\frac{220 \sqrt{5 x+3}}{49 \sqrt{1-2 x} (3 x+2)^{7/2}}-\frac{65672 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{823543}+\frac{98642 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{823543} \]

[Out]

(220*Sqrt[3 + 5*x])/(49*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2)) - (4545*Sqrt[1 - 2*x]*Sqr
t[3 + 5*x])/(2401*(2 + 3*x)^(7/2)) - (11433*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(16807*
(2 + 3*x)^(5/2)) - (33778*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(117649*(2 + 3*x)^(3/2))
- (98642*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(823543*Sqrt[2 + 3*x]) + (11*(3 + 5*x)^(3/
2))/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^(7/2)) + (98642*Sqrt[11/3]*EllipticE[ArcSin[Sq
rt[3/7]*Sqrt[1 - 2*x]], 35/33])/823543 - (65672*Sqrt[11/3]*EllipticF[ArcSin[Sqrt
[3/7]*Sqrt[1 - 2*x]], 35/33])/823543

_______________________________________________________________________________________

Rubi [A]  time = 0.592929, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{7/2}}-\frac{98642 \sqrt{1-2 x} \sqrt{5 x+3}}{823543 \sqrt{3 x+2}}-\frac{33778 \sqrt{1-2 x} \sqrt{5 x+3}}{117649 (3 x+2)^{3/2}}-\frac{11433 \sqrt{1-2 x} \sqrt{5 x+3}}{16807 (3 x+2)^{5/2}}-\frac{4545 \sqrt{1-2 x} \sqrt{5 x+3}}{2401 (3 x+2)^{7/2}}+\frac{220 \sqrt{5 x+3}}{49 \sqrt{1-2 x} (3 x+2)^{7/2}}-\frac{65672 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{823543}+\frac{98642 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{823543} \]

Antiderivative was successfully verified.

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

[Out]

(220*Sqrt[3 + 5*x])/(49*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2)) - (4545*Sqrt[1 - 2*x]*Sqr
t[3 + 5*x])/(2401*(2 + 3*x)^(7/2)) - (11433*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(16807*
(2 + 3*x)^(5/2)) - (33778*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(117649*(2 + 3*x)^(3/2))
- (98642*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(823543*Sqrt[2 + 3*x]) + (11*(3 + 5*x)^(3/
2))/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^(7/2)) + (98642*Sqrt[11/3]*EllipticE[ArcSin[Sq
rt[3/7]*Sqrt[1 - 2*x]], 35/33])/823543 - (65672*Sqrt[11/3]*EllipticF[ArcSin[Sqrt
[3/7]*Sqrt[1 - 2*x]], 35/33])/823543

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 53.1269, size = 230, normalized size = 0.91 \[ - \frac{98642 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{823543 \sqrt{3 x + 2}} + \frac{98642 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{2470629} - \frac{722392 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{28824005} + \frac{67556 \sqrt{5 x + 3}}{352947 \sqrt{- 2 x + 1} \sqrt{3 x + 2}} - \frac{10912 \sqrt{5 x + 3}}{50421 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{781 \sqrt{5 x + 3}}{2401 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}}} + \frac{25 \sqrt{5 x + 3}}{343 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{7}{2}}} + \frac{11 \left (5 x + 3\right )^{\frac{3}{2}}}{21 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-98642*sqrt(-2*x + 1)*sqrt(5*x + 3)/(823543*sqrt(3*x + 2)) + 98642*sqrt(33)*elli
ptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/2470629 - 722392*sqrt(35)*ellipti
c_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/28824005 + 67556*sqrt(5*x + 3)/(352
947*sqrt(-2*x + 1)*sqrt(3*x + 2)) - 10912*sqrt(5*x + 3)/(50421*sqrt(-2*x + 1)*(3
*x + 2)**(3/2)) - 781*sqrt(5*x + 3)/(2401*sqrt(-2*x + 1)*(3*x + 2)**(5/2)) + 25*
sqrt(5*x + 3)/(343*sqrt(-2*x + 1)*(3*x + 2)**(7/2)) + 11*(5*x + 3)**(3/2)/(21*(-
2*x + 1)**(3/2)*(3*x + 2)**(7/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.439288, size = 113, normalized size = 0.45 \[ \frac{2 \left (\frac{\sqrt{5 x+3} \left (-15980004 x^5-28748088 x^4-7681599 x^3+10746933 x^2+6524789 x+866085\right )}{(1-2 x)^{3/2} (3 x+2)^{7/2}}+\sqrt{2} \left (591115 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-49321 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{2470629} \]

Antiderivative was successfully verified.

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

[Out]

(2*((Sqrt[3 + 5*x]*(866085 + 6524789*x + 10746933*x^2 - 7681599*x^3 - 28748088*x
^4 - 15980004*x^5))/((1 - 2*x)^(3/2)*(2 + 3*x)^(7/2)) + Sqrt[2]*(-49321*Elliptic
E[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 591115*EllipticF[ArcSin[Sqrt[2/11]*
Sqrt[3 + 5*x]], -33/2])))/2470629

_______________________________________________________________________________________

Maple [C]  time = 0.037, size = 621, normalized size = 2.5 \[ -{\frac{2}{2470629\, \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 31920210\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{4}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-2663334\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{4}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}-3995001\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-887778\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-11822300\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+986420\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,{x}^{6}-4728920\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +394568\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}-30689868\,{x}^{3}-64864744\,{x}^{2}-23904792\,x-2598255 \right ) \left ( 2+3\,x \right ) ^{-{\frac{7}{2}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/2470629*(1-2*x)^(1/2)*(31920210*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^4*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)
^(1/2)-2663334*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^4*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)[phone]*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*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)-3995001*2^(1/2)*EllipticE(1/11*1
1^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))*x^3*(1-2*x)^(1/2)*
(3+5*x)^(1/2)*(2+3*x)^(1/2)[phone]*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^2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*
x)^(1/2)-887778*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^2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-11822300*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)+986420*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^6-4728920*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))+394568*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+124652259*x^4-30689868*x^3-64864744*x^2-23904792*x-2598255)/(2+3*x)^(7/2)/(-1+
2*x)^2/(3+5*x)^(1/2)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3}}{{\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((25*x^2 + 30*x + 9)*sqrt(5*x + 3)/((324*x^6 + 540*x^5 + 81*x^4 - 264*x^
3 - 104*x^2 + 32*x + 16)*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((3+5*x)**(5/2)/(1-2*x)**(5/2)/(2+3*x)**(9/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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