[next] [prev] [prev-tail] [tail] [up]

3.2936 \(\int \frac{\sqrt{3+5 x}}{(1-2 x)^{5/2} (2+3 x)^{7/2}} \, dx\)

Optimal. Leaf size=220 \[ \frac{189368 \sqrt{1-2 x} \sqrt{5 x+3}}{924385 \sqrt{3 x+2}}-\frac{5438 \sqrt{1-2 x} \sqrt{5 x+3}}{132055 (3 x+2)^{3/2}}-\frac{2818 \sqrt{1-2 x} \sqrt{5 x+3}}{18865 (3 x+2)^{5/2}}+\frac{458 \sqrt{5 x+3}}{1617 \sqrt{1-2 x} (3 x+2)^{5/2}}+\frac{2 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^{5/2}}-\frac{2092 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035}-\frac{189368 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035 \sqrt{33}} \]

[Out]

(2*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)) + (458*Sqrt[3 + 5*x])/(16
17*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)) - (2818*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(18865*(2
 + 3*x)^(5/2)) - (5438*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(132055*(2 + 3*x)^(3/2)) + (
189368*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(924385*Sqrt[2 + 3*x]) - (189368*EllipticE[A
rcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(84035*Sqrt[33]) - (2092*Sqrt[11/3]*Elli
pticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/84035

_______________________________________________________________________________________

Rubi [A]  time = 0.505976, antiderivative size = 220, 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 \[ \frac{189368 \sqrt{1-2 x} \sqrt{5 x+3}}{924385 \sqrt{3 x+2}}-\frac{5438 \sqrt{1-2 x} \sqrt{5 x+3}}{132055 (3 x+2)^{3/2}}-\frac{2818 \sqrt{1-2 x} \sqrt{5 x+3}}{18865 (3 x+2)^{5/2}}+\frac{458 \sqrt{5 x+3}}{1617 \sqrt{1-2 x} (3 x+2)^{5/2}}+\frac{2 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^{5/2}}-\frac{2092 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035}-\frac{189368 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)) + (458*Sqrt[3 + 5*x])/(16
17*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)) - (2818*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(18865*(2
 + 3*x)^(5/2)) - (5438*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(132055*(2 + 3*x)^(3/2)) + (
189368*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(924385*Sqrt[2 + 3*x]) - (189368*EllipticE[A
rcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(84035*Sqrt[33]) - (2092*Sqrt[11/3]*Elli
pticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/84035

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 44.553, size = 201, normalized size = 0.91 \[ \frac{189368 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{924385 \sqrt{3 x + 2}} - \frac{5438 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{132055 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{189368 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{2773155} - \frac{2092 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{252105} + \frac{5636 \sqrt{5 x + 3}}{56595 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{16 \sqrt{5 x + 3}}{245 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}}} + \frac{2 \sqrt{5 x + 3}}{21 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

189368*sqrt(-2*x + 1)*sqrt(5*x + 3)/(924385*sqrt(3*x + 2)) - 5438*sqrt(-2*x + 1)
*sqrt(5*x + 3)/(132055*(3*x + 2)**(3/2)) - 189368*sqrt(33)*elliptic_e(asin(sqrt(
21)*sqrt(-2*x + 1)/7), 35/33)/2773155 - 2092*sqrt(33)*elliptic_f(asin(sqrt(21)*s
qrt(-2*x + 1)/7), 35/33)/252105 + 5636*sqrt(5*x + 3)/(56595*sqrt(-2*x + 1)*(3*x
+ 2)**(3/2)) - 16*sqrt(5*x + 3)/(245*sqrt(-2*x + 1)*(3*x + 2)**(5/2)) + 2*sqrt(5
*x + 3)/(21*(-2*x + 1)**(3/2)*(3*x + 2)**(5/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.300575, size = 108, normalized size = 0.49 \[ \frac{2 \left (\frac{\sqrt{5 x+3} \left (10225872 x^4+2723436 x^3-7133292 x^2-807691 x+1339677\right )}{(1-2 x)^{3/2} (3 x+2)^{5/2}}+\sqrt{2} \left (95165 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+94684 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{2773155} \]

Antiderivative was successfully verified.

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

[Out]

(2*((Sqrt[3 + 5*x]*(1339677 - 807691*x - 7133292*x^2 + 2723436*x^3 + 10225872*x^
4))/((1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)) + Sqrt[2]*(94684*EllipticE[ArcSin[Sqrt[2/1
1]*Sqrt[3 + 5*x]], -33/2] + 95165*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -3
3/2])))/2773155

_______________________________________________________________________________________

Maple [C]  time = 0.036, size = 502, normalized size = 2.3 \[ -{\frac{2}{2773155\, \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 1712970\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}+1704312\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}+1427475\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+1420260\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-380660\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-378736\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-380660\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -378736\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -51129360\,{x}^{5}-44294796\,{x}^{4}+27496152\,{x}^{3}+25438331\,{x}^{2}-4275312\,x-4019031 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/2773155*(1-2*x)^(1/2)*(1712970*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)+1704312*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*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)+1427475*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)+1420260*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)-380660*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)-378736*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)-380660*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))-378736*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))-51129360*x^5-44294796*x^4+27496152*x^3+25438331*x^2-4275312*x-4019031)
/(2+3*x)^(5/2)/(-1+2*x)^2/(3+5*x)^(1/2)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{5 \, x + 3}}{{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(5*x + 3)/((108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*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)**(1/2)/(1-2*x)**(5/2)/(2+3*x)**(7/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[next] [prev] [prev-tail] [front] [up]