3.2849 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=158 \[ \frac{20644 \sqrt{1-2 x} \sqrt{5 x+3}}{1715 \sqrt{3 x+2}}+\frac{296 \sqrt{1-2 x} \sqrt{5 x+3}}{245 (3 x+2)^{3/2}}+\frac{6 \sqrt{1-2 x} \sqrt{5 x+3}}{35 (3 x+2)^{5/2}}-\frac{6856 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1715 \sqrt{33}}-\frac{20644 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1715} \]

[Out]

(6*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(35*(2 + 3*x)^(5/2)) + (296*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(245*(2 + 3*x)^(3/2)) + (20644*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1715*Sqrt[
2 + 3*x]) - (20644*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])
/1715 - (6856*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1715*Sqrt[33])

_______________________________________________________________________________________

Rubi [A]  time = 0.346188, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{20644 \sqrt{1-2 x} \sqrt{5 x+3}}{1715 \sqrt{3 x+2}}+\frac{296 \sqrt{1-2 x} \sqrt{5 x+3}}{245 (3 x+2)^{3/2}}+\frac{6 \sqrt{1-2 x} \sqrt{5 x+3}}{35 (3 x+2)^{5/2}}-\frac{6856 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1715 \sqrt{33}}-\frac{20644 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1715} \]

Antiderivative was successfully verified.

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

[Out]

(6*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(35*(2 + 3*x)^(5/2)) + (296*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(245*(2 + 3*x)^(3/2)) + (20644*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1715*Sqrt[
2 + 3*x]) - (20644*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])
/1715 - (6856*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1715*Sqrt[33])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 32.3054, size = 143, normalized size = 0.91 \[ \frac{20644 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1715 \sqrt{3 x + 2}} + \frac{296 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{245 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{6 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{35 \left (3 x + 2\right )^{\frac{5}{2}}} - \frac{20644 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{5145} - \frac{6856 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{56595} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

20644*sqrt(-2*x + 1)*sqrt(5*x + 3)/(1715*sqrt(3*x + 2)) + 296*sqrt(-2*x + 1)*sqr
t(5*x + 3)/(245*(3*x + 2)**(3/2)) + 6*sqrt(-2*x + 1)*sqrt(5*x + 3)/(35*(3*x + 2)
**(5/2)) - 20644*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/514
5 - 6856*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/56595

_______________________________________________________________________________________

Mathematica [A]  time = 0.315014, size = 101, normalized size = 0.64 \[ \frac{4 \left (\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (92898 x^2+126972 x+43507\right )}{2 (3 x+2)^{5/2}}+\sqrt{2} \left (5161 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-2590 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{5145} \]

Antiderivative was successfully verified.

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

[Out]

(4*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(43507 + 126972*x + 92898*x^2))/(2*(2 + 3*x)^
(5/2)) + Sqrt[2]*(5161*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 2590
*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/5145

_______________________________________________________________________________________

Maple [C]  time = 0.033, size = 386, normalized size = 2.4 \[ -{\frac{2}{51450\,{x}^{2}+5145\,x-15435} \left ( 92898\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-46620\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+123864\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-62160\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+41288\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -20720\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -2786940\,{x}^{4}-4087854\,{x}^{3}-850044\,{x}^{2}+1012227\,x+391563 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/5145*(92898*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)-46620*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)+123864*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)-62160*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)+41
288*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))-20720*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))-2786940*x^4-4087854*x^3-850044*x^2+1012227*x+391563)*(
3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(5/2)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/((27*x^3 + 54*x^2 + 36*x + 8)*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(1/(2+3*x)**(7/2)/(1-2*x)**(1/2)/(3+5*x)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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