3.2664 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^{7/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=158 \[ \frac{6388 \sqrt{1-2 x} \sqrt{5 x+3}}{245 \sqrt{3 x+2}}+\frac{92 \sqrt{1-2 x} \sqrt{5 x+3}}{35 (3 x+2)^{3/2}}+\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{5 (3 x+2)^{5/2}}-\frac{64}{245} \sqrt{33} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{6388}{245} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(5*(2 + 3*x)^(5/2)) + (92*Sqrt[1 - 2*x]*Sqrt[3 +
5*x])/(35*(2 + 3*x)^(3/2)) + (6388*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(245*Sqrt[2 + 3
*x]) - (6388*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/245 -
(64*Sqrt[33]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/245

_______________________________________________________________________________________

Rubi [A]  time = 0.343588, 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 \[ \frac{6388 \sqrt{1-2 x} \sqrt{5 x+3}}{245 \sqrt{3 x+2}}+\frac{92 \sqrt{1-2 x} \sqrt{5 x+3}}{35 (3 x+2)^{3/2}}+\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{5 (3 x+2)^{5/2}}-\frac{64}{245} \sqrt{33} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{6388}{245} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(5*(2 + 3*x)^(5/2)) + (92*Sqrt[1 - 2*x]*Sqrt[3 +
5*x])/(35*(2 + 3*x)^(3/2)) + (6388*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(245*Sqrt[2 + 3
*x]) - (6388*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/245 -
(64*Sqrt[33]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/245

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 33.269, size = 143, normalized size = 0.91 \[ \frac{6388 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{245 \sqrt{3 x + 2}} + \frac{92 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{35 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{2 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{5 \left (3 x + 2\right )^{\frac{5}{2}}} - \frac{6388 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{735} - \frac{2112 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{8575} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6388*sqrt(-2*x + 1)*sqrt(5*x + 3)/(245*sqrt(3*x + 2)) + 92*sqrt(-2*x + 1)*sqrt(5
*x + 3)/(35*(3*x + 2)**(3/2)) + 2*sqrt(-2*x + 1)*sqrt(5*x + 3)/(5*(3*x + 2)**(5/
2)) - 6388*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/735 - 211
2*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/8575

_______________________________________________________________________________________

Mathematica [A]  time = 0.312761, size = 101, normalized size = 0.64 \[ \frac{4}{735} \left (\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (28746 x^2+39294 x+13469\right )}{2 (3 x+2)^{5/2}}+\sqrt{2} \left (1597 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-805 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[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]*(13469 + 39294*x + 28746*x^2))/(2*(2 + 3*x)^(
5/2)) + Sqrt[2]*(1597*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 805*E
llipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/735

_______________________________________________________________________________________

Maple [C]  time = 0.03, size = 386, normalized size = 2.4 \[{\frac{2}{7350\,{x}^{2}+735\,x-2205} \left ( 14490\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-28746\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+19320\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-38328\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+6440\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -12776\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +862380\,{x}^{4}+1265058\,{x}^{3}+263238\,{x}^{2}-313239\,x-121221 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/735*(14490*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)-28746*2^(1/2)*E
llipticE(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)+19320*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)-38328*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)+6440*
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))-12776*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))+862380*x^4+1265058*x^3+263238*x^2-313239*x-121221)*(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. \[ \int \frac{\sqrt{-2 \, x + 1}}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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