3.2969 \(\int \frac{1}{(1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=156 \[ -\frac{41570 \sqrt{1-2 x} \sqrt{3 x+2}}{195657 \sqrt{5 x+3}}+\frac{824 \sqrt{3 x+2}}{17787 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{4 \sqrt{3 x+2}}{231 (1-2 x)^{3/2} \sqrt{5 x+3}}-\frac{824 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5929 \sqrt{33}}+\frac{8314 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5929 \sqrt{33}} \]

[Out]

(4*Sqrt[2 + 3*x])/(231*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) + (824*Sqrt[2 + 3*x])/(177
87*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (41570*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(195657*Sq
rt[3 + 5*x]) + (8314*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(5929*Sq
rt[33]) - (824*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(5929*Sqrt[33]
)

_______________________________________________________________________________________

Rubi [A]  time = 0.349203, antiderivative size = 156, 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 \[ -\frac{41570 \sqrt{1-2 x} \sqrt{3 x+2}}{195657 \sqrt{5 x+3}}+\frac{824 \sqrt{3 x+2}}{17787 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{4 \sqrt{3 x+2}}{231 (1-2 x)^{3/2} \sqrt{5 x+3}}-\frac{824 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5929 \sqrt{33}}+\frac{8314 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5929 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[2 + 3*x])/(231*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) + (824*Sqrt[2 + 3*x])/(177
87*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (41570*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(195657*Sq
rt[3 + 5*x]) + (8314*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(5929*Sq
rt[33]) - (824*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(5929*Sqrt[33]
)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 30.5882, size = 143, normalized size = 0.92 \[ \frac{8314 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{195657} - \frac{824 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{195657} + \frac{16628 \sqrt{3 x + 2} \sqrt{5 x + 3}}{195657 \sqrt{- 2 x + 1}} - \frac{1070 \sqrt{3 x + 2}}{2541 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} + \frac{4 \sqrt{3 x + 2}}{231 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8314*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/195657 - 824*sq
rt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/195657 + 16628*sqrt(3*
x + 2)*sqrt(5*x + 3)/(195657*sqrt(-2*x + 1)) - 1070*sqrt(3*x + 2)/(2541*sqrt(-2*
x + 1)*sqrt(5*x + 3)) + 4*sqrt(3*x + 2)/(231*(-2*x + 1)**(3/2)*sqrt(5*x + 3))

_______________________________________________________________________________________

Mathematica [A]  time = 0.245662, size = 98, normalized size = 0.63 \[ \frac{2 \left (\frac{\sqrt{3 x+2} \left (-83140 x^2+74076 x-14559\right )}{(1-2 x)^{3/2} \sqrt{5 x+3}}+\sqrt{2} \left (10955 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-4157 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{195657} \]

Antiderivative was successfully verified.

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

[Out]

(2*((Sqrt[2 + 3*x]*(-14559 + 74076*x - 83140*x^2))/((1 - 2*x)^(3/2)*Sqrt[3 + 5*x
]) + Sqrt[2]*(-4157*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 10955*E
llipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/195657

_______________________________________________________________________________________

Maple [C]  time = 0.036, size = 276, normalized size = 1.8 \[ -{\frac{2}{ \left ( 2934855\,{x}^{2}+3717483\,x+1173942 \right ) \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 21910\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-8314\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-10955\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +4157\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +249420\,{x}^{3}-55948\,{x}^{2}-104475\,x+29118 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/195657*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(21910*2^(1/2)*EllipticF(1/1
1*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)-8314*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)-10955*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))+4157*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))+249420*x^3-55948*x^2-104475*x+29118)/(15*x^2+19*x
+6)/(-1+2*x)^2

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/((20*x^3 - 8*x^2 - 7*x + 3)*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/(1-2*x)**(5/2)/(3+5*x)**(3/2)/(2+3*x)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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