3.2734 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=158 \[ \frac{556 \sqrt{1-2 x} \sqrt{3 x+2}}{3 \sqrt{5 x+3}}-\frac{92 \sqrt{1-2 x} \sqrt{3 x+2}}{3 (5 x+3)^{3/2}}+\frac{14 \sqrt{1-2 x}}{3 \sqrt{3 x+2} (5 x+3)^{3/2}}-\frac{184 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5 \sqrt{33}}-\frac{556}{5} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(14*Sqrt[1 - 2*x])/(3*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (92*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x])/(3*(3 + 5*x)^(3/2)) + (556*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(3*Sqrt[3 + 5*x]
) - (556*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (184*
EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(5*Sqrt[33])

_______________________________________________________________________________________

Rubi [A]  time = 0.338069, 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 \[ \frac{556 \sqrt{1-2 x} \sqrt{3 x+2}}{3 \sqrt{5 x+3}}-\frac{92 \sqrt{1-2 x} \sqrt{3 x+2}}{3 (5 x+3)^{3/2}}+\frac{14 \sqrt{1-2 x}}{3 \sqrt{3 x+2} (5 x+3)^{3/2}}-\frac{184 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5 \sqrt{33}}-\frac{556}{5} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(14*Sqrt[1 - 2*x])/(3*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (92*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x])/(3*(3 + 5*x)^(3/2)) + (556*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(3*Sqrt[3 + 5*x]
) - (556*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (184*
EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(5*Sqrt[33])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 32.5199, size = 143, normalized size = 0.91 \[ \frac{556 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{3 \sqrt{5 x + 3}} - \frac{92 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{3 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{14 \sqrt{- 2 x + 1}}{3 \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{556 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{15} - \frac{184 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{175} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

556*sqrt(-2*x + 1)*sqrt(3*x + 2)/(3*sqrt(5*x + 3)) - 92*sqrt(-2*x + 1)*sqrt(3*x
+ 2)/(3*(5*x + 3)**(3/2)) + 14*sqrt(-2*x + 1)/(3*sqrt(3*x + 2)*(5*x + 3)**(3/2))
- 556*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/15 - 184*sqrt
(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/175

_______________________________________________________________________________________

Mathematica [A]  time = 0.212619, size = 100, normalized size = 0.63 \[ \frac{2 \sqrt{1-2 x} \left (4170 x^2+5144 x+1583\right )}{3 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{4}{15} \sqrt{2} \left (139 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-70 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 - 2*x]*(1583 + 5144*x + 4170*x^2))/(3*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) +
(4*Sqrt[2]*(139*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 70*Ellipti
cF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/15

_______________________________________________________________________________________

Maple [C]  time = 0.033, size = 267, normalized size = 1.7 \[{\frac{2}{90\,{x}^{2}+15\,x-30}\sqrt{1-2\,x}\sqrt{2+3\,x} \left ( 700\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-1390\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+420\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -834\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +41700\,{x}^{3}+30590\,{x}^{2}-9890\,x-7915 \right ) \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(700*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)-1390*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)+420*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))-834*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))+41700*x^3+30590*x^2-9890*x-7915)/(3+5*x)^(3/2)/(6*x^2+x-2)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((-2*x + 1)^(3/2)/((75*x^3 + 140*x^2 + 87*x + 18)*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)**(3/2)/(2+3*x)**(3/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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