3.2736 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=222 \[ \frac{301304 \sqrt{1-2 x} \sqrt{3 x+2}}{21 \sqrt{5 x+3}}-\frac{16616 \sqrt{1-2 x} \sqrt{3 x+2}}{7 (5 x+3)^{3/2}}+\frac{111884 \sqrt{1-2 x}}{315 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{536 \sqrt{1-2 x}}{45 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac{14 \sqrt{1-2 x}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac{33232}{35} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{301304}{35} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(14*Sqrt[1 - 2*x])/(15*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (536*Sqrt[1 - 2*x])/(4
5*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (111884*Sqrt[1 - 2*x])/(315*Sqrt[2 + 3*x]*(
3 + 5*x)^(3/2)) - (16616*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(7*(3 + 5*x)^(3/2)) + (301
304*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(21*Sqrt[3 + 5*x]) - (301304*Sqrt[11/3]*Ellipti
cE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/35 - (33232*Sqrt[3/11]*EllipticF[Arc
Sin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/35

_______________________________________________________________________________________

Rubi [A]  time = 0.518788, antiderivative size = 222, 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 \[ \frac{301304 \sqrt{1-2 x} \sqrt{3 x+2}}{21 \sqrt{5 x+3}}-\frac{16616 \sqrt{1-2 x} \sqrt{3 x+2}}{7 (5 x+3)^{3/2}}+\frac{111884 \sqrt{1-2 x}}{315 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{536 \sqrt{1-2 x}}{45 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac{14 \sqrt{1-2 x}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac{33232}{35} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{301304}{35} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(14*Sqrt[1 - 2*x])/(15*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (536*Sqrt[1 - 2*x])/(4
5*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (111884*Sqrt[1 - 2*x])/(315*Sqrt[2 + 3*x]*(
3 + 5*x)^(3/2)) - (16616*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(7*(3 + 5*x)^(3/2)) + (301
304*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(21*Sqrt[3 + 5*x]) - (301304*Sqrt[11/3]*Ellipti
cE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/35 - (33232*Sqrt[3/11]*EllipticF[Arc
Sin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/35

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 48.8799, size = 201, normalized size = 0.91 \[ \frac{301304 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{35 \sqrt{3 x + 2}} + \frac{4336 \sqrt{- 2 x + 1}}{3 \sqrt{3 x + 2} \sqrt{5 x + 3}} - \frac{1076 \sqrt{- 2 x + 1}}{9 \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{536 \sqrt{- 2 x + 1}}{45 \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{14 \sqrt{- 2 x + 1}}{15 \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{301304 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{105} - \frac{99696 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{1225} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

301304*sqrt(-2*x + 1)*sqrt(5*x + 3)/(35*sqrt(3*x + 2)) + 4336*sqrt(-2*x + 1)/(3*
sqrt(3*x + 2)*sqrt(5*x + 3)) - 1076*sqrt(-2*x + 1)/(9*sqrt(3*x + 2)*(5*x + 3)**(
3/2)) + 536*sqrt(-2*x + 1)/(45*(3*x + 2)**(3/2)*(5*x + 3)**(3/2)) + 14*sqrt(-2*x
+ 1)/(15*(3*x + 2)**(5/2)*(5*x + 3)**(3/2)) - 301304*sqrt(33)*elliptic_e(asin(s
qrt(21)*sqrt(-2*x + 1)/7), 35/33)/105 - 99696*sqrt(35)*elliptic_f(asin(sqrt(55)*
sqrt(-2*x + 1)/11), 33/35)/1225

_______________________________________________________________________________________

Mathematica [A]  time = 0.361923, size = 109, normalized size = 0.49 \[ \frac{2}{105} \left (\frac{\sqrt{1-2 x} \left (101690100 x^4+261029520 x^3+251053266 x^2+107221804 x+17157169\right )}{(3 x+2)^{5/2} (5 x+3)^{3/2}}+4 \sqrt{2} \left (37663 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-18970 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*((Sqrt[1 - 2*x]*(17157169 + 107221804*x + 251053266*x^2 + 261029520*x^3 + 101
690100*x^4))/((2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + 4*Sqrt[2]*(37663*EllipticE[ArcS
in[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 18970*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3
+ 5*x]], -33/2])))/105

_______________________________________________________________________________________

Maple [C]  time = 0.037, size = 502, normalized size = 2.3 \[ -{\frac{2}{-105+210\,x}\sqrt{1-2\,x} \left ( 6779340\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}-3414600\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-6601560\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+8436512\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-4249280\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+1807824\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -910560\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -203380200\,{x}^{5}-420368940\,{x}^{4}-241077012\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+17157169 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/105*(1-2*x)^(1/2)*(6779340*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
)-3414600*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)[phone]*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)-6601560*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)+8436512*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)
-4249280*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)+1807824*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))-910560*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))-203380200*x^5-420368940*x^4-241077012*x^3+36609658*x^2+72907466*x+1715
7169)/(2+3*x)^(5/2)/(3+5*x)^(3/2)/(-1+2*x)

_______________________________________________________________________________________

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

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)^(7/2)),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}, x\right ) \]

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)^(7/2)),x, algorithm="fricas")

[Out]

integral((-2*x + 1)^(3/2)/((675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 7
2)*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)**(7/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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

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)^(7/2)),x, algorithm="giac")

[Out]

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