3.2778 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^{13/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=249 \[ \frac{14 \sqrt{5 x+3} (1-2 x)^{3/2}}{33 (3 x+2)^{11/2}}+\frac{23204503328 \sqrt{5 x+3} \sqrt{1-2 x}}{2139291 \sqrt{3 x+2}}+\frac{333930952 \sqrt{5 x+3} \sqrt{1-2 x}}{305613 (3 x+2)^{3/2}}+\frac{7173272 \sqrt{5 x+3} \sqrt{1-2 x}}{43659 (3 x+2)^{5/2}}+\frac{171004 \sqrt{5 x+3} \sqrt{1-2 x}}{6237 (3 x+2)^{7/2}}+\frac{4508 \sqrt{5 x+3} \sqrt{1-2 x}}{891 (3 x+2)^{9/2}}-\frac{697995152 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{194481 \sqrt{33}}-\frac{23204503328 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{194481 \sqrt{33}} \]

[Out]

(14*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(33*(2 + 3*x)^(11/2)) + (4508*Sqrt[1 - 2*x]*S
qrt[3 + 5*x])/(891*(2 + 3*x)^(9/2)) + (171004*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(6237
*(2 + 3*x)^(7/2)) + (7173272*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(43659*(2 + 3*x)^(5/2)
) + (333930952*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(305613*(2 + 3*x)^(3/2)) + (23204503
328*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2139291*Sqrt[2 + 3*x]) - (23204503328*Elliptic
E[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(194481*Sqrt[33]) - (697995152*Ellipt
icF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(194481*Sqrt[33])

_______________________________________________________________________________________

Rubi [A]  time = 0.600311, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{14 \sqrt{5 x+3} (1-2 x)^{3/2}}{33 (3 x+2)^{11/2}}+\frac{23204503328 \sqrt{5 x+3} \sqrt{1-2 x}}{2139291 \sqrt{3 x+2}}+\frac{333930952 \sqrt{5 x+3} \sqrt{1-2 x}}{305613 (3 x+2)^{3/2}}+\frac{7173272 \sqrt{5 x+3} \sqrt{1-2 x}}{43659 (3 x+2)^{5/2}}+\frac{171004 \sqrt{5 x+3} \sqrt{1-2 x}}{6237 (3 x+2)^{7/2}}+\frac{4508 \sqrt{5 x+3} \sqrt{1-2 x}}{891 (3 x+2)^{9/2}}-\frac{697995152 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{194481 \sqrt{33}}-\frac{23204503328 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{194481 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - 2*x)^(5/2)/((2 + 3*x)^(13/2)*Sqrt[3 + 5*x]),x]

[Out]

(14*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(33*(2 + 3*x)^(11/2)) + (4508*Sqrt[1 - 2*x]*S
qrt[3 + 5*x])/(891*(2 + 3*x)^(9/2)) + (171004*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(6237
*(2 + 3*x)^(7/2)) + (7173272*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(43659*(2 + 3*x)^(5/2)
) + (333930952*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(305613*(2 + 3*x)^(3/2)) + (23204503
328*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2139291*Sqrt[2 + 3*x]) - (23204503328*Elliptic
E[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(194481*Sqrt[33]) - (697995152*Ellipt
icF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(194481*Sqrt[33])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 57.4079, size = 230, normalized size = 0.92 \[ \frac{14 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{33 \left (3 x + 2\right )^{\frac{11}{2}}} + \frac{23204503328 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2139291 \sqrt{3 x + 2}} + \frac{333930952 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{305613 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{7173272 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{43659 \left (3 x + 2\right )^{\frac{5}{2}}} + \frac{171004 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{6237 \left (3 x + 2\right )^{\frac{7}{2}}} + \frac{4508 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{891 \left (3 x + 2\right )^{\frac{9}{2}}} - \frac{23204503328 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{6417873} - \frac{697995152 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{6806835} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)/(2+3*x)**(13/2)/(3+5*x)**(1/2),x)

[Out]

14*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(33*(3*x + 2)**(11/2)) + 23204503328*sqrt(-2*
x + 1)*sqrt(5*x + 3)/(2139291*sqrt(3*x + 2)) + 333930952*sqrt(-2*x + 1)*sqrt(5*x
+ 3)/(305613*(3*x + 2)**(3/2)) + 7173272*sqrt(-2*x + 1)*sqrt(5*x + 3)/(43659*(3
*x + 2)**(5/2)) + 171004*sqrt(-2*x + 1)*sqrt(5*x + 3)/(6237*(3*x + 2)**(7/2)) +
4508*sqrt(-2*x + 1)*sqrt(5*x + 3)/(891*(3*x + 2)**(9/2)) - 23204503328*sqrt(33)*
elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/6417873 - 697995152*sqrt(35)*
elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/6806835

_______________________________________________________________________________________

Mathematica [A]  time = 0.477696, size = 115, normalized size = 0.46 \[ \frac{\frac{12 \sqrt{1-2 x} \sqrt{5 x+3} \left (2819347154352 x^5+9492493272732 x^4+12787628716260 x^3+8615827181322 x^2+2903435279352 x+391506734113\right )}{(3 x+2)^{11/2}}+16 \sqrt{2} \left (2900562916 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-1460947915 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{12835746} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^(13/2)*Sqrt[3 + 5*x]),x]

[Out]

((12*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(391506734113 + 2903435279352*x + 8615827181322
*x^2 + 12787628716260*x^3 + 9492493272732*x^4 + 2819347154352*x^5))/(2 + 3*x)^(1
1/2) + 16*Sqrt[2]*(2900562916*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]
- 1460947915*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/12835746

_______________________________________________________________________________________

Maple [C]  time = 0.033, size = 743, normalized size = 3. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)/(2+3*x)^(13/2)/(3+5*x)^(1/2),x)

[Out]

2/6417873*(1420041373380*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^5*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-281
9347154352*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^5*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)[phone]*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^4*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-9397823847840*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^4*(3+5*x)
^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)[phone]*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)-12530431797120*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)[phone]*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)-835362
1198080*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)[phone]*x^7+1
402509998400*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)-2784540399360*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)[phone]*x^6+187001333120*
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))-371272053248*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))[phone]*x^5+211405262133852*x^4-21381185218
14*x^3-57086936770452*x^2-24956397311829*x-3523560607017)*(3+5*x)^(1/2)*(1-2*x)^
(1/2)/(10*x^2+x-3)/(2+3*x)^(11/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^(13/2)),x, algorithm="maxima")

[Out]

integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^(13/2)), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{-2 \, x + 1}}{{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^(13/2)),x, algorithm="fricas")

[Out]

integral((4*x^2 - 4*x + 1)*sqrt(-2*x + 1)/((729*x^6 + 2916*x^5 + 4860*x^4 + 4320
*x^3 + 2160*x^2 + 576*x + 64)*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)**(5/2)/(2+3*x)**(13/2)/(3+5*x)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^(13/2)),x, algorithm="giac")

[Out]

integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^(13/2)), x)