3.1971 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^2} \, dx\)

Optimal. Leaf size=154 \[ \frac{7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)}+\frac{6649 \sqrt{1-2 x}}{27 (3 x+2) (5 x+3)}+\frac{917 \sqrt{1-2 x}}{54 (3 x+2)^2 (5 x+3)}-\frac{44545 \sqrt{1-2 x}}{18 (5 x+3)}-\frac{307295 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3 \sqrt{21}}+3014 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-44545*Sqrt[1 - 2*x])/(18*(3 + 5*x)) + (7*(1 - 2*x)^(3/2))/(9*(2 + 3*x)^3*(3 +
5*x)) + (917*Sqrt[1 - 2*x])/(54*(2 + 3*x)^2*(3 + 5*x)) + (6649*Sqrt[1 - 2*x])/(2
7*(2 + 3*x)*(3 + 5*x)) - (307295*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(3*Sqrt[21])
+ 3014*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

_______________________________________________________________________________________

Rubi [A]  time = 0.31887, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)}+\frac{6649 \sqrt{1-2 x}}{27 (3 x+2) (5 x+3)}+\frac{917 \sqrt{1-2 x}}{54 (3 x+2)^2 (5 x+3)}-\frac{44545 \sqrt{1-2 x}}{18 (5 x+3)}-\frac{307295 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3 \sqrt{21}}+3014 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 - 2*x)^(5/2)/((2 + 3*x)^4*(3 + 5*x)^2),x]

[Out]

(-44545*Sqrt[1 - 2*x])/(18*(3 + 5*x)) + (7*(1 - 2*x)^(3/2))/(9*(2 + 3*x)^3*(3 +
5*x)) + (917*Sqrt[1 - 2*x])/(54*(2 + 3*x)^2*(3 + 5*x)) + (6649*Sqrt[1 - 2*x])/(2
7*(2 + 3*x)*(3 + 5*x)) - (307295*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(3*Sqrt[21])
+ 3014*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 34.2833, size = 133, normalized size = 0.86 \[ \frac{7 \left (- 2 x + 1\right )^{\frac{3}{2}}}{9 \left (3 x + 2\right )^{3} \left (5 x + 3\right )} - \frac{44545 \sqrt{- 2 x + 1}}{18 \left (5 x + 3\right )} + \frac{6649 \sqrt{- 2 x + 1}}{27 \left (3 x + 2\right ) \left (5 x + 3\right )} + \frac{917 \sqrt{- 2 x + 1}}{54 \left (3 x + 2\right )^{2} \left (5 x + 3\right )} - \frac{307295 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{63} + 3014 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)/(2+3*x)**4/(3+5*x)**2,x)

[Out]

7*(-2*x + 1)**(3/2)/(9*(3*x + 2)**3*(5*x + 3)) - 44545*sqrt(-2*x + 1)/(18*(5*x +
3)) + 6649*sqrt(-2*x + 1)/(27*(3*x + 2)*(5*x + 3)) + 917*sqrt(-2*x + 1)/(54*(3*
x + 2)**2*(5*x + 3)) - 307295*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/63 + 301
4*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)

_______________________________________________________________________________________

Mathematica [A]  time = 0.229, size = 95, normalized size = 0.62 \[ -\frac{\sqrt{1-2 x} \left (400905 x^3+788512 x^2+516513 x+112668\right )}{6 (3 x+2)^3 (5 x+3)}-\frac{307295 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3 \sqrt{21}}+3014 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^4*(3 + 5*x)^2),x]

[Out]

-(Sqrt[1 - 2*x]*(112668 + 516513*x + 788512*x^2 + 400905*x^3))/(6*(2 + 3*x)^3*(3
+ 5*x)) - (307295*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(3*Sqrt[21]) + 3014*Sqrt[55
]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

_______________________________________________________________________________________

Maple [A]  time = 0.021, size = 91, normalized size = 0.6 \[ 108\,{\frac{1}{ \left ( -4-6\,x \right ) ^{3}} \left ({\frac{6731\, \left ( 1-2\,x \right ) ^{5/2}}{36}}-{\frac{71365\, \left ( 1-2\,x \right ) ^{3/2}}{81}}+{\frac{336385\,\sqrt{1-2\,x}}{324}} \right ) }-{\frac{307295\,\sqrt{21}}{63}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+242\,{\frac{\sqrt{1-2\,x}}{-6/5-2\,x}}+3014\,{\it Artanh} \left ( 1/11\,\sqrt{55}\sqrt{1-2\,x} \right ) \sqrt{55} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)/(2+3*x)^4/(3+5*x)^2,x)

[Out]

108*(6731/36*(1-2*x)^(5/2)-71365/81*(1-2*x)^(3/2)+336385/324*(1-2*x)^(1/2))/(-4-
6*x)^3-307295/63*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+242*(1-2*x)^(1/2)/
(-6/5-2*x)+3014*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.4911, size = 197, normalized size = 1.28 \[ -1507 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{307295}{126} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{400905 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 2779739 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 6422815 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 4945325 \, \sqrt{-2 \, x + 1}}{3 \,{\left (135 \,{\left (2 \, x - 1\right )}^{4} + 1242 \,{\left (2 \, x - 1\right )}^{3} + 4284 \,{\left (2 \, x - 1\right )}^{2} + 13132 \, x - 2793\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(5/2)/((5*x + 3)^2*(3*x + 2)^4),x, algorithm="maxima")

[Out]

-1507*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)))
+ 307295/126*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*
x + 1))) + 1/3*(400905*(-2*x + 1)^(7/2) - 2779739*(-2*x + 1)^(5/2) + 6422815*(-2
*x + 1)^(3/2) - 4945325*sqrt(-2*x + 1))/(135*(2*x - 1)^4 + 1242*(2*x - 1)^3 + 42
84*(2*x - 1)^2 + 13132*x - 2793)

_______________________________________________________________________________________

Fricas [A]  time = 0.219914, size = 215, normalized size = 1.4 \[ \frac{\sqrt{21}{\left (9042 \, \sqrt{55} \sqrt{21}{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (\frac{5 \, x - \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) - \sqrt{21}{\left (400905 \, x^{3} + 788512 \, x^{2} + 516513 \, x + 112668\right )} \sqrt{-2 \, x + 1} + 307295 \,{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{126 \,{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(5/2)/((5*x + 3)^2*(3*x + 2)^4),x, algorithm="fricas")

[Out]

1/126*sqrt(21)*(9042*sqrt(55)*sqrt(21)*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24
)*log((5*x - sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) - sqrt(21)*(400905*x^3 + 78
8512*x^2 + 516513*x + 112668)*sqrt(-2*x + 1) + 307295*(135*x^4 + 351*x^3 + 342*x
^2 + 148*x + 24)*log((sqrt(21)*(3*x - 5) + 21*sqrt(-2*x + 1))/(3*x + 2)))/(135*x
^4 + 351*x^3 + 342*x^2 + 148*x + 24)

_______________________________________________________________________________________

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)**4/(3+5*x)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.217571, size = 188, normalized size = 1.22 \[ -1507 \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{307295}{126} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{605 \, \sqrt{-2 \, x + 1}}{5 \, x + 3} - \frac{60579 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 285460 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 336385 \, \sqrt{-2 \, x + 1}}{24 \,{\left (3 \, x + 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(5/2)/((5*x + 3)^2*(3*x + 2)^4),x, algorithm="giac")

[Out]

-1507*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2
*x + 1))) + 307295/126*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt
(21) + 3*sqrt(-2*x + 1))) - 605*sqrt(-2*x + 1)/(5*x + 3) - 1/24*(60579*(2*x - 1)
^2*sqrt(-2*x + 1) - 285460*(-2*x + 1)^(3/2) + 336385*sqrt(-2*x + 1))/(3*x + 2)^3