3.1936 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=148 \[ \frac{83 (1-2 x)^{7/2}}{2646 (3 x+2)^5}-\frac{(1-2 x)^{7/2}}{378 (3 x+2)^6}-\frac{263 (1-2 x)^{5/2}}{1176 (3 x+2)^4}+\frac{1315 (1-2 x)^{3/2}}{10584 (3 x+2)^3}+\frac{1315 \sqrt{1-2 x}}{148176 (3 x+2)}-\frac{1315 \sqrt{1-2 x}}{21168 (3 x+2)^2}+\frac{1315 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{74088 \sqrt{21}} \]

[Out]

-(1 - 2*x)^(7/2)/(378*(2 + 3*x)^6) + (83*(1 - 2*x)^(7/2))/(2646*(2 + 3*x)^5) - (
263*(1 - 2*x)^(5/2))/(1176*(2 + 3*x)^4) + (1315*(1 - 2*x)^(3/2))/(10584*(2 + 3*x
)^3) - (1315*Sqrt[1 - 2*x])/(21168*(2 + 3*x)^2) + (1315*Sqrt[1 - 2*x])/(148176*(
2 + 3*x)) + (1315*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(74088*Sqrt[21])

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Rubi [A]  time = 0.174873, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{83 (1-2 x)^{7/2}}{2646 (3 x+2)^5}-\frac{(1-2 x)^{7/2}}{378 (3 x+2)^6}-\frac{263 (1-2 x)^{5/2}}{1176 (3 x+2)^4}+\frac{1315 (1-2 x)^{3/2}}{10584 (3 x+2)^3}+\frac{1315 \sqrt{1-2 x}}{148176 (3 x+2)}-\frac{1315 \sqrt{1-2 x}}{21168 (3 x+2)^2}+\frac{1315 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{74088 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

-(1 - 2*x)^(7/2)/(378*(2 + 3*x)^6) + (83*(1 - 2*x)^(7/2))/(2646*(2 + 3*x)^5) - (
263*(1 - 2*x)^(5/2))/(1176*(2 + 3*x)^4) + (1315*(1 - 2*x)^(3/2))/(10584*(2 + 3*x
)^3) - (1315*Sqrt[1 - 2*x])/(21168*(2 + 3*x)^2) + (1315*Sqrt[1 - 2*x])/(148176*(
2 + 3*x)) + (1315*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(74088*Sqrt[21])

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Rubi in Sympy [A]  time = 17.5621, size = 131, normalized size = 0.89 \[ \frac{83 \left (- 2 x + 1\right )^{\frac{7}{2}}}{2646 \left (3 x + 2\right )^{5}} - \frac{\left (- 2 x + 1\right )^{\frac{7}{2}}}{378 \left (3 x + 2\right )^{6}} - \frac{263 \left (- 2 x + 1\right )^{\frac{5}{2}}}{1176 \left (3 x + 2\right )^{4}} + \frac{1315 \left (- 2 x + 1\right )^{\frac{3}{2}}}{10584 \left (3 x + 2\right )^{3}} + \frac{1315 \sqrt{- 2 x + 1}}{148176 \left (3 x + 2\right )} - \frac{1315 \sqrt{- 2 x + 1}}{21168 \left (3 x + 2\right )^{2}} + \frac{1315 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{1555848} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

83*(-2*x + 1)**(7/2)/(2646*(3*x + 2)**5) - (-2*x + 1)**(7/2)/(378*(3*x + 2)**6)
- 263*(-2*x + 1)**(5/2)/(1176*(3*x + 2)**4) + 1315*(-2*x + 1)**(3/2)/(10584*(3*x
+ 2)**3) + 1315*sqrt(-2*x + 1)/(148176*(3*x + 2)) - 1315*sqrt(-2*x + 1)/(21168*
(3*x + 2)**2) + 1315*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/1555848

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Mathematica [A]  time = 0.133995, size = 73, normalized size = 0.49 \[ \frac{\frac{189 \sqrt{1-2 x} \left (319545 x^5-1979115 x^4-2360850 x^3-587502 x^2-106808 x-81568\right )}{(3 x+2)^6}+23670 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{28005264} \]

Antiderivative was successfully verified.

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

[Out]

((189*Sqrt[1 - 2*x]*(-81568 - 106808*x - 587502*x^2 - 2360850*x^3 - 1979115*x^4
+ 319545*x^5))/(2 + 3*x)^6 + 23670*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/28
005264

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Maple [A]  time = 0.019, size = 84, normalized size = 0.6 \[ 23328\,{\frac{1}{ \left ( -4-6\,x \right ) ^{6}} \left ( -{\frac{1315\, \left ( 1-2\,x \right ) ^{11/2}}{7112448}}-{\frac{112405\, \left ( 1-2\,x \right ) ^{9/2}}{82301184}}+{\frac{8345\, \left ( 1-2\,x \right ) ^{7/2}}{653184}}-{\frac{2893\, \left ( 1-2\,x \right ) ^{5/2}}{93312}}+{\frac{156485\, \left ( 1-2\,x \right ) ^{3/2}}{5038848}}-{\frac{64435\,\sqrt{1-2\,x}}{5038848}} \right ) }+{\frac{1315\,\sqrt{21}}{1555848}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

23328*(-1315/7112448*(1-2*x)^(11/2)-112405/82301184*(1-2*x)^(9/2)+8345/653184*(1
-2*x)^(7/2)-2893/93312*(1-2*x)^(5/2)+156485/5038848*(1-2*x)^(3/2)-64435/5038848*
(1-2*x)^(1/2))/(-4-6*x)^6+1315/1555848*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1
/2)

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Maxima [A]  time = 1.52881, size = 197, normalized size = 1.33 \[ -\frac{1315}{3111696} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{319545 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + 2360505 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 22080870 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 53584146 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 53674355 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 22101205 \, \sqrt{-2 \, x + 1}}{74088 \,{\left (729 \,{\left (2 \, x - 1\right )}^{6} + 10206 \,{\left (2 \, x - 1\right )}^{5} + 59535 \,{\left (2 \, x - 1\right )}^{4} + 185220 \,{\left (2 \, x - 1\right )}^{3} + 324135 \,{\left (2 \, x - 1\right )}^{2} + 605052 \, x - 184877\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^2*(-2*x + 1)^(5/2)/(3*x + 2)^7,x, algorithm="maxima")

[Out]

-1315/3111696*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*
x + 1))) - 1/74088*(319545*(-2*x + 1)^(11/2) + 2360505*(-2*x + 1)^(9/2) - 220808
70*(-2*x + 1)^(7/2) + 53584146*(-2*x + 1)^(5/2) - 53674355*(-2*x + 1)^(3/2) + 22
101205*sqrt(-2*x + 1))/(729*(2*x - 1)^6 + 10206*(2*x - 1)^5 + 59535*(2*x - 1)^4
+ 185220*(2*x - 1)^3 + 324135*(2*x - 1)^2 + 605052*x - 184877)

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Fricas [A]  time = 0.216313, size = 181, normalized size = 1.22 \[ \frac{\sqrt{21}{\left (\sqrt{21}{\left (319545 \, x^{5} - 1979115 \, x^{4} - 2360850 \, x^{3} - 587502 \, x^{2} - 106808 \, x - 81568\right )} \sqrt{-2 \, x + 1} + 1315 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{3111696 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^2*(-2*x + 1)^(5/2)/(3*x + 2)^7,x, algorithm="fricas")

[Out]

1/3111696*sqrt(21)*(sqrt(21)*(319545*x^5 - 1979115*x^4 - 2360850*x^3 - 587502*x^
2 - 106808*x - 81568)*sqrt(-2*x + 1) + 1315*(729*x^6 + 2916*x^5 + 4860*x^4 + 432
0*x^3 + 2160*x^2 + 576*x + 64)*log((sqrt(21)*(3*x - 5) - 21*sqrt(-2*x + 1))/(3*x
+ 2)))/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

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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)*(3+5*x)**2/(2+3*x)**7,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216735, size = 178, normalized size = 1.2 \[ -\frac{1315}{3111696} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{319545 \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - 2360505 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - 22080870 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - 53584146 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + 53674355 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 22101205 \, \sqrt{-2 \, x + 1}}{4741632 \,{\left (3 \, x + 2\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^2*(-2*x + 1)^(5/2)/(3*x + 2)^7,x, algorithm="giac")

[Out]

-1315/3111696*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*
sqrt(-2*x + 1))) + 1/4741632*(319545*(2*x - 1)^5*sqrt(-2*x + 1) - 2360505*(2*x -
1)^4*sqrt(-2*x + 1) - 22080870*(2*x - 1)^3*sqrt(-2*x + 1) - 53584146*(2*x - 1)^
2*sqrt(-2*x + 1) + 53674355*(-2*x + 1)^(3/2) - 22101205*sqrt(-2*x + 1))/(3*x + 2
)^6