3.1946 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=147 \[ -\frac{275 \sqrt{1-2 x} (5 x+3)^3}{9 (3 x+2)}+\frac{55 (1-2 x)^{3/2} (5 x+3)^3}{27 (3 x+2)^2}-\frac{(1-2 x)^{5/2} (5 x+3)^3}{9 (3 x+2)^3}+\frac{1441}{27} \sqrt{1-2 x} (5 x+3)^2-\frac{22}{243} \sqrt{1-2 x} (1885 x+578)-\frac{41360 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{243 \sqrt{21}} \]

[Out]

(1441*Sqrt[1 - 2*x]*(3 + 5*x)^2)/27 - ((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(9*(2 + 3*x)
^3) + (55*(1 - 2*x)^(3/2)*(3 + 5*x)^3)/(27*(2 + 3*x)^2) - (275*Sqrt[1 - 2*x]*(3
+ 5*x)^3)/(9*(2 + 3*x)) - (22*Sqrt[1 - 2*x]*(578 + 1885*x))/243 - (41360*ArcTanh
[Sqrt[3/7]*Sqrt[1 - 2*x]])/(243*Sqrt[21])

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Rubi [A]  time = 0.288188, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ -\frac{275 \sqrt{1-2 x} (5 x+3)^3}{9 (3 x+2)}+\frac{55 (1-2 x)^{3/2} (5 x+3)^3}{27 (3 x+2)^2}-\frac{(1-2 x)^{5/2} (5 x+3)^3}{9 (3 x+2)^3}+\frac{1441}{27} \sqrt{1-2 x} (5 x+3)^2-\frac{22}{243} \sqrt{1-2 x} (1885 x+578)-\frac{41360 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{243 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(1441*Sqrt[1 - 2*x]*(3 + 5*x)^2)/27 - ((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(9*(2 + 3*x)
^3) + (55*(1 - 2*x)^(3/2)*(3 + 5*x)^3)/(27*(2 + 3*x)^2) - (275*Sqrt[1 - 2*x]*(3
+ 5*x)^3)/(9*(2 + 3*x)) - (22*Sqrt[1 - 2*x]*(578 + 1885*x))/243 - (41360*ArcTanh
[Sqrt[3/7]*Sqrt[1 - 2*x]])/(243*Sqrt[21])

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Rubi in Sympy [A]  time = 21.0823, size = 117, normalized size = 0.8 \[ \frac{11 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (12600 x + 9180\right )}{23814 \left (3 x + 2\right )} - \frac{55 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{2}}{189 \left (3 x + 2\right )^{2}} - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{3}}{9 \left (3 x + 2\right )^{3}} + \frac{41360 \left (- 2 x + 1\right )^{\frac{3}{2}}}{11907} + \frac{41360 \sqrt{- 2 x + 1}}{1701} - \frac{41360 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{5103} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

11*(-2*x + 1)**(5/2)*(12600*x + 9180)/(23814*(3*x + 2)) - 55*(-2*x + 1)**(5/2)*(
5*x + 3)**2/(189*(3*x + 2)**2) - (-2*x + 1)**(5/2)*(5*x + 3)**3/(9*(3*x + 2)**3)
+ 41360*(-2*x + 1)**(3/2)/11907 + 41360*sqrt(-2*x + 1)/1701 - 41360*sqrt(21)*at
anh(sqrt(21)*sqrt(-2*x + 1)/7)/5103

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Mathematica [A]  time = 0.136946, size = 73, normalized size = 0.5 \[ \frac{\frac{21 \sqrt{1-2 x} \left (16200 x^5-20700 x^4+87030 x^3+289719 x^2+229336 x+56141\right )}{(3 x+2)^3}-41360 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{5103} \]

Antiderivative was successfully verified.

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

[Out]

((21*Sqrt[1 - 2*x]*(56141 + 229336*x + 289719*x^2 + 87030*x^3 - 20700*x^4 + 1620
0*x^5))/(2 + 3*x)^3 - 41360*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/5103

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Maple [A]  time = 0.019, size = 84, normalized size = 0.6 \[{\frac{50}{81} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{2050}{729} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{16570}{729}\sqrt{1-2\,x}}+{\frac{2}{27\, \left ( -4-6\,x \right ) ^{3}} \left ( -{\frac{4153}{3} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{172130}{27} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{198205}{27}\sqrt{1-2\,x}} \right ) }-{\frac{41360\,\sqrt{21}}{5103}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

50/81*(1-2*x)^(5/2)+2050/729*(1-2*x)^(3/2)+16570/729*(1-2*x)^(1/2)+2/27*(-4153/3
*(1-2*x)^(5/2)+172130/27*(1-2*x)^(3/2)-198205/27*(1-2*x)^(1/2))/(-4-6*x)^3-41360
/5103*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.49995, size = 161, normalized size = 1.1 \[ \frac{50}{81} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{2050}{729} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{20680}{5103} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{16570}{729} \, \sqrt{-2 \, x + 1} + \frac{2 \,{\left (37377 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 172130 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 198205 \, \sqrt{-2 \, x + 1}\right )}}{729 \,{\left (27 \,{\left (2 \, x - 1\right )}^{3} + 189 \,{\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

50/81*(-2*x + 1)^(5/2) + 2050/729*(-2*x + 1)^(3/2) + 20680/5103*sqrt(21)*log(-(s
qrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 16570/729*sqrt(-2*x
+ 1) + 2/729*(37377*(-2*x + 1)^(5/2) - 172130*(-2*x + 1)^(3/2) + 198205*sqrt(-2
*x + 1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)

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Fricas [A]  time = 0.211639, size = 140, normalized size = 0.95 \[ \frac{\sqrt{21}{\left (\sqrt{21}{\left (16200 \, x^{5} - 20700 \, x^{4} + 87030 \, x^{3} + 289719 \, x^{2} + 229336 \, x + 56141\right )} \sqrt{-2 \, x + 1} + 20680 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{5103 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5103*sqrt(21)*(sqrt(21)*(16200*x^5 - 20700*x^4 + 87030*x^3 + 289719*x^2 + 2293
36*x + 56141)*sqrt(-2*x + 1) + 20680*(27*x^3 + 54*x^2 + 36*x + 8)*log((sqrt(21)*
(3*x - 5) + 21*sqrt(-2*x + 1))/(3*x + 2)))/(27*x^3 + 54*x^2 + 36*x + 8)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220987, size = 159, normalized size = 1.08 \[ \frac{50}{81} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{2050}{729} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{20680}{5103} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{16570}{729} \, \sqrt{-2 \, x + 1} + \frac{37377 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 172130 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 198205 \, \sqrt{-2 \, x + 1}}{2916 \,{\left (3 \, x + 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

50/81*(2*x - 1)^2*sqrt(-2*x + 1) + 2050/729*(-2*x + 1)^(3/2) + 20680/5103*sqrt(2
1)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1
6570/729*sqrt(-2*x + 1) + 1/2916*(37377*(2*x - 1)^2*sqrt(-2*x + 1) - 172130*(-2*
x + 1)^(3/2) + 198205*sqrt(-2*x + 1))/(3*x + 2)^3