3.2175 \(\int \frac{(2+3 x)^4}{(1-2 x)^{5/2} (3+5 x)^3} \, dx\)

Optimal. Leaf size=107 \[ \frac{7 (3 x+2)^3}{33 (1-2 x)^{3/2} (5 x+3)^2}-\frac{73 (3 x+2)^2}{3630 \sqrt{1-2 x} (5 x+3)^2}-\frac{2133933 x+1287116}{2196150 \sqrt{1-2 x} (5 x+3)}-\frac{14423 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{366025 \sqrt{55}} \]

[Out]

(-73*(2 + 3*x)^2)/(3630*Sqrt[1 - 2*x]*(3 + 5*x)^2) + (7*(2 + 3*x)^3)/(33*(1 - 2*
x)^(3/2)*(3 + 5*x)^2) - (1287116 + 2133933*x)/(2196150*Sqrt[1 - 2*x]*(3 + 5*x))
- (14423*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(366025*Sqrt[55])

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Rubi [A]  time = 0.18054, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{7 (3 x+2)^3}{33 (1-2 x)^{3/2} (5 x+3)^2}-\frac{73 (3 x+2)^2}{3630 \sqrt{1-2 x} (5 x+3)^2}-\frac{2133933 x+1287116}{2196150 \sqrt{1-2 x} (5 x+3)}-\frac{14423 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{366025 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(-73*(2 + 3*x)^2)/(3630*Sqrt[1 - 2*x]*(3 + 5*x)^2) + (7*(2 + 3*x)^3)/(33*(1 - 2*
x)^(3/2)*(3 + 5*x)^2) - (1287116 + 2133933*x)/(2196150*Sqrt[1 - 2*x]*(3 + 5*x))
- (14423*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(366025*Sqrt[55])

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Rubi in Sympy [A]  time = 19.2301, size = 95, normalized size = 0.89 \[ - \frac{14423 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{20131375} - \frac{73 \left (3 x + 2\right )^{2}}{3630 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{2}} - \frac{2133933 x + 1287116}{2196150 \sqrt{- 2 x + 1} \left (5 x + 3\right )} + \frac{7 \left (3 x + 2\right )^{3}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-14423*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/20131375 - 73*(3*x + 2)**2/(36
30*sqrt(-2*x + 1)*(5*x + 3)**2) - (2133933*x + 1287116)/(2196150*sqrt(-2*x + 1)*
(5*x + 3)) + 7*(3*x + 2)**3/(33*(-2*x + 1)**(3/2)*(5*x + 3)**2)

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Mathematica [A]  time = 0.118551, size = 66, normalized size = 0.62 \[ \frac{\frac{55 \sqrt{1-2 x} \left (34712250 x^3+40823468 x^2+11479257 x-311208\right )}{\left (10 x^2+x-3\right )^2}-86538 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{120788250} \]

Antiderivative was successfully verified.

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

[Out]

((55*Sqrt[1 - 2*x]*(-311208 + 11479257*x + 40823468*x^2 + 34712250*x^3))/(-3 + x
+ 10*x^2)^2 - 86538*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/120788250

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Maple [A]  time = 0.023, size = 66, normalized size = 0.6 \[{\frac{2401}{7986} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{9261}{29282}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{100}{14641\, \left ( -6-10\,x \right ) ^{2}} \left ({\frac{11}{20} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{3047}{2500}\sqrt{1-2\,x}} \right ) }-{\frac{14423\,\sqrt{55}}{20131375}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2401/7986/(1-2*x)^(3/2)-9261/29282/(1-2*x)^(1/2)+100/14641*(11/20*(1-2*x)^(3/2)-
3047/2500*(1-2*x)^(1/2))/(-6-10*x)^2-14423/20131375*arctanh(1/11*55^(1/2)*(1-2*x
)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.51396, size = 124, normalized size = 1.16 \[ \frac{14423}{40262750} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{17356125 \,{\left (2 \, x - 1\right )}^{3} + 92891843 \,{\left (2 \, x - 1\right )}^{2} + 313347650 \, x - 76780550}{2196150 \,{\left (25 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 110 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 121 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

14423/40262750*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2
*x + 1))) + 1/2196150*(17356125*(2*x - 1)^3 + 92891843*(2*x - 1)^2 + 313347650*x
- 76780550)/(25*(-2*x + 1)^(7/2) - 110*(-2*x + 1)^(5/2) + 121*(-2*x + 1)^(3/2))

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Fricas [A]  time = 0.22077, size = 138, normalized size = 1.29 \[ \frac{\sqrt{55}{\left (43269 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) - \sqrt{55}{\left (34712250 \, x^{3} + 40823468 \, x^{2} + 11479257 \, x - 311208\right )}\right )}}{120788250 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/120788250*sqrt(55)*(43269*(50*x^3 + 35*x^2 - 12*x - 9)*sqrt(-2*x + 1)*log((sqr
t(55)*(5*x - 8) + 55*sqrt(-2*x + 1))/(5*x + 3)) - sqrt(55)*(34712250*x^3 + 40823
468*x^2 + 11479257*x - 311208))/((50*x^3 + 35*x^2 - 12*x - 9)*sqrt(-2*x + 1))

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.215695, size = 120, normalized size = 1.12 \[ \frac{14423}{40262750} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{343 \,{\left (81 \, x - 2\right )}}{43923 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} + \frac{125 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 277 \, \sqrt{-2 \, x + 1}}{133100 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

14423/40262750*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) +
5*sqrt(-2*x + 1))) - 343/43923*(81*x - 2)/((2*x - 1)*sqrt(-2*x + 1)) + 1/133100*
(125*(-2*x + 1)^(3/2) - 277*sqrt(-2*x + 1))/(5*x + 3)^2