3.1964 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^3}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=121 \[ -\frac{(1-2 x)^{5/2} (3 x+2)^3}{5 (5 x+3)}+\frac{11}{75} (1-2 x)^{5/2} (3 x+2)^2+\frac{188 (1-2 x)^{3/2}}{9375}-\frac{2 (1-2 x)^{5/2} (2850 x+6191)}{65625}+\frac{2068 \sqrt{1-2 x}}{15625}-\frac{2068 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15625} \]

[Out]

(2068*Sqrt[1 - 2*x])/15625 + (188*(1 - 2*x)^(3/2))/9375 + (11*(1 - 2*x)^(5/2)*(2
+ 3*x)^2)/75 - ((1 - 2*x)^(5/2)*(2 + 3*x)^3)/(5*(3 + 5*x)) - (2*(1 - 2*x)^(5/2)
*(6191 + 2850*x))/65625 - (2068*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/15
625

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Rubi [A]  time = 0.192115, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{(1-2 x)^{5/2} (3 x+2)^3}{5 (5 x+3)}+\frac{11}{75} (1-2 x)^{5/2} (3 x+2)^2+\frac{188 (1-2 x)^{3/2}}{9375}-\frac{2 (1-2 x)^{5/2} (2850 x+6191)}{65625}+\frac{2068 \sqrt{1-2 x}}{15625}-\frac{2068 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15625} \]

Antiderivative was successfully verified.

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

[Out]

(2068*Sqrt[1 - 2*x])/15625 + (188*(1 - 2*x)^(3/2))/9375 + (11*(1 - 2*x)^(5/2)*(2
+ 3*x)^2)/75 - ((1 - 2*x)^(5/2)*(2 + 3*x)^3)/(5*(3 + 5*x)) - (2*(1 - 2*x)^(5/2)
*(6191 + 2850*x))/65625 - (2068*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/15
625

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Rubi in Sympy [A]  time = 21.935, size = 102, normalized size = 0.84 \[ - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{3}}{5 \left (5 x + 3\right )} + \frac{11 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{2}}{75} - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (17100 x + 37146\right )}{196875} + \frac{188 \left (- 2 x + 1\right )^{\frac{3}{2}}}{9375} + \frac{2068 \sqrt{- 2 x + 1}}{15625} - \frac{2068 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{78125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(5/2)*(3*x + 2)**3/(5*(5*x + 3)) + 11*(-2*x + 1)**(5/2)*(3*x + 2)**
2/75 - (-2*x + 1)**(5/2)*(17100*x + 37146)/196875 + 188*(-2*x + 1)**(3/2)/9375 +
2068*sqrt(-2*x + 1)/15625 - 2068*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/781
25

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Mathematica [A]  time = 0.12092, size = 73, normalized size = 0.6 \[ \frac{\frac{5 \sqrt{1-2 x} \left (1575000 x^5+427500 x^4-1858950 x^3+152105 x^2+680930 x+16794\right )}{5 x+3}-43428 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1640625} \]

Antiderivative was successfully verified.

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

[Out]

((5*Sqrt[1 - 2*x]*(16794 + 680930*x + 152105*x^2 - 1858950*x^3 + 427500*x^4 + 15
75000*x^5))/(3 + 5*x) - 43428*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/164062
5

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Maple [A]  time = 0.016, size = 81, normalized size = 0.7 \[{\frac{3}{50} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}-{\frac{351}{1750} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{18}{3125} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{194}{9375} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{418}{3125}\sqrt{1-2\,x}}+{\frac{242}{78125}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{2068\,\sqrt{55}}{78125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/50*(1-2*x)^(9/2)-351/1750*(1-2*x)^(7/2)+18/3125*(1-2*x)^(5/2)+194/9375*(1-2*x)
^(3/2)+418/3125*(1-2*x)^(1/2)+242/78125*(1-2*x)^(1/2)/(-6/5-2*x)-2068/78125*arct
anh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.53054, size = 132, normalized size = 1.09 \[ \frac{3}{50} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{351}{1750} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{18}{3125} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{194}{9375} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1034}{78125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{418}{3125} \, \sqrt{-2 \, x + 1} - \frac{121 \, \sqrt{-2 \, x + 1}}{15625 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/50*(-2*x + 1)^(9/2) - 351/1750*(-2*x + 1)^(7/2) + 18/3125*(-2*x + 1)^(5/2) + 1
94/9375*(-2*x + 1)^(3/2) + 1034/78125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1)
)/(sqrt(55) + 5*sqrt(-2*x + 1))) + 418/3125*sqrt(-2*x + 1) - 121/15625*sqrt(-2*x
+ 1)/(5*x + 3)

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Fricas [A]  time = 0.212319, size = 122, normalized size = 1.01 \[ \frac{\sqrt{5}{\left (21714 \, \sqrt{11}{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{5}{\left (1575000 \, x^{5} + 427500 \, x^{4} - 1858950 \, x^{3} + 152105 \, x^{2} + 680930 \, x + 16794\right )} \sqrt{-2 \, x + 1}\right )}}{1640625 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1640625*sqrt(5)*(21714*sqrt(11)*(5*x + 3)*log((sqrt(5)*(5*x - 8) + 5*sqrt(11)*
sqrt(-2*x + 1))/(5*x + 3)) + sqrt(5)*(1575000*x^5 + 427500*x^4 - 1858950*x^3 + 1
52105*x^2 + 680930*x + 16794)*sqrt(-2*x + 1))/(5*x + 3)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.217818, size = 165, normalized size = 1.36 \[ \frac{3}{50} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{351}{1750} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{18}{3125} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{194}{9375} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1034}{78125} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{418}{3125} \, \sqrt{-2 \, x + 1} - \frac{121 \, \sqrt{-2 \, x + 1}}{15625 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/50*(2*x - 1)^4*sqrt(-2*x + 1) + 351/1750*(2*x - 1)^3*sqrt(-2*x + 1) + 18/3125*
(2*x - 1)^2*sqrt(-2*x + 1) + 194/9375*(-2*x + 1)^(3/2) + 1034/78125*sqrt(55)*ln(
1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 418/31
25*sqrt(-2*x + 1) - 121/15625*sqrt(-2*x + 1)/(5*x + 3)