3.1965 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=102 \[ -\frac{(1-2 x)^{7/2}}{275 (5 x+3)}-\frac{9}{175} (1-2 x)^{7/2}+\frac{122 (1-2 x)^{5/2}}{6875}+\frac{122 (1-2 x)^{3/2}}{1875}+\frac{1342 \sqrt{1-2 x}}{3125}-\frac{1342 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125} \]

[Out]

(1342*Sqrt[1 - 2*x])/3125 + (122*(1 - 2*x)^(3/2))/1875 + (122*(1 - 2*x)^(5/2))/6
875 - (9*(1 - 2*x)^(7/2))/175 - (1 - 2*x)^(7/2)/(275*(3 + 5*x)) - (1342*Sqrt[11/
5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/3125

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Rubi [A]  time = 0.131915, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{(1-2 x)^{7/2}}{275 (5 x+3)}-\frac{9}{175} (1-2 x)^{7/2}+\frac{122 (1-2 x)^{5/2}}{6875}+\frac{122 (1-2 x)^{3/2}}{1875}+\frac{1342 \sqrt{1-2 x}}{3125}-\frac{1342 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125} \]

Antiderivative was successfully verified.

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

[Out]

(1342*Sqrt[1 - 2*x])/3125 + (122*(1 - 2*x)^(3/2))/1875 + (122*(1 - 2*x)^(5/2))/6
875 - (9*(1 - 2*x)^(7/2))/175 - (1 - 2*x)^(7/2)/(275*(3 + 5*x)) - (1342*Sqrt[11/
5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/3125

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Rubi in Sympy [A]  time = 12.2443, size = 85, normalized size = 0.83 \[ - \frac{9 \left (- 2 x + 1\right )^{\frac{7}{2}}}{175} - \frac{\left (- 2 x + 1\right )^{\frac{7}{2}}}{275 \left (5 x + 3\right )} + \frac{122 \left (- 2 x + 1\right )^{\frac{5}{2}}}{6875} + \frac{122 \left (- 2 x + 1\right )^{\frac{3}{2}}}{1875} + \frac{1342 \sqrt{- 2 x + 1}}{3125} - \frac{1342 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{15625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9*(-2*x + 1)**(7/2)/175 - (-2*x + 1)**(7/2)/(275*(5*x + 3)) + 122*(-2*x + 1)**(
5/2)/6875 + 122*(-2*x + 1)**(3/2)/1875 + 1342*sqrt(-2*x + 1)/3125 - 1342*sqrt(55
)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/15625

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Mathematica [A]  time = 0.10638, size = 68, normalized size = 0.67 \[ \frac{\frac{5 \sqrt{1-2 x} \left (135000 x^4-96300 x^3-75130 x^2+173795 x+90486\right )}{5 x+3}-28182 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{328125} \]

Antiderivative was successfully verified.

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

[Out]

((5*Sqrt[1 - 2*x]*(90486 + 173795*x - 75130*x^2 - 96300*x^3 + 135000*x^4))/(3 +
5*x) - 28182*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/328125

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Maple [A]  time = 0.015, size = 72, normalized size = 0.7 \[ -{\frac{9}{175} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{12}{625} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{128}{1875} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{1364}{3125}\sqrt{1-2\,x}}+{\frac{242}{15625}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{1342\,\sqrt{55}}{15625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/175*(1-2*x)^(7/2)+12/625*(1-2*x)^(5/2)+128/1875*(1-2*x)^(3/2)+1364/3125*(1-2*
x)^(1/2)+242/15625*(1-2*x)^(1/2)/(-6/5-2*x)-1342/15625*arctanh(1/11*55^(1/2)*(1-
2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.48869, size = 120, normalized size = 1.18 \[ -\frac{9}{175} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{12}{625} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{128}{1875} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{671}{15625} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{1364}{3125} \, \sqrt{-2 \, x + 1} - \frac{121 \, \sqrt{-2 \, x + 1}}{3125 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/175*(-2*x + 1)^(7/2) + 12/625*(-2*x + 1)^(5/2) + 128/1875*(-2*x + 1)^(3/2) +
671/15625*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x +
1))) + 1364/3125*sqrt(-2*x + 1) - 121/3125*sqrt(-2*x + 1)/(5*x + 3)

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Fricas [A]  time = 0.214823, size = 115, normalized size = 1.13 \[ \frac{\sqrt{5}{\left (14091 \, \sqrt{11}{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{5}{\left (135000 \, x^{4} - 96300 \, x^{3} - 75130 \, x^{2} + 173795 \, x + 90486\right )} \sqrt{-2 \, x + 1}\right )}}{328125 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/328125*sqrt(5)*(14091*sqrt(11)*(5*x + 3)*log((sqrt(5)*(5*x - 8) + 5*sqrt(11)*s
qrt(-2*x + 1))/(5*x + 3)) + sqrt(5)*(135000*x^4 - 96300*x^3 - 75130*x^2 + 173795
*x + 90486)*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)**2/(3+5*x)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.212434, size = 143, normalized size = 1.4 \[ \frac{9}{175} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{12}{625} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{128}{1875} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{671}{15625} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{1364}{3125} \, \sqrt{-2 \, x + 1} - \frac{121 \, \sqrt{-2 \, x + 1}}{3125 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/175*(2*x - 1)^3*sqrt(-2*x + 1) + 12/625*(2*x - 1)^2*sqrt(-2*x + 1) + 128/1875*
(-2*x + 1)^(3/2) + 671/15625*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1)
)/(sqrt(55) + 5*sqrt(-2*x + 1))) + 1364/3125*sqrt(-2*x + 1) - 121/3125*sqrt(-2*x
+ 1)/(5*x + 3)