3.2178 \(\int \frac{2+3 x}{(1-2 x)^{5/2} (3+5 x)^3} \, dx\)

Optimal. Leaf size=96 \[ \frac{365}{14641 \sqrt{1-2 x}}-\frac{73}{1210 (1-2 x)^{3/2} (5 x+3)}+\frac{73}{3993 (1-2 x)^{3/2}}-\frac{1}{110 (1-2 x)^{3/2} (5 x+3)^2}-\frac{365 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{14641} \]

[Out]

73/(3993*(1 - 2*x)^(3/2)) + 365/(14641*Sqrt[1 - 2*x]) - 1/(110*(1 - 2*x)^(3/2)*(
3 + 5*x)^2) - 73/(1210*(1 - 2*x)^(3/2)*(3 + 5*x)) - (365*Sqrt[5/11]*ArcTanh[Sqrt
[5/11]*Sqrt[1 - 2*x]])/14641

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Rubi [A]  time = 0.106192, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{365}{14641 \sqrt{1-2 x}}-\frac{73}{1210 (1-2 x)^{3/2} (5 x+3)}+\frac{73}{3993 (1-2 x)^{3/2}}-\frac{1}{110 (1-2 x)^{3/2} (5 x+3)^2}-\frac{365 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{14641} \]

Antiderivative was successfully verified.

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

[Out]

73/(3993*(1 - 2*x)^(3/2)) + 365/(14641*Sqrt[1 - 2*x]) - 1/(110*(1 - 2*x)^(3/2)*(
3 + 5*x)^2) - 73/(1210*(1 - 2*x)^(3/2)*(3 + 5*x)) - (365*Sqrt[5/11]*ArcTanh[Sqrt
[5/11]*Sqrt[1 - 2*x]])/14641

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Rubi in Sympy [A]  time = 10.0831, size = 83, normalized size = 0.86 \[ - \frac{365 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{161051} + \frac{365}{14641 \sqrt{- 2 x + 1}} + \frac{73}{3993 \left (- 2 x + 1\right )^{\frac{3}{2}}} - \frac{73}{1210 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )} - \frac{1}{110 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-365*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/161051 + 365/(14641*sqrt(-2*x +
1)) + 73/(3993*(-2*x + 1)**(3/2)) - 73/(1210*(-2*x + 1)**(3/2)*(5*x + 3)) - 1/(1
10*(-2*x + 1)**(3/2)*(5*x + 3)**2)

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Mathematica [A]  time = 0.122551, size = 66, normalized size = 0.69 \[ \frac{\frac{11 \sqrt{1-2 x} \left (-109500 x^3-36500 x^2+47961 x+17466\right )}{\left (10 x^2+x-3\right )^2}-2190 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{966306} \]

Antiderivative was successfully verified.

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

[Out]

((11*Sqrt[1 - 2*x]*(17466 + 47961*x - 36500*x^2 - 109500*x^3))/(-3 + x + 10*x^2)
^2 - 2190*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/966306

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Maple [A]  time = 0.02, size = 66, normalized size = 0.7 \[{\frac{28}{3993} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{288}{14641}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{500}{14641\, \left ( -6-10\,x \right ) ^{2}} \left ({\frac{77}{20} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{869}{100}\sqrt{1-2\,x}} \right ) }-{\frac{365\,\sqrt{55}}{161051}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

28/3993/(1-2*x)^(3/2)+288/14641/(1-2*x)^(1/2)+500/14641*(77/20*(1-2*x)^(3/2)-869
/100*(1-2*x)^(1/2))/(-6-10*x)^2-365/161051*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*
55^(1/2)

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Maxima [A]  time = 1.49373, size = 124, normalized size = 1.29 \[ \frac{365}{322102} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{27375 \,{\left (2 \, x - 1\right )}^{3} + 100375 \,{\left (2 \, x - 1\right )}^{2} + 141328 \, x - 107932}{43923 \,{\left (25 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 110 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 121 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

365/322102*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x +
1))) - 1/43923*(27375*(2*x - 1)^3 + 100375*(2*x - 1)^2 + 141328*x - 107932)/(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 = 0.227733, size = 144, normalized size = 1.5 \[ \frac{\sqrt{11}{\left (1095 \, \sqrt{5}{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{11}{\left (109500 \, x^{3} + 36500 \, x^{2} - 47961 \, x - 17466\right )}\right )}}{966306 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/966306*sqrt(11)*(1095*sqrt(5)*(50*x^3 + 35*x^2 - 12*x - 9)*sqrt(-2*x + 1)*log(
(sqrt(11)*(5*x - 8) + 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) + sqrt(11)*(109500*x
^3 + 36500*x^2 - 47961*x - 17466))/((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)/(1-2*x)**(5/2)/(3+5*x)**3,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.245379, size = 120, normalized size = 1.25 \[ \frac{365}{322102} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{4 \,{\left (432 \, x - 293\right )}}{43923 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} + \frac{5 \,{\left (35 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 79 \, \sqrt{-2 \, x + 1}\right )}}{5324 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

365/322102*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sq
rt(-2*x + 1))) + 4/43923*(432*x - 293)/((2*x - 1)*sqrt(-2*x + 1)) + 5/5324*(35*(
-2*x + 1)^(3/2) - 79*sqrt(-2*x + 1))/(5*x + 3)^2