3.2157 \(\int \frac{2+3 x}{(1-2 x)^{5/2} (3+5 x)} \, dx\)

Optimal. Leaf size=56 \[ \frac{2}{121 \sqrt{1-2 x}}+\frac{7}{33 (1-2 x)^{3/2}}-\frac{2}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

7/(33*(1 - 2*x)^(3/2)) + 2/(121*Sqrt[1 - 2*x]) - (2*Sqrt[5/11]*ArcTanh[Sqrt[5/11
]*Sqrt[1 - 2*x]])/121

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Rubi [A]  time = 0.0695774, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2}{121 \sqrt{1-2 x}}+\frac{7}{33 (1-2 x)^{3/2}}-\frac{2}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

7/(33*(1 - 2*x)^(3/2)) + 2/(121*Sqrt[1 - 2*x]) - (2*Sqrt[5/11]*ArcTanh[Sqrt[5/11
]*Sqrt[1 - 2*x]])/121

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Rubi in Sympy [A]  time = 6.81272, size = 48, normalized size = 0.86 \[ - \frac{2 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{1331} + \frac{2}{121 \sqrt{- 2 x + 1}} + \frac{7}{33 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/1331 + 2/(121*sqrt(-2*x + 1)) + 7/
(33*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.0780064, size = 52, normalized size = 0.93 \[ -\frac{132 x+6 \sqrt{55} (1-2 x)^{3/2} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-913}{3993 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(-913 + 132*x + 6*Sqrt[55]*(1 - 2*x)^(3/2)*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(
3993*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.013, size = 38, normalized size = 0.7 \[{\frac{7}{33} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{2\,\sqrt{55}}{1331}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{2}{121}{\frac{1}{\sqrt{1-2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7/33/(1-2*x)^(3/2)-2/1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+2/121/(1
-2*x)^(1/2)

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Maxima [A]  time = 1.50383, size = 69, normalized size = 1.23 \[ \frac{1}{1331} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{12 \, x - 83}{363 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))
) - 1/363*(12*x - 83)/(-2*x + 1)^(3/2)

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Fricas [A]  time = 0.250135, size = 104, normalized size = 1.86 \[ \frac{\sqrt{11}{\left (3 \, \sqrt{5}{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{11}{\left (12 \, x - 83\right )}\right )}}{3993 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3993*sqrt(11)*(3*sqrt(5)*(2*x - 1)*sqrt(-2*x + 1)*log((sqrt(11)*(5*x - 8) + 11
*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) + sqrt(11)*(12*x - 83))/((2*x - 1)*sqrt(-2*x
+ 1))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{3 x + 2}{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((3*x + 2)/((-2*x + 1)**(5/2)*(5*x + 3)), x)

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GIAC/XCAS [A]  time = 0.22136, size = 82, normalized size = 1.46 \[ \frac{1}{1331} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{12 \, x - 83}{363 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1331*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-
2*x + 1))) + 1/363*(12*x - 83)/((2*x - 1)*sqrt(-2*x + 1))