3.2168 \(\int \frac{2+3 x}{(1-2 x)^{5/2} (3+5 x)^2} \, dx\)

Optimal. Leaf size=76 \[ \frac{76}{1331 \sqrt{1-2 x}}-\frac{1}{55 (1-2 x)^{3/2} (5 x+3)}+\frac{76}{1815 (1-2 x)^{3/2}}-\frac{76 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]

[Out]

76/(1815*(1 - 2*x)^(3/2)) + 76/(1331*Sqrt[1 - 2*x]) - 1/(55*(1 - 2*x)^(3/2)*(3 +
5*x)) - (76*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1331

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Rubi [A]  time = 0.0875819, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{76}{1331 \sqrt{1-2 x}}-\frac{1}{55 (1-2 x)^{3/2} (5 x+3)}+\frac{76}{1815 (1-2 x)^{3/2}}-\frac{76 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]

Antiderivative was successfully verified.

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

[Out]

76/(1815*(1 - 2*x)^(3/2)) + 76/(1331*Sqrt[1 - 2*x]) - 1/(55*(1 - 2*x)^(3/2)*(3 +
5*x)) - (76*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1331

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Rubi in Sympy [A]  time = 8.29191, size = 65, normalized size = 0.86 \[ - \frac{76 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{14641} + \frac{76}{1331 \sqrt{- 2 x + 1}} + \frac{76}{1815 \left (- 2 x + 1\right )^{\frac{3}{2}}} - \frac{1}{55 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-76*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/14641 + 76/(1331*sqrt(-2*x + 1))
+ 76/(1815*(-2*x + 1)**(3/2)) - 1/(55*(-2*x + 1)**(3/2)*(5*x + 3))

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Mathematica [A]  time = 0.111674, size = 58, normalized size = 0.76 \[ \frac{\frac{11 \left (-2280 x^2+608 x+1113\right )}{(1-2 x)^{3/2} (5 x+3)}-228 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{43923} \]

Antiderivative was successfully verified.

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

[Out]

((11*(1113 + 608*x - 2280*x^2))/((1 - 2*x)^(3/2)*(3 + 5*x)) - 228*Sqrt[55]*ArcTa
nh[Sqrt[5/11]*Sqrt[1 - 2*x]])/43923

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Maple [A]  time = 0.019, size = 54, normalized size = 0.7 \[{\frac{14}{363} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{74}{1331}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{2}{1331}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{76\,\sqrt{55}}{14641}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

14/363/(1-2*x)^(3/2)+74/1331/(1-2*x)^(1/2)+2/1331*(1-2*x)^(1/2)/(-6/5-2*x)-76/14
641*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.49684, size = 100, normalized size = 1.32 \[ \frac{38}{14641} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2 \,{\left (570 \,{\left (2 \, x - 1\right )}^{2} + 1672 \, x - 1683\right )}}{3993 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 11 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

38/14641*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1
))) + 2/3993*(570*(2*x - 1)^2 + 1672*x - 1683)/(5*(-2*x + 1)^(5/2) - 11*(-2*x +
1)^(3/2))

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Fricas [A]  time = 0.214208, size = 119, normalized size = 1.57 \[ \frac{\sqrt{11}{\left (114 \, \sqrt{5}{\left (10 \, x^{2} + x - 3\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{11}{\left (2280 \, x^{2} - 608 \, x - 1113\right )}\right )}}{43923 \,{\left (10 \, x^{2} + x - 3\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/43923*sqrt(11)*(114*sqrt(5)*(10*x^2 + x - 3)*sqrt(-2*x + 1)*log((sqrt(11)*(5*x
- 8) + 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) + sqrt(11)*(2280*x^2 - 608*x - 111
3))/((10*x^2 + x - 3)*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)**2,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.222182, size = 104, normalized size = 1.37 \[ \frac{38}{14641} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{4 \,{\left (111 \, x - 94\right )}}{3993 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} - \frac{5 \, \sqrt{-2 \, x + 1}}{1331 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

38/14641*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt
(-2*x + 1))) + 4/3993*(111*x - 94)/((2*x - 1)*sqrt(-2*x + 1)) - 5/1331*sqrt(-2*x
+ 1)/(5*x + 3)