3.2105 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx\)

Optimal. Leaf size=92 \[ \frac{78}{847 \sqrt{1-2 x}}-\frac{5}{11 \sqrt{1-2 x} (5 x+3)}-\frac{18}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{300}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

78/(847*Sqrt[1 - 2*x]) - 5/(11*Sqrt[1 - 2*x]*(3 + 5*x)) - (18*Sqrt[3/7]*ArcTanh[
Sqrt[3/7]*Sqrt[1 - 2*x]])/7 + (300*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])
/121

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Rubi [A]  time = 0.210876, antiderivative size = 92, 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 \[ \frac{78}{847 \sqrt{1-2 x}}-\frac{5}{11 \sqrt{1-2 x} (5 x+3)}-\frac{18}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{300}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

78/(847*Sqrt[1 - 2*x]) - 5/(11*Sqrt[1 - 2*x]*(3 + 5*x)) - (18*Sqrt[3/7]*ArcTanh[
Sqrt[3/7]*Sqrt[1 - 2*x]])/7 + (300*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])
/121

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Rubi in Sympy [A]  time = 21.5036, size = 78, normalized size = 0.85 \[ - \frac{18 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{49} + \frac{300 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{1331} + \frac{78}{847 \sqrt{- 2 x + 1}} - \frac{5}{11 \sqrt{- 2 x + 1} \left (5 x + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-18*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/49 + 300*sqrt(55)*atanh(sqrt(55)*s
qrt(-2*x + 1)/11)/1331 + 78/(847*sqrt(-2*x + 1)) - 5/(11*sqrt(-2*x + 1)*(5*x + 3
))

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Mathematica [A]  time = 0.251795, size = 86, normalized size = 0.93 \[ \frac{\frac{11 \sqrt{1-2 x} (151-390 x)}{10 x^2+x-3}+2100 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{9317}-\frac{18}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-18*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 + ((11*(151 - 390*x)*Sqrt[1 -
2*x])/(-3 + x + 10*x^2) + 2100*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/9317

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Maple [A]  time = 0.02, size = 63, normalized size = 0.7 \[{\frac{8}{847}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{18\,\sqrt{21}}{49}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{10}{121}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}+{\frac{300\,\sqrt{55}}{1331}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8/847/(1-2*x)^(1/2)-18/49*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+10/121*(1
-2*x)^(1/2)/(-6/5-2*x)+300/1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.50086, size = 136, normalized size = 1.48 \[ -\frac{150}{1331} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{9}{49} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2 \,{\left (390 \, x - 151\right )}}{847 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-150/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x +
1))) + 9/49*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) - 2/847*(390*x - 151)/(5*(-2*x + 1)^(3/2) - 11*sqrt(-2*x + 1))

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Fricas [A]  time = 0.242311, size = 185, normalized size = 2.01 \[ \frac{\sqrt{11} \sqrt{7}{\left (1050 \, \sqrt{7} \sqrt{5}{\left (5 \, x + 3\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} - 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 1089 \, \sqrt{11} \sqrt{3}{\left (5 \, x + 3\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} + 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{11} \sqrt{7}{\left (390 \, x - 151\right )}\right )}}{65219 \,{\left (5 \, x + 3\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/65219*sqrt(11)*sqrt(7)*(1050*sqrt(7)*sqrt(5)*(5*x + 3)*sqrt(-2*x + 1)*log((sqr
t(11)*(5*x - 8) - 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) + 1089*sqrt(11)*sqrt(3)*
(5*x + 3)*sqrt(-2*x + 1)*log((sqrt(7)*(3*x - 5) + 7*sqrt(3)*sqrt(-2*x + 1))/(3*x
+ 2)) + sqrt(11)*sqrt(7)*(390*x - 151))/((5*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(1/(1-2*x)**(3/2)/(2+3*x)/(3+5*x)**2,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.221161, size = 144, normalized size = 1.57 \[ -\frac{150}{1331} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{9}{49} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{2 \,{\left (390 \, x - 151\right )}}{847 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-150/1331*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqr
t(-2*x + 1))) + 9/49*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(2
1) + 3*sqrt(-2*x + 1))) - 2/847*(390*x - 151)/(5*(-2*x + 1)^(3/2) - 11*sqrt(-2*x
+ 1))