3.2115 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^3} \, dx\)

Optimal. Leaf size=112 \[ -\frac{2049}{9317 \sqrt{1-2 x}}+\frac{305}{242 \sqrt{1-2 x} (5 x+3)}-\frac{5}{22 \sqrt{1-2 x} (5 x+3)^2}+\frac{54}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{9975 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]

[Out]

-2049/(9317*Sqrt[1 - 2*x]) - 5/(22*Sqrt[1 - 2*x]*(3 + 5*x)^2) + 305/(242*Sqrt[1
- 2*x]*(3 + 5*x)) + (54*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 - (9975*Sq
rt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1331

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Rubi [A]  time = 0.280695, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{2049}{9317 \sqrt{1-2 x}}+\frac{305}{242 \sqrt{1-2 x} (5 x+3)}-\frac{5}{22 \sqrt{1-2 x} (5 x+3)^2}+\frac{54}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{9975 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]

Antiderivative was successfully verified.

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

[Out]

-2049/(9317*Sqrt[1 - 2*x]) - 5/(22*Sqrt[1 - 2*x]*(3 + 5*x)^2) + 305/(242*Sqrt[1
- 2*x]*(3 + 5*x)) + (54*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 - (9975*Sq
rt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1331

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Rubi in Sympy [A]  time = 27.9507, size = 97, normalized size = 0.87 \[ \frac{54 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{49} - \frac{9975 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{14641} - \frac{2049}{9317 \sqrt{- 2 x + 1}} + \frac{305}{242 \sqrt{- 2 x + 1} \left (5 x + 3\right )} - \frac{5}{22 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{2}} \]

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)**3,x)

[Out]

54*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/49 - 9975*sqrt(55)*atanh(sqrt(55)*s
qrt(-2*x + 1)/11)/14641 - 2049/(9317*sqrt(-2*x + 1)) + 305/(242*sqrt(-2*x + 1)*(
5*x + 3)) - 5/(22*sqrt(-2*x + 1)*(5*x + 3)**2)

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Mathematica [A]  time = 0.39825, size = 88, normalized size = 0.79 \[ \frac{-\frac{11 \left (102450 x^2+5515 x-29338\right )}{\sqrt{1-2 x} (5 x+3)^2}-139650 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{204974}+\frac{54}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(54*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 + ((-11*(-29338 + 5515*x + 102
450*x^2))/(Sqrt[1 - 2*x]*(3 + 5*x)^2) - 139650*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[
1 - 2*x]])/204974

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Maple [A]  time = 0.02, size = 75, normalized size = 0.7 \[{\frac{16}{9317}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{54\,\sqrt{21}}{49}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{1250}{1331\, \left ( -6-10\,x \right ) ^{2}} \left ( -{\frac{59}{10} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{627}{50}\sqrt{1-2\,x}} \right ) }-{\frac{9975\,\sqrt{55}}{14641}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

16/9317/(1-2*x)^(1/2)+54/49*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+1250/13
31*(-59/10*(1-2*x)^(3/2)+627/50*(1-2*x)^(1/2))/(-6-10*x)^2-9975/14641*arctanh(1/
11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.50306, size = 161, normalized size = 1.44 \[ \frac{9975}{29282} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{27}{49} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{51225 \,{\left (2 \, x - 1\right )}^{2} + 215930 \, x - 109901}{9317 \,{\left (25 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 110 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 121 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9975/29282*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x +
1))) - 27/49*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*
x + 1))) - 1/9317*(51225*(2*x - 1)^2 + 215930*x - 109901)/(25*(-2*x + 1)^(5/2) -
110*(-2*x + 1)^(3/2) + 121*sqrt(-2*x + 1))

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Fricas [A]  time = 0.243497, size = 213, normalized size = 1.9 \[ \frac{\sqrt{11} \sqrt{7}{\left (69825 \, \sqrt{7} \sqrt{5}{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 71874 \, \sqrt{11} \sqrt{3}{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} - 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{11} \sqrt{7}{\left (102450 \, x^{2} + 5515 \, x - 29338\right )}\right )}}{1434818 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1434818*sqrt(11)*sqrt(7)*(69825*sqrt(7)*sqrt(5)*(25*x^2 + 30*x + 9)*sqrt(-2*x
+ 1)*log((sqrt(11)*(5*x - 8) + 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) + 71874*sqr
t(11)*sqrt(3)*(25*x^2 + 30*x + 9)*sqrt(-2*x + 1)*log((sqrt(7)*(3*x - 5) - 7*sqrt
(3)*sqrt(-2*x + 1))/(3*x + 2)) - sqrt(11)*sqrt(7)*(102450*x^2 + 5515*x - 29338))
/((25*x^2 + 30*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(1/(1-2*x)**(3/2)/(2+3*x)/(3+5*x)**3,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.227042, size = 157, normalized size = 1.4 \[ \frac{9975}{29282} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{27}{49} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{16}{9317 \, \sqrt{-2 \, x + 1}} - \frac{25 \,{\left (295 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 627 \, \sqrt{-2 \, x + 1}\right )}}{5324 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9975/29282*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sq
rt(-2*x + 1))) - 27/49*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt
(21) + 3*sqrt(-2*x + 1))) + 16/9317/sqrt(-2*x + 1) - 25/5324*(295*(-2*x + 1)^(3/
2) - 627*sqrt(-2*x + 1))/(5*x + 3)^2