3.2097 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x)^4 (3+5 x)} \, dx\)

Optimal. Leaf size=132 \[ -\frac{12790}{3773 \sqrt{1-2 x}}+\frac{565}{49 \sqrt{1-2 x} (3 x+2)}+\frac{8}{7 \sqrt{1-2 x} (3 x+2)^2}+\frac{1}{7 \sqrt{1-2 x} (3 x+2)^3}+\frac{40140}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{1250}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

-12790/(3773*Sqrt[1 - 2*x]) + 1/(7*Sqrt[1 - 2*x]*(2 + 3*x)^3) + 8/(7*Sqrt[1 - 2*
x]*(2 + 3*x)^2) + 565/(49*Sqrt[1 - 2*x]*(2 + 3*x)) + (40140*Sqrt[3/7]*ArcTanh[Sq
rt[3/7]*Sqrt[1 - 2*x]])/343 - (1250*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
)/11

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Rubi [A]  time = 0.350243, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{12790}{3773 \sqrt{1-2 x}}+\frac{565}{49 \sqrt{1-2 x} (3 x+2)}+\frac{8}{7 \sqrt{1-2 x} (3 x+2)^2}+\frac{1}{7 \sqrt{1-2 x} (3 x+2)^3}+\frac{40140}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{1250}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/((1 - 2*x)^(3/2)*(2 + 3*x)^4*(3 + 5*x)),x]

[Out]

-12790/(3773*Sqrt[1 - 2*x]) + 1/(7*Sqrt[1 - 2*x]*(2 + 3*x)^3) + 8/(7*Sqrt[1 - 2*
x]*(2 + 3*x)^2) + 565/(49*Sqrt[1 - 2*x]*(2 + 3*x)) + (40140*Sqrt[3/7]*ArcTanh[Sq
rt[3/7]*Sqrt[1 - 2*x]])/343 - (1250*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
)/11

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Rubi in Sympy [A]  time = 34.3174, size = 116, normalized size = 0.88 \[ \frac{40140 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{2401} - \frac{1250 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{121} - \frac{12790}{3773 \sqrt{- 2 x + 1}} + \frac{565}{49 \sqrt{- 2 x + 1} \left (3 x + 2\right )} + \frac{8}{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}} + \frac{1}{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1-2*x)**(3/2)/(2+3*x)**4/(3+5*x),x)

[Out]

40140*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/2401 - 1250*sqrt(55)*atanh(sqrt(
55)*sqrt(-2*x + 1)/11)/121 - 12790/(3773*sqrt(-2*x + 1)) + 565/(49*sqrt(-2*x + 1
)*(3*x + 2)) + 8/(7*sqrt(-2*x + 1)*(3*x + 2)**2) + 1/(7*sqrt(-2*x + 1)*(3*x + 2)
**3)

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Mathematica [A]  time = 0.17226, size = 94, normalized size = 0.71 \[ \frac{-345330 x^3-299115 x^2+74556 x+80863}{3773 \sqrt{1-2 x} (3 x+2)^3}+\frac{40140}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{1250}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((1 - 2*x)^(3/2)*(2 + 3*x)^4*(3 + 5*x)),x]

[Out]

(80863 + 74556*x - 299115*x^2 - 345330*x^3)/(3773*Sqrt[1 - 2*x]*(2 + 3*x)^3) + (
40140*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (1250*Sqrt[5/11]*ArcTanh
[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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Maple [A]  time = 0.021, size = 84, normalized size = 0.6 \[{\frac{32}{26411}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{486}{2401\, \left ( -4-6\,x \right ) ^{3}} \left ({\frac{1357}{3} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{57806}{27} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{68453}{27}\sqrt{1-2\,x}} \right ) }+{\frac{40140\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{1250\,\sqrt{55}}{121}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-2*x)^(3/2)/(2+3*x)^4/(3+5*x),x)

[Out]

32/26411/(1-2*x)^(1/2)-486/2401*(1357/3*(1-2*x)^(5/2)-57806/27*(1-2*x)^(3/2)+684
53/27*(1-2*x)^(1/2))/(-4-6*x)^3+40140/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*2
1^(1/2)-1250/121*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.5193, size = 185, normalized size = 1.4 \[ \frac{625}{121} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{20070}{2401} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2 \,{\left (172665 \,{\left (2 \, x - 1\right )}^{3} + 817110 \,{\left (2 \, x - 1\right )}^{2} + 1934226 \, x - 967897\right )}}{3773 \,{\left (27 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 189 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 441 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 343 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((5*x + 3)*(3*x + 2)^4*(-2*x + 1)^(3/2)),x, algorithm="maxima")

[Out]

625/121*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)
)) - 20070/2401*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-
2*x + 1))) + 2/3773*(172665*(2*x - 1)^3 + 817110*(2*x - 1)^2 + 1934226*x - 96789
7)/(27*(-2*x + 1)^(7/2) - 189*(-2*x + 1)^(5/2) + 441*(-2*x + 1)^(3/2) - 343*sqrt
(-2*x + 1))

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Fricas [A]  time = 0.267547, size = 240, normalized size = 1.82 \[ \frac{\sqrt{11} \sqrt{7}{\left (214375 \, \sqrt{7} \sqrt{5}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 220770 \, \sqrt{11} \sqrt{3}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} - 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{11} \sqrt{7}{\left (345330 \, x^{3} + 299115 \, x^{2} - 74556 \, x - 80863\right )}\right )}}{290521 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((5*x + 3)*(3*x + 2)^4*(-2*x + 1)^(3/2)),x, algorithm="fricas")

[Out]

1/290521*sqrt(11)*sqrt(7)*(214375*sqrt(7)*sqrt(5)*(27*x^3 + 54*x^2 + 36*x + 8)*s
qrt(-2*x + 1)*log((sqrt(11)*(5*x - 8) + 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) +
220770*sqrt(11)*sqrt(3)*(27*x^3 + 54*x^2 + 36*x + 8)*sqrt(-2*x + 1)*log((sqrt(7)
*(3*x - 5) - 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2)) - sqrt(11)*sqrt(7)*(345330*x^3
+ 299115*x^2 - 74556*x - 80863))/((27*x^3 + 54*x^2 + 36*x + 8)*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)**4/(3+5*x),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.234571, size = 178, normalized size = 1.35 \[ \frac{625}{121} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{20070}{2401} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{32}{26411 \, \sqrt{-2 \, x + 1}} + \frac{9 \,{\left (12213 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 57806 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 68453 \, \sqrt{-2 \, x + 1}\right )}}{9604 \,{\left (3 \, x + 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((5*x + 3)*(3*x + 2)^4*(-2*x + 1)^(3/2)),x, algorithm="giac")

[Out]

625/121*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(
-2*x + 1))) - 20070/2401*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sq
rt(21) + 3*sqrt(-2*x + 1))) + 32/26411/sqrt(-2*x + 1) + 9/9604*(12213*(2*x - 1)^
2*sqrt(-2*x + 1) - 57806*(-2*x + 1)^(3/2) + 68453*sqrt(-2*x + 1))/(3*x + 2)^3