3.2161 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)} \, dx\)

Optimal. Leaf size=125 \[ -\frac{12295}{41503 \sqrt{1-2 x}}+\frac{33}{14 (1-2 x)^{3/2} (3 x+2)}-\frac{1115}{1617 (1-2 x)^{3/2}}+\frac{3}{14 (1-2 x)^{3/2} (3 x+2)^2}+\frac{3645}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{1250}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

-1115/(1617*(1 - 2*x)^(3/2)) - 12295/(41503*Sqrt[1 - 2*x]) + 3/(14*(1 - 2*x)^(3/
2)*(2 + 3*x)^2) + 33/(14*(1 - 2*x)^(3/2)*(2 + 3*x)) + (3645*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]]
)/121

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Rubi [A]  time = 0.351474, antiderivative size = 125, 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 \[ -\frac{12295}{41503 \sqrt{1-2 x}}+\frac{33}{14 (1-2 x)^{3/2} (3 x+2)}-\frac{1115}{1617 (1-2 x)^{3/2}}+\frac{3}{14 (1-2 x)^{3/2} (3 x+2)^2}+\frac{3645}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{1250}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-1115/(1617*(1 - 2*x)^(3/2)) - 12295/(41503*Sqrt[1 - 2*x]) + 3/(14*(1 - 2*x)^(3/
2)*(2 + 3*x)^2) + 33/(14*(1 - 2*x)^(3/2)*(2 + 3*x)) + (3645*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]]
)/121

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Rubi in Sympy [A]  time = 35.6632, size = 109, normalized size = 0.87 \[ \frac{3645 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{2401} - \frac{1250 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{1331} - \frac{12295}{41503 \sqrt{- 2 x + 1}} - \frac{1115}{1617 \left (- 2 x + 1\right )^{\frac{3}{2}}} + \frac{33}{14 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )} + \frac{3}{14 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3645*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/2401 - 1250*sqrt(55)*atanh(sqrt(5
5)*sqrt(-2*x + 1)/11)/1331 - 12295/(41503*sqrt(-2*x + 1)) - 1115/(1617*(-2*x + 1
)**(3/2)) + 33/(14*(-2*x + 1)**(3/2)*(3*x + 2)) + 3/(14*(-2*x + 1)**(3/2)*(3*x +
2)**2)

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Mathematica [A]  time = 0.166521, size = 97, normalized size = 0.78 \[ \frac{\sqrt{1-2 x} \left (1327860 x^3-438840 x^2-594687 x+245383\right )}{249018 \left (6 x^2+x-2\right )^2}+\frac{3645}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{1250}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(245383 - 594687*x - 438840*x^2 + 1327860*x^3))/(249018*(-2 + x +
6*x^2)^2) + (3645*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (1250*Sqrt[
5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Maple [A]  time = 0.023, size = 84, normalized size = 0.7 \[{\frac{16}{11319} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{2144}{290521}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{486}{2401\, \left ( -4-6\,x \right ) ^{2}} \left ({\frac{27}{2} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{581}{18}\sqrt{1-2\,x}} \right ) }+{\frac{3645\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{1250\,\sqrt{55}}{1331}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

16/11319/(1-2*x)^(3/2)+2144/290521/(1-2*x)^(1/2)-486/2401*(27/2*(1-2*x)^(3/2)-58
1/18*(1-2*x)^(1/2))/(-4-6*x)^2+3645/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^
(1/2)-1250/1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.503, size = 173, normalized size = 1.38 \[ \frac{625}{1331} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{3645}{4802} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{331965 \,{\left (2 \, x - 1\right )}^{3} + 776475 \,{\left (2 \, x - 1\right )}^{2} - 75264 \, x + 46256}{124509 \,{\left (9 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 42 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 49 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

625/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1
))) - 3645/4802*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-
2*x + 1))) + 1/124509*(331965*(2*x - 1)^3 + 776475*(2*x - 1)^2 - 75264*x + 46256
)/(9*(-2*x + 1)^(7/2) - 42*(-2*x + 1)^(5/2) + 49*(-2*x + 1)^(3/2))

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Fricas [A]  time = 0.230846, size = 240, normalized size = 1.92 \[ \frac{\sqrt{11} \sqrt{7}{\left (1286250 \, \sqrt{7} \sqrt{5}{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 1323135 \, \sqrt{11} \sqrt{3}{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} - 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{11} \sqrt{7}{\left (1327860 \, x^{3} - 438840 \, x^{2} - 594687 \, x + 245383\right )}\right )}}{19174386 \,{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/19174386*sqrt(11)*sqrt(7)*(1286250*sqrt(7)*sqrt(5)*(18*x^3 + 15*x^2 - 4*x - 4)
*sqrt(-2*x + 1)*log((sqrt(11)*(5*x - 8) + 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3))
+ 1323135*sqrt(11)*sqrt(3)*(18*x^3 + 15*x^2 - 4*x - 4)*sqrt(-2*x + 1)*log((sqrt(
7)*(3*x - 5) - 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2)) - sqrt(11)*sqrt(7)*(1327860*
x^3 - 438840*x^2 - 594687*x + 245383))/((18*x^3 + 15*x^2 - 4*x - 4)*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)**(5/2)/(2+3*x)**3/(3+5*x),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.239316, size = 173, normalized size = 1.38 \[ \frac{625}{1331} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{3645}{4802} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{16 \,{\left (804 \, x - 479\right )}}{871563 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} - \frac{27 \,{\left (243 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 581 \, \sqrt{-2 \, x + 1}\right )}}{9604 \,{\left (3 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

625/1331*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt
(-2*x + 1))) - 3645/4802*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sq
rt(21) + 3*sqrt(-2*x + 1))) + 16/871563*(804*x - 479)/((2*x - 1)*sqrt(-2*x + 1))
- 27/9604*(243*(-2*x + 1)^(3/2) - 581*sqrt(-2*x + 1))/(3*x + 2)^2