3.2171 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^2} \, dx\)

Optimal. Leaf size=132 \[ \frac{159800}{456533 \sqrt{1-2 x}}+\frac{3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)}-\frac{340}{77 (1-2 x)^{3/2} (5 x+3)}+\frac{13900}{17787 (1-2 x)^{3/2}}-\frac{4050}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{15250 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]

[Out]

13900/(17787*(1 - 2*x)^(3/2)) + 159800/(456533*Sqrt[1 - 2*x]) - 340/(77*(1 - 2*x
)^(3/2)*(3 + 5*x)) + 3/(7*(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)) - (4050*Sqrt[3/7]
*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 + (15250*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sq
rt[1 - 2*x]])/1331

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Rubi [A]  time = 0.344383, 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 \[ \frac{159800}{456533 \sqrt{1-2 x}}+\frac{3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)}-\frac{340}{77 (1-2 x)^{3/2} (5 x+3)}+\frac{13900}{17787 (1-2 x)^{3/2}}-\frac{4050}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{15250 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]

Antiderivative was successfully verified.

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

[Out]

13900/(17787*(1 - 2*x)^(3/2)) + 159800/(456533*Sqrt[1 - 2*x]) - 340/(77*(1 - 2*x
)^(3/2)*(3 + 5*x)) + 3/(7*(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)) - (4050*Sqrt[3/7]
*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 + (15250*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sq
rt[1 - 2*x]])/1331

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Rubi in Sympy [A]  time = 34.7321, size = 112, normalized size = 0.85 \[ - \frac{4050 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{2401} + \frac{15250 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{14641} + \frac{159800}{456533 \sqrt{- 2 x + 1}} + \frac{13900}{17787 \left (- 2 x + 1\right )^{\frac{3}{2}}} - \frac{204}{77 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )} - \frac{5}{11 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right ) \left (5 x + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4050*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/2401 + 15250*sqrt(55)*atanh(sqrt
(55)*sqrt(-2*x + 1)/11)/14641 + 159800/(456533*sqrt(-2*x + 1)) + 13900/(17787*(-
2*x + 1)**(3/2)) - 204/(77*(-2*x + 1)**(3/2)*(3*x + 2)) - 5/(11*(-2*x + 1)**(3/2
)*(3*x + 2)*(5*x + 3))

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Mathematica [A]  time = 0.181032, size = 101, normalized size = 0.77 \[ \frac{-14382000 x^3+5028300 x^2+5548760 x-2209989}{1369599 (1-2 x)^{3/2} (3 x+2) (5 x+3)}-\frac{4050}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{15250 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]

Antiderivative was successfully verified.

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

[Out]

(-2209989 + 5548760*x + 5028300*x^2 - 14382000*x^3)/(1369599*(1 - 2*x)^(3/2)*(2
+ 3*x)*(3 + 5*x)) - (4050*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 + (152
50*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1331

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Maple [A]  time = 0.026, size = 88, normalized size = 0.7 \[{\frac{16}{17787} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{2176}{456533}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{54}{343}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}-{\frac{4050\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{250}{1331}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}+{\frac{15250\,\sqrt{55}}{14641}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

16/17787/(1-2*x)^(3/2)+2176/456533/(1-2*x)^(1/2)+54/343*(1-2*x)^(1/2)/(-4/3-2*x)
-4050/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+250/1331*(1-2*x)^(1/2)/(
-6/5-2*x)+15250/14641*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.51464, size = 173, normalized size = 1.31 \[ -\frac{7625}{14641} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2025}{2401} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{4 \,{\left (1797750 \,{\left (2 \, x - 1\right )}^{3} + 4136175 \,{\left (2 \, x - 1\right )}^{2} + 209440 \, x - 128436\right )}}{1369599 \,{\left (15 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 68 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 77 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7625/14641*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x
+ 1))) + 2025/2401*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqr
t(-2*x + 1))) - 4/1369599*(1797750*(2*x - 1)^3 + 4136175*(2*x - 1)^2 + 209440*x
- 128436)/(15*(-2*x + 1)^(7/2) - 68*(-2*x + 1)^(5/2) + 77*(-2*x + 1)^(3/2))

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Fricas [A]  time = 0.23213, size = 239, normalized size = 1.81 \[ \frac{\sqrt{11} \sqrt{7}{\left (7846125 \, \sqrt{7} \sqrt{5}{\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} - 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 8085825 \, \sqrt{11} \sqrt{3}{\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} + 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{11} \sqrt{7}{\left (14382000 \, x^{3} - 5028300 \, x^{2} - 5548760 \, x + 2209989\right )}\right )}}{105459123 \,{\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/105459123*sqrt(11)*sqrt(7)*(7846125*sqrt(7)*sqrt(5)*(30*x^3 + 23*x^2 - 7*x - 6
)*sqrt(-2*x + 1)*log((sqrt(11)*(5*x - 8) - 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3))
+ 8085825*sqrt(11)*sqrt(3)*(30*x^3 + 23*x^2 - 7*x - 6)*sqrt(-2*x + 1)*log((sqrt
(7)*(3*x - 5) + 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2)) + sqrt(11)*sqrt(7)*(1438200
0*x^3 - 5028300*x^2 - 5548760*x + 2209989))/((30*x^3 + 23*x^2 - 7*x - 6)*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)**2/(3+5*x)**2,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.21977, size = 185, normalized size = 1.4 \[ -\frac{7625}{14641} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{2025}{2401} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{4 \,{\left (591090 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1343273 \, \sqrt{-2 \, x + 1}\right )}}{456533 \,{\left (15 \,{\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}} + \frac{16 \,{\left (816 \, x - 485\right )}}{1369599 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7625/14641*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*s
qrt(-2*x + 1))) + 2025/2401*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/
(sqrt(21) + 3*sqrt(-2*x + 1))) + 4/456533*(591090*(-2*x + 1)^(3/2) - 1343273*sqr
t(-2*x + 1))/(15*(2*x - 1)^2 + 136*x + 9) + 16/1369599*(816*x - 485)/((2*x - 1)*
sqrt(-2*x + 1))