3.2160 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)} \, dx\)

Optimal. Leaf size=105 \[ -\frac{1370}{41503 \sqrt{1-2 x}}+\frac{3}{7 (1-2 x)^{3/2} (3 x+2)}-\frac{190}{1617 (1-2 x)^{3/2}}+\frac{720}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{250}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

-190/(1617*(1 - 2*x)^(3/2)) - 1370/(41503*Sqrt[1 - 2*x]) + 3/(7*(1 - 2*x)^(3/2)*
(2 + 3*x)) + (720*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (250*Sqrt[5/
11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Rubi [A]  time = 0.281731, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{1370}{41503 \sqrt{1-2 x}}+\frac{3}{7 (1-2 x)^{3/2} (3 x+2)}-\frac{190}{1617 (1-2 x)^{3/2}}+\frac{720}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{250}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-190/(1617*(1 - 2*x)^(3/2)) - 1370/(41503*Sqrt[1 - 2*x]) + 3/(7*(1 - 2*x)^(3/2)*
(2 + 3*x)) + (720*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (250*Sqrt[5/
11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Rubi in Sympy [A]  time = 28.6037, size = 90, normalized size = 0.86 \[ \frac{720 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{2401} - \frac{250 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{1331} - \frac{1370}{41503 \sqrt{- 2 x + 1}} - \frac{190}{1617 \left (- 2 x + 1\right )^{\frac{3}{2}}} + \frac{3}{7 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )} \]

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),x)

[Out]

720*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/2401 - 250*sqrt(55)*atanh(sqrt(55)
*sqrt(-2*x + 1)/11)/1331 - 1370/(41503*sqrt(-2*x + 1)) - 190/(1617*(-2*x + 1)**(
3/2)) + 3/(7*(-2*x + 1)**(3/2)*(3*x + 2))

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Mathematica [A]  time = 0.282215, size = 105, normalized size = 1. \[ \frac{11 \left (24660 x^2-39780 x+15881\right )+257250 \sqrt{55} \sqrt{1-2 x} \left (6 x^2+x-2\right ) \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1369599 (1-2 x)^{3/2} (3 x+2)}+\frac{720}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(720*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 + (11*(15881 - 39780*x + 24
660*x^2) + 257250*Sqrt[55]*Sqrt[1 - 2*x]*(-2 + x + 6*x^2)*ArcTanh[Sqrt[5/11]*Sqr
t[1 - 2*x]])/(1369599*(1 - 2*x)^(3/2)*(2 + 3*x))

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Maple [A]  time = 0.022, size = 72, normalized size = 0.7 \[{\frac{8}{1617} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{808}{41503}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{18}{343}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}+{\frac{720\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{250\,\sqrt{55}}{1331}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8/1617/(1-2*x)^(3/2)+808/41503/(1-2*x)^(1/2)-18/343*(1-2*x)^(1/2)/(-4/3-2*x)+720
/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-250/1331*arctanh(1/11*55^(1/2
)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.507, size = 149, normalized size = 1.42 \[ \frac{125}{1331} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{360}{2401} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2 \,{\left (6165 \,{\left (2 \, x - 1\right )}^{2} - 15120 \, x + 9716\right )}}{124509 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 7 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1
))) - 360/2401*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2
*x + 1))) - 2/124509*(6165*(2*x - 1)^2 - 15120*x + 9716)/(3*(-2*x + 1)^(5/2) - 7
*(-2*x + 1)^(3/2))

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Fricas [A]  time = 0.229615, size = 205, normalized size = 1.95 \[ \frac{\sqrt{11} \sqrt{7}{\left (128625 \, \sqrt{7} \sqrt{5}{\left (6 \, x^{2} + x - 2\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 130680 \, \sqrt{11} \sqrt{3}{\left (6 \, x^{2} + x - 2\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} - 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{11} \sqrt{7}{\left (24660 \, x^{2} - 39780 \, x + 15881\right )}\right )}}{9587193 \,{\left (6 \, x^{2} + x - 2\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9587193*sqrt(11)*sqrt(7)*(128625*sqrt(7)*sqrt(5)*(6*x^2 + x - 2)*sqrt(-2*x + 1
)*log((sqrt(11)*(5*x - 8) + 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) + 130680*sqrt(
11)*sqrt(3)*(6*x^2 + x - 2)*sqrt(-2*x + 1)*log((sqrt(7)*(3*x - 5) - 7*sqrt(3)*sq
rt(-2*x + 1))/(3*x + 2)) - sqrt(11)*sqrt(7)*(24660*x^2 - 39780*x + 15881))/((6*x
^2 + x - 2)*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),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.217633, size = 157, normalized size = 1.5 \[ \frac{125}{1331} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{360}{2401} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{16 \,{\left (303 \, x - 190\right )}}{124509 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} + \frac{27 \, \sqrt{-2 \, x + 1}}{343 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125/1331*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt
(-2*x + 1))) - 360/2401*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqr
t(21) + 3*sqrt(-2*x + 1))) + 16/124509*(303*x - 190)/((2*x - 1)*sqrt(-2*x + 1))
+ 27/343*sqrt(-2*x + 1)/(3*x + 2)