3.2182 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^3} \, dx\)

Optimal. Leaf size=179 \[ -\frac{77527480}{5021863 \sqrt{1-2 x}}+\frac{167960}{847 (1-2 x)^{3/2} (5 x+3)}-\frac{6845810}{195657 (1-2 x)^{3/2}}+\frac{9}{2 (1-2 x)^{3/2} (3 x+2) (5 x+3)^2}-\frac{5165}{154 (1-2 x)^{3/2} (5 x+3)^2}+\frac{3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^2}+\frac{182655}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{7570625 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{14641} \]

[Out]

-6845810/(195657*(1 - 2*x)^(3/2)) - 77527480/(5021863*Sqrt[1 - 2*x]) - 5165/(154
*(1 - 2*x)^(3/2)*(3 + 5*x)^2) + 3/(14*(1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^2) +
9/(2*(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)^2) + 167960/(847*(1 - 2*x)^(3/2)*(3 +
5*x)) + (182655*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (7570625*Sqrt[
5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/14641

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Rubi [A]  time = 0.485601, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{77527480}{5021863 \sqrt{1-2 x}}+\frac{167960}{847 (1-2 x)^{3/2} (5 x+3)}-\frac{6845810}{195657 (1-2 x)^{3/2}}+\frac{9}{2 (1-2 x)^{3/2} (3 x+2) (5 x+3)^2}-\frac{5165}{154 (1-2 x)^{3/2} (5 x+3)^2}+\frac{3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^2}+\frac{182655}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{7570625 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{14641} \]

Antiderivative was successfully verified.

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

[Out]

-6845810/(195657*(1 - 2*x)^(3/2)) - 77527480/(5021863*Sqrt[1 - 2*x]) - 5165/(154
*(1 - 2*x)^(3/2)*(3 + 5*x)^2) + 3/(14*(1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^2) +
9/(2*(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)^2) + 167960/(847*(1 - 2*x)^(3/2)*(3 +
5*x)) + (182655*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (7570625*Sqrt[
5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/14641

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Rubi in Sympy [A]  time = 50.5649, size = 158, normalized size = 0.88 \[ \frac{182655 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{2401} - \frac{7570625 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{161051} - \frac{77527480}{5021863 \sqrt{- 2 x + 1}} - \frac{6845810}{195657 \left (- 2 x + 1\right )^{\frac{3}{2}}} + \frac{100776}{847 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )} + \frac{19857}{1694 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2}} + \frac{625}{242 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2} \left (5 x + 3\right )} - \frac{5}{22 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{2}} \]

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)**3,x)

[Out]

182655*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/2401 - 7570625*sqrt(55)*atanh(s
qrt(55)*sqrt(-2*x + 1)/11)/161051 - 77527480/(5021863*sqrt(-2*x + 1)) - 6845810/
(195657*(-2*x + 1)**(3/2)) + 100776/(847*(-2*x + 1)**(3/2)*(3*x + 2)) + 19857/(1
694*(-2*x + 1)**(3/2)*(3*x + 2)**2) + 625/(242*(-2*x + 1)**(3/2)*(3*x + 2)**2*(5
*x + 3)) - 5/(22*(-2*x + 1)**(3/2)*(3*x + 2)**2*(5*x + 3)**2)

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Mathematica [A]  time = 0.253301, size = 111, normalized size = 0.62 \[ \frac{209324196000 x^5+188418548700 x^4-93885376440 x^3-99160158305 x^2+9944654283 x+13236365823}{30131178 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^2}+\frac{182655}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{7570625 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{14641} \]

Antiderivative was successfully verified.

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

[Out]

(13236365823 + 9944654283*x - 99160158305*x^2 - 93885376440*x^3 + 188418548700*x
^4 + 209324196000*x^5)/(30131178*(1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^2) + (182
655*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (7570625*Sqrt[5/11]*ArcTan
h[Sqrt[5/11]*Sqrt[1 - 2*x]])/14641

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Maple [A]  time = 0.027, size = 112, normalized size = 0.6 \[{\frac{64}{1369599} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{13056}{35153041}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{26244}{2401\, \left ( -4-6\,x \right ) ^{2}} \left ({\frac{221}{36} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{1561}{108}\sqrt{1-2\,x}} \right ) }+{\frac{182655\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{312500}{14641\, \left ( -6-10\,x \right ) ^{2}} \left ( -{\frac{187}{20} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{407}{20}\sqrt{1-2\,x}} \right ) }-{\frac{7570625\,\sqrt{55}}{161051}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

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)^3,x)

[Out]

64/1369599/(1-2*x)^(3/2)+13056/35153041/(1-2*x)^(1/2)-26244/2401*(221/36*(1-2*x)
^(3/2)-1561/108*(1-2*x)^(1/2))/(-4-6*x)^2+182655/2401*arctanh(1/7*21^(1/2)*(1-2*
x)^(1/2))*21^(1/2)+312500/14641*(-187/20*(1-2*x)^(3/2)+407/20*(1-2*x)^(1/2))/(-6
-10*x)^2-7570625/161051*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.48088, size = 221, normalized size = 1.23 \[ \frac{7570625}{322102} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{182655}{4802} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2 \,{\left (26165524500 \,{\left (2 \, x - 1\right )}^{5} + 177932259675 \,{\left (2 \, x - 1\right )}^{4} + 403131105480 \,{\left (2 \, x - 1\right )}^{3} + 304294845085 \,{\left (2 \, x - 1\right )}^{2} - 25803008 \, x + 14988512\right )}}{15065589 \,{\left (225 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - 2040 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + 6934 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 10472 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 5929 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7570625/322102*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2
*x + 1))) - 182655/4802*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) +
3*sqrt(-2*x + 1))) + 2/15065589*(26165524500*(2*x - 1)^5 + 177932259675*(2*x - 1
)^4 + 403131105480*(2*x - 1)^3 + 304294845085*(2*x - 1)^2 - 25803008*x + 1498851
2)/(225*(-2*x + 1)^(11/2) - 2040*(-2*x + 1)^(9/2) + 6934*(-2*x + 1)^(7/2) - 1047
2*(-2*x + 1)^(5/2) + 5929*(-2*x + 1)^(3/2))

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Fricas [A]  time = 0.225709, size = 294, normalized size = 1.64 \[ \frac{\sqrt{11} \sqrt{7}{\left (7790173125 \, \sqrt{7} \sqrt{5}{\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 8022755565 \, \sqrt{11} \sqrt{3}{\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} - 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{11} \sqrt{7}{\left (209324196000 \, x^{5} + 188418548700 \, x^{4} - 93885376440 \, x^{3} - 99160158305 \, x^{2} + 9944654283 \, x + 13236365823\right )}\right )}}{2320100706 \,{\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2320100706*sqrt(11)*sqrt(7)*(7790173125*sqrt(7)*sqrt(5)*(450*x^5 + 915*x^4 + 5
12*x^3 - 85*x^2 - 156*x - 36)*sqrt(-2*x + 1)*log((sqrt(11)*(5*x - 8) + 11*sqrt(5
)*sqrt(-2*x + 1))/(5*x + 3)) + 8022755565*sqrt(11)*sqrt(3)*(450*x^5 + 915*x^4 +
512*x^3 - 85*x^2 - 156*x - 36)*sqrt(-2*x + 1)*log((sqrt(7)*(3*x - 5) - 7*sqrt(3)
*sqrt(-2*x + 1))/(3*x + 2)) - sqrt(11)*sqrt(7)*(209324196000*x^5 + 188418548700*
x^4 - 93885376440*x^3 - 99160158305*x^2 + 9944654283*x + 13236365823))/((450*x^5
+ 915*x^4 + 512*x^3 - 85*x^2 - 156*x - 36)*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)**3,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.21895, size = 228, normalized size = 1.27 \[ \frac{7570625}{322102} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{182655}{4802} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{64 \,{\left (1224 \, x - 689\right )}}{105459123 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} + \frac{2 \,{\left (5550396300 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 37744400445 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 85516621432 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 64553088299 \, \sqrt{-2 \, x + 1}\right )}}{3195731 \,{\left (15 \,{\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7570625/322102*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) +
5*sqrt(-2*x + 1))) - 182655/4802*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x +
1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 64/105459123*(1224*x - 689)/((2*x - 1)*sqr
t(-2*x + 1)) + 2/3195731*(5550396300*(2*x - 1)^3*sqrt(-2*x + 1) + 37744400445*(2
*x - 1)^2*sqrt(-2*x + 1) - 85516621432*(-2*x + 1)^(3/2) + 64553088299*sqrt(-2*x
+ 1))/(15*(2*x - 1)^2 + 136*x + 9)^2