3.2107 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^2} \, dx\)

Optimal. Leaf size=146 \[ \frac{245865}{41503 \sqrt{1-2 x}}-\frac{36175}{1078 \sqrt{1-2 x} (5 x+3)}+\frac{165}{49 \sqrt{1-2 x} (3 x+2) (5 x+3)}+\frac{3}{14 \sqrt{1-2 x} (3 x+2)^2 (5 x+3)}-\frac{70065}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{24000}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

245865/(41503*Sqrt[1 - 2*x]) - 36175/(1078*Sqrt[1 - 2*x]*(3 + 5*x)) + 3/(14*Sqrt
[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)) + 165/(49*Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)) -
(70065*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 + (24000*Sqrt[5/11]*ArcTa
nh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Rubi [A]  time = 0.357459, antiderivative size = 146, 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 \[ \frac{245865}{41503 \sqrt{1-2 x}}-\frac{36175}{1078 \sqrt{1-2 x} (5 x+3)}+\frac{165}{49 \sqrt{1-2 x} (3 x+2) (5 x+3)}+\frac{3}{14 \sqrt{1-2 x} (3 x+2)^2 (5 x+3)}-\frac{70065}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{24000}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

245865/(41503*Sqrt[1 - 2*x]) - 36175/(1078*Sqrt[1 - 2*x]*(3 + 5*x)) + 3/(14*Sqrt
[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)) + 165/(49*Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)) -
(70065*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 + (24000*Sqrt[5/11]*ArcTa
nh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Rubi in Sympy [A]  time = 34.9193, size = 121, normalized size = 0.83 \[ - \frac{70065 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{2401} + \frac{24000 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{1331} + \frac{245865}{41503 \sqrt{- 2 x + 1}} - \frac{21705}{1078 \sqrt{- 2 x + 1} \left (3 x + 2\right )} - \frac{309}{154 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}} - \frac{5}{11 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2} \left (5 x + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-70065*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/2401 + 24000*sqrt(55)*atanh(sqr
t(55)*sqrt(-2*x + 1)/11)/1331 + 245865/(41503*sqrt(-2*x + 1)) - 21705/(1078*sqrt
(-2*x + 1)*(3*x + 2)) - 309/(154*sqrt(-2*x + 1)*(3*x + 2)**2) - 5/(11*sqrt(-2*x
+ 1)*(3*x + 2)**2*(5*x + 3))

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Mathematica [A]  time = 0.171697, size = 104, normalized size = 0.71 \[ \frac{\sqrt{1-2 x} \left (-22127850 x^3-17711235 x^2+5050290 x+4664333\right )}{83006 (3 x+2)^2 \left (10 x^2+x-3\right )}-\frac{70065}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{24000}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(4664333 + 5050290*x - 17711235*x^2 - 22127850*x^3))/(83006*(2 +
3*x)^2*(-3 + x + 10*x^2)) - (70065*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/3
43 + (24000*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Maple [A]  time = 0.024, size = 91, normalized size = 0.6 \[{\frac{32}{41503}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{486}{343\, \left ( -4-6\,x \right ) ^{2}} \left ({\frac{49}{2} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{1043}{18}\sqrt{1-2\,x}} \right ) }-{\frac{70065\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{250}{121}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}+{\frac{24000\,\sqrt{55}}{1331}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

32/41503/(1-2*x)^(1/2)+486/343*(49/2*(1-2*x)^(3/2)-1043/18*(1-2*x)^(1/2))/(-4-6*
x)^2-70065/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+250/121*(1-2*x)^(1/
2)/(-6/5-2*x)+24000/1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.49674, size = 185, normalized size = 1.27 \[ -\frac{12000}{1331} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{70065}{4802} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{11063925 \,{\left (2 \, x - 1\right )}^{3} + 50903010 \,{\left (2 \, x - 1\right )}^{2} + 117027330 \, x - 58496417}{41503 \,{\left (45 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 309 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 707 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 539 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-12000/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x
+ 1))) + 70065/4802*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sq
rt(-2*x + 1))) - 1/41503*(11063925*(2*x - 1)^3 + 50903010*(2*x - 1)^2 + 11702733
0*x - 58496417)/(45*(-2*x + 1)^(7/2) - 309*(-2*x + 1)^(5/2) + 707*(-2*x + 1)^(3/
2) - 539*sqrt(-2*x + 1))

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Fricas [A]  time = 0.239975, size = 239, normalized size = 1.64 \[ \frac{\sqrt{11} \sqrt{7}{\left (8232000 \, \sqrt{7} \sqrt{5}{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} - 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 8477865 \, \sqrt{11} \sqrt{3}{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} + 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{11} \sqrt{7}{\left (22127850 \, x^{3} + 17711235 \, x^{2} - 5050290 \, x - 4664333\right )}\right )}}{6391462 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6391462*sqrt(11)*sqrt(7)*(8232000*sqrt(7)*sqrt(5)*(45*x^3 + 87*x^2 + 56*x + 12
)*sqrt(-2*x + 1)*log((sqrt(11)*(5*x - 8) - 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3))
+ 8477865*sqrt(11)*sqrt(3)*(45*x^3 + 87*x^2 + 56*x + 12)*sqrt(-2*x + 1)*log((sq
rt(7)*(3*x - 5) + 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2)) + sqrt(11)*sqrt(7)*(22127
850*x^3 + 17711235*x^2 - 5050290*x - 4664333))/((45*x^3 + 87*x^2 + 56*x + 12)*sq
rt(-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)**3/(3+5*x)**2,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.260598, size = 182, normalized size = 1.25 \[ -\frac{12000}{1331} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{70065}{4802} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{2 \,{\left (428910 \, x - 214279\right )}}{41503 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}} + \frac{27 \,{\left (63 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 149 \, \sqrt{-2 \, x + 1}\right )}}{196 \,{\left (3 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-12000/1331*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*s
qrt(-2*x + 1))) + 70065/4802*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))
/(sqrt(21) + 3*sqrt(-2*x + 1))) - 2/41503*(428910*x - 214279)/(5*(-2*x + 1)^(3/2
) - 11*sqrt(-2*x + 1)) + 27/196*(63*(-2*x + 1)^(3/2) - 149*sqrt(-2*x + 1))/(3*x
+ 2)^2