3.2108 \(\int \frac{(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)^3} \, dx\)

Optimal. Leaf size=147 \[ \frac{7 (3 x+2)^5}{11 \sqrt{1-2 x} (5 x+3)^2}-\frac{71 \sqrt{1-2 x} (3 x+2)^4}{1210 (5 x+3)^2}-\frac{2721 \sqrt{1-2 x} (3 x+2)^3}{66550 (5 x+3)}+\frac{377748 \sqrt{1-2 x} (3 x+2)^2}{831875}+\frac{63 \sqrt{1-2 x} (831375 x+2492512)}{8318750}-\frac{33873 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{4159375 \sqrt{55}} \]

[Out]

(377748*Sqrt[1 - 2*x]*(2 + 3*x)^2)/831875 - (71*Sqrt[1 - 2*x]*(2 + 3*x)^4)/(1210
*(3 + 5*x)^2) + (7*(2 + 3*x)^5)/(11*Sqrt[1 - 2*x]*(3 + 5*x)^2) - (2721*Sqrt[1 -
2*x]*(2 + 3*x)^3)/(66550*(3 + 5*x)) + (63*Sqrt[1 - 2*x]*(2492512 + 831375*x))/83
18750 - (33873*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(4159375*Sqrt[55])

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Rubi [A]  time = 0.286171, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{7 (3 x+2)^5}{11 \sqrt{1-2 x} (5 x+3)^2}-\frac{71 \sqrt{1-2 x} (3 x+2)^4}{1210 (5 x+3)^2}-\frac{2721 \sqrt{1-2 x} (3 x+2)^3}{66550 (5 x+3)}+\frac{377748 \sqrt{1-2 x} (3 x+2)^2}{831875}+\frac{63 \sqrt{1-2 x} (831375 x+2492512)}{8318750}-\frac{33873 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{4159375 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + 3*x)^6/((1 - 2*x)^(3/2)*(3 + 5*x)^3),x]

[Out]

(377748*Sqrt[1 - 2*x]*(2 + 3*x)^2)/831875 - (71*Sqrt[1 - 2*x]*(2 + 3*x)^4)/(1210
*(3 + 5*x)^2) + (7*(2 + 3*x)^5)/(11*Sqrt[1 - 2*x]*(3 + 5*x)^2) - (2721*Sqrt[1 -
2*x]*(2 + 3*x)^3)/(66550*(3 + 5*x)) + (63*Sqrt[1 - 2*x]*(2492512 + 831375*x))/83
18750 - (33873*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(4159375*Sqrt[55])

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Rubi in Sympy [A]  time = 32.9209, size = 131, normalized size = 0.89 \[ - \frac{71 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{4}}{1210 \left (5 x + 3\right )^{2}} - \frac{2721 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}}{66550 \left (5 x + 3\right )} + \frac{377748 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}}{831875} + \frac{\sqrt{- 2 x + 1} \left (785649375 x + 2355423840\right )}{124781250} - \frac{33873 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{228765625} + \frac{7 \left (3 x + 2\right )^{5}}{11 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**6/(1-2*x)**(3/2)/(3+5*x)**3,x)

[Out]

-71*sqrt(-2*x + 1)*(3*x + 2)**4/(1210*(5*x + 3)**2) - 2721*sqrt(-2*x + 1)*(3*x +
2)**3/(66550*(5*x + 3)) + 377748*sqrt(-2*x + 1)*(3*x + 2)**2/831875 + sqrt(-2*x
+ 1)*(785649375*x + 2355423840)/124781250 - 33873*sqrt(55)*atanh(sqrt(55)*sqrt(
-2*x + 1)/11)/228765625 + 7*(3*x + 2)**5/(11*sqrt(-2*x + 1)*(5*x + 3)**2)

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Mathematica [A]  time = 0.21866, size = 73, normalized size = 0.5 \[ \frac{-\frac{55 \left (242574750 x^5+1423105200 x^4+5682717810 x^3+762244410 x^2-4150263077 x-1702670584\right )}{\sqrt{1-2 x} (5 x+3)^2}-67746 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{457531250} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + 3*x)^6/((1 - 2*x)^(3/2)*(3 + 5*x)^3),x]

[Out]

((-55*(-1702670584 - 4150263077*x + 762244410*x^2 + 5682717810*x^3 + 1423105200*
x^4 + 242574750*x^5))/(Sqrt[1 - 2*x]*(3 + 5*x)^2) - 67746*Sqrt[55]*ArcTanh[Sqrt[
5/11]*Sqrt[1 - 2*x]])/457531250

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Maple [A]  time = 0.02, size = 84, normalized size = 0.6 \[{\frac{729}{5000} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{8991}{5000} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{333639}{25000}\sqrt{1-2\,x}}+{\frac{117649}{10648}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{2}{166375\, \left ( -6-10\,x \right ) ^{2}} \left ({\frac{403}{10} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{891}{10}\sqrt{1-2\,x}} \right ) }-{\frac{33873\,\sqrt{55}}{228765625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^6/(1-2*x)^(3/2)/(3+5*x)^3,x)

[Out]

729/5000*(1-2*x)^(5/2)-8991/5000*(1-2*x)^(3/2)+333639/25000*(1-2*x)^(1/2)+117649
/10648/(1-2*x)^(1/2)+2/166375*(403/10*(1-2*x)^(3/2)-891/10*(1-2*x)^(1/2))/(-6-10
*x)^2-33873/228765625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.51133, size = 149, normalized size = 1.01 \[ \frac{729}{5000} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{8991}{5000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{33873}{457531250} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{333639}{25000} \, \sqrt{-2 \, x + 1} + \frac{1838268849 \,{\left (2 \, x - 1\right )}^{2} + 16176751756 \, x + 808829747}{6655000 \,{\left (25 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 110 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 121 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^6/((5*x + 3)^3*(-2*x + 1)^(3/2)),x, algorithm="maxima")

[Out]

729/5000*(-2*x + 1)^(5/2) - 8991/5000*(-2*x + 1)^(3/2) + 33873/457531250*sqrt(55
)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 333639/250
00*sqrt(-2*x + 1) + 1/6655000*(1838268849*(2*x - 1)^2 + 16176751756*x + 80882974
7)/(25*(-2*x + 1)^(5/2) - 110*(-2*x + 1)^(3/2) + 121*sqrt(-2*x + 1))

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Fricas [A]  time = 0.239535, size = 138, normalized size = 0.94 \[ \frac{\sqrt{55}{\left (33873 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) - \sqrt{55}{\left (242574750 \, x^{5} + 1423105200 \, x^{4} + 5682717810 \, x^{3} + 762244410 \, x^{2} - 4150263077 \, x - 1702670584\right )}\right )}}{457531250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^6/((5*x + 3)^3*(-2*x + 1)^(3/2)),x, algorithm="fricas")

[Out]

1/457531250*sqrt(55)*(33873*(25*x^2 + 30*x + 9)*sqrt(-2*x + 1)*log((sqrt(55)*(5*
x - 8) + 55*sqrt(-2*x + 1))/(5*x + 3)) - sqrt(55)*(242574750*x^5 + 1423105200*x^
4 + 5682717810*x^3 + 762244410*x^2 - 4150263077*x - 1702670584))/((25*x^2 + 30*x
+ 9)*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((2+3*x)**6/(1-2*x)**(3/2)/(3+5*x)**3,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.232537, size = 150, normalized size = 1.02 \[ \frac{729}{5000} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{8991}{5000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{33873}{457531250} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{333639}{25000} \, \sqrt{-2 \, x + 1} + \frac{117649}{10648 \, \sqrt{-2 \, x + 1}} + \frac{403 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 891 \, \sqrt{-2 \, x + 1}}{3327500 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^6/((5*x + 3)^3*(-2*x + 1)^(3/2)),x, algorithm="giac")

[Out]

729/5000*(2*x - 1)^2*sqrt(-2*x + 1) - 8991/5000*(-2*x + 1)^(3/2) + 33873/4575312
50*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x
+ 1))) + 333639/25000*sqrt(-2*x + 1) + 117649/10648/sqrt(-2*x + 1) + 1/3327500*(
403*(-2*x + 1)^(3/2) - 891*sqrt(-2*x + 1))/(5*x + 3)^2