3.2030 \(\int \frac{(2+3 x)^6}{\sqrt{1-2 x} (3+5 x)^2} \, dx\)

Optimal. Leaf size=133 \[ -\frac{\sqrt{1-2 x} (3 x+2)^5}{55 (5 x+3)}-\frac{8}{275} \sqrt{1-2 x} (3 x+2)^4-\frac{1717 \sqrt{1-2 x} (3 x+2)^3}{9625}-\frac{26352 \sqrt{1-2 x} (3 x+2)^2}{34375}-\frac{3 \sqrt{1-2 x} (615875 x+1847824)}{171875}-\frac{398 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{171875 \sqrt{55}} \]

[Out]

(-26352*Sqrt[1 - 2*x]*(2 + 3*x)^2)/34375 - (1717*Sqrt[1 - 2*x]*(2 + 3*x)^3)/9625
- (8*Sqrt[1 - 2*x]*(2 + 3*x)^4)/275 - (Sqrt[1 - 2*x]*(2 + 3*x)^5)/(55*(3 + 5*x)
) - (3*Sqrt[1 - 2*x]*(1847824 + 615875*x))/171875 - (398*ArcTanh[Sqrt[5/11]*Sqrt
[1 - 2*x]])/(171875*Sqrt[55])

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Rubi [A]  time = 0.28194, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{\sqrt{1-2 x} (3 x+2)^5}{55 (5 x+3)}-\frac{8}{275} \sqrt{1-2 x} (3 x+2)^4-\frac{1717 \sqrt{1-2 x} (3 x+2)^3}{9625}-\frac{26352 \sqrt{1-2 x} (3 x+2)^2}{34375}-\frac{3 \sqrt{1-2 x} (615875 x+1847824)}{171875}-\frac{398 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{171875 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(-26352*Sqrt[1 - 2*x]*(2 + 3*x)^2)/34375 - (1717*Sqrt[1 - 2*x]*(2 + 3*x)^3)/9625
- (8*Sqrt[1 - 2*x]*(2 + 3*x)^4)/275 - (Sqrt[1 - 2*x]*(2 + 3*x)^5)/(55*(3 + 5*x)
) - (3*Sqrt[1 - 2*x]*(1847824 + 615875*x))/171875 - (398*ArcTanh[Sqrt[5/11]*Sqrt
[1 - 2*x]])/(171875*Sqrt[55])

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Rubi in Sympy [A]  time = 33.008, size = 117, normalized size = 0.88 \[ - \frac{\sqrt{- 2 x + 1} \left (3 x + 2\right )^{5}}{55 \left (5 x + 3\right )} - \frac{8 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{4}}{275} - \frac{1717 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}}{9625} - \frac{26352 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}}{34375} - \frac{\sqrt{- 2 x + 1} \left (1746005625 x + 5238581040\right )}{162421875} - \frac{398 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{9453125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-2*x + 1)*(3*x + 2)**5/(55*(5*x + 3)) - 8*sqrt(-2*x + 1)*(3*x + 2)**4/275
- 1717*sqrt(-2*x + 1)*(3*x + 2)**3/9625 - 26352*sqrt(-2*x + 1)*(3*x + 2)**2/3437
5 - sqrt(-2*x + 1)*(1746005625*x + 5238581040)/162421875 - 398*sqrt(55)*atanh(sq
rt(55)*sqrt(-2*x + 1)/11)/9453125

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Mathematica [A]  time = 0.131079, size = 73, normalized size = 0.55 \[ \frac{-\frac{55 \sqrt{1-2 x} \left (19490625 x^5+92998125 x^4+200942775 x^3+273540465 x^2+334366065 x+135011752\right )}{5 x+3}-2786 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{66171875} \]

Antiderivative was successfully verified.

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

[Out]

((-55*Sqrt[1 - 2*x]*(135011752 + 334366065*x + 273540465*x^2 + 200942775*x^3 + 9
2998125*x^4 + 19490625*x^5))/(3 + 5*x) - 2786*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1
- 2*x]])/66171875

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Maple [A]  time = 0.016, size = 81, normalized size = 0.6 \[ -{\frac{81}{400} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}+{\frac{2187}{875} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{315171}{25000} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{105228}{3125} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{607689}{10000}\sqrt{1-2\,x}}+{\frac{2}{859375}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{398\,\sqrt{55}}{9453125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-81/400*(1-2*x)^(9/2)+2187/875*(1-2*x)^(7/2)-315171/25000*(1-2*x)^(5/2)+105228/3
125*(1-2*x)^(3/2)-607689/10000*(1-2*x)^(1/2)+2/859375*(1-2*x)^(1/2)/(-6/5-2*x)-3
98/9453125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.49439, size = 132, normalized size = 0.99 \[ -\frac{81}{400} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{2187}{875} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{315171}{25000} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{105228}{3125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{199}{9453125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{607689}{10000} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{171875 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-81/400*(-2*x + 1)^(9/2) + 2187/875*(-2*x + 1)^(7/2) - 315171/25000*(-2*x + 1)^(
5/2) + 105228/3125*(-2*x + 1)^(3/2) + 199/9453125*sqrt(55)*log(-(sqrt(55) - 5*sq
rt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 607689/10000*sqrt(-2*x + 1) - 1/1
71875*sqrt(-2*x + 1)/(5*x + 3)

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Fricas [A]  time = 0.217784, size = 113, normalized size = 0.85 \[ -\frac{\sqrt{55}{\left (\sqrt{55}{\left (19490625 \, x^{5} + 92998125 \, x^{4} + 200942775 \, x^{3} + 273540465 \, x^{2} + 334366065 \, x + 135011752\right )} \sqrt{-2 \, x + 1} - 1393 \,{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{66171875 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/66171875*sqrt(55)*(sqrt(55)*(19490625*x^5 + 92998125*x^4 + 200942775*x^3 + 27
3540465*x^2 + 334366065*x + 135011752)*sqrt(-2*x + 1) - 1393*(5*x + 3)*log((sqrt
(55)*(5*x - 8) + 55*sqrt(-2*x + 1))/(5*x + 3)))/(5*x + 3)

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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/(3+5*x)**2/(1-2*x)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.222229, size = 165, normalized size = 1.24 \[ -\frac{81}{400} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{2187}{875} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{315171}{25000} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{105228}{3125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{199}{9453125} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{607689}{10000} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{171875 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-81/400*(2*x - 1)^4*sqrt(-2*x + 1) - 2187/875*(2*x - 1)^3*sqrt(-2*x + 1) - 31517
1/25000*(2*x - 1)^2*sqrt(-2*x + 1) + 105228/3125*(-2*x + 1)^(3/2) + 199/9453125*
sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1
))) - 607689/10000*sqrt(-2*x + 1) - 1/171875*sqrt(-2*x + 1)/(5*x + 3)