3.2173 \(\int \frac{(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^3} \, dx\)

Optimal. Leaf size=147 \[ \frac{7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^2}-\frac{73 (3 x+2)^4}{3630 \sqrt{1-2 x} (5 x+3)^2}-\frac{3269 (3 x+2)^3}{199650 \sqrt{1-2 x} (5 x+3)}-\frac{256172 (3 x+2)^2}{366025 \sqrt{1-2 x}}-\frac{21 \sqrt{1-2 x} (736875 x+2211616)}{3660250}-\frac{6937 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1830125 \sqrt{55}} \]

[Out]

(-256172*(2 + 3*x)^2)/(366025*Sqrt[1 - 2*x]) - (73*(2 + 3*x)^4)/(3630*Sqrt[1 - 2
*x]*(3 + 5*x)^2) + (7*(2 + 3*x)^5)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^2) - (3269*(2 +
 3*x)^3)/(199650*Sqrt[1 - 2*x]*(3 + 5*x)) - (21*Sqrt[1 - 2*x]*(2211616 + 736875*
x))/3660250 - (6937*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1830125*Sqrt[55])

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Rubi [A]  time = 0.307643, 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 \[ \frac{7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^2}-\frac{73 (3 x+2)^4}{3630 \sqrt{1-2 x} (5 x+3)^2}-\frac{3269 (3 x+2)^3}{199650 \sqrt{1-2 x} (5 x+3)}-\frac{256172 (3 x+2)^2}{366025 \sqrt{1-2 x}}-\frac{21 \sqrt{1-2 x} (736875 x+2211616)}{3660250}-\frac{6937 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1830125 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(-256172*(2 + 3*x)^2)/(366025*Sqrt[1 - 2*x]) - (73*(2 + 3*x)^4)/(3630*Sqrt[1 - 2
*x]*(3 + 5*x)^2) + (7*(2 + 3*x)^5)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^2) - (3269*(2 +
 3*x)^3)/(199650*Sqrt[1 - 2*x]*(3 + 5*x)) - (21*Sqrt[1 - 2*x]*(2211616 + 736875*
x))/3660250 - (6937*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1830125*Sqrt[55])

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Rubi in Sympy [A]  time = 32.0532, size = 133, normalized size = 0.9 \[ - \frac{\sqrt{- 2 x + 1} \left (696346875 x + 2089977120\right )}{164711250} - \frac{6937 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{100656875} - \frac{73 \left (3 x + 2\right )^{4}}{3630 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{2}} - \frac{3269 \left (3 x + 2\right )^{3}}{199650 \sqrt{- 2 x + 1} \left (5 x + 3\right )} - \frac{256172 \left (3 x + 2\right )^{2}}{366025 \sqrt{- 2 x + 1}} + \frac{7 \left (3 x + 2\right )^{5}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-2*x + 1)*(696346875*x + 2089977120)/164711250 - 6937*sqrt(55)*atanh(sqrt(
55)*sqrt(-2*x + 1)/11)/100656875 - 73*(3*x + 2)**4/(3630*sqrt(-2*x + 1)*(5*x + 3
)**2) - 3269*(3*x + 2)**3/(199650*sqrt(-2*x + 1)*(5*x + 3)) - 256172*(3*x + 2)**
2/(366025*sqrt(-2*x + 1)) + 7*(3*x + 2)**5/(33*(-2*x + 1)**(3/2)*(5*x + 3)**2)

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Mathematica [A]  time = 0.148009, size = 76, normalized size = 0.52 \[ \frac{-\frac{55 \sqrt{1-2 x} \left (533664450 x^5+5763576060 x^4-6510290070 x^3-9509366452 x^2+253794537 x+1463964312\right )}{\left (10 x^2+x-3\right )^2}-41622 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{603941250} \]

Antiderivative was successfully verified.

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

[Out]

((-55*Sqrt[1 - 2*x]*(1463964312 + 253794537*x - 9509366452*x^2 - 6510290070*x^3
+ 5763576060*x^4 + 533664450*x^5))/(-3 + x + 10*x^2)^2 - 41622*Sqrt[55]*ArcTanh[
Sqrt[5/11]*Sqrt[1 - 2*x]])/603941250

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Maple [A]  time = 0.023, size = 84, normalized size = 0.6 \[{\frac{243}{1000} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{26973}{5000}\sqrt{1-2\,x}}+{\frac{117649}{31944} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{1563051}{117128}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{2}{366025\, \left ( -6-10\,x \right ) ^{2}} \left ({\frac{407}{10} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{4499}{50}\sqrt{1-2\,x}} \right ) }-{\frac{6937\,\sqrt{55}}{100656875}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

243/1000*(1-2*x)^(3/2)-26973/5000*(1-2*x)^(1/2)+117649/31944/(1-2*x)^(3/2)-15630
51/117128/(1-2*x)^(1/2)+2/366025*(407/10*(1-2*x)^(3/2)-4499/50*(1-2*x)^(1/2))/(-
6-10*x)^2-6937/100656875*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.51909, size = 149, normalized size = 1.01 \[ \frac{243}{1000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{6937}{201313750} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{26973}{5000} \, \sqrt{-2 \, x + 1} + \frac{73267966785 \,{\left (2 \, x - 1\right )}^{3} + 342600082649 \,{\left (2 \, x - 1\right )}^{2} + 887178503750 \, x - 345719990000}{219615000 \,{\left (25 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 110 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 121 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

243/1000*(-2*x + 1)^(3/2) + 6937/201313750*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x
 + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 26973/5000*sqrt(-2*x + 1) + 1/219615000*
(73267966785*(2*x - 1)^3 + 342600082649*(2*x - 1)^2 + 887178503750*x - 345719990
000)/(25*(-2*x + 1)^(7/2) - 110*(-2*x + 1)^(5/2) + 121*(-2*x + 1)^(3/2))

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Fricas [A]  time = 0.221216, size = 150, normalized size = 1.02 \[ \frac{\sqrt{55}{\left (20811 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{55}{\left (533664450 \, x^{5} + 5763576060 \, x^{4} - 6510290070 \, x^{3} - 9509366452 \, x^{2} + 253794537 \, x + 1463964312\right )}\right )}}{603941250 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/603941250*sqrt(55)*(20811*(50*x^3 + 35*x^2 - 12*x - 9)*sqrt(-2*x + 1)*log((sqr
t(55)*(5*x - 8) + 55*sqrt(-2*x + 1))/(5*x + 3)) + sqrt(55)*(533664450*x^5 + 5763
576060*x^4 - 6510290070*x^3 - 9509366452*x^2 + 253794537*x + 1463964312))/((50*x
^3 + 35*x^2 - 12*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)**(5/2)/(3+5*x)**3,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.217143, size = 144, normalized size = 0.98 \[ \frac{243}{1000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{6937}{201313750} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{26973}{5000} \, \sqrt{-2 \, x + 1} - \frac{16807 \,{\left (279 \, x - 101\right )}}{175692 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} + \frac{185 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 409 \, \sqrt{-2 \, x + 1}}{3327500 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

243/1000*(-2*x + 1)^(3/2) + 6937/201313750*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*
sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 26973/5000*sqrt(-2*x + 1) - 168
07/175692*(279*x - 101)/((2*x - 1)*sqrt(-2*x + 1)) + 1/3327500*(185*(-2*x + 1)^(
3/2) - 409*sqrt(-2*x + 1))/(5*x + 3)^2