3.2147 \(\int \frac{(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^2} \, dx\)

Optimal. Leaf size=80 \[ \frac{11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)}-\frac{10 (1450 x+969)}{1029 \sqrt{1-2 x} (3 x+2)}-\frac{200 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}} \]

[Out]

(11*(3 + 5*x)^2)/(21*(1 - 2*x)^(3/2)*(2 + 3*x)) - (10*(969 + 1450*x))/(1029*Sqrt
[1 - 2*x]*(2 + 3*x)) - (200*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1029*Sqrt[21])

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Rubi [A]  time = 0.119727, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)}-\frac{10 (1450 x+969)}{1029 \sqrt{1-2 x} (3 x+2)}-\frac{200 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(11*(3 + 5*x)^2)/(21*(1 - 2*x)^(3/2)*(2 + 3*x)) - (10*(969 + 1450*x))/(1029*Sqrt
[1 - 2*x]*(2 + 3*x)) - (200*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1029*Sqrt[21])

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Rubi in Sympy [A]  time = 12.9208, size = 68, normalized size = 0.85 \[ - \frac{200 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21609} - \frac{43500 x + 29070}{3087 \sqrt{- 2 x + 1} \left (3 x + 2\right )} + \frac{11 \left (5 x + 3\right )^{2}}{21 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-200*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/21609 - (43500*x + 29070)/(3087*s
qrt(-2*x + 1)*(3*x + 2)) + 11*(5*x + 3)**2/(21*(-2*x + 1)**(3/2)*(3*x + 2))

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Mathematica [A]  time = 0.125909, size = 58, normalized size = 0.72 \[ \frac{\frac{21 \left (42475 x^2+21050 x-4839\right )}{(1-2 x)^{3/2} (3 x+2)}-200 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21609} \]

Antiderivative was successfully verified.

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

[Out]

((21*(-4839 + 21050*x + 42475*x^2))/((1 - 2*x)^(3/2)*(2 + 3*x)) - 200*Sqrt[21]*A
rcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/21609

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Maple [A]  time = 0.022, size = 54, normalized size = 0.7 \[{\frac{1331}{294} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{4719}{686}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{2}{3087}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}-{\frac{200\,\sqrt{21}}{21609}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1331/294/(1-2*x)^(3/2)-4719/686/(1-2*x)^(1/2)-2/3087*(1-2*x)^(1/2)/(-4/3-2*x)-20
0/21609*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.49368, size = 100, normalized size = 1.25 \[ \frac{100}{21609} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{42475 \,{\left (2 \, x - 1\right )}^{2} + 254100 \, x - 61831}{2058 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 7 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

100/21609*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x +
1))) - 1/2058*(42475*(2*x - 1)^2 + 254100*x - 61831)/(3*(-2*x + 1)^(5/2) - 7*(-2
*x + 1)^(3/2))

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Fricas [A]  time = 0.215973, size = 112, normalized size = 1.4 \[ \frac{\sqrt{21}{\left (100 \,{\left (6 \, x^{2} + x - 2\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{21}{\left (42475 \, x^{2} + 21050 \, x - 4839\right )}\right )}}{21609 \,{\left (6 \, x^{2} + x - 2\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/21609*sqrt(21)*(100*(6*x^2 + x - 2)*sqrt(-2*x + 1)*log((sqrt(21)*(3*x - 5) + 2
1*sqrt(-2*x + 1))/(3*x + 2)) - sqrt(21)*(42475*x^2 + 21050*x - 4839))/((6*x^2 +
x - 2)*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((3+5*x)**3/(1-2*x)**(5/2)/(2+3*x)**2,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.219024, size = 104, normalized size = 1.3 \[ \frac{100}{21609} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{121 \,{\left (117 \, x - 20\right )}}{1029 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} + \frac{\sqrt{-2 \, x + 1}}{1029 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

100/21609*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt
(-2*x + 1))) - 121/1029*(117*x - 20)/((2*x - 1)*sqrt(-2*x + 1)) + 1/1029*sqrt(-2
*x + 1)/(3*x + 2)