3.2100 \(\int \frac{(2+3 x)^4}{(1-2 x)^{3/2} (3+5 x)^2} \, dx\)

Optimal. Leaf size=100 \[ \frac{7 (3 x+2)^3}{11 \sqrt{1-2 x} (5 x+3)}-\frac{36 \sqrt{1-2 x} (3 x+2)^2}{605 (5 x+3)}+\frac{27 \sqrt{1-2 x} (265 x+792)}{3025}-\frac{54 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3025 \sqrt{55}} \]

[Out]

(-36*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(605*(3 + 5*x)) + (7*(2 + 3*x)^3)/(11*Sqrt[1 - 2
*x]*(3 + 5*x)) + (27*Sqrt[1 - 2*x]*(792 + 265*x))/3025 - (54*ArcTanh[Sqrt[5/11]*
Sqrt[1 - 2*x]])/(3025*Sqrt[55])

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Rubi [A]  time = 0.168531, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{7 (3 x+2)^3}{11 \sqrt{1-2 x} (5 x+3)}-\frac{36 \sqrt{1-2 x} (3 x+2)^2}{605 (5 x+3)}+\frac{27 \sqrt{1-2 x} (265 x+792)}{3025}-\frac{54 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3025 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + 3*x)^4/((1 - 2*x)^(3/2)*(3 + 5*x)^2),x]

[Out]

(-36*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(605*(3 + 5*x)) + (7*(2 + 3*x)^3)/(11*Sqrt[1 - 2
*x]*(3 + 5*x)) + (27*Sqrt[1 - 2*x]*(792 + 265*x))/3025 - (54*ArcTanh[Sqrt[5/11]*
Sqrt[1 - 2*x]])/(3025*Sqrt[55])

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Rubi in Sympy [A]  time = 18.2623, size = 85, normalized size = 0.85 \[ - \frac{36 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}}{605 \left (5 x + 3\right )} + \frac{\sqrt{- 2 x + 1} \left (107325 x + 320760\right )}{45375} - \frac{54 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{166375} + \frac{7 \left (3 x + 2\right )^{3}}{11 \sqrt{- 2 x + 1} \left (5 x + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**4/(1-2*x)**(3/2)/(3+5*x)**2,x)

[Out]

-36*sqrt(-2*x + 1)*(3*x + 2)**2/(605*(5*x + 3)) + sqrt(-2*x + 1)*(107325*x + 320
760)/45375 - 54*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/166375 + 7*(3*x + 2)*
*3/(11*sqrt(-2*x + 1)*(5*x + 3))

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Mathematica [A]  time = 0.118535, size = 66, normalized size = 0.66 \[ \frac{\frac{55 \sqrt{1-2 x} \left (16335 x^3+114345 x^2-68661 x-78832\right )}{10 x^2+x-3}-54 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{166375} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + 3*x)^4/((1 - 2*x)^(3/2)*(3 + 5*x)^2),x]

[Out]

((55*Sqrt[1 - 2*x]*(-78832 - 68661*x + 114345*x^2 + 16335*x^3))/(-3 + x + 10*x^2
) - 54*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/166375

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Maple [A]  time = 0.019, size = 63, normalized size = 0.6 \[ -{\frac{27}{100} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{999}{250}\sqrt{1-2\,x}}+{\frac{2401}{484}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{2}{75625}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{54\,\sqrt{55}}{166375}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^4/(1-2*x)^(3/2)/(3+5*x)^2,x)

[Out]

-27/100*(1-2*x)^(3/2)+999/250*(1-2*x)^(1/2)+2401/484/(1-2*x)^(1/2)+2/75625*(1-2*
x)^(1/2)/(-6/5-2*x)-54/166375*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.49778, size = 112, normalized size = 1.12 \[ -\frac{27}{100} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{27}{166375} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{999}{250} \, \sqrt{-2 \, x + 1} - \frac{1500633 \, x + 900371}{30250 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^4/((5*x + 3)^2*(-2*x + 1)^(3/2)),x, algorithm="maxima")

[Out]

-27/100*(-2*x + 1)^(3/2) + 27/166375*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))
/(sqrt(55) + 5*sqrt(-2*x + 1))) + 999/250*sqrt(-2*x + 1) - 1/30250*(1500633*x +
900371)/(5*(-2*x + 1)^(3/2) - 11*sqrt(-2*x + 1))

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Fricas [A]  time = 0.248313, size = 111, normalized size = 1.11 \[ \frac{\sqrt{55}{\left (27 \,{\left (5 \, x + 3\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) - \sqrt{55}{\left (16335 \, x^{3} + 114345 \, x^{2} - 68661 \, x - 78832\right )}\right )}}{166375 \,{\left (5 \, x + 3\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^4/((5*x + 3)^2*(-2*x + 1)^(3/2)),x, algorithm="fricas")

[Out]

1/166375*sqrt(55)*(27*(5*x + 3)*sqrt(-2*x + 1)*log((sqrt(55)*(5*x - 8) + 55*sqrt
(-2*x + 1))/(5*x + 3)) - sqrt(55)*(16335*x^3 + 114345*x^2 - 68661*x - 78832))/((
5*x + 3)*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)**4/(1-2*x)**(3/2)/(3+5*x)**2,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.222137, size = 116, normalized size = 1.16 \[ -\frac{27}{100} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{27}{166375} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{999}{250} \, \sqrt{-2 \, x + 1} - \frac{1500633 \, x + 900371}{30250 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^4/((5*x + 3)^2*(-2*x + 1)^(3/2)),x, algorithm="giac")

[Out]

-27/100*(-2*x + 1)^(3/2) + 27/166375*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-
2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 999/250*sqrt(-2*x + 1) - 1/30250*(150
0633*x + 900371)/(5*(-2*x + 1)^(3/2) - 11*sqrt(-2*x + 1))