3.2154 \(\int \frac{(2+3 x)^4}{(1-2 x)^{5/2} (3+5 x)} \, dx\)

Optimal. Leaf size=80 \[ \frac{27}{40} (1-2 x)^{3/2}-\frac{2889}{200} \sqrt{1-2 x}-\frac{33271}{968 \sqrt{1-2 x}}+\frac{2401}{264 (1-2 x)^{3/2}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3025 \sqrt{55}} \]

[Out]

2401/(264*(1 - 2*x)^(3/2)) - 33271/(968*Sqrt[1 - 2*x]) - (2889*Sqrt[1 - 2*x])/20
0 + (27*(1 - 2*x)^(3/2))/40 - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(3025*Sqrt[5
5])

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Rubi [A]  time = 0.118381, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{27}{40} (1-2 x)^{3/2}-\frac{2889}{200} \sqrt{1-2 x}-\frac{33271}{968 \sqrt{1-2 x}}+\frac{2401}{264 (1-2 x)^{3/2}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3025 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

2401/(264*(1 - 2*x)^(3/2)) - 33271/(968*Sqrt[1 - 2*x]) - (2889*Sqrt[1 - 2*x])/20
0 + (27*(1 - 2*x)^(3/2))/40 - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(3025*Sqrt[5
5])

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Rubi in Sympy [A]  time = 12.1933, size = 71, normalized size = 0.89 \[ \frac{27 \left (- 2 x + 1\right )^{\frac{3}{2}}}{40} - \frac{2889 \sqrt{- 2 x + 1}}{200} - \frac{2 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{166375} - \frac{33271}{968 \sqrt{- 2 x + 1}} + \frac{2401}{264 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27*(-2*x + 1)**(3/2)/40 - 2889*sqrt(-2*x + 1)/200 - 2*sqrt(55)*atanh(sqrt(55)*sq
rt(-2*x + 1)/11)/166375 - 33271/(968*sqrt(-2*x + 1)) + 2401/(264*(-2*x + 1)**(3/
2))

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Mathematica [A]  time = 0.15108, size = 56, normalized size = 0.7 \[ \frac{-\frac{55 \left (49005 x^3+450846 x^2-1111431 x+354344\right )}{(1-2 x)^{3/2}}-6 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{499125} \]

Antiderivative was successfully verified.

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

[Out]

((-55*(354344 - 1111431*x + 450846*x^2 + 49005*x^3))/(1 - 2*x)^(3/2) - 6*Sqrt[55
]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/499125

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Maple [A]  time = 0.016, size = 56, normalized size = 0.7 \[{\frac{2401}{264} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{27}{40} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{2\,\sqrt{55}}{166375}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }-{\frac{33271}{968}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{2889}{200}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2401/264/(1-2*x)^(3/2)+27/40*(1-2*x)^(3/2)-2/166375*arctanh(1/11*55^(1/2)*(1-2*x
)^(1/2))*55^(1/2)-33271/968/(1-2*x)^(1/2)-2889/200*(1-2*x)^(1/2)

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Maxima [A]  time = 1.49253, size = 93, normalized size = 1.16 \[ \frac{27}{40} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{166375} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2889}{200} \, \sqrt{-2 \, x + 1} + \frac{343 \,{\left (291 \, x - 107\right )}}{1452 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27/40*(-2*x + 1)^(3/2) + 1/166375*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(s
qrt(55) + 5*sqrt(-2*x + 1))) - 2889/200*sqrt(-2*x + 1) + 343/1452*(291*x - 107)/
(-2*x + 1)^(3/2)

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Fricas [A]  time = 0.2233, size = 109, normalized size = 1.36 \[ \frac{\sqrt{55}{\left (3 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{55}{\left (49005 \, x^{3} + 450846 \, x^{2} - 1111431 \, x + 354344\right )}\right )}}{499125 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/499125*sqrt(55)*(3*(2*x - 1)*sqrt(-2*x + 1)*log((sqrt(55)*(5*x - 8) + 55*sqrt(
-2*x + 1))/(5*x + 3)) + sqrt(55)*(49005*x^3 + 450846*x^2 - 1111431*x + 354344))/
((2*x - 1)*sqrt(-2*x + 1))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (3 x + 2\right )^{4}}{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((3*x + 2)**4/((-2*x + 1)**(5/2)*(5*x + 3)), x)

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GIAC/XCAS [A]  time = 0.224883, size = 107, normalized size = 1.34 \[ \frac{27}{40} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{166375} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{2889}{200} \, \sqrt{-2 \, x + 1} - \frac{343 \,{\left (291 \, x - 107\right )}}{1452 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27/40*(-2*x + 1)^(3/2) + 1/166375*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x
+ 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 2889/200*sqrt(-2*x + 1) - 343/1452*(291*
x - 107)/((2*x - 1)*sqrt(-2*x + 1))