3.2165 \(\int \frac{(2+3 x)^4}{(1-2 x)^{5/2} (3+5 x)^2} \, dx\)

Optimal. Leaf size=100 \[ \frac{7 (3 x+2)^3}{33 (1-2 x)^{3/2} (5 x+3)}-\frac{38 (3 x+2)^2}{1815 \sqrt{1-2 x} (5 x+3)}-\frac{3 (40912-24739 x)}{33275 \sqrt{1-2 x}}-\frac{274 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{33275 \sqrt{55}} \]

[Out]

(-3*(40912 - 24739*x))/(33275*Sqrt[1 - 2*x]) - (38*(2 + 3*x)^2)/(1815*Sqrt[1 - 2
*x]*(3 + 5*x)) + (7*(2 + 3*x)^3)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)) - (274*ArcTanh[S
qrt[5/11]*Sqrt[1 - 2*x]])/(33275*Sqrt[55])

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Rubi [A]  time = 0.179053, 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 \[ \frac{7 (3 x+2)^3}{33 (1-2 x)^{3/2} (5 x+3)}-\frac{38 (3 x+2)^2}{1815 \sqrt{1-2 x} (5 x+3)}-\frac{3 (40912-24739 x)}{33275 \sqrt{1-2 x}}-\frac{274 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{33275 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(40912 - 24739*x))/(33275*Sqrt[1 - 2*x]) - (38*(2 + 3*x)^2)/(1815*Sqrt[1 - 2
*x]*(3 + 5*x)) + (7*(2 + 3*x)^3)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)) - (274*ArcTanh[S
qrt[5/11]*Sqrt[1 - 2*x]])/(33275*Sqrt[55])

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Rubi in Sympy [A]  time = 18.8154, size = 87, normalized size = 0.87 \[ - \frac{- 222651 x + 368208}{99825 \sqrt{- 2 x + 1}} - \frac{274 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{1830125} - \frac{38 \left (3 x + 2\right )^{2}}{1815 \sqrt{- 2 x + 1} \left (5 x + 3\right )} + \frac{7 \left (3 x + 2\right )^{3}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-222651*x + 368208)/(99825*sqrt(-2*x + 1)) - 274*sqrt(55)*atanh(sqrt(55)*sqrt(
-2*x + 1)/11)/1830125 - 38*(3*x + 2)**2/(1815*sqrt(-2*x + 1)*(5*x + 3)) + 7*(3*x
+ 2)**3/(33*(-2*x + 1)**(3/2)*(5*x + 3))

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Mathematica [A]  time = 0.161337, size = 63, normalized size = 0.63 \[ \frac{-\frac{55 \left (1617165 x^3-4634229 x^2-1790101 x+943584\right )}{(1-2 x)^{3/2} (5 x+3)}-822 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{5490375} \]

Antiderivative was successfully verified.

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

[Out]

((-55*(943584 - 1790101*x - 4634229*x^2 + 1617165*x^3))/((1 - 2*x)^(3/2)*(3 + 5*
x)) - 822*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/5490375

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Maple [A]  time = 0.02, size = 63, normalized size = 0.6 \[ -{\frac{81}{100}\sqrt{1-2\,x}}+{\frac{2401}{1452} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{10633}{2662}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{2}{166375}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{274\,\sqrt{55}}{1830125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-81/100*(1-2*x)^(1/2)+2401/1452/(1-2*x)^(3/2)-10633/2662/(1-2*x)^(1/2)+2/166375*
(1-2*x)^(1/2)/(-6/5-2*x)-274/1830125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/
2)

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Maxima [A]  time = 1.49654, size = 112, normalized size = 1.12 \[ \frac{137}{1830125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{81}{100} \, \sqrt{-2 \, x + 1} - \frac{3987363 \,{\left (2 \, x - 1\right )}^{2} + 20845825 \, x - 6791400}{199650 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 11 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

137/1830125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x
+ 1))) - 81/100*sqrt(-2*x + 1) - 1/199650*(3987363*(2*x - 1)^2 + 20845825*x - 67
91400)/(5*(-2*x + 1)^(5/2) - 11*(-2*x + 1)^(3/2))

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Fricas [A]  time = 0.219369, size = 117, normalized size = 1.17 \[ \frac{\sqrt{55}{\left (411 \,{\left (10 \, x^{2} + x - 3\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{55}{\left (1617165 \, x^{3} - 4634229 \, x^{2} - 1790101 \, x + 943584\right )}\right )}}{5490375 \,{\left (10 \, x^{2} + x - 3\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5490375*sqrt(55)*(411*(10*x^2 + x - 3)*sqrt(-2*x + 1)*log((sqrt(55)*(5*x - 8)
+ 55*sqrt(-2*x + 1))/(5*x + 3)) + sqrt(55)*(1617165*x^3 - 4634229*x^2 - 1790101*
x + 943584))/((10*x^2 + 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)**(5/2)/(3+5*x)**2,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.214994, size = 116, normalized size = 1.16 \[ \frac{137}{1830125} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{81}{100} \, \sqrt{-2 \, x + 1} - \frac{343 \,{\left (372 \, x - 109\right )}}{15972 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} - \frac{\sqrt{-2 \, x + 1}}{33275 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

137/1830125*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*s
qrt(-2*x + 1))) - 81/100*sqrt(-2*x + 1) - 343/15972*(372*x - 109)/((2*x - 1)*sqr
t(-2*x + 1)) - 1/33275*sqrt(-2*x + 1)/(5*x + 3)