3.2019 \(\int \frac{(2+3 x)^5}{\sqrt{1-2 x} (3+5 x)} \, dx\)

Optimal. Leaf size=93 \[ -\frac{27}{80} (1-2 x)^{9/2}+\frac{5751 (1-2 x)^{7/2}}{1400}-\frac{51057 (1-2 x)^{5/2}}{2500}+\frac{268707 (1-2 x)^{3/2}}{5000}-\frac{4774713 \sqrt{1-2 x}}{50000}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125 \sqrt{55}} \]

[Out]

(-4774713*Sqrt[1 - 2*x])/50000 + (268707*(1 - 2*x)^(3/2))/5000 - (51057*(1 - 2*x
)^(5/2))/2500 + (5751*(1 - 2*x)^(7/2))/1400 - (27*(1 - 2*x)^(9/2))/80 - (2*ArcTa
nh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(3125*Sqrt[55])

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Rubi [A]  time = 0.0971523, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{27}{80} (1-2 x)^{9/2}+\frac{5751 (1-2 x)^{7/2}}{1400}-\frac{51057 (1-2 x)^{5/2}}{2500}+\frac{268707 (1-2 x)^{3/2}}{5000}-\frac{4774713 \sqrt{1-2 x}}{50000}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(-4774713*Sqrt[1 - 2*x])/50000 + (268707*(1 - 2*x)^(3/2))/5000 - (51057*(1 - 2*x
)^(5/2))/2500 + (5751*(1 - 2*x)^(7/2))/1400 - (27*(1 - 2*x)^(9/2))/80 - (2*ArcTa
nh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(3125*Sqrt[55])

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Rubi in Sympy [A]  time = 9.93523, size = 83, normalized size = 0.89 \[ - \frac{27 \left (- 2 x + 1\right )^{\frac{9}{2}}}{80} + \frac{5751 \left (- 2 x + 1\right )^{\frac{7}{2}}}{1400} - \frac{51057 \left (- 2 x + 1\right )^{\frac{5}{2}}}{2500} + \frac{268707 \left (- 2 x + 1\right )^{\frac{3}{2}}}{5000} - \frac{4774713 \sqrt{- 2 x + 1}}{50000} - \frac{2 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{171875} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27*(-2*x + 1)**(9/2)/80 + 5751*(-2*x + 1)**(7/2)/1400 - 51057*(-2*x + 1)**(5/2)
/2500 + 268707*(-2*x + 1)**(3/2)/5000 - 4774713*sqrt(-2*x + 1)/50000 - 2*sqrt(55
)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/171875

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Mathematica [A]  time = 0.115263, size = 61, normalized size = 0.66 \[ -\frac{3 \sqrt{1-2 x} \left (39375 x^4+160875 x^3+295290 x^2+348095 x+425872\right )}{21875}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*Sqrt[1 - 2*x]*(425872 + 348095*x + 295290*x^2 + 160875*x^3 + 39375*x^4))/218
75 - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(3125*Sqrt[55])

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Maple [A]  time = 0.01, size = 65, normalized size = 0.7 \[{\frac{268707}{5000} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{51057}{2500} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{5751}{1400} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{27}{80} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}-{\frac{2\,\sqrt{55}}{171875}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }-{\frac{4774713}{50000}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

268707/5000*(1-2*x)^(3/2)-51057/2500*(1-2*x)^(5/2)+5751/1400*(1-2*x)^(7/2)-27/80
*(1-2*x)^(9/2)-2/171875*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-4774713/50
000*(1-2*x)^(1/2)

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Maxima [A]  time = 1.54709, size = 111, normalized size = 1.19 \[ -\frac{27}{80} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{5751}{1400} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{51057}{2500} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{268707}{5000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{171875} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{4774713}{50000} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27/80*(-2*x + 1)^(9/2) + 5751/1400*(-2*x + 1)^(7/2) - 51057/2500*(-2*x + 1)^(5/
2) + 268707/5000*(-2*x + 1)^(3/2) + 1/171875*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2
*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 4774713/50000*sqrt(-2*x + 1)

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Fricas [A]  time = 0.226179, size = 92, normalized size = 0.99 \[ -\frac{1}{1203125} \, \sqrt{55}{\left (3 \, \sqrt{55}{\left (39375 \, x^{4} + 160875 \, x^{3} + 295290 \, x^{2} + 348095 \, x + 425872\right )} \sqrt{-2 \, x + 1} - 7 \, \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1203125*sqrt(55)*(3*sqrt(55)*(39375*x^4 + 160875*x^3 + 295290*x^2 + 348095*x
+ 425872)*sqrt(-2*x + 1) - 7*log((sqrt(55)*(5*x - 8) + 55*sqrt(-2*x + 1))/(5*x +
3)))

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Sympy [A]  time = 7.3983, size = 126, normalized size = 1.35 \[ - \frac{27 \left (- 2 x + 1\right )^{\frac{9}{2}}}{80} + \frac{5751 \left (- 2 x + 1\right )^{\frac{7}{2}}}{1400} - \frac{51057 \left (- 2 x + 1\right )^{\frac{5}{2}}}{2500} + \frac{268707 \left (- 2 x + 1\right )^{\frac{3}{2}}}{5000} - \frac{4774713 \sqrt{- 2 x + 1}}{50000} + \frac{2 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55}}{5 \sqrt{- 2 x + 1}} \right )}}{55} & \text{for}\: \frac{1}{- 2 x + 1} > \frac{5}{11} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55}}{5 \sqrt{- 2 x + 1}} \right )}}{55} & \text{for}\: \frac{1}{- 2 x + 1} < \frac{5}{11} \end{cases}\right )}{3125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27*(-2*x + 1)**(9/2)/80 + 5751*(-2*x + 1)**(7/2)/1400 - 51057*(-2*x + 1)**(5/2)
/2500 + 268707*(-2*x + 1)**(3/2)/5000 - 4774713*sqrt(-2*x + 1)/50000 + 2*Piecewi
se((-sqrt(55)*acoth(sqrt(55)/(5*sqrt(-2*x + 1)))/55, 1/(-2*x + 1) > 5/11), (-sqr
t(55)*atanh(sqrt(55)/(5*sqrt(-2*x + 1)))/55, 1/(-2*x + 1) < 5/11))/3125

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GIAC/XCAS [A]  time = 0.215382, size = 143, normalized size = 1.54 \[ -\frac{27}{80} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{5751}{1400} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{51057}{2500} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{268707}{5000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{171875} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{4774713}{50000} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27/80*(2*x - 1)^4*sqrt(-2*x + 1) - 5751/1400*(2*x - 1)^3*sqrt(-2*x + 1) - 51057
/2500*(2*x - 1)^2*sqrt(-2*x + 1) + 268707/5000*(-2*x + 1)^(3/2) + 1/171875*sqrt(
55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) -
4774713/50000*sqrt(-2*x + 1)