3.2153 \(\int \frac{(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)} \, dx\)

Optimal. Leaf size=93 \[ -\frac{243}{400} (1-2 x)^{5/2}+\frac{1917}{200} (1-2 x)^{3/2}-\frac{51057}{500} \sqrt{1-2 x}-\frac{156065}{968 \sqrt{1-2 x}}+\frac{16807}{528 (1-2 x)^{3/2}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15125 \sqrt{55}} \]

[Out]

16807/(528*(1 - 2*x)^(3/2)) - 156065/(968*Sqrt[1 - 2*x]) - (51057*Sqrt[1 - 2*x])
/500 + (1917*(1 - 2*x)^(3/2))/200 - (243*(1 - 2*x)^(5/2))/400 - (2*ArcTanh[Sqrt[
5/11]*Sqrt[1 - 2*x]])/(15125*Sqrt[55])

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Rubi [A]  time = 0.152038, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{243}{400} (1-2 x)^{5/2}+\frac{1917}{200} (1-2 x)^{3/2}-\frac{51057}{500} \sqrt{1-2 x}-\frac{156065}{968 \sqrt{1-2 x}}+\frac{16807}{528 (1-2 x)^{3/2}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15125 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

16807/(528*(1 - 2*x)^(3/2)) - 156065/(968*Sqrt[1 - 2*x]) - (51057*Sqrt[1 - 2*x])
/500 + (1917*(1 - 2*x)^(3/2))/200 - (243*(1 - 2*x)^(5/2))/400 - (2*ArcTanh[Sqrt[
5/11]*Sqrt[1 - 2*x]])/(15125*Sqrt[55])

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Rubi in Sympy [A]  time = 14.23, size = 83, normalized size = 0.89 \[ - \frac{243 \left (- 2 x + 1\right )^{\frac{5}{2}}}{400} + \frac{1917 \left (- 2 x + 1\right )^{\frac{3}{2}}}{200} - \frac{51057 \sqrt{- 2 x + 1}}{500} - \frac{2 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{831875} - \frac{156065}{968 \sqrt{- 2 x + 1}} + \frac{16807}{528 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-243*(-2*x + 1)**(5/2)/400 + 1917*(-2*x + 1)**(3/2)/200 - 51057*sqrt(-2*x + 1)/5
00 - 2*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/831875 - 156065/(968*sqrt(-2*x
+ 1)) + 16807/(528*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.154653, size = 61, normalized size = 0.66 \[ \frac{-\frac{55 \left (441045 x^4+2597265 x^3+13976226 x^2-30775791 x+10097264\right )}{(1-2 x)^{3/2}}-6 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{2495625} \]

Antiderivative was successfully verified.

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

[Out]

((-55*(10097264 - 30775791*x + 13976226*x^2 + 2597265*x^3 + 441045*x^4))/(1 - 2*
x)^(3/2) - 6*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/2495625

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Maple [A]  time = 0.016, size = 65, normalized size = 0.7 \[{\frac{16807}{528} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{1917}{200} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{243}{400} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{2\,\sqrt{55}}{831875}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }-{\frac{156065}{968}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{51057}{500}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

16807/528/(1-2*x)^(3/2)+1917/200*(1-2*x)^(3/2)-243/400*(1-2*x)^(5/2)-2/831875*ar
ctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-156065/968/(1-2*x)^(1/2)-51057/500*(
1-2*x)^(1/2)

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Maxima [A]  time = 1.48581, size = 105, normalized size = 1.13 \[ -\frac{243}{400} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{1917}{200} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{831875} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{51057}{500} \, \sqrt{-2 \, x + 1} + \frac{2401 \,{\left (780 \, x - 313\right )}}{5808 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-243/400*(-2*x + 1)^(5/2) + 1917/200*(-2*x + 1)^(3/2) + 1/831875*sqrt(55)*log(-(
sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 51057/500*sqrt(-2*
x + 1) + 2401/5808*(780*x - 313)/(-2*x + 1)^(3/2)

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Fricas [A]  time = 0.216562, size = 116, normalized size = 1.25 \[ \frac{\sqrt{55}{\left (3 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{55}{\left (441045 \, x^{4} + 2597265 \, x^{3} + 13976226 \, x^{2} - 30775791 \, x + 10097264\right )}\right )}}{2495625 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2495625*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)*(441045*x^4 + 2597265*x^3 + 13976226*x^2 - 307
75791*x + 10097264))/((2*x - 1)*sqrt(-2*x + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.217719, size = 128, normalized size = 1.38 \[ -\frac{243}{400} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{1917}{200} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{831875} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{51057}{500} \, \sqrt{-2 \, x + 1} - \frac{2401 \,{\left (780 \, x - 313\right )}}{5808 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-243/400*(2*x - 1)^2*sqrt(-2*x + 1) + 1917/200*(-2*x + 1)^(3/2) + 1/831875*sqrt(
55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) -
51057/500*sqrt(-2*x + 1) - 2401/5808*(780*x - 313)/((2*x - 1)*sqrt(-2*x + 1))