3.2085 \(\int \frac{(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^4} \, dx\)

Optimal. Leaf size=100 \[ \frac{11 (5 x+3)^2}{7 \sqrt{1-2 x} (3 x+2)^3}+\frac{2 \sqrt{1-2 x} (470 x+297)}{441 (3 x+2)^3}-\frac{4660 \sqrt{1-2 x}}{3087 (3 x+2)}-\frac{9320 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3087 \sqrt{21}} \]

[Out]

(-4660*Sqrt[1 - 2*x])/(3087*(2 + 3*x)) + (11*(3 + 5*x)^2)/(7*Sqrt[1 - 2*x]*(2 +
3*x)^3) + (2*Sqrt[1 - 2*x]*(297 + 470*x))/(441*(2 + 3*x)^3) - (9320*ArcTanh[Sqrt
[3/7]*Sqrt[1 - 2*x]])/(3087*Sqrt[21])

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Rubi [A]  time = 0.133371, 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 \[ \frac{11 (5 x+3)^2}{7 \sqrt{1-2 x} (3 x+2)^3}+\frac{2 \sqrt{1-2 x} (470 x+297)}{441 (3 x+2)^3}-\frac{4660 \sqrt{1-2 x}}{3087 (3 x+2)}-\frac{9320 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3087 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(-4660*Sqrt[1 - 2*x])/(3087*(2 + 3*x)) + (11*(3 + 5*x)^2)/(7*Sqrt[1 - 2*x]*(2 +
3*x)^3) + (2*Sqrt[1 - 2*x]*(297 + 470*x))/(441*(2 + 3*x)^3) - (9320*ArcTanh[Sqrt
[3/7]*Sqrt[1 - 2*x]])/(3087*Sqrt[21])

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Rubi in Sympy [A]  time = 13.9115, size = 87, normalized size = 0.87 \[ - \frac{4660 \sqrt{- 2 x + 1}}{3087 \left (3 x + 2\right )} + \frac{\sqrt{- 2 x + 1} \left (39480 x + 24948\right )}{18522 \left (3 x + 2\right )^{3}} - \frac{9320 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{64827} + \frac{11 \left (5 x + 3\right )^{2}}{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4660*sqrt(-2*x + 1)/(3087*(3*x + 2)) + sqrt(-2*x + 1)*(39480*x + 24948)/(18522*
(3*x + 2)**3) - 9320*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/64827 + 11*(5*x +
3)**2/(7*sqrt(-2*x + 1)*(3*x + 2)**3)

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Mathematica [A]  time = 0.161347, size = 63, normalized size = 0.63 \[ \frac{\frac{21 \left (83880 x^3+178015 x^2+125154 x+29177\right )}{\sqrt{1-2 x} (3 x+2)^3}-9320 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{64827} \]

Antiderivative was successfully verified.

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

[Out]

((21*(29177 + 125154*x + 178015*x^2 + 83880*x^3))/(Sqrt[1 - 2*x]*(2 + 3*x)^3) -
9320*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/64827

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Maple [A]  time = 0.02, size = 66, normalized size = 0.7 \[{\frac{2662}{2401}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{54}{2401\, \left ( -4-6\,x \right ) ^{3}} \left ( -{\frac{3317}{27} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{137186}{243} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{157633}{243}\sqrt{1-2\,x}} \right ) }-{\frac{9320\,\sqrt{21}}{64827}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2662/2401/(1-2*x)^(1/2)+54/2401*(-3317/27*(1-2*x)^(5/2)+137186/243*(1-2*x)^(3/2)
-157633/243*(1-2*x)^(1/2))/(-4-6*x)^3-9320/64827*arctanh(1/7*21^(1/2)*(1-2*x)^(1
/2))*21^(1/2)

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Maxima [A]  time = 1.48082, size = 136, normalized size = 1.36 \[ \frac{4660}{64827} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2 \,{\left (41940 \,{\left (2 \, x - 1\right )}^{3} + 303835 \,{\left (2 \, x - 1\right )}^{2} + 1464316 \, x - 145187\right )}}{3087 \,{\left (27 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 189 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 441 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 343 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4660/64827*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x +
1))) - 2/3087*(41940*(2*x - 1)^3 + 303835*(2*x - 1)^2 + 1464316*x - 145187)/(27
*(-2*x + 1)^(7/2) - 189*(-2*x + 1)^(5/2) + 441*(-2*x + 1)^(3/2) - 343*sqrt(-2*x
+ 1))

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Fricas [A]  time = 0.224021, size = 136, normalized size = 1.36 \[ \frac{\sqrt{21}{\left (4660 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{21}{\left (83880 \, x^{3} + 178015 \, x^{2} + 125154 \, x + 29177\right )}\right )}}{64827 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/64827*sqrt(21)*(4660*(27*x^3 + 54*x^2 + 36*x + 8)*sqrt(-2*x + 1)*log((sqrt(21)
*(3*x - 5) + 21*sqrt(-2*x + 1))/(3*x + 2)) + sqrt(21)*(83880*x^3 + 178015*x^2 +
125154*x + 29177))/((27*x^3 + 54*x^2 + 36*x + 8)*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((3+5*x)**3/(1-2*x)**(3/2)/(2+3*x)**4,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.232806, size = 126, normalized size = 1.26 \[ \frac{4660}{64827} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{2662}{2401 \, \sqrt{-2 \, x + 1}} + \frac{29853 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 137186 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 157633 \, \sqrt{-2 \, x + 1}}{86436 \,{\left (3 \, x + 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4660/64827*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqr
t(-2*x + 1))) + 2662/2401/sqrt(-2*x + 1) + 1/86436*(29853*(2*x - 1)^2*sqrt(-2*x
+ 1) - 137186*(-2*x + 1)^(3/2) + 157633*sqrt(-2*x + 1))/(3*x + 2)^3