3.2086 \(\int \frac{(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^5} \, dx\)

Optimal. Leaf size=120 \[ \frac{11 (5 x+3)^2}{7 \sqrt{1-2 x} (3 x+2)^4}+\frac{\sqrt{1-2 x} (3789 x+2395)}{1764 (3 x+2)^4}-\frac{39185 \sqrt{1-2 x}}{57624 (3 x+2)}-\frac{39185 \sqrt{1-2 x}}{24696 (3 x+2)^2}-\frac{39185 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{28812 \sqrt{21}} \]

[Out]

(-39185*Sqrt[1 - 2*x])/(24696*(2 + 3*x)^2) - (39185*Sqrt[1 - 2*x])/(57624*(2 + 3
*x)) + (11*(3 + 5*x)^2)/(7*Sqrt[1 - 2*x]*(2 + 3*x)^4) + (Sqrt[1 - 2*x]*(2395 + 3
789*x))/(1764*(2 + 3*x)^4) - (39185*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(28812*Sqr
t[21])

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Rubi [A]  time = 0.157384, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{11 (5 x+3)^2}{7 \sqrt{1-2 x} (3 x+2)^4}+\frac{\sqrt{1-2 x} (3789 x+2395)}{1764 (3 x+2)^4}-\frac{39185 \sqrt{1-2 x}}{57624 (3 x+2)}-\frac{39185 \sqrt{1-2 x}}{24696 (3 x+2)^2}-\frac{39185 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{28812 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(-39185*Sqrt[1 - 2*x])/(24696*(2 + 3*x)^2) - (39185*Sqrt[1 - 2*x])/(57624*(2 + 3
*x)) + (11*(3 + 5*x)^2)/(7*Sqrt[1 - 2*x]*(2 + 3*x)^4) + (Sqrt[1 - 2*x]*(2395 + 3
789*x))/(1764*(2 + 3*x)^4) - (39185*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(28812*Sqr
t[21])

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Rubi in Sympy [A]  time = 15.8774, size = 105, normalized size = 0.88 \[ - \frac{39185 \sqrt{- 2 x + 1}}{57624 \left (3 x + 2\right )} - \frac{39185 \sqrt{- 2 x + 1}}{24696 \left (3 x + 2\right )^{2}} + \frac{\sqrt{- 2 x + 1} \left (79569 x + 50295\right )}{37044 \left (3 x + 2\right )^{4}} - \frac{39185 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{605052} + \frac{11 \left (5 x + 3\right )^{2}}{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{4}} \]

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)**5,x)

[Out]

-39185*sqrt(-2*x + 1)/(57624*(3*x + 2)) - 39185*sqrt(-2*x + 1)/(24696*(3*x + 2)*
*2) + sqrt(-2*x + 1)*(79569*x + 50295)/(37044*(3*x + 2)**4) - 39185*sqrt(21)*ata
nh(sqrt(21)*sqrt(-2*x + 1)/7)/605052 + 11*(5*x + 3)**2/(7*sqrt(-2*x + 1)*(3*x +
2)**4)

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Mathematica [A]  time = 0.17323, size = 68, normalized size = 0.57 \[ \frac{\frac{21 \left (2115990 x^4+4819755 x^3+4093057 x^2+1534434 x+213998\right )}{\sqrt{1-2 x} (3 x+2)^4}-78370 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1210104} \]

Antiderivative was successfully verified.

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

[Out]

((21*(213998 + 1534434*x + 4093057*x^2 + 4819755*x^3 + 2115990*x^4))/(Sqrt[1 - 2
*x]*(2 + 3*x)^4) - 78370*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/1210104

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Maple [A]  time = 0.02, size = 75, normalized size = 0.6 \[{\frac{5324}{16807}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{324}{16807\, \left ( -4-6\,x \right ) ^{4}} \left ({\frac{82631}{144} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{5020939}{1296} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{33905795}{3888} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{25445455}{3888}\sqrt{1-2\,x}} \right ) }-{\frac{39185\,\sqrt{21}}{605052}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5324/16807/(1-2*x)^(1/2)+324/16807*(82631/144*(1-2*x)^(7/2)-5020939/1296*(1-2*x)
^(5/2)[phone]/3888*(1-2*x)^(3/2)-25445455/3888*(1-2*x)^(1/2))/(-4-6*x)^4-39185
/605052*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.49407, size = 161, normalized size = 1.34 \[ \frac{39185}{1210104} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{1057995 \,{\left (2 \, x - 1\right )}^{4} + 9051735 \,{\left (2 \, x - 1\right )}^{3} + 28993349 \,{\left (2 \, x - 1\right )}^{2} + 82402418 \, x - 19287625}{28812 \,{\left (81 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 756 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 2646 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 4116 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 2401 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

39185/1210104*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*
x + 1))) + 1/28812*(1057995*(2*x - 1)^4 + 9051735*(2*x - 1)^3 + 28993349*(2*x -
1)^2 + 82402418*x - 19287625)/(81*(-2*x + 1)^(9/2) - 756*(-2*x + 1)^(7/2) + 2646
*(-2*x + 1)^(5/2) - 4116*(-2*x + 1)^(3/2) + 2401*sqrt(-2*x + 1))

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Fricas [A]  time = 0.218555, size = 157, normalized size = 1.31 \[ \frac{\sqrt{21}{\left (39185 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{21}{\left (2115990 \, x^{4} + 4819755 \, x^{3} + 4093057 \, x^{2} + 1534434 \, x + 213998\right )}\right )}}{1210104 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1210104*sqrt(21)*(39185*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*sqrt(-2*x + 1
)*log((sqrt(21)*(3*x - 5) + 21*sqrt(-2*x + 1))/(3*x + 2)) + sqrt(21)*(2115990*x^
4 + 4819755*x^3 + 4093057*x^2 + 1534434*x + 213998))/((81*x^4 + 216*x^3 + 216*x^
2 + 96*x + 16)*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)**5,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.219484, size = 147, normalized size = 1.22 \[ \frac{39185}{1210104} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{5324}{16807 \, \sqrt{-2 \, x + 1}} - \frac{2231037 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 15062817 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 33905795 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 25445455 \, \sqrt{-2 \, x + 1}}{3226944 \,{\left (3 \, x + 2\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

39185/1210104*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*
sqrt(-2*x + 1))) + 5324/16807/sqrt(-2*x + 1) - 1/3226944*(2231037*(2*x - 1)^3*sq
rt(-2*x + 1) + 15062817*(2*x - 1)^2*sqrt(-2*x + 1) - 33905795*(-2*x + 1)^(3/2) +
25445455*sqrt(-2*x + 1))/(3*x + 2)^4