3.2076 \(\int \frac{(3+5 x)^2}{(1-2 x)^{3/2} (2+3 x)^5} \, dx\)

Optimal. Leaf size=128 \[ -\frac{9145 \sqrt{1-2 x}}{57624 (3 x+2)}-\frac{9145 \sqrt{1-2 x}}{24696 (3 x+2)^2}-\frac{1829 \sqrt{1-2 x}}{1764 (3 x+2)^3}-\frac{2179 \sqrt{1-2 x}}{588 (3 x+2)^4}+\frac{121}{14 \sqrt{1-2 x} (3 x+2)^4}-\frac{9145 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{28812 \sqrt{21}} \]

[Out]

121/(14*Sqrt[1 - 2*x]*(2 + 3*x)^4) - (2179*Sqrt[1 - 2*x])/(588*(2 + 3*x)^4) - (1
829*Sqrt[1 - 2*x])/(1764*(2 + 3*x)^3) - (9145*Sqrt[1 - 2*x])/(24696*(2 + 3*x)^2)
- (9145*Sqrt[1 - 2*x])/(57624*(2 + 3*x)) - (9145*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x
]])/(28812*Sqrt[21])

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Rubi [A]  time = 0.157534, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{9145 \sqrt{1-2 x}}{57624 (3 x+2)}-\frac{9145 \sqrt{1-2 x}}{24696 (3 x+2)^2}-\frac{1829 \sqrt{1-2 x}}{1764 (3 x+2)^3}-\frac{2179 \sqrt{1-2 x}}{588 (3 x+2)^4}+\frac{121}{14 \sqrt{1-2 x} (3 x+2)^4}-\frac{9145 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{28812 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

121/(14*Sqrt[1 - 2*x]*(2 + 3*x)^4) - (2179*Sqrt[1 - 2*x])/(588*(2 + 3*x)^4) - (1
829*Sqrt[1 - 2*x])/(1764*(2 + 3*x)^3) - (9145*Sqrt[1 - 2*x])/(24696*(2 + 3*x)^2)
- (9145*Sqrt[1 - 2*x])/(57624*(2 + 3*x)) - (9145*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x
]])/(28812*Sqrt[21])

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Rubi in Sympy [A]  time = 13.6688, size = 114, normalized size = 0.89 \[ - \frac{9145 \sqrt{- 2 x + 1}}{57624 \left (3 x + 2\right )} - \frac{9145 \sqrt{- 2 x + 1}}{24696 \left (3 x + 2\right )^{2}} - \frac{1829 \sqrt{- 2 x + 1}}{1764 \left (3 x + 2\right )^{3}} - \frac{2179 \sqrt{- 2 x + 1}}{588 \left (3 x + 2\right )^{4}} - \frac{9145 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{605052} + \frac{121}{14 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9145*sqrt(-2*x + 1)/(57624*(3*x + 2)) - 9145*sqrt(-2*x + 1)/(24696*(3*x + 2)**2
) - 1829*sqrt(-2*x + 1)/(1764*(3*x + 2)**3) - 2179*sqrt(-2*x + 1)/(588*(3*x + 2)
**4) - 9145*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/605052 + 121/(14*sqrt(-2*x
+ 1)*(3*x + 2)**4)

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Mathematica [A]  time = 0.176661, size = 68, normalized size = 0.53 \[ \frac{\frac{21 \left (493830 x^4+1124835 x^3+843169 x^2+218578 x+6486\right )}{\sqrt{1-2 x} (3 x+2)^4}-18290 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1210104} \]

Antiderivative was successfully verified.

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

[Out]

((21*(6486 + 218578*x + 843169*x^2 + 1124835*x^3 + 493830*x^4))/(Sqrt[1 - 2*x]*(
2 + 3*x)^4) - 18290*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 \[{\frac{968}{16807}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{648}{16807\, \left ( -4-6\,x \right ) ^{4}} \left ({\frac{29167}{288} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{2001923}{2592} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{15060395}{7776} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{12452615}{7776}\sqrt{1-2\,x}} \right ) }-{\frac{9145\,\sqrt{21}}{605052}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

968/16807/(1-2*x)^(1/2)+648/16807*(29167/288*(1-2*x)^(7/2)-2001923/2592*(1-2*x)^
(5/2)[phone]/7776*(1-2*x)^(3/2)-12452615/7776*(1-2*x)^(1/2))/(-4-6*x)^4-9145/6
05052*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.50552, size = 161, normalized size = 1.26 \[ \frac{9145}{1210104} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{246915 \,{\left (2 \, x - 1\right )}^{4} + 2112495 \,{\left (2 \, x - 1\right )}^{3} + 6542333 \,{\left (2 \, x - 1\right )}^{2} + 17218306 \, x - 4624865}{28812 \,{\left (81 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 756 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 2646 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 4116 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 2401 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9145/1210104*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) + 1/28812*(246915*(2*x - 1)^4 + 2112495*(2*x - 1)^3 + 6542333*(2*x - 1)^
2 + 17218306*x - 4624865)/(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 = 0.224804, size = 157, normalized size = 1.23 \[ \frac{\sqrt{21}{\left (9145 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{21}{\left (493830 \, x^{4} + 1124835 \, x^{3} + 843169 \, x^{2} + 218578 \, x + 6486\right )}\right )}}{1210104 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1210104*sqrt(21)*(9145*(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)*(493830*x^4
+ 1124835*x^3 + 843169*x^2 + 218578*x + 6486))/((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)**2/(1-2*x)**(3/2)/(2+3*x)**5,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.223905, size = 147, normalized size = 1.15 \[ \frac{9145}{1210104} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{968}{16807 \, \sqrt{-2 \, x + 1}} - \frac{787509 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 6005769 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 15060395 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 12452615 \, \sqrt{-2 \, x + 1}}{3226944 \,{\left (3 \, x + 2\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9145/1210104*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*s
qrt(-2*x + 1))) + 968/16807/sqrt(-2*x + 1) - 1/3226944*(787509*(2*x - 1)^3*sqrt(
-2*x + 1) + 6005769*(2*x - 1)^2*sqrt(-2*x + 1) - 15060395*(-2*x + 1)^(3/2) + 124
52615*sqrt(-2*x + 1))/(3*x + 2)^4