3.2138 \(\int \frac{(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^4} \, dx\)

Optimal. Leaf size=128 \[ -\frac{1415 \sqrt{1-2 x}}{4802 (3 x+2)}-\frac{1415 \sqrt{1-2 x}}{2058 (3 x+2)^2}+\frac{566}{441 \sqrt{1-2 x} (3 x+2)^2}-\frac{1091}{882 \sqrt{1-2 x} (3 x+2)^3}+\frac{121}{42 (1-2 x)^{3/2} (3 x+2)^3}-\frac{1415 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2401 \sqrt{21}} \]

[Out]

121/(42*(1 - 2*x)^(3/2)*(2 + 3*x)^3) - 1091/(882*Sqrt[1 - 2*x]*(2 + 3*x)^3) + 56
6/(441*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (1415*Sqrt[1 - 2*x])/(2058*(2 + 3*x)^2) - (1
415*Sqrt[1 - 2*x])/(4802*(2 + 3*x)) - (1415*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(2
401*Sqrt[21])

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Rubi [A]  time = 0.159669, 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 \[ -\frac{1415 \sqrt{1-2 x}}{4802 (3 x+2)}-\frac{1415 \sqrt{1-2 x}}{2058 (3 x+2)^2}+\frac{566}{441 \sqrt{1-2 x} (3 x+2)^2}-\frac{1091}{882 \sqrt{1-2 x} (3 x+2)^3}+\frac{121}{42 (1-2 x)^{3/2} (3 x+2)^3}-\frac{1415 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2401 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

121/(42*(1 - 2*x)^(3/2)*(2 + 3*x)^3) - 1091/(882*Sqrt[1 - 2*x]*(2 + 3*x)^3) + 56
6/(441*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (1415*Sqrt[1 - 2*x])/(2058*(2 + 3*x)^2) - (1
415*Sqrt[1 - 2*x])/(4802*(2 + 3*x)) - (1415*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(2
401*Sqrt[21])

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Rubi in Sympy [A]  time = 14.3617, size = 114, normalized size = 0.89 \[ - \frac{1415 \sqrt{- 2 x + 1}}{4802 \left (3 x + 2\right )} - \frac{1415 \sqrt{- 2 x + 1}}{2058 \left (3 x + 2\right )^{2}} - \frac{1415 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{50421} + \frac{566}{441 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}} - \frac{1091}{882 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}} + \frac{121}{42 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1415*sqrt(-2*x + 1)/(4802*(3*x + 2)) - 1415*sqrt(-2*x + 1)/(2058*(3*x + 2)**2)
- 1415*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/50421 + 566/(441*sqrt(-2*x + 1)
*(3*x + 2)**2) - 1091/(882*sqrt(-2*x + 1)*(3*x + 2)**3) + 121/(42*(-2*x + 1)**(3
/2)*(3*x + 2)**3)

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Mathematica [A]  time = 0.161477, size = 68, normalized size = 0.53 \[ \frac{-\frac{7 \left (152820 x^4+169800 x^3-26319 x^2-83655 x-23872\right )}{(1-2 x)^{3/2} (3 x+2)^3}-2830 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{100842} \]

Antiderivative was successfully verified.

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

[Out]

((-7*(-23872 - 83655*x - 26319*x^2 + 169800*x^3 + 152820*x^4))/((1 - 2*x)^(3/2)*
(2 + 3*x)^3) - 2830*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/100842

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Maple [A]  time = 0.022, size = 75, normalized size = 0.6 \[{\frac{484}{7203} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{2728}{16807}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{108}{16807\, \left ( -4-6\,x \right ) ^{3}} \left ({\frac{1721}{12} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{17395}{27} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{78155}{108}\sqrt{1-2\,x}} \right ) }-{\frac{1415\,\sqrt{21}}{50421}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

484/7203/(1-2*x)^(3/2)+2728/16807/(1-2*x)^(1/2)+108/16807*(1721/12*(1-2*x)^(5/2)
-17395/27*(1-2*x)^(3/2)+78155/108*(1-2*x)^(1/2))/(-4-6*x)^3-1415/50421*arctanh(1
/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.48803, size = 149, normalized size = 1.16 \[ \frac{1415}{100842} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{38205 \,{\left (2 \, x - 1\right )}^{4} + 237720 \,{\left (2 \, x - 1\right )}^{3} + 457611 \,{\left (2 \, x - 1\right )}^{2} + 375144 \, x - 353584}{7203 \,{\left (27 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 189 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 441 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 343 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1415/100842*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) + 1/7203*(38205*(2*x - 1)^4 + 237720*(2*x - 1)^3 + 457611*(2*x - 1)^2 + 3
75144*x - 353584)/(27*(-2*x + 1)^(9/2) - 189*(-2*x + 1)^(7/2) + 441*(-2*x + 1)^(
5/2) - 343*(-2*x + 1)^(3/2))

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Fricas [A]  time = 0.215118, size = 157, normalized size = 1.23 \[ \frac{\sqrt{21}{\left (4245 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{21}{\left (152820 \, x^{4} + 169800 \, x^{3} - 26319 \, x^{2} - 83655 \, x - 23872\right )}\right )}}{302526 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/302526*sqrt(21)*(4245*(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*sqrt(-2*x + 1)*log
((sqrt(21)*(3*x - 5) + 21*sqrt(-2*x + 1))/(3*x + 2)) + sqrt(21)*(152820*x^4 + 16
9800*x^3 - 26319*x^2 - 83655*x - 23872))/((54*x^4 + 81*x^3 + 18*x^2 - 20*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)**2/(1-2*x)**(5/2)/(2+3*x)**4,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.213834, size = 128, normalized size = 1. \[ \frac{1415}{100842} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{38205 \,{\left (2 \, x - 1\right )}^{4} + 237720 \,{\left (2 \, x - 1\right )}^{3} + 457611 \,{\left (2 \, x - 1\right )}^{2} + 375144 \, x - 353584}{7203 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 7 \, \sqrt{-2 \, x + 1}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1415/100842*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sq
rt(-2*x + 1))) + 1/7203*(38205*(2*x - 1)^4 + 237720*(2*x - 1)^3 + 457611*(2*x -
1)^2 + 375144*x - 353584)/(3*(-2*x + 1)^(3/2) - 7*sqrt(-2*x + 1))^3