3.2126 \(\int \frac{3+5 x}{(1-2 x)^{5/2} (2+3 x)^3} \, dx\)

Optimal. Leaf size=96 \[ \frac{45}{343 \sqrt{1-2 x}}-\frac{3}{14 (1-2 x)^{3/2} (3 x+2)}+\frac{5}{49 (1-2 x)^{3/2}}+\frac{1}{42 (1-2 x)^{3/2} (3 x+2)^2}-\frac{45}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

[Out]

5/(49*(1 - 2*x)^(3/2)) + 45/(343*Sqrt[1 - 2*x]) + 1/(42*(1 - 2*x)^(3/2)*(2 + 3*x
)^2) - 3/(14*(1 - 2*x)^(3/2)*(2 + 3*x)) - (45*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1
- 2*x]])/343

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Rubi [A]  time = 0.106526, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{45}{343 \sqrt{1-2 x}}-\frac{3}{14 (1-2 x)^{3/2} (3 x+2)}+\frac{5}{49 (1-2 x)^{3/2}}+\frac{1}{42 (1-2 x)^{3/2} (3 x+2)^2}-\frac{45}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

5/(49*(1 - 2*x)^(3/2)) + 45/(343*Sqrt[1 - 2*x]) + 1/(42*(1 - 2*x)^(3/2)*(2 + 3*x
)^2) - 3/(14*(1 - 2*x)^(3/2)*(2 + 3*x)) - (45*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1
- 2*x]])/343

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Rubi in Sympy [A]  time = 10.2509, size = 83, normalized size = 0.86 \[ - \frac{45 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{2401} + \frac{45}{343 \sqrt{- 2 x + 1}} + \frac{5}{49 \left (- 2 x + 1\right )^{\frac{3}{2}}} - \frac{3}{14 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )} + \frac{1}{42 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-45*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/2401 + 45/(343*sqrt(-2*x + 1)) + 5
/(49*(-2*x + 1)**(3/2)) - 3/(14*(-2*x + 1)**(3/2)*(3*x + 2)) + 1/(42*(-2*x + 1)*
*(3/2)*(3*x + 2)**2)

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Mathematica [A]  time = 0.124331, size = 66, normalized size = 0.69 \[ \frac{\frac{7 \sqrt{1-2 x} \left (-4860 x^3-2160 x^2+2277 x+1087\right )}{\left (6 x^2+x-2\right )^2}-270 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{14406} \]

Antiderivative was successfully verified.

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

[Out]

((7*Sqrt[1 - 2*x]*(1087 + 2277*x - 2160*x^2 - 4860*x^3))/(-2 + x + 6*x^2)^2 - 27
0*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/14406

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Maple [A]  time = 0.02, size = 66, normalized size = 0.7 \[{\frac{44}{1029} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{256}{2401}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{324}{2401\, \left ( -4-6\,x \right ) ^{2}} \left ({\frac{59}{36} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{133}{36}\sqrt{1-2\,x}} \right ) }-{\frac{45\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

44/1029/(1-2*x)^(3/2)+256/2401/(1-2*x)^(1/2)+324/2401*(59/36*(1-2*x)^(3/2)-133/3
6*(1-2*x)^(1/2))/(-4-6*x)^2-45/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.50197, size = 124, normalized size = 1.29 \[ \frac{45}{4802} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{1215 \,{\left (2 \, x - 1\right )}^{3} + 4725 \,{\left (2 \, x - 1\right )}^{2} + 7056 \, x - 5684}{1029 \,{\left (9 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 42 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 49 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

45/4802*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1)
)) - 1/1029*(1215*(2*x - 1)^3 + 4725*(2*x - 1)^2 + 7056*x - 5684)/(9*(-2*x + 1)^
(7/2) - 42*(-2*x + 1)^(5/2) + 49*(-2*x + 1)^(3/2))

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Fricas [A]  time = 0.2176, size = 144, normalized size = 1.5 \[ \frac{\sqrt{7}{\left (135 \, \sqrt{3}{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} + 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{7}{\left (4860 \, x^{3} + 2160 \, x^{2} - 2277 \, x - 1087\right )}\right )}}{14406 \,{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/14406*sqrt(7)*(135*sqrt(3)*(18*x^3 + 15*x^2 - 4*x - 4)*sqrt(-2*x + 1)*log((sqr
t(7)*(3*x - 5) + 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2)) + sqrt(7)*(4860*x^3 + 2160
*x^2 - 2277*x - 1087))/((18*x^3 + 15*x^2 - 4*x - 4)*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)/(1-2*x)**(5/2)/(2+3*x)**3,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.234125, size = 120, normalized size = 1.25 \[ \frac{45}{4802} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{4 \,{\left (384 \, x - 269\right )}}{7203 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} + \frac{9 \,{\left (59 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 133 \, \sqrt{-2 \, x + 1}\right )}}{9604 \,{\left (3 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

45/4802*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-
2*x + 1))) + 4/7203*(384*x - 269)/((2*x - 1)*sqrt(-2*x + 1)) + 9/9604*(59*(-2*x
+ 1)^(3/2) - 133*sqrt(-2*x + 1))/(3*x + 2)^2