3.2064 \(\int \frac{3+5 x}{(1-2 x)^{3/2} (2+3 x)^5} \, dx\)

Optimal. Leaf size=128 \[ -\frac{655 \sqrt{1-2 x}}{19208 (3 x+2)}-\frac{655 \sqrt{1-2 x}}{8232 (3 x+2)^2}-\frac{131 \sqrt{1-2 x}}{588 (3 x+2)^3}+\frac{131}{294 \sqrt{1-2 x} (3 x+2)^3}+\frac{1}{84 \sqrt{1-2 x} (3 x+2)^4}-\frac{655 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9604 \sqrt{21}} \]

[Out]

1/(84*Sqrt[1 - 2*x]*(2 + 3*x)^4) + 131/(294*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (131*Sq
rt[1 - 2*x])/(588*(2 + 3*x)^3) - (655*Sqrt[1 - 2*x])/(8232*(2 + 3*x)^2) - (655*S
qrt[1 - 2*x])/(19208*(2 + 3*x)) - (655*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(9604*S
qrt[21])

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Rubi [A]  time = 0.140805, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{655 \sqrt{1-2 x}}{19208 (3 x+2)}-\frac{655 \sqrt{1-2 x}}{8232 (3 x+2)^2}-\frac{131 \sqrt{1-2 x}}{588 (3 x+2)^3}+\frac{131}{294 \sqrt{1-2 x} (3 x+2)^3}+\frac{1}{84 \sqrt{1-2 x} (3 x+2)^4}-\frac{655 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9604 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

1/(84*Sqrt[1 - 2*x]*(2 + 3*x)^4) + 131/(294*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (131*Sq
rt[1 - 2*x])/(588*(2 + 3*x)^3) - (655*Sqrt[1 - 2*x])/(8232*(2 + 3*x)^2) - (655*S
qrt[1 - 2*x])/(19208*(2 + 3*x)) - (655*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(9604*S
qrt[21])

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Rubi in Sympy [A]  time = 13.837, size = 114, normalized size = 0.89 \[ - \frac{655 \sqrt{- 2 x + 1}}{19208 \left (3 x + 2\right )} - \frac{655 \sqrt{- 2 x + 1}}{8232 \left (3 x + 2\right )^{2}} - \frac{655 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{201684} + \frac{131}{882 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}} - \frac{131}{1764 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}} + \frac{1}{84 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-655*sqrt(-2*x + 1)/(19208*(3*x + 2)) - 655*sqrt(-2*x + 1)/(8232*(3*x + 2)**2) -
655*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/201684 + 131/(882*sqrt(-2*x + 1)*
(3*x + 2)**2) - 131/(1764*sqrt(-2*x + 1)*(3*x + 2)**3) + 1/(84*sqrt(-2*x + 1)*(3
*x + 2)**4)

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Mathematica [A]  time = 0.177045, size = 68, normalized size = 0.53 \[ \frac{\frac{21 \left (35370 x^4+80565 x^3+60391 x^2+10742 x-2566\right )}{\sqrt{1-2 x} (3 x+2)^4}-1310 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{403368} \]

Antiderivative was successfully verified.

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

[Out]

((21*(-2566 + 10742*x + 60391*x^2 + 80565*x^3 + 35370*x^4))/(Sqrt[1 - 2*x]*(2 +
3*x)^4) - 1310*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/403368

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Maple [A]  time = 0.019, size = 75, normalized size = 0.6 \[{\frac{176}{16807}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{1296}{16807\, \left ( -4-6\,x \right ) ^{4}} \left ({\frac{2473}{192} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{175637}{1728} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{1417325}{5184} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{142345}{576}\sqrt{1-2\,x}} \right ) }-{\frac{655\,\sqrt{21}}{201684}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

176/16807/(1-2*x)^(1/2)+1296/16807*(2473/192*(1-2*x)^(7/2)-175637/1728*(1-2*x)^(
5/2)+1417325/5184*(1-2*x)^(3/2)-142345/576*(1-2*x)^(1/2))/(-4-6*x)^4-655/201684*
arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.57874, size = 161, normalized size = 1.26 \[ \frac{655}{403368} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{17685 \,{\left (2 \, x - 1\right )}^{4} + 151305 \,{\left (2 \, x - 1\right )}^{3} + 468587 \,{\left (2 \, x - 1\right )}^{2} + 1193934 \, x - 355495}{9604 \,{\left (81 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 756 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 2646 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 4116 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 2401 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

655/403368*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x +
1))) + 1/9604*(17685*(2*x - 1)^4 + 151305*(2*x - 1)^3 + 468587*(2*x - 1)^2 + 11
93934*x - 355495)/(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 = 0.255312, size = 157, normalized size = 1.23 \[ \frac{\sqrt{21}{\left (655 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{21}{\left (35370 \, x^{4} + 80565 \, x^{3} + 60391 \, x^{2} + 10742 \, x - 2566\right )}\right )}}{403368 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/403368*sqrt(21)*(655*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*sqrt(-2*x + 1)*l
og((sqrt(21)*(3*x - 5) + 21*sqrt(-2*x + 1))/(3*x + 2)) + sqrt(21)*(35370*x^4 + 8
0565*x^3 + 60391*x^2 + 10742*x - 2566))/((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)/(1-2*x)**(3/2)/(2+3*x)**5,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.215566, size = 147, normalized size = 1.15 \[ \frac{655}{403368} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{176}{16807 \, \sqrt{-2 \, x + 1}} - \frac{66771 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 526911 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 1417325 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 1281105 \, \sqrt{-2 \, x + 1}}{1075648 \,{\left (3 \, x + 2\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

655/403368*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqr
t(-2*x + 1))) + 176/16807/sqrt(-2*x + 1) - 1/1075648*(66771*(2*x - 1)^3*sqrt(-2*
x + 1) + 526911*(2*x - 1)^2*sqrt(-2*x + 1) - 1417325*(-2*x + 1)^(3/2) + 1281105*
sqrt(-2*x + 1))/(3*x + 2)^4