3.2127 \(\int \frac{3+5 x}{(1-2 x)^{5/2} (2+3 x)^4} \, dx\)

Optimal. Leaf size=130 \[ -\frac{240 \sqrt{1-2 x}}{2401 (3 x+2)}-\frac{80 \sqrt{1-2 x}}{343 (3 x+2)^2}+\frac{64}{147 \sqrt{1-2 x} (3 x+2)^2}+\frac{64}{441 (1-2 x)^{3/2} (3 x+2)^2}+\frac{1}{63 (1-2 x)^{3/2} (3 x+2)^3}-\frac{160 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2401} \]

[Out]

1/(63*(1 - 2*x)^(3/2)*(2 + 3*x)^3) + 64/(441*(1 - 2*x)^(3/2)*(2 + 3*x)^2) + 64/(
147*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (80*Sqrt[1 - 2*x])/(343*(2 + 3*x)^2) - (240*Sqr
t[1 - 2*x])/(2401*(2 + 3*x)) - (160*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/
2401

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Rubi [A]  time = 0.128117, antiderivative size = 130, 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 \[ -\frac{240 \sqrt{1-2 x}}{2401 (3 x+2)}-\frac{80 \sqrt{1-2 x}}{343 (3 x+2)^2}+\frac{64}{147 \sqrt{1-2 x} (3 x+2)^2}+\frac{64}{441 (1-2 x)^{3/2} (3 x+2)^2}+\frac{1}{63 (1-2 x)^{3/2} (3 x+2)^3}-\frac{160 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2401} \]

Antiderivative was successfully verified.

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

[Out]

1/(63*(1 - 2*x)^(3/2)*(2 + 3*x)^3) + 64/(441*(1 - 2*x)^(3/2)*(2 + 3*x)^2) + 64/(
147*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (80*Sqrt[1 - 2*x])/(343*(2 + 3*x)^2) - (240*Sqr
t[1 - 2*x])/(2401*(2 + 3*x)) - (160*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/
2401

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Rubi in Sympy [A]  time = 12.2645, size = 102, normalized size = 0.78 \[ - \frac{160 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{16807} + \frac{160}{2401 \sqrt{- 2 x + 1}} + \frac{160}{3087 \left (- 2 x + 1\right )^{\frac{3}{2}}} - \frac{16}{147 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )} - \frac{16}{147 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2}} + \frac{1}{63 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-160*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/16807 + 160/(2401*sqrt(-2*x + 1))
+ 160/(3087*(-2*x + 1)**(3/2)) - 16/(147*(-2*x + 1)**(3/2)*(3*x + 2)) - 16/(147
*(-2*x + 1)**(3/2)*(3*x + 2)**2) + 1/(63*(-2*x + 1)**(3/2)*(3*x + 2)**3)

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Mathematica [A]  time = 0.155191, size = 68, normalized size = 0.52 \[ \frac{\frac{7 \left (-25920 x^4-28800 x^3+4464 x^2+11280 x+2237\right )}{(1-2 x)^{3/2} (3 x+2)^3}-480 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{50421} \]

Antiderivative was successfully verified.

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

[Out]

((7*(2237 + 11280*x + 4464*x^2 - 28800*x^3 - 25920*x^4))/((1 - 2*x)^(3/2)*(2 + 3
*x)^3) - 480*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/50421

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Maple [A]  time = 0.02, size = 75, normalized size = 0.6 \[{\frac{88}{7203} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{776}{16807}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{648}{16807\, \left ( -4-6\,x \right ) ^{3}} \left ({\frac{43}{3} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{1960}{27} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{2450}{27}\sqrt{1-2\,x}} \right ) }-{\frac{160\,\sqrt{21}}{16807}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

88/7203/(1-2*x)^(3/2)+776/16807/(1-2*x)^(1/2)+648/16807*(43/3*(1-2*x)^(5/2)-1960
/27*(1-2*x)^(3/2)+2450/27*(1-2*x)^(1/2))/(-4-6*x)^3-160/16807*arctanh(1/7*21^(1/
2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.49705, size = 149, normalized size = 1.15 \[ \frac{80}{16807} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{8 \,{\left (1620 \,{\left (2 \, x - 1\right )}^{4} + 10080 \,{\left (2 \, x - 1\right )}^{3} + 19404 \,{\left (2 \, x - 1\right )}^{2} + 18816 \, x - 13181\right )}}{7203 \,{\left (27 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 189 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 441 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 343 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

80/16807*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1
))) + 8/7203*(1620*(2*x - 1)^4 + 10080*(2*x - 1)^3 + 19404*(2*x - 1)^2 + 18816*x
- 13181)/(27*(-2*x + 1)^(9/2) - 189*(-2*x + 1)^(7/2) + 441*(-2*x + 1)^(5/2) - 3
43*(-2*x + 1)^(3/2))

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Fricas [A]  time = 0.220115, size = 165, normalized size = 1.27 \[ \frac{\sqrt{7}{\left (240 \, \sqrt{3}{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} + 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{7}{\left (25920 \, x^{4} + 28800 \, x^{3} - 4464 \, x^{2} - 11280 \, x - 2237\right )}\right )}}{50421 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/50421*sqrt(7)*(240*sqrt(3)*(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*sqrt(-2*x + 1
)*log((sqrt(7)*(3*x - 5) + 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2)) + sqrt(7)*(25920
*x^4 + 28800*x^3 - 4464*x^2 - 11280*x - 2237))/((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)/(1-2*x)**(5/2)/(2+3*x)**4,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.238239, size = 128, normalized size = 0.98 \[ \frac{80}{16807} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{8 \,{\left (1620 \,{\left (2 \, x - 1\right )}^{4} + 10080 \,{\left (2 \, x - 1\right )}^{3} + 19404 \,{\left (2 \, x - 1\right )}^{2} + 18816 \, x - 13181\right )}}{7203 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 7 \, \sqrt{-2 \, x + 1}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

80/16807*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(
-2*x + 1))) + 8/7203*(1620*(2*x - 1)^4 + 10080*(2*x - 1)^3 + 19404*(2*x - 1)^2 +
18816*x - 13181)/(3*(-2*x + 1)^(3/2) - 7*sqrt(-2*x + 1))^3