3.2110 \(\int \frac{(2+3 x)^4}{(1-2 x)^{3/2} (3+5 x)^3} \, dx\)

Optimal. Leaf size=107 \[ \frac{7 (3 x+2)^3}{11 \sqrt{1-2 x} (5 x+3)^2}-\frac{71 \sqrt{1-2 x} (3 x+2)^2}{1210 (5 x+3)^2}+\frac{9 \sqrt{1-2 x} (5093 x+3044)}{13310 (5 x+3)}-\frac{111 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331 \sqrt{55}} \]

[Out]

(-71*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(1210*(3 + 5*x)^2) + (7*(2 + 3*x)^3)/(11*Sqrt[1
- 2*x]*(3 + 5*x)^2) + (9*Sqrt[1 - 2*x]*(3044 + 5093*x))/(13310*(3 + 5*x)) - (111
*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1331*Sqrt[55])

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Rubi [A]  time = 0.172062, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{7 (3 x+2)^3}{11 \sqrt{1-2 x} (5 x+3)^2}-\frac{71 \sqrt{1-2 x} (3 x+2)^2}{1210 (5 x+3)^2}+\frac{9 \sqrt{1-2 x} (5093 x+3044)}{13310 (5 x+3)}-\frac{111 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(-71*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(1210*(3 + 5*x)^2) + (7*(2 + 3*x)^3)/(11*Sqrt[1
- 2*x]*(3 + 5*x)^2) + (9*Sqrt[1 - 2*x]*(3044 + 5093*x))/(13310*(3 + 5*x)) - (111
*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1331*Sqrt[55])

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Rubi in Sympy [A]  time = 19.1288, size = 94, normalized size = 0.88 \[ - \frac{71 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}}{1210 \left (5 x + 3\right )^{2}} + \frac{\sqrt{- 2 x + 1} \left (1145925 x + 684900\right )}{332750 \left (5 x + 3\right )} - \frac{111 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{73205} + \frac{7 \left (3 x + 2\right )^{3}}{11 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-71*sqrt(-2*x + 1)*(3*x + 2)**2/(1210*(5*x + 3)**2) + sqrt(-2*x + 1)*(1145925*x
+ 684900)/(332750*(5*x + 3)) - 111*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/73
205 + 7*(3*x + 2)**3/(11*sqrt(-2*x + 1)*(5*x + 3)**2)

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Mathematica [A]  time = 0.186293, size = 63, normalized size = 0.59 \[ \frac{\frac{11 \left (-215622 x^3+149298 x^2+411911 x+146824\right )}{\sqrt{1-2 x} (5 x+3)^2}-222 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{146410} \]

Antiderivative was successfully verified.

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

[Out]

((11*(146824 + 411911*x + 149298*x^2 - 215622*x^3))/(Sqrt[1 - 2*x]*(3 + 5*x)^2)
- 222*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/146410

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Maple [A]  time = 0.02, size = 66, normalized size = 0.6 \[{\frac{81}{250}\sqrt{1-2\,x}}+{\frac{2401}{2662}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{4}{6655\, \left ( -6-10\,x \right ) ^{2}} \left ({\frac{271}{20} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{3003}{100}\sqrt{1-2\,x}} \right ) }-{\frac{111\,\sqrt{55}}{73205}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

81/250*(1-2*x)^(1/2)+2401/2662/(1-2*x)^(1/2)+4/6655*(271/20*(1-2*x)^(3/2)-3003/1
00*(1-2*x)^(1/2))/(-6-10*x)^2-111/73205*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^
(1/2)

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Maxima [A]  time = 1.50872, size = 124, normalized size = 1.16 \[ \frac{111}{146410} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{81}{250} \, \sqrt{-2 \, x + 1} + \frac{7505835 \,{\left (2 \, x - 1\right )}^{2} + 66039512 \, x + 3295369}{332750 \,{\left (25 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 110 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 121 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

111/146410*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x +
1))) + 81/250*sqrt(-2*x + 1) + 1/332750*(7505835*(2*x - 1)^2 + 66039512*x + 329
5369)/(25*(-2*x + 1)^(5/2) - 110*(-2*x + 1)^(3/2) + 121*sqrt(-2*x + 1))

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Fricas [A]  time = 0.224368, size = 124, normalized size = 1.16 \[ \frac{\sqrt{55}{\left (555 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) - \sqrt{55}{\left (215622 \, x^{3} - 149298 \, x^{2} - 411911 \, x - 146824\right )}\right )}}{732050 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/732050*sqrt(55)*(555*(25*x^2 + 30*x + 9)*sqrt(-2*x + 1)*log((sqrt(55)*(5*x - 8
) + 55*sqrt(-2*x + 1))/(5*x + 3)) - sqrt(55)*(215622*x^3 - 149298*x^2 - 411911*x
- 146824))/((25*x^2 + 30*x + 9)*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((2+3*x)**4/(1-2*x)**(3/2)/(3+5*x)**3,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.21966, size = 116, normalized size = 1.08 \[ \frac{111}{146410} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{81}{250} \, \sqrt{-2 \, x + 1} + \frac{2401}{2662 \, \sqrt{-2 \, x + 1}} + \frac{1355 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 3003 \, \sqrt{-2 \, x + 1}}{665500 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

111/146410*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sq
rt(-2*x + 1))) + 81/250*sqrt(-2*x + 1) + 2401/2662/sqrt(-2*x + 1) + 1/665500*(13
55*(-2*x + 1)^(3/2) - 3003*sqrt(-2*x + 1))/(5*x + 3)^2