3.2032 \(\int \frac{(2+3 x)^4}{\sqrt{1-2 x} (3+5 x)^2} \, dx\)

Optimal. Leaf size=93 \[ -\frac{\sqrt{1-2 x} (3 x+2)^3}{55 (5 x+3)}-\frac{84 \sqrt{1-2 x} (3 x+2)^2}{1375}-\frac{21 \sqrt{1-2 x} (375 x+1144)}{6875}-\frac{266 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{6875 \sqrt{55}} \]

[Out]

(-84*Sqrt[1 - 2*x]*(2 + 3*x)^2)/1375 - (Sqrt[1 - 2*x]*(2 + 3*x)^3)/(55*(3 + 5*x)
) - (21*Sqrt[1 - 2*x]*(1144 + 375*x))/6875 - (266*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*
x]])/(6875*Sqrt[55])

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Rubi [A]  time = 0.161259, antiderivative size = 93, 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 \[ -\frac{\sqrt{1-2 x} (3 x+2)^3}{55 (5 x+3)}-\frac{84 \sqrt{1-2 x} (3 x+2)^2}{1375}-\frac{21 \sqrt{1-2 x} (375 x+1144)}{6875}-\frac{266 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{6875 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(-84*Sqrt[1 - 2*x]*(2 + 3*x)^2)/1375 - (Sqrt[1 - 2*x]*(2 + 3*x)^3)/(55*(3 + 5*x)
) - (21*Sqrt[1 - 2*x]*(1144 + 375*x))/6875 - (266*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*
x]])/(6875*Sqrt[55])

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Rubi in Sympy [A]  time = 18.9843, size = 80, normalized size = 0.86 \[ - \frac{\sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}}{55 \left (5 x + 3\right )} - \frac{84 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}}{1375} - \frac{\sqrt{- 2 x + 1} \left (118125 x + 360360\right )}{103125} - \frac{266 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{378125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-2*x + 1)*(3*x + 2)**3/(55*(5*x + 3)) - 84*sqrt(-2*x + 1)*(3*x + 2)**2/137
5 - sqrt(-2*x + 1)*(118125*x + 360360)/103125 - 266*sqrt(55)*atanh(sqrt(55)*sqrt
(-2*x + 1)/11)/378125

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Mathematica [A]  time = 0.113201, size = 63, normalized size = 0.68 \[ \frac{-\frac{55 \sqrt{1-2 x} \left (22275 x^3+82665 x^2+171765 x+78112\right )}{5 x+3}-266 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{378125} \]

Antiderivative was successfully verified.

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

[Out]

((-55*Sqrt[1 - 2*x]*(78112 + 171765*x + 82665*x^2 + 22275*x^3))/(3 + 5*x) - 266*
Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/378125

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Maple [A]  time = 0.016, size = 63, normalized size = 0.7 \[ -{\frac{81}{500} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{333}{250} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{12393}{2500}\sqrt{1-2\,x}}+{\frac{2}{34375}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{266\,\sqrt{55}}{378125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-81/500*(1-2*x)^(5/2)+333/250*(1-2*x)^(3/2)-12393/2500*(1-2*x)^(1/2)+2/34375*(1-
2*x)^(1/2)/(-6/5-2*x)-266/378125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.49908, size = 108, normalized size = 1.16 \[ -\frac{81}{500} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{333}{250} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{133}{378125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{12393}{2500} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{6875 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-81/500*(-2*x + 1)^(5/2) + 333/250*(-2*x + 1)^(3/2) + 133/378125*sqrt(55)*log(-(
sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 12393/2500*sqrt(-2
*x + 1) - 1/6875*sqrt(-2*x + 1)/(5*x + 3)

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Fricas [A]  time = 0.251424, size = 100, normalized size = 1.08 \[ -\frac{\sqrt{55}{\left (\sqrt{55}{\left (22275 \, x^{3} + 82665 \, x^{2} + 171765 \, x + 78112\right )} \sqrt{-2 \, x + 1} - 133 \,{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{378125 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/378125*sqrt(55)*(sqrt(55)*(22275*x^3 + 82665*x^2 + 171765*x + 78112)*sqrt(-2*
x + 1) - 133*(5*x + 3)*log((sqrt(55)*(5*x - 8) + 55*sqrt(-2*x + 1))/(5*x + 3)))/
(5*x + 3)

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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/(3+5*x)**2/(1-2*x)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.214161, size = 122, normalized size = 1.31 \[ -\frac{81}{500} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{333}{250} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{133}{378125} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{12393}{2500} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{6875 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-81/500*(2*x - 1)^2*sqrt(-2*x + 1) + 333/250*(-2*x + 1)^(3/2) + 133/378125*sqrt(
55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) -
12393/2500*sqrt(-2*x + 1) - 1/6875*sqrt(-2*x + 1)/(5*x + 3)