3.2016 \(\int \frac{(3+5 x)^3}{\sqrt{1-2 x} (2+3 x)^5} \, dx\)

Optimal. Leaf size=100 \[ \frac{\sqrt{1-2 x} (5 x+3)^2}{84 (3 x+2)^4}+\frac{\sqrt{1-2 x} (4955 x+3168)}{10584 (3 x+2)^3}-\frac{42995 \sqrt{1-2 x}}{74088 (3 x+2)}-\frac{42995 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{37044 \sqrt{21}} \]

[Out]

(-42995*Sqrt[1 - 2*x])/(74088*(2 + 3*x)) + (Sqrt[1 - 2*x]*(3 + 5*x)^2)/(84*(2 +
3*x)^4) + (Sqrt[1 - 2*x]*(3168 + 4955*x))/(10584*(2 + 3*x)^3) - (42995*ArcTanh[S
qrt[3/7]*Sqrt[1 - 2*x]])/(37044*Sqrt[21])

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Rubi [A]  time = 0.130912, antiderivative size = 100, 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 \[ \frac{\sqrt{1-2 x} (5 x+3)^2}{84 (3 x+2)^4}+\frac{\sqrt{1-2 x} (4955 x+3168)}{10584 (3 x+2)^3}-\frac{42995 \sqrt{1-2 x}}{74088 (3 x+2)}-\frac{42995 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{37044 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(-42995*Sqrt[1 - 2*x])/(74088*(2 + 3*x)) + (Sqrt[1 - 2*x]*(3 + 5*x)^2)/(84*(2 +
3*x)^4) + (Sqrt[1 - 2*x]*(3168 + 4955*x))/(10584*(2 + 3*x)^3) - (42995*ArcTanh[S
qrt[3/7]*Sqrt[1 - 2*x]])/(37044*Sqrt[21])

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Rubi in Sympy [A]  time = 13.5307, size = 85, normalized size = 0.85 \[ - \frac{42995 \sqrt{- 2 x + 1}}{74088 \left (3 x + 2\right )} + \frac{\sqrt{- 2 x + 1} \left (104055 x + 66528\right )}{222264 \left (3 x + 2\right )^{3}} + \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{2}}{84 \left (3 x + 2\right )^{4}} - \frac{42995 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{777924} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-42995*sqrt(-2*x + 1)/(74088*(3*x + 2)) + sqrt(-2*x + 1)*(104055*x + 66528)/(222
264*(3*x + 2)**3) + sqrt(-2*x + 1)*(5*x + 3)**2/(84*(3*x + 2)**4) - 42995*sqrt(2
1)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/777924

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Mathematica [A]  time = 0.11309, size = 63, normalized size = 0.63 \[ \frac{-\frac{21 \sqrt{1-2 x} \left (1160865 x^3+2195625 x^2+1385462 x+291670\right )}{(3 x+2)^4}-85990 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1555848} \]

Antiderivative was successfully verified.

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

[Out]

((-21*Sqrt[1 - 2*x]*(291670 + 1385462*x + 2195625*x^2 + 1160865*x^3))/(2 + 3*x)^
4 - 85990*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/1555848

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Maple [A]  time = 0.017, size = 66, normalized size = 0.7 \[ -324\,{\frac{1}{ \left ( -4-6\,x \right ) ^{4}} \left ( -{\frac{42995\, \left ( 1-2\,x \right ) ^{7/2}}{444528}}+{\frac{374945\, \left ( 1-2\,x \right ) ^{5/2}}{571536}}-{\frac{363407\, \left ( 1-2\,x \right ) ^{3/2}}{244944}}+{\frac{274027\,\sqrt{1-2\,x}}{244944}} \right ) }-{\frac{42995\,\sqrt{21}}{777924}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-324*(-42995/444528*(1-2*x)^(7/2)+374945/571536*(1-2*x)^(5/2)-363407/244944*(1-2
*x)^(3/2)+274027/244944*(1-2*x)^(1/2))/(-4-6*x)^4-42995/777924*arctanh(1/7*21^(1
/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.53148, size = 149, normalized size = 1.49 \[ \frac{42995}{1555848} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{1160865 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 7873845 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 17806943 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 13427323 \, \sqrt{-2 \, x + 1}}{37044 \,{\left (81 \,{\left (2 \, x - 1\right )}^{4} + 756 \,{\left (2 \, x - 1\right )}^{3} + 2646 \,{\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

42995/1555848*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*
x + 1))) + 1/37044*(1160865*(-2*x + 1)^(7/2) - 7873845*(-2*x + 1)^(5/2) + 178069
43*(-2*x + 1)^(3/2) - 13427323*sqrt(-2*x + 1))/(81*(2*x - 1)^4 + 756*(2*x - 1)^3
+ 2646*(2*x - 1)^2 + 8232*x - 1715)

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Fricas [A]  time = 0.245313, size = 140, normalized size = 1.4 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (1160865 \, x^{3} + 2195625 \, x^{2} + 1385462 \, x + 291670\right )} \sqrt{-2 \, x + 1} - 42995 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{1555848 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1555848*sqrt(21)*(sqrt(21)*(1160865*x^3 + 2195625*x^2 + 1385462*x + 291670)*s
qrt(-2*x + 1) - 42995*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((sqrt(21)*(3*
x - 5) + 21*sqrt(-2*x + 1))/(3*x + 2)))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.213583, size = 135, normalized size = 1.35 \[ \frac{42995}{1555848} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{1160865 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 7873845 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 17806943 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 13427323 \, \sqrt{-2 \, x + 1}}{592704 \,{\left (3 \, x + 2\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

42995/1555848*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*
sqrt(-2*x + 1))) - 1/592704*(1160865*(2*x - 1)^3*sqrt(-2*x + 1) + 7873845*(2*x -
1)^2*sqrt(-2*x + 1) - 17806943*(-2*x + 1)^(3/2) + 13427323*sqrt(-2*x + 1))/(3*x
+ 2)^4