3.2014 \(\int \frac{(3+5 x)^3}{\sqrt{1-2 x} (2+3 x)^3} \, dx\)

Optimal. Leaf size=80 \[ \frac{\sqrt{1-2 x} (5 x+3)^2}{42 (3 x+2)^2}-\frac{\sqrt{1-2 x} (12425 x+8329)}{882 (3 x+2)}+\frac{2381 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{441 \sqrt{21}} \]

[Out]

(Sqrt[1 - 2*x]*(3 + 5*x)^2)/(42*(2 + 3*x)^2) - (Sqrt[1 - 2*x]*(8329 + 12425*x))/
(882*(2 + 3*x)) + (2381*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(441*Sqrt[21])

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Rubi [A]  time = 0.111224, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\sqrt{1-2 x} (5 x+3)^2}{42 (3 x+2)^2}-\frac{\sqrt{1-2 x} (12425 x+8329)}{882 (3 x+2)}+\frac{2381 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{441 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(3 + 5*x)^2)/(42*(2 + 3*x)^2) - (Sqrt[1 - 2*x]*(8329 + 12425*x))/
(882*(2 + 3*x)) + (2381*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(441*Sqrt[21])

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Rubi in Sympy [A]  time = 12.1511, size = 66, normalized size = 0.82 \[ - \frac{\sqrt{- 2 x + 1} \left (37275 x + 24987\right )}{2646 \left (3 x + 2\right )} + \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{2}}{42 \left (3 x + 2\right )^{2}} + \frac{2381 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{9261} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-2*x + 1)*(37275*x + 24987)/(2646*(3*x + 2)) + sqrt(-2*x + 1)*(5*x + 3)**2
/(42*(3*x + 2)**2) + 2381*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/9261

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Mathematica [A]  time = 0.114521, size = 58, normalized size = 0.72 \[ \frac{2381 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{441 \sqrt{21}}-\frac{\sqrt{1-2 x} \left (36750 x^2+49207 x+16469\right )}{882 (3 x+2)^2} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(16469 + 49207*x + 36750*x^2))/(882*(2 + 3*x)^2) + (2381*ArcTanh
[Sqrt[3/7]*Sqrt[1 - 2*x]])/(441*Sqrt[21])

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Maple [A]  time = 0.016, size = 57, normalized size = 0.7 \[ -{\frac{125}{27}\sqrt{1-2\,x}}-{\frac{2}{3\, \left ( -4-6\,x \right ) ^{2}} \left ( -{\frac{69}{98} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{205}{126}\sqrt{1-2\,x}} \right ) }+{\frac{2381\,\sqrt{21}}{9261}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-125/27*(1-2*x)^(1/2)-2/3*(-69/98*(1-2*x)^(3/2)+205/126*(1-2*x)^(1/2))/(-4-6*x)^
2+2381/9261*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.52617, size = 112, normalized size = 1.4 \[ -\frac{2381}{18522} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{125}{27} \, \sqrt{-2 \, x + 1} + \frac{621 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1435 \, \sqrt{-2 \, x + 1}}{1323 \,{\left (9 \,{\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2381/18522*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) - 125/27*sqrt(-2*x + 1) + 1/1323*(621*(-2*x + 1)^(3/2) - 1435*sqrt(-2*x +
1))/(9*(2*x - 1)^2 + 84*x + 7)

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Fricas [A]  time = 0.238761, size = 107, normalized size = 1.34 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (36750 \, x^{2} + 49207 \, x + 16469\right )} \sqrt{-2 \, x + 1} - 2381 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{18522 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/18522*sqrt(21)*(sqrt(21)*(36750*x^2 + 49207*x + 16469)*sqrt(-2*x + 1) - 2381*
(9*x^2 + 12*x + 4)*log((sqrt(21)*(3*x - 5) - 21*sqrt(-2*x + 1))/(3*x + 2)))/(9*x
^2 + 12*x + 4)

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.211114, size = 104, normalized size = 1.3 \[ -\frac{2381}{18522} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{125}{27} \, \sqrt{-2 \, x + 1} + \frac{621 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1435 \, \sqrt{-2 \, x + 1}}{5292 \,{\left (3 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2381/18522*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sq
rt(-2*x + 1))) - 125/27*sqrt(-2*x + 1) + 1/5292*(621*(-2*x + 1)^(3/2) - 1435*sqr
t(-2*x + 1))/(3*x + 2)^2