3.2013 \(\int \frac{(3+5 x)^3}{\sqrt{1-2 x} (2+3 x)^2} \, dx\)

Optimal. Leaf size=73 \[ \frac{\sqrt{1-2 x} (5 x+3)^2}{21 (3 x+2)}-\frac{10}{189} \sqrt{1-2 x} (95 x+214)-\frac{208 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{189 \sqrt{21}} \]

[Out]

(Sqrt[1 - 2*x]*(3 + 5*x)^2)/(21*(2 + 3*x)) - (10*Sqrt[1 - 2*x]*(214 + 95*x))/189
- (208*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(189*Sqrt[21])

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Rubi [A]  time = 0.107423, antiderivative size = 73, 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 \[ \frac{\sqrt{1-2 x} (5 x+3)^2}{21 (3 x+2)}-\frac{10}{189} \sqrt{1-2 x} (95 x+214)-\frac{208 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{189 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(3 + 5*x)^2)/(21*(2 + 3*x)) - (10*Sqrt[1 - 2*x]*(214 + 95*x))/189
- (208*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(189*Sqrt[21])

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Rubi in Sympy [A]  time = 11.7547, size = 60, normalized size = 0.82 \[ - \frac{\sqrt{- 2 x + 1} \left (2850 x + 6420\right )}{567} + \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{2}}{21 \left (3 x + 2\right )} - \frac{208 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{3969} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-2*x + 1)*(2850*x + 6420)/567 + sqrt(-2*x + 1)*(5*x + 3)**2/(21*(3*x + 2))
- 208*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/3969

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Mathematica [A]  time = 0.0951492, size = 58, normalized size = 0.79 \[ \frac{-\frac{21 \sqrt{1-2 x} \left (2625 x^2+8050 x+4199\right )}{3 x+2}-208 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3969} \]

Antiderivative was successfully verified.

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

[Out]

((-21*Sqrt[1 - 2*x]*(4199 + 8050*x + 2625*x^2))/(2 + 3*x) - 208*Sqrt[21]*ArcTanh
[Sqrt[3/7]*Sqrt[1 - 2*x]])/3969

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Maple [A]  time = 0.015, size = 54, normalized size = 0.7 \[{\frac{125}{54} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{725}{54}\sqrt{1-2\,x}}-{\frac{2}{567}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}-{\frac{208\,\sqrt{21}}{3969}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125/54*(1-2*x)^(3/2)-725/54*(1-2*x)^(1/2)-2/567*(1-2*x)^(1/2)/(-4/3-2*x)-208/396
9*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.50784, size = 96, normalized size = 1.32 \[ \frac{125}{54} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{104}{3969} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{725}{54} \, \sqrt{-2 \, x + 1} + \frac{\sqrt{-2 \, x + 1}}{189 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125/54*(-2*x + 1)^(3/2) + 104/3969*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(
sqrt(21) + 3*sqrt(-2*x + 1))) - 725/54*sqrt(-2*x + 1) + 1/189*sqrt(-2*x + 1)/(3*
x + 2)

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Fricas [A]  time = 0.239547, size = 93, normalized size = 1.27 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (2625 \, x^{2} + 8050 \, x + 4199\right )} \sqrt{-2 \, x + 1} - 104 \,{\left (3 \, x + 2\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{3969 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3969*sqrt(21)*(sqrt(21)*(2625*x^2 + 8050*x + 4199)*sqrt(-2*x + 1) - 104*(3*x
+ 2)*log((sqrt(21)*(3*x - 5) + 21*sqrt(-2*x + 1))/(3*x + 2)))/(3*x + 2)

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.217925, size = 100, normalized size = 1.37 \[ \frac{125}{54} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{104}{3969} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{725}{54} \, \sqrt{-2 \, x + 1} + \frac{\sqrt{-2 \, x + 1}}{189 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125/54*(-2*x + 1)^(3/2) + 104/3969*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x
+ 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 725/54*sqrt(-2*x + 1) + 1/189*sqrt(-2*x
+ 1)/(3*x + 2)