3.2111 \(\int \frac{(2+3 x)^3}{(1-2 x)^{3/2} (3+5 x)^3} \, dx\)

Optimal. Leaf size=80 \[ \frac{7 (3 x+2)^2}{11 \sqrt{1-2 x} (5 x+3)^2}-\frac{\sqrt{1-2 x} (24825 x+15676)}{66550 (5 x+3)^2}-\frac{7143 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{33275 \sqrt{55}} \]

[Out]

(7*(2 + 3*x)^2)/(11*Sqrt[1 - 2*x]*(3 + 5*x)^2) - (Sqrt[1 - 2*x]*(15676 + 24825*x
))/(66550*(3 + 5*x)^2) - (7143*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(33275*Sqrt[55
])

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Rubi [A]  time = 0.110009, 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 \[ \frac{7 (3 x+2)^2}{11 \sqrt{1-2 x} (5 x+3)^2}-\frac{\sqrt{1-2 x} (24825 x+15676)}{66550 (5 x+3)^2}-\frac{7143 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{33275 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(7*(2 + 3*x)^2)/(11*Sqrt[1 - 2*x]*(3 + 5*x)^2) - (Sqrt[1 - 2*x]*(15676 + 24825*x
))/(66550*(3 + 5*x)^2) - (7143*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(33275*Sqrt[55
])

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Rubi in Sympy [A]  time = 11.8903, size = 71, normalized size = 0.89 \[ - \frac{\sqrt{- 2 x + 1} \left (24825 x + 15676\right )}{66550 \left (5 x + 3\right )^{2}} - \frac{7143 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{1830125} + \frac{7 \left (3 x + 2\right )^{2}}{11 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-2*x + 1)*(24825*x + 15676)/(66550*(5*x + 3)**2) - 7143*sqrt(55)*atanh(sqr
t(55)*sqrt(-2*x + 1)/11)/1830125 + 7*(3*x + 2)**2/(11*sqrt(-2*x + 1)*(5*x + 3)**
2)

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Mathematica [A]  time = 0.127547, size = 58, normalized size = 0.72 \[ \frac{\frac{55 \left (430800 x^2+514727 x+153724\right )}{\sqrt{1-2 x} (5 x+3)^2}-14286 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3660250} \]

Antiderivative was successfully verified.

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

[Out]

((55*(153724 + 514727*x + 430800*x^2))/(Sqrt[1 - 2*x]*(3 + 5*x)^2) - 14286*Sqrt[
55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/3660250

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Maple [A]  time = 0.02, size = 57, normalized size = 0.7 \[{\frac{343}{1331}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{50}{1331\, \left ( -6-10\,x \right ) ^{2}} \left ({\frac{41}{50} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{2277}{1250}\sqrt{1-2\,x}} \right ) }-{\frac{7143\,\sqrt{55}}{1830125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

343/1331/(1-2*x)^(1/2)+50/1331*(41/50*(1-2*x)^(3/2)-2277/1250*(1-2*x)^(1/2))/(-6
-10*x)^2-7143/1830125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.50857, size = 112, normalized size = 1.4 \[ \frac{7143}{3660250} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2 \,{\left (107700 \,{\left (2 \, x - 1\right )}^{2} + 945527 \, x + 46024\right )}}{33275 \,{\left (25 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 110 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 121 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7143/3660250*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x
+ 1))) + 2/33275*(107700*(2*x - 1)^2 + 945527*x + 46024)/(25*(-2*x + 1)^(5/2) -
110*(-2*x + 1)^(3/2) + 121*sqrt(-2*x + 1))

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Fricas [A]  time = 0.222368, size = 116, normalized size = 1.45 \[ \frac{\sqrt{55}{\left (7143 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{55}{\left (430800 \, x^{2} + 514727 \, x + 153724\right )}\right )}}{3660250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3660250*sqrt(55)*(7143*(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)*(430800*x^2 + 514727*x + 153724))
/((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)**3/(1-2*x)**(3/2)/(3+5*x)**3,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.22034, size = 104, normalized size = 1.3 \[ \frac{7143}{3660250} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{343}{1331 \, \sqrt{-2 \, x + 1}} + \frac{1025 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2277 \, \sqrt{-2 \, x + 1}}{133100 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7143/3660250*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*
sqrt(-2*x + 1))) + 343/1331/sqrt(-2*x + 1) + 1/133100*(1025*(-2*x + 1)^(3/2) - 2
277*sqrt(-2*x + 1))/(5*x + 3)^2