3.2102 \(\int \frac{(2+3 x)^2}{(1-2 x)^{3/2} (3+5 x)^2} \, dx\)

Optimal. Leaf size=68 \[ -\frac{1227 \sqrt{1-2 x}}{1210 (5 x+3)}+\frac{49}{22 \sqrt{1-2 x} (5 x+3)}-\frac{138 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{605 \sqrt{55}} \]

[Out]

49/(22*Sqrt[1 - 2*x]*(3 + 5*x)) - (1227*Sqrt[1 - 2*x])/(1210*(3 + 5*x)) - (138*A
rcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(605*Sqrt[55])

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Rubi [A]  time = 0.0936686, antiderivative size = 68, 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 \[ -\frac{1227 \sqrt{1-2 x}}{1210 (5 x+3)}+\frac{49}{22 \sqrt{1-2 x} (5 x+3)}-\frac{138 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{605 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

49/(22*Sqrt[1 - 2*x]*(3 + 5*x)) - (1227*Sqrt[1 - 2*x])/(1210*(3 + 5*x)) - (138*A
rcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(605*Sqrt[55])

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Rubi in Sympy [A]  time = 8.41024, size = 53, normalized size = 0.78 \[ - \frac{138 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{33275} + \frac{1227}{3025 \sqrt{- 2 x + 1}} - \frac{1}{275 \sqrt{- 2 x + 1} \left (5 x + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-138*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/33275 + 1227/(3025*sqrt(-2*x + 1
)) - 1/(275*sqrt(-2*x + 1)*(5*x + 3))

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Mathematica [A]  time = 0.106869, size = 56, normalized size = 0.82 \[ -\frac{\sqrt{1-2 x} (1227 x+734)}{605 \left (10 x^2+x-3\right )}-\frac{138 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{605 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(734 + 1227*x))/(605*(-3 + x + 10*x^2)) - (138*ArcTanh[Sqrt[5/11
]*Sqrt[1 - 2*x]])/(605*Sqrt[55])

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Maple [A]  time = 0.019, size = 45, normalized size = 0.7 \[{\frac{49}{121}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{2}{3025}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{138\,\sqrt{55}}{33275}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

49/121/(1-2*x)^(1/2)+2/3025*(1-2*x)^(1/2)/(-6/5-2*x)-138/33275*arctanh(1/11*55^(
1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.50358, size = 88, normalized size = 1.29 \[ \frac{69}{33275} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2 \,{\left (1227 \, x + 734\right )}}{605 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

69/33275*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1
))) - 2/605*(1227*x + 734)/(5*(-2*x + 1)^(3/2) - 11*sqrt(-2*x + 1))

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Fricas [A]  time = 0.216766, size = 96, normalized size = 1.41 \[ \frac{\sqrt{55}{\left (69 \,{\left (5 \, x + 3\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{55}{\left (1227 \, x + 734\right )}\right )}}{33275 \,{\left (5 \, x + 3\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/33275*sqrt(55)*(69*(5*x + 3)*sqrt(-2*x + 1)*log((sqrt(55)*(5*x - 8) + 55*sqrt(
-2*x + 1))/(5*x + 3)) + sqrt(55)*(1227*x + 734))/((5*x + 3)*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)**2/(1-2*x)**(3/2)/(3+5*x)**2,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.222694, size = 92, normalized size = 1.35 \[ \frac{69}{33275} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{2 \,{\left (1227 \, x + 734\right )}}{605 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

69/33275*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt
(-2*x + 1))) - 2/605*(1227*x + 734)/(5*(-2*x + 1)^(3/2) - 11*sqrt(-2*x + 1))