3.2604 \(\int \frac{(2+3 x)^2}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{793 \sqrt{5 x+3}}{19965 \sqrt{1-2 x}}-\frac{1237}{3630 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{49}{66 (1-2 x)^{3/2} \sqrt{5 x+3}} \]

[Out]

49/(66*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) - 1237/(3630*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])
- (793*Sqrt[3 + 5*x])/(19965*Sqrt[1 - 2*x])

_______________________________________________________________________________________

Rubi [A]  time = 0.0910496, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{793 \sqrt{5 x+3}}{19965 \sqrt{1-2 x}}-\frac{1237}{3630 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{49}{66 (1-2 x)^{3/2} \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

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

[Out]

49/(66*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) - 1237/(3630*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])
- (793*Sqrt[3 + 5*x])/(19965*Sqrt[1 - 2*x])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.08141, size = 60, normalized size = 0.9 \[ - \frac{793 \sqrt{5 x + 3}}{19965 \sqrt{- 2 x + 1}} + \frac{1237 \sqrt{5 x + 3}}{9075 \left (- 2 x + 1\right )^{\frac{3}{2}}} - \frac{2}{275 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-793*sqrt(5*x + 3)/(19965*sqrt(-2*x + 1)) + 1237*sqrt(5*x + 3)/(9075*(-2*x + 1)*
*(3/2)) - 2/(275*(-2*x + 1)**(3/2)*sqrt(5*x + 3))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0515915, size = 32, normalized size = 0.48 \[ \frac{2 \left (793 x^2+1440 x+564\right )}{3993 (1-2 x)^{3/2} \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(564 + 1440*x + 793*x^2))/(3993*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 27, normalized size = 0.4 \[{\frac{1586\,{x}^{2}+2880\,x+1128}{3993} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3993*(793*x^2+1440*x+564)/(3+5*x)^(1/2)/(1-2*x)^(3/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.32976, size = 86, normalized size = 1.28 \[ -\frac{793 \, x}{3993 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{3673}{7986 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{49}{66 \,{\left (2 \, \sqrt{-10 \, x^{2} - x + 3} x - \sqrt{-10 \, x^{2} - x + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-793/3993*x/sqrt(-10*x^2 - x + 3) - 3673/7986/sqrt(-10*x^2 - x + 3) - 49/66/(2*s
qrt(-10*x^2 - x + 3)*x - sqrt(-10*x^2 - x + 3))

_______________________________________________________________________________________

Fricas [A]  time = 0.21601, size = 58, normalized size = 0.87 \[ \frac{2 \,{\left (793 \, x^{2} + 1440 \, x + 564\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{3993 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3993*(793*x^2 + 1440*x + 564)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(20*x^3 - 8*x^2 - 7
*x + 3)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (3 x + 2\right )^{2}}{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.270602, size = 135, normalized size = 2.01 \[ -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{13310 \, \sqrt{5 \, x + 3}} + \frac{14 \,{\left (23 \, \sqrt{5}{\left (5 \, x + 3\right )} + 66 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{99825 \,{\left (2 \, x - 1\right )}^{2}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{6655 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/13310*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 14/99825*
(23*sqrt(5)*(5*x + 3) + 66*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 +
2/6655*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))