3.2613 \(\int \frac{(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=96 \[ \frac{2 (3 x+2)^3}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{8182 \sqrt{1-2 x}}{219615 \sqrt{5 x+3}}-\frac{3679 \sqrt{1-2 x}}{19965 (5 x+3)^{3/2}}+\frac{49}{121 \sqrt{1-2 x} (5 x+3)^{3/2}} \]

[Out]

49/(121*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (3679*Sqrt[1 - 2*x])/(19965*(3 + 5*x)^(
3/2)) + (2*(2 + 3*x)^3)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) - (8182*Sqrt[1 - 2*
x])/(219615*Sqrt[3 + 5*x])

_______________________________________________________________________________________

Rubi [A]  time = 0.134302, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 (3 x+2)^3}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{8182 \sqrt{1-2 x}}{219615 \sqrt{5 x+3}}-\frac{3679 \sqrt{1-2 x}}{19965 (5 x+3)^{3/2}}+\frac{49}{121 \sqrt{1-2 x} (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

49/(121*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (3679*Sqrt[1 - 2*x])/(19965*(3 + 5*x)^(
3/2)) + (2*(2 + 3*x)^3)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) - (8182*Sqrt[1 - 2*
x])/(219615*Sqrt[3 + 5*x])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.3508, size = 87, normalized size = 0.91 \[ - \frac{11102 \sqrt{5 x + 3}}{219615 \sqrt{- 2 x + 1}} - \frac{2 \left (3 x + 2\right )^{3}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{17318 \sqrt{5 x + 3}}{99825 \left (- 2 x + 1\right )^{\frac{3}{2}}} - \frac{28}{3025 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-11102*sqrt(5*x + 3)/(219615*sqrt(-2*x + 1)) - 2*(3*x + 2)**3/(33*(-2*x + 1)**(3
/2)*(5*x + 3)**(3/2)) + 17318*sqrt(5*x + 3)/(99825*(-2*x + 1)**(3/2)) - 28/(3025
*(-2*x + 1)**(3/2)*sqrt(5*x + 3))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0541571, size = 37, normalized size = 0.39 \[ \frac{2 \left (19573 x^3+62232 x^2+52044 x+13040\right )}{43923 (1-2 x)^{3/2} (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(13040 + 52044*x + 62232*x^2 + 19573*x^3))/(43923*(1 - 2*x)^(3/2)*(3 + 5*x)^(
3/2))

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 32, normalized size = 0.3 \[{\frac{39146\,{x}^{3}+124464\,{x}^{2}+104088\,x+26080}{43923} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/43923*(19573*x^3+62232*x^2+52044*x+13040)/(3+5*x)^(3/2)/(1-2*x)^(3/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.34686, size = 103, normalized size = 1.07 \[ -\frac{19573 \, x}{219615 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{27 \, x^{2}}{10 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{19573}{4392300 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{95567 \, x}{36300 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{22039}{36300 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-19573/219615*x/sqrt(-10*x^2 - x + 3) + 27/10*x^2/(-10*x^2 - x + 3)^(3/2) - 1957
3/4392300/sqrt(-10*x^2 - x + 3) + 95567/36300*x/(-10*x^2 - x + 3)^(3/2) + 22039/
36300/(-10*x^2 - x + 3)^(3/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.219934, size = 72, normalized size = 0.75 \[ \frac{2 \,{\left (19573 \, x^{3} + 62232 \, x^{2} + 52044 \, x + 13040\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{43923 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/43923*(19573*x^3 + 62232*x^2 + 52044*x + 13040)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(
100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.266256, size = 223, normalized size = 2.32 \[ -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{17569200 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} - \frac{19 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{133100 \, \sqrt{5 \, x + 3}} + \frac{98 \,{\left (17 \, \sqrt{5}{\left (5 \, x + 3\right )} + 99 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{1098075 \,{\left (2 \, x - 1\right )}^{2}} + \frac{{\left (\frac{627 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{1098075 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/17569200*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) - 19
/133100*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 98/1098075
*(17*sqrt(5)*(5*x + 3) + 99*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 +
1/1098075*(627*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4*sq
rt(10))*(5*x + 3)^(3/2)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3