3.2631 \(\int \frac {(2+3 x)^2}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=89 \[ -\frac {3298 \sqrt {1-2 x}}{43923 \sqrt {5 x+3}}-\frac {1649 \sqrt {1-2 x}}{7986 (5 x+3)^{3/2}}+\frac {14}{121 (5 x+3)^{3/2} \sqrt {1-2 x}}+\frac {49}{66 (5 x+3)^{3/2} (1-2 x)^{3/2}} \]

[Out]

49/66/(1-2*x)^(3/2)/(3+5*x)^(3/2)+14/121/(3+5*x)^(3/2)/(1-2*x)^(1/2)-1649/7986*(1-2*x)^(1/2)/(3+5*x)^(3/2)-329
8/43923*(1-2*x)^(1/2)/(3+5*x)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {89, 78, 45, 37} \[ -\frac {3298 \sqrt {1-2 x}}{43923 \sqrt {5 x+3}}-\frac {1649 \sqrt {1-2 x}}{7986 (5 x+3)^{3/2}}+\frac {14}{121 (5 x+3)^{3/2} \sqrt {1-2 x}}+\frac {49}{66 (5 x+3)^{3/2} (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

49/(66*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + 14/(121*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (1649*Sqrt[1 - 2*x])/(7986*
(3 + 5*x)^(3/2)) - (3298*Sqrt[1 - 2*x])/(43923*Sqrt[3 + 5*x])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rubi steps

\begin {align*} \int \frac {(2+3 x)^2}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx &=\frac {49}{66 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1}{66} \int \frac {-\frac {381}{2}+297 x}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=\frac {49}{66 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {14}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {1649}{484} \int \frac {1}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {49}{66 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {14}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {1649 \sqrt {1-2 x}}{7986 (3+5 x)^{3/2}}+\frac {1649 \int \frac {1}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx}{3993}\\ &=\frac {49}{66 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {14}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {1649 \sqrt {1-2 x}}{7986 (3+5 x)^{3/2}}-\frac {3298 \sqrt {1-2 x}}{43923 \sqrt {3+5 x}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 37, normalized size = 0.42 \[ \frac {-65960 x^3-9894 x^2+49200 x+18728}{43923 (1-2 x)^{3/2} (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(18728 + 49200*x - 9894*x^2 - 65960*x^3)/(43923*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))

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fricas [A]  time = 1.04, size = 53, normalized size = 0.60 \[ -\frac {2 \, {\left (32980 \, x^{3} + 4947 \, x^{2} - 24600 \, x - 9364\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{43923 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/43923*(32980*x^3 + 4947*x^2 - 24600*x - 9364)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(100*x^4 + 20*x^3 - 59*x^2 - 6*x
+ 9)

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giac [B]  time = 1.39, size = 160, normalized size = 1.80 \[ -\frac {1}{3513840} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {1716 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} - \frac {14 \, {\left (164 \, \sqrt {5} {\left (5 \, x + 3\right )} - 1287 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1098075 \, {\left (2 \, x - 1\right )}^{2}} + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {429 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{219615 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3513840*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 1716*(sqrt(2)*sqrt(-10*x + 5) -
sqrt(22))/sqrt(5*x + 3)) - 14/1098075*(164*sqrt(5)*(5*x + 3) - 1287*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*
x - 1)^2 + 1/219615*sqrt(10)*(5*x + 3)^(3/2)*(429*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4)/(sqrt(
2)*sqrt(-10*x + 5) - sqrt(22))^3

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maple [A]  time = 0.00, size = 32, normalized size = 0.36 \[ -\frac {2 \left (32980 x^{3}+4947 x^{2}-24600 x -9364\right )}{43923 \left (5 x +3\right )^{\frac {3}{2}} \left (-2 x +1\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/43923*(32980*x^3+4947*x^2-24600*x-9364)/(5*x+3)^(3/2)/(-2*x+1)^(3/2)

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maxima [A]  time = 0.86, size = 59, normalized size = 0.66 \[ \frac {6596 \, x}{43923 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1649}{219615 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1229 \, x}{1815 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {733}{1815 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6596/43923*x/sqrt(-10*x^2 - x + 3) + 1649/219615/sqrt(-10*x^2 - x + 3) + 1229/1815*x/(-10*x^2 - x + 3)^(3/2) +
733/1815/(-10*x^2 - x + 3)^(3/2)

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mupad [B]  time = 2.52, size = 69, normalized size = 0.78 \[ \frac {\sqrt {5\,x+3}\,\left (-\frac {6596\,x^3}{219615}-\frac {1649\,x^2}{366025}+\frac {328\,x}{14641}+\frac {9364}{1098075}\right )}{\frac {6\,x\,\sqrt {1-2\,x}}{25}+\frac {9\,\sqrt {1-2\,x}}{50}-\frac {7\,x^2\,\sqrt {1-2\,x}}{10}-x^3\,\sqrt {1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((5*x + 3)^(1/2)*((328*x)/14641 - (1649*x^2)/366025 - (6596*x^3)/219615 + 9364/1098075))/((6*x*(1 - 2*x)^(1/2)
)/25 + (9*(1 - 2*x)^(1/2))/50 - (7*x^2*(1 - 2*x)^(1/2))/10 - x^3*(1 - 2*x)^(1/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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