3.2632 \(\int \frac {2+3 x}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=89 \[ -\frac {2960 \sqrt {1-2 x}}{43923 \sqrt {5 x+3}}+\frac {296}{3993 \sqrt {5 x+3} \sqrt {1-2 x}}+\frac {74}{1815 \sqrt {5 x+3} (1-2 x)^{3/2}}-\frac {2}{165 (5 x+3)^{3/2} (1-2 x)^{3/2}} \]

[Out]

-2/165/(1-2*x)^(3/2)/(3+5*x)^(3/2)+74/1815/(1-2*x)^(3/2)/(3+5*x)^(1/2)+296/3993/(1-2*x)^(1/2)/(3+5*x)^(1/2)-29
60/43923*(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 = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {78, 45, 37} \[ -\frac {2960 \sqrt {1-2 x}}{43923 \sqrt {5 x+3}}+\frac {296}{3993 \sqrt {5 x+3} \sqrt {1-2 x}}+\frac {74}{1815 \sqrt {5 x+3} (1-2 x)^{3/2}}-\frac {2}{165 (5 x+3)^{3/2} (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(165*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + 74/(1815*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) + 296/(3993*Sqrt[1 - 2*x]*S
qrt[3 + 5*x]) - (2960*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]))))

Rubi steps

\begin {align*} \int \frac {2+3 x}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx &=-\frac {2}{165 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {37}{55} \int \frac {1}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx\\ &=-\frac {2}{165 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {74}{1815 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {148}{363} \int \frac {1}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=-\frac {2}{165 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {74}{1815 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {296}{3993 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {1480 \int \frac {1}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx}{3993}\\ &=-\frac {2}{165 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {74}{1815 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {296}{3993 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {2960 \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 \[ \frac {-59200 x^3-8880 x^2+26418 x+5728}{43923 (1-2 x)^{3/2} (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5728 + 26418*x - 8880*x^2 - 59200*x^3)/(43923*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))

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fricas [A]  time = 1.00, size = 53, normalized size = 0.60 \[ -\frac {2 \, {\left (29600 \, x^{3} + 4440 \, x^{2} - 13209 \, x - 2864\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{43923 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/43923*(29600*x^3 + 4440*x^2 - 13209*x - 2864)*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 = 1.36, size = 160, normalized size = 1.80 \[ -\frac {1}{702768} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {924 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} - \frac {8 \, {\left (181 \, \sqrt {5} {\left (5 \, x + 3\right )} - 1188 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1098075 \, {\left (2 \, x - 1\right )}^{2}} + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {231 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{43923 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/702768*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 924*(sqrt(2)*sqrt(-10*x + 5) - sq
rt(22))/sqrt(5*x + 3)) - 8/1098075*(181*sqrt(5)*(5*x + 3) - 1188*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x -
1)^2 + 1/43923*sqrt(10)*(5*x + 3)^(3/2)*(231*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4)/(sqrt(2)*s
qrt(-10*x + 5) - sqrt(22))^3

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maple [A]  time = 0.00, size = 32, normalized size = 0.36 \[ -\frac {2 \left (29600 x^{3}+4440 x^{2}-13209 x -2864\right )}{43923 \left (5 x +3\right )^{\frac {3}{2}} \left (-2 x +1\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/43923*(29600*x^3+4440*x^2-13209*x-2864)/(5*x+3)^(3/2)/(-2*x+1)^(3/2)

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maxima [A]  time = 0.58, size = 59, normalized size = 0.66 \[ \frac {5920 \, x}{43923 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {296}{43923 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {74 \, x}{363 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {40}{363 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5920/43923*x/sqrt(-10*x^2 - x + 3) + 296/43923/sqrt(-10*x^2 - x + 3) + 74/363*x/(-10*x^2 - x + 3)^(3/2) + 40/3
63/(-10*x^2 - x + 3)^(3/2)

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mupad [B]  time = 0.27, size = 69, normalized size = 0.78 \[ \frac {\sqrt {5\,x+3}\,\left (-\frac {1184\,x^3}{43923}-\frac {296\,x^2}{73205}+\frac {4403\,x}{366025}+\frac {2864}{1098075}\right )}{\frac {6\,x\,\sqrt {1-2\,x}}{25}+\frac {9\,\sqrt {1-2\,x}}{50}-\frac {7\,x^2\,\sqrt {1-2\,x}}{10}-x^3\,\sqrt {1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((5*x + 3)^(1/2)*((4403*x)/366025 - (296*x^2)/73205 - (1184*x^3)/43923 + 2864/1098075))/((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]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {3 x + 2}{\left (1 - 2 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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