3.2565 \(\int \frac{2+3 x}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=45 \[ \frac{74 \sqrt{5 x+3}}{605 \sqrt{1-2 x}}-\frac{2}{55 \sqrt{1-2 x} \sqrt{5 x+3}} \]

[Out]

-2/(55*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (74*Sqrt[3 + 5*x])/(605*Sqrt[1 - 2*x])

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Rubi [A]  time = 0.0058102, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {78, 37} \[ \frac{74 \sqrt{5 x+3}}{605 \sqrt{1-2 x}}-\frac{2}{55 \sqrt{1-2 x} \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(55*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (74*Sqrt[3 + 5*x])/(605*Sqrt[1 - 2*x])

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 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]

Rubi steps

\begin{align*} \int \frac{2+3 x}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx &=-\frac{2}{55 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{37}{55} \int \frac{1}{(1-2 x)^{3/2} \sqrt{3+5 x}} \, dx\\ &=-\frac{2}{55 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{74 \sqrt{3+5 x}}{605 \sqrt{1-2 x}}\\ \end{align*}

Mathematica [A]  time = 0.0081187, size = 27, normalized size = 0.6 \[ \frac{2 (37 x+20)}{121 \sqrt{1-2 x} \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(20 + 37*x))/(121*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])

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Maple [A]  time = 0.003, size = 22, normalized size = 0.5 \begin{align*}{\frac{74\,x+40}{121}{\frac{1}{\sqrt{1-2\,x}}}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/121*(37*x+20)/(3+5*x)^(1/2)/(1-2*x)^(1/2)

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Maxima [A]  time = 3.3092, size = 41, normalized size = 0.91 \begin{align*} \frac{74 \, x}{121 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{40}{121 \, \sqrt{-10 \, x^{2} - x + 3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

74/121*x/sqrt(-10*x^2 - x + 3) + 40/121/sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.44313, size = 89, normalized size = 1.98 \begin{align*} -\frac{2 \,{\left (37 \, x + 20\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{121 \,{\left (10 \, x^{2} + x - 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/121*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.99393, size = 117, normalized size = 2.6 \begin{align*} -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{1210 \, \sqrt{5 \, x + 3}} - \frac{14 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{605 \,{\left (2 \, x - 1\right )}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{605 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1210*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 14/605*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x
+ 5)/(2*x - 1) + 2/605*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))