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

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

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Rubi [A]  time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {74 \sqrt {5 x+3}}{605 \sqrt {1-2 x}}-\frac {2}{55 \sqrt {1-2 x} \sqrt {5 x+3}} \end {gather*}

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 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 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)^{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*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 0.60 \begin {gather*} \frac {2 (37 x+20)}{121 \sqrt {1-2 x} \sqrt {5 x+3}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.08, size = 37, normalized size = 0.82 \begin {gather*} -\frac {2 \sqrt {5 x+3} \left (\frac {1-2 x}{5 x+3}-7\right )}{121 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[3 + 5*x]*(-7 + (1 - 2*x)/(3 + 5*x)))/(121*Sqrt[1 - 2*x])

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fricas [A]  time = 1.07, size = 31, normalized size = 0.69 \begin {gather*} -\frac {2 \, {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{121 \, {\left (10 \, x^{2} + x - 3\right )}} \end {gather*}

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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giac [B]  time = 1.04, size = 87, normalized size = 1.93 \begin {gather*} -\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 {gather*}

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))

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maple [A]  time = 0.00, size = 22, normalized size = 0.49 \begin {gather*} \frac {\frac {74 x}{121}+\frac {40}{121}}{\sqrt {-2 x +1}\, \sqrt {5 x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.47, size = 30, normalized size = 0.67 \begin {gather*} \frac {74 \, x}{121 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {40}{121 \, \sqrt {-10 \, x^{2} - x + 3}} \end {gather*}

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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mupad [B]  time = 0.25, size = 34, normalized size = 0.76 \begin {gather*} \frac {\sqrt {5\,x+3}\,\left (\frac {74\,x}{605}+\frac {8}{121}\right )}{x\,\sqrt {1-2\,x}+\frac {3\,\sqrt {1-2\,x}}{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((5*x + 3)^(1/2)*((74*x)/605 + 8/121))/(x*(1 - 2*x)^(1/2) + (3*(1 - 2*x)^(1/2))/5)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 x + 2}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \end {gather*}

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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