3.2612 \(\int \frac{2+3 x}{(1-2 x)^{5/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=45 \[ \frac{7 \sqrt{5 x+3}}{33 (1-2 x)^{3/2}}-\frac{29 \sqrt{5 x+3}}{363 \sqrt{1-2 x}} \]

[Out]

(7*Sqrt[3 + 5*x])/(33*(1 - 2*x)^(3/2)) - (29*Sqrt[3 + 5*x])/(363*Sqrt[1 - 2*x])

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Rubi [A]  time = 0.0053075, 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} \[ \frac{7 \sqrt{5 x+3}}{33 (1-2 x)^{3/2}}-\frac{29 \sqrt{5 x+3}}{363 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7*Sqrt[3 + 5*x])/(33*(1 - 2*x)^(3/2)) - (29*Sqrt[3 + 5*x])/(363*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)^{5/2} \sqrt{3+5 x}} \, dx &=\frac{7 \sqrt{3+5 x}}{33 (1-2 x)^{3/2}}-\frac{29}{66} \int \frac{1}{(1-2 x)^{3/2} \sqrt{3+5 x}} \, dx\\ &=\frac{7 \sqrt{3+5 x}}{33 (1-2 x)^{3/2}}-\frac{29 \sqrt{3+5 x}}{363 \sqrt{1-2 x}}\\ \end{align*}

Mathematica [A]  time = 0.0096052, size = 27, normalized size = 0.6 \[ \frac{2 \sqrt{5 x+3} (29 x+24)}{363 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[3 + 5*x]*(24 + 29*x))/(363*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.003, size = 22, normalized size = 0.5 \begin{align*}{\frac{58\,x+48}{363}\sqrt{3+5\,x} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/363*(3+5*x)^(1/2)*(29*x+24)/(1-2*x)^(3/2)

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Maxima [A]  time = 1.6299, size = 65, normalized size = 1.44 \begin{align*} \frac{7 \, \sqrt{-10 \, x^{2} - x + 3}}{33 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{29 \, \sqrt{-10 \, x^{2} - x + 3}}{363 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7/33*sqrt(-10*x^2 - x + 3)/(4*x^2 - 4*x + 1) + 29/363*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 1.48573, size = 89, normalized size = 1.98 \begin{align*} \frac{2 \,{\left (29 \, x + 24\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{363 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/363*(29*x + 24)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

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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{5}{2}} \sqrt{5 x + 3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.98092, size = 53, normalized size = 1.18 \begin{align*} \frac{2 \,{\left (29 \, \sqrt{5}{\left (5 \, x + 3\right )} + 33 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{9075 \,{\left (2 \, x - 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9075*(29*sqrt(5)*(5*x + 3) + 33*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2