3.2511 \(\int \frac{2+3 x}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=45 \[ -\frac{206 \sqrt{1-2 x}}{1815 \sqrt{5 x+3}}-\frac{2 \sqrt{1-2 x}}{165 (5 x+3)^{3/2}} \]

[Out]

(-2*Sqrt[1 - 2*x])/(165*(3 + 5*x)^(3/2)) - (206*Sqrt[1 - 2*x])/(1815*Sqrt[3 + 5*x])

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Rubi [A]  time = 0.0056482, 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} \[ -\frac{206 \sqrt{1-2 x}}{1815 \sqrt{5 x+3}}-\frac{2 \sqrt{1-2 x}}{165 (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x])/(165*(3 + 5*x)^(3/2)) - (206*Sqrt[1 - 2*x])/(1815*Sqrt[3 + 5*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}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx &=-\frac{2 \sqrt{1-2 x}}{165 (3+5 x)^{3/2}}+\frac{103}{165} \int \frac{1}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac{2 \sqrt{1-2 x}}{165 (3+5 x)^{3/2}}-\frac{206 \sqrt{1-2 x}}{1815 \sqrt{3+5 x}}\\ \end{align*}

Mathematica [A]  time = 0.0078725, size = 27, normalized size = 0.6 \[ -\frac{2 \sqrt{1-2 x} (103 x+64)}{363 (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x]*(64 + 103*x))/(363*(3 + 5*x)^(3/2))

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Maple [A]  time = 0.003, size = 22, normalized size = 0.5 \begin{align*} -{\frac{206\,x+128}{363}\sqrt{1-2\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/363*(103*x+64)/(3+5*x)^(3/2)*(1-2*x)^(1/2)

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Maxima [A]  time = 2.25692, size = 65, normalized size = 1.44 \begin{align*} -\frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{165 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac{206 \, \sqrt{-10 \, x^{2} - x + 3}}{1815 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/165*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x + 9) - 206/1815*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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Fricas [A]  time = 1.61772, size = 95, normalized size = 2.11 \begin{align*} -\frac{2 \,{\left (103 \, x + 64\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{363 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/363*(103*x + 64)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(25*x^2 + 30*x + 9)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 2.57601, size = 170, normalized size = 3.78 \begin{align*} -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{145200 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} - \frac{69 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{12100 \, \sqrt{5 \, x + 3}} + \frac{{\left (\frac{207 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{9075 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/145200*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) - 69/12100*sqrt(10)*(sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22))/sqrt(5*x + 3) + 1/9075*(207*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4
*sqrt(10))*(5*x + 3)^(3/2)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3