3.615 \(\int \frac{1}{x^{9/2} \sqrt{2+b x}} \, dx\)

Optimal. Leaf size=80 \[ -\frac{2 b^2 \sqrt{b x+2}}{35 x^{3/2}}+\frac{2 b^3 \sqrt{b x+2}}{35 \sqrt{x}}+\frac{3 b \sqrt{b x+2}}{35 x^{5/2}}-\frac{\sqrt{b x+2}}{7 x^{7/2}} \]

[Out]

-Sqrt[2 + b*x]/(7*x^(7/2)) + (3*b*Sqrt[2 + b*x])/(35*x^(5/2)) - (2*b^2*Sqrt[2 + b*x])/(35*x^(3/2)) + (2*b^3*Sq
rt[2 + b*x])/(35*Sqrt[x])

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Rubi [A]  time = 0.0123568, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {45, 37} \[ -\frac{2 b^2 \sqrt{b x+2}}{35 x^{3/2}}+\frac{2 b^3 \sqrt{b x+2}}{35 \sqrt{x}}+\frac{3 b \sqrt{b x+2}}{35 x^{5/2}}-\frac{\sqrt{b x+2}}{7 x^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(9/2)*Sqrt[2 + b*x]),x]

[Out]

-Sqrt[2 + b*x]/(7*x^(7/2)) + (3*b*Sqrt[2 + b*x])/(35*x^(5/2)) - (2*b^2*Sqrt[2 + b*x])/(35*x^(3/2)) + (2*b^3*Sq
rt[2 + b*x])/(35*Sqrt[x])

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 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{1}{x^{9/2} \sqrt{2+b x}} \, dx &=-\frac{\sqrt{2+b x}}{7 x^{7/2}}-\frac{1}{7} (3 b) \int \frac{1}{x^{7/2} \sqrt{2+b x}} \, dx\\ &=-\frac{\sqrt{2+b x}}{7 x^{7/2}}+\frac{3 b \sqrt{2+b x}}{35 x^{5/2}}+\frac{1}{35} \left (6 b^2\right ) \int \frac{1}{x^{5/2} \sqrt{2+b x}} \, dx\\ &=-\frac{\sqrt{2+b x}}{7 x^{7/2}}+\frac{3 b \sqrt{2+b x}}{35 x^{5/2}}-\frac{2 b^2 \sqrt{2+b x}}{35 x^{3/2}}-\frac{1}{35} \left (2 b^3\right ) \int \frac{1}{x^{3/2} \sqrt{2+b x}} \, dx\\ &=-\frac{\sqrt{2+b x}}{7 x^{7/2}}+\frac{3 b \sqrt{2+b x}}{35 x^{5/2}}-\frac{2 b^2 \sqrt{2+b x}}{35 x^{3/2}}+\frac{2 b^3 \sqrt{2+b x}}{35 \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.0090322, size = 40, normalized size = 0.5 \[ \frac{\sqrt{b x+2} \left (2 b^3 x^3-2 b^2 x^2+3 b x-5\right )}{35 x^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(9/2)*Sqrt[2 + b*x]),x]

[Out]

(Sqrt[2 + b*x]*(-5 + 3*b*x - 2*b^2*x^2 + 2*b^3*x^3))/(35*x^(7/2))

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Maple [A]  time = 0.005, size = 35, normalized size = 0.4 \begin{align*}{\frac{2\,{b}^{3}{x}^{3}-2\,{b}^{2}{x}^{2}+3\,bx-5}{35}\sqrt{bx+2}{x}^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(9/2)/(b*x+2)^(1/2),x)

[Out]

1/35*(b*x+2)^(1/2)*(2*b^3*x^3-2*b^2*x^2+3*b*x-5)/x^(7/2)

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Maxima [A]  time = 1.08572, size = 76, normalized size = 0.95 \begin{align*} \frac{\sqrt{b x + 2} b^{3}}{8 \, \sqrt{x}} - \frac{{\left (b x + 2\right )}^{\frac{3}{2}} b^{2}}{8 \, x^{\frac{3}{2}}} + \frac{3 \,{\left (b x + 2\right )}^{\frac{5}{2}} b}{40 \, x^{\frac{5}{2}}} - \frac{{\left (b x + 2\right )}^{\frac{7}{2}}}{56 \, x^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(9/2)/(b*x+2)^(1/2),x, algorithm="maxima")

[Out]

1/8*sqrt(b*x + 2)*b^3/sqrt(x) - 1/8*(b*x + 2)^(3/2)*b^2/x^(3/2) + 3/40*(b*x + 2)^(5/2)*b/x^(5/2) - 1/56*(b*x +
2)^(7/2)/x^(7/2)

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Fricas [A]  time = 1.75463, size = 86, normalized size = 1.08 \begin{align*} \frac{{\left (2 \, b^{3} x^{3} - 2 \, b^{2} x^{2} + 3 \, b x - 5\right )} \sqrt{b x + 2}}{35 \, x^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(9/2)/(b*x+2)^(1/2),x, algorithm="fricas")

[Out]

1/35*(2*b^3*x^3 - 2*b^2*x^2 + 3*b*x - 5)*sqrt(b*x + 2)/x^(7/2)

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Sympy [B]  time = 104.571, size = 374, normalized size = 4.68 \begin{align*} \frac{2 b^{\frac{31}{2}} x^{6} \sqrt{1 + \frac{2}{b x}}}{35 b^{12} x^{6} + 210 b^{11} x^{5} + 420 b^{10} x^{4} + 280 b^{9} x^{3}} + \frac{10 b^{\frac{29}{2}} x^{5} \sqrt{1 + \frac{2}{b x}}}{35 b^{12} x^{6} + 210 b^{11} x^{5} + 420 b^{10} x^{4} + 280 b^{9} x^{3}} + \frac{15 b^{\frac{27}{2}} x^{4} \sqrt{1 + \frac{2}{b x}}}{35 b^{12} x^{6} + 210 b^{11} x^{5} + 420 b^{10} x^{4} + 280 b^{9} x^{3}} + \frac{5 b^{\frac{25}{2}} x^{3} \sqrt{1 + \frac{2}{b x}}}{35 b^{12} x^{6} + 210 b^{11} x^{5} + 420 b^{10} x^{4} + 280 b^{9} x^{3}} - \frac{10 b^{\frac{23}{2}} x^{2} \sqrt{1 + \frac{2}{b x}}}{35 b^{12} x^{6} + 210 b^{11} x^{5} + 420 b^{10} x^{4} + 280 b^{9} x^{3}} - \frac{36 b^{\frac{21}{2}} x \sqrt{1 + \frac{2}{b x}}}{35 b^{12} x^{6} + 210 b^{11} x^{5} + 420 b^{10} x^{4} + 280 b^{9} x^{3}} - \frac{40 b^{\frac{19}{2}} \sqrt{1 + \frac{2}{b x}}}{35 b^{12} x^{6} + 210 b^{11} x^{5} + 420 b^{10} x^{4} + 280 b^{9} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(9/2)/(b*x+2)**(1/2),x)

[Out]

2*b**(31/2)*x**6*sqrt(1 + 2/(b*x))/(35*b**12*x**6 + 210*b**11*x**5 + 420*b**10*x**4 + 280*b**9*x**3) + 10*b**(
29/2)*x**5*sqrt(1 + 2/(b*x))/(35*b**12*x**6 + 210*b**11*x**5 + 420*b**10*x**4 + 280*b**9*x**3) + 15*b**(27/2)*
x**4*sqrt(1 + 2/(b*x))/(35*b**12*x**6 + 210*b**11*x**5 + 420*b**10*x**4 + 280*b**9*x**3) + 5*b**(25/2)*x**3*sq
rt(1 + 2/(b*x))/(35*b**12*x**6 + 210*b**11*x**5 + 420*b**10*x**4 + 280*b**9*x**3) - 10*b**(23/2)*x**2*sqrt(1 +
2/(b*x))/(35*b**12*x**6 + 210*b**11*x**5 + 420*b**10*x**4 + 280*b**9*x**3) - 36*b**(21/2)*x*sqrt(1 + 2/(b*x))
/(35*b**12*x**6 + 210*b**11*x**5 + 420*b**10*x**4 + 280*b**9*x**3) - 40*b**(19/2)*sqrt(1 + 2/(b*x))/(35*b**12*
x**6 + 210*b**11*x**5 + 420*b**10*x**4 + 280*b**9*x**3)

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Giac [A]  time = 1.10613, size = 92, normalized size = 1.15 \begin{align*} -\frac{{\left (35 \, b^{7} -{\left (35 \, b^{7} + 2 \,{\left ({\left (b x + 2\right )} b^{7} - 7 \, b^{7}\right )}{\left (b x + 2\right )}\right )}{\left (b x + 2\right )}\right )} \sqrt{b x + 2} b}{35 \,{\left ({\left (b x + 2\right )} b - 2 \, b\right )}^{\frac{7}{2}}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(9/2)/(b*x+2)^(1/2),x, algorithm="giac")

[Out]

-1/35*(35*b^7 - (35*b^7 + 2*((b*x + 2)*b^7 - 7*b^7)*(b*x + 2))*(b*x + 2))*sqrt(b*x + 2)*b/(((b*x + 2)*b - 2*b)
^(7/2)*abs(b))