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

Optimal. Leaf size=59 \[ -\frac{2 b^2 (b x+2)^{3/2}}{105 x^{3/2}}+\frac{2 b (b x+2)^{3/2}}{35 x^{5/2}}-\frac{(b x+2)^{3/2}}{7 x^{7/2}} \]

[Out]

-(2 + b*x)^(3/2)/(7*x^(7/2)) + (2*b*(2 + b*x)^(3/2))/(35*x^(5/2)) - (2*b^2*(2 + b*x)^(3/2))/(105*x^(3/2))

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Rubi [A]  time = 0.0082292, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {45, 37} \[ -\frac{2 b^2 (b x+2)^{3/2}}{105 x^{3/2}}+\frac{2 b (b x+2)^{3/2}}{35 x^{5/2}}-\frac{(b x+2)^{3/2}}{7 x^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[2 + b*x]/x^(9/2),x]

[Out]

-(2 + b*x)^(3/2)/(7*x^(7/2)) + (2*b*(2 + b*x)^(3/2))/(35*x^(5/2)) - (2*b^2*(2 + b*x)^(3/2))/(105*x^(3/2))

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{\sqrt{2+b x}}{x^{9/2}} \, dx &=-\frac{(2+b x)^{3/2}}{7 x^{7/2}}-\frac{1}{7} (2 b) \int \frac{\sqrt{2+b x}}{x^{7/2}} \, dx\\ &=-\frac{(2+b x)^{3/2}}{7 x^{7/2}}+\frac{2 b (2+b x)^{3/2}}{35 x^{5/2}}+\frac{1}{35} \left (2 b^2\right ) \int \frac{\sqrt{2+b x}}{x^{5/2}} \, dx\\ &=-\frac{(2+b x)^{3/2}}{7 x^{7/2}}+\frac{2 b (2+b x)^{3/2}}{35 x^{5/2}}-\frac{2 b^2 (2+b x)^{3/2}}{105 x^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0121579, size = 32, normalized size = 0.54 \[ -\frac{(b x+2)^{3/2} \left (2 b^2 x^2-6 b x+15\right )}{105 x^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[2 + b*x]/x^(9/2),x]

[Out]

-((2 + b*x)^(3/2)*(15 - 6*b*x + 2*b^2*x^2))/(105*x^(7/2))

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Maple [A]  time = 0.003, size = 27, normalized size = 0.5 \begin{align*} -{\frac{2\,{b}^{2}{x}^{2}-6\,bx+15}{105} \left ( bx+2 \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(b*x+2)^(3/2)*(2*b^2*x^2-6*b*x+15)/x^(7/2)

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Maxima [A]  time = 0.995352, size = 55, normalized size = 0.93 \begin{align*} -\frac{{\left (b x + 2\right )}^{\frac{3}{2}} b^{2}}{12 \, x^{\frac{3}{2}}} + \frac{{\left (b x + 2\right )}^{\frac{5}{2}} b}{10 \, x^{\frac{5}{2}}} - \frac{{\left (b x + 2\right )}^{\frac{7}{2}}}{28 \, x^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(b*x + 2)^(3/2)*b^2/x^(3/2) + 1/10*(b*x + 2)^(5/2)*b/x^(5/2) - 1/28*(b*x + 2)^(7/2)/x^(7/2)

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Fricas [A]  time = 1.55872, size = 90, normalized size = 1.53 \begin{align*} -\frac{{\left (2 \, b^{3} x^{3} - 2 \, b^{2} x^{2} + 3 \, b x + 30\right )} \sqrt{b x + 2}}{105 \, x^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(2*b^3*x^3 - 2*b^2*x^2 + 3*b*x + 30)*sqrt(b*x + 2)/x^(7/2)

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Sympy [B]  time = 71.3997, size = 270, normalized size = 4.58 \begin{align*} - \frac{2 b^{\frac{19}{2}} x^{5} \sqrt{1 + \frac{2}{b x}}}{105 b^{6} x^{5} + 420 b^{5} x^{4} + 420 b^{4} x^{3}} - \frac{6 b^{\frac{17}{2}} x^{4} \sqrt{1 + \frac{2}{b x}}}{105 b^{6} x^{5} + 420 b^{5} x^{4} + 420 b^{4} x^{3}} - \frac{3 b^{\frac{15}{2}} x^{3} \sqrt{1 + \frac{2}{b x}}}{105 b^{6} x^{5} + 420 b^{5} x^{4} + 420 b^{4} x^{3}} - \frac{34 b^{\frac{13}{2}} x^{2} \sqrt{1 + \frac{2}{b x}}}{105 b^{6} x^{5} + 420 b^{5} x^{4} + 420 b^{4} x^{3}} - \frac{132 b^{\frac{11}{2}} x \sqrt{1 + \frac{2}{b x}}}{105 b^{6} x^{5} + 420 b^{5} x^{4} + 420 b^{4} x^{3}} - \frac{120 b^{\frac{9}{2}} \sqrt{1 + \frac{2}{b x}}}{105 b^{6} x^{5} + 420 b^{5} x^{4} + 420 b^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*b**(19/2)*x**5*sqrt(1 + 2/(b*x))/(105*b**6*x**5 + 420*b**5*x**4 + 420*b**4*x**3) - 6*b**(17/2)*x**4*sqrt(1
+ 2/(b*x))/(105*b**6*x**5 + 420*b**5*x**4 + 420*b**4*x**3) - 3*b**(15/2)*x**3*sqrt(1 + 2/(b*x))/(105*b**6*x**5
+ 420*b**5*x**4 + 420*b**4*x**3) - 34*b**(13/2)*x**2*sqrt(1 + 2/(b*x))/(105*b**6*x**5 + 420*b**5*x**4 + 420*b
**4*x**3) - 132*b**(11/2)*x*sqrt(1 + 2/(b*x))/(105*b**6*x**5 + 420*b**5*x**4 + 420*b**4*x**3) - 120*b**(9/2)*s
qrt(1 + 2/(b*x))/(105*b**6*x**5 + 420*b**5*x**4 + 420*b**4*x**3)

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Giac [A]  time = 1.22253, size = 74, normalized size = 1.25 \begin{align*} -\frac{{\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 )}^{\frac{3}{2}} b}{105 \,{\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((b*x+2)^(1/2)/x^(9/2),x, algorithm="giac")

[Out]

-1/105*(35*b^7 + 2*((b*x + 2)*b^7 - 7*b^7)*(b*x + 2))*(b*x + 2)^(3/2)*b/(((b*x + 2)*b - 2*b)^(7/2)*abs(b))