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

Optimal. Leaf size=62 \[ -\frac{2 b^2 (2-b x)^{3/2}}{105 x^{3/2}}-\frac{2 b (2-b x)^{3/2}}{35 x^{5/2}}-\frac{(2-b x)^{3/2}}{7 x^{7/2}} \]

[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 = 0.0080486, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {45, 37} \[ -\frac{2 b^2 (2-b x)^{3/2}}{105 x^{3/2}}-\frac{2 b (2-b x)^{3/2}}{35 x^{5/2}}-\frac{(2-b x)^{3/2}}{7 x^{7/2}} \]

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 = 0.0119495, size = 33, normalized size = 0.53 \[ -\frac{(2-b x)^{3/2} \left (2 b^2 x^2+6 b x+15\right )}{105 x^{7/2}} \]

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 = 28, 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*(2*b^2*x^2+6*b*x+15)*(-b*x+2)^(3/2)/x^(7/2)

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Maxima [A]  time = 1.04392, size = 59, normalized size = 0.95 \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 = 1.56232, size = 90, normalized size = 1.45 \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 = 67.6331, size = 554, normalized size = 8.94 \begin{align*} \begin{cases} \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}} & \text{for}\: \frac{2}{\left |{b x}\right |} > 1 \\\frac{2 i 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 i 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 i 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 i 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 i 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 i 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}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((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)*sqrt(-1 + 2/(b*x))/(105*b**6*x**5 - 420*b**5*x**4 + 420*b**4*x**3), 2/Abs(b*x) > 1), (2*I*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*I*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*I*b**(15/2)*x**3*sqrt(1 - 2/(b*x))/(105*b**6*x**5 - 42
0*b**5*x**4 + 420*b**4*x**3) - 34*I*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*I*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*I*b**(9/2)*
sqrt(1 - 2/(b*x))/(105*b**6*x**5 - 420*b**5*x**4 + 420*b**4*x**3), True))

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Giac [A]  time = 1.27835, size = 82, normalized size = 1.32 \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 )} \sqrt{-b x + 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)*sqrt(-b*x + 2)*b/(((b*x - 2)*b + 2*b)^(7/2)*abs
(b))