3.519 \(\int \frac{\sqrt{2-b x}}{x^{7/2}} \, dx\)

Optimal. Leaf size=40 \[ -\frac{b (2-b x)^{3/2}}{15 x^{3/2}}-\frac{(2-b x)^{3/2}}{5 x^{5/2}} \]

[Out]

-(2 - b*x)^(3/2)/(5*x^(5/2)) - (b*(2 - b*x)^(3/2))/(15*x^(3/2))

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Rubi [A]  time = 0.0041434, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {45, 37} \[ -\frac{b (2-b x)^{3/2}}{15 x^{3/2}}-\frac{(2-b x)^{3/2}}{5 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[2 - b*x]/x^(7/2),x]

[Out]

-(2 - b*x)^(3/2)/(5*x^(5/2)) - (b*(2 - b*x)^(3/2))/(15*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^{7/2}} \, dx &=-\frac{(2-b x)^{3/2}}{5 x^{5/2}}+\frac{1}{5} b \int \frac{\sqrt{2-b x}}{x^{5/2}} \, dx\\ &=-\frac{(2-b x)^{3/2}}{5 x^{5/2}}-\frac{b (2-b x)^{3/2}}{15 x^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.009253, size = 24, normalized size = 0.6 \[ -\frac{(2-b x)^{3/2} (b x+3)}{15 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[2 - b*x]/x^(7/2),x]

[Out]

-((2 - b*x)^(3/2)*(3 + b*x))/(15*x^(5/2))

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Maple [A]  time = 0.003, size = 19, normalized size = 0.5 \begin{align*} -{\frac{bx+3}{15} \left ( -bx+2 \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b*x+2)^(1/2)/x^(7/2),x)

[Out]

-1/15*(b*x+3)*(-b*x+2)^(3/2)/x^(5/2)

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Maxima [A]  time = 1.08431, size = 38, normalized size = 0.95 \begin{align*} -\frac{{\left (-b x + 2\right )}^{\frac{3}{2}} b}{6 \, x^{\frac{3}{2}}} - \frac{{\left (-b x + 2\right )}^{\frac{5}{2}}}{10 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x+2)^(1/2)/x^(7/2),x, algorithm="maxima")

[Out]

-1/6*(-b*x + 2)^(3/2)*b/x^(3/2) - 1/10*(-b*x + 2)^(5/2)/x^(5/2)

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Fricas [A]  time = 1.56244, size = 66, normalized size = 1.65 \begin{align*} \frac{{\left (b^{2} x^{2} + b x - 6\right )} \sqrt{-b x + 2}}{15 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x+2)^(1/2)/x^(7/2),x, algorithm="fricas")

[Out]

1/15*(b^2*x^2 + b*x - 6)*sqrt(-b*x + 2)/x^(5/2)

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Sympy [A]  time = 18.4888, size = 194, normalized size = 4.85 \begin{align*} \begin{cases} \frac{b^{\frac{5}{2}} \sqrt{-1 + \frac{2}{b x}}}{15} + \frac{b^{\frac{3}{2}} \sqrt{-1 + \frac{2}{b x}}}{15 x} - \frac{2 \sqrt{b} \sqrt{-1 + \frac{2}{b x}}}{5 x^{2}} & \text{for}\: \frac{2}{\left |{b x}\right |} > 1 \\\frac{i b^{\frac{9}{2}} x^{2} \sqrt{1 - \frac{2}{b x}}}{15 b^{2} x^{2} - 30 b x} - \frac{i b^{\frac{7}{2}} x \sqrt{1 - \frac{2}{b x}}}{15 b^{2} x^{2} - 30 b x} - \frac{8 i b^{\frac{5}{2}} \sqrt{1 - \frac{2}{b x}}}{15 b^{2} x^{2} - 30 b x} + \frac{12 i b^{\frac{3}{2}} \sqrt{1 - \frac{2}{b x}}}{x \left (15 b^{2} x^{2} - 30 b x\right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x+2)**(1/2)/x**(7/2),x)

[Out]

Piecewise((b**(5/2)*sqrt(-1 + 2/(b*x))/15 + b**(3/2)*sqrt(-1 + 2/(b*x))/(15*x) - 2*sqrt(b)*sqrt(-1 + 2/(b*x))/
(5*x**2), 2/Abs(b*x) > 1), (I*b**(9/2)*x**2*sqrt(1 - 2/(b*x))/(15*b**2*x**2 - 30*b*x) - I*b**(7/2)*x*sqrt(1 -
2/(b*x))/(15*b**2*x**2 - 30*b*x) - 8*I*b**(5/2)*sqrt(1 - 2/(b*x))/(15*b**2*x**2 - 30*b*x) + 12*I*b**(3/2)*sqrt
(1 - 2/(b*x))/(x*(15*b**2*x**2 - 30*b*x)), True))

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Giac [A]  time = 1.17405, size = 65, normalized size = 1.62 \begin{align*} \frac{{\left ({\left (b x - 2\right )} b^{5} + 5 \, b^{5}\right )}{\left (b x - 2\right )} \sqrt{-b x + 2} b}{15 \,{\left ({\left (b x - 2\right )} b + 2 \, b\right )}^{\frac{5}{2}}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x+2)^(1/2)/x^(7/2),x, algorithm="giac")

[Out]

1/15*((b*x - 2)*b^5 + 5*b^5)*(b*x - 2)*sqrt(-b*x + 2)*b/(((b*x - 2)*b + 2*b)^(5/2)*abs(b))