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

Optimal. Leaf size=71 \[ -\frac {16 b^2 (a-b x)^{3/2}}{105 a^3 x^{3/2}}-\frac {8 b (a-b x)^{3/2}}{35 a^2 x^{5/2}}-\frac {2 (a-b x)^{3/2}}{7 a x^{7/2}} \]

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Rubi [A]  time = 0.01, antiderivative size = 71, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {16 b^2 (a-b x)^{3/2}}{105 a^3 x^{3/2}}-\frac {8 b (a-b x)^{3/2}}{35 a^2 x^{5/2}}-\frac {2 (a-b x)^{3/2}}{7 a x^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a - b*x]/x^(9/2),x]

[Out]

(-2*(a - b*x)^(3/2))/(7*a*x^(7/2)) - (8*b*(a - b*x)^(3/2))/(35*a^2*x^(5/2)) - (16*b^2*(a - b*x)^(3/2))/(105*a^
3*x^(3/2))

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]

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])

Rubi steps

\begin {align*} \int \frac {\sqrt {a-b x}}{x^{9/2}} \, dx &=-\frac {2 (a-b x)^{3/2}}{7 a x^{7/2}}+\frac {(4 b) \int \frac {\sqrt {a-b x}}{x^{7/2}} \, dx}{7 a}\\ &=-\frac {2 (a-b x)^{3/2}}{7 a x^{7/2}}-\frac {8 b (a-b x)^{3/2}}{35 a^2 x^{5/2}}+\frac {\left (8 b^2\right ) \int \frac {\sqrt {a-b x}}{x^{5/2}} \, dx}{35 a^2}\\ &=-\frac {2 (a-b x)^{3/2}}{7 a x^{7/2}}-\frac {8 b (a-b x)^{3/2}}{35 a^2 x^{5/2}}-\frac {16 b^2 (a-b x)^{3/2}}{105 a^3 x^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 41, normalized size = 0.58 \begin {gather*} -\frac {2 (a-b x)^{3/2} \left (15 a^2+12 a b x+8 b^2 x^2\right )}{105 a^3 x^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a - b*x]/x^(9/2),x]

[Out]

(-2*(a - b*x)^(3/2)*(15*a^2 + 12*a*b*x + 8*b^2*x^2))/(105*a^3*x^(7/2))

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IntegrateAlgebraic [A]  time = 0.11, size = 52, normalized size = 0.73 \begin {gather*} \frac {2 \sqrt {a-b x} \left (-15 a^3+3 a^2 b x+4 a b^2 x^2+8 b^3 x^3\right )}{105 a^3 x^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[a - b*x]/x^(9/2),x]

[Out]

(2*Sqrt[a - b*x]*(-15*a^3 + 3*a^2*b*x + 4*a*b^2*x^2 + 8*b^3*x^3))/(105*a^3*x^(7/2))

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fricas [A]  time = 0.96, size = 46, normalized size = 0.65 \begin {gather*} \frac {2 \, {\left (8 \, b^{3} x^{3} + 4 \, a b^{2} x^{2} + 3 \, a^{2} b x - 15 \, a^{3}\right )} \sqrt {-b x + a}}{105 \, a^{3} x^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(8*b^3*x^3 + 4*a*b^2*x^2 + 3*a^2*b*x - 15*a^3)*sqrt(-b*x + a)/(a^3*x^(7/2))

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giac [A]  time = 1.34, size = 79, normalized size = 1.11 \begin {gather*} \frac {2 \, {\left (\frac {35 \, b^{7}}{a} + 4 \, {\left (\frac {2 \, {\left (b x - a\right )} b^{7}}{a^{3}} + \frac {7 \, b^{7}}{a^{2}}\right )} {\left (b x - a\right )}\right )} {\left (b x - a\right )} \sqrt {-b x + a} b}{105 \, {\left ({\left (b x - a\right )} b + a b\right )}^{\frac {7}{2}} {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(35*b^7/a + 4*(2*(b*x - a)*b^7/a^3 + 7*b^7/a^2)*(b*x - a))*(b*x - a)*sqrt(-b*x + a)*b/(((b*x - a)*b + a*
b)^(7/2)*abs(b))

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maple [A]  time = 0.00, size = 36, normalized size = 0.51 \begin {gather*} -\frac {2 \left (-b x +a \right )^{\frac {3}{2}} \left (8 b^{2} x^{2}+12 a b x +15 a^{2}\right )}{105 a^{3} x^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/105*(-b*x+a)^(3/2)*(8*b^2*x^2+12*a*b*x+15*a^2)/x^(7/2)/a^3

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maxima [A]  time = 1.35, size = 49, normalized size = 0.69 \begin {gather*} -\frac {2 \, {\left (\frac {35 \, {\left (-b x + a\right )}^{\frac {3}{2}} b^{2}}{x^{\frac {3}{2}}} + \frac {42 \, {\left (-b x + a\right )}^{\frac {5}{2}} b}{x^{\frac {5}{2}}} + \frac {15 \, {\left (-b x + a\right )}^{\frac {7}{2}}}{x^{\frac {7}{2}}}\right )}}{105 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/105*(35*(-b*x + a)^(3/2)*b^2/x^(3/2) + 42*(-b*x + a)^(5/2)*b/x^(5/2) + 15*(-b*x + a)^(7/2)/x^(7/2))/a^3

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mupad [B]  time = 0.27, size = 43, normalized size = 0.61 \begin {gather*} \frac {\sqrt {a-b\,x}\,\left (\frac {8\,b^2\,x^2}{105\,a^2}+\frac {16\,b^3\,x^3}{105\,a^3}+\frac {2\,b\,x}{35\,a}-\frac {2}{7}\right )}{x^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a - b*x)^(1/2)/x^(9/2),x)

[Out]

((a - b*x)^(1/2)*((8*b^2*x^2)/(105*a^2) + (16*b^3*x^3)/(105*a^3) + (2*b*x)/(35*a) - 2/7))/x^(7/2)

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sympy [B]  time = 26.81, size = 707, normalized size = 9.96 \begin {gather*} \begin {cases} \frac {30 a^{5} b^{\frac {9}{2}} \sqrt {\frac {a}{b x} - 1}}{- 105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} - 105 a^{3} b^{6} x^{5}} - \frac {66 a^{4} b^{\frac {11}{2}} x \sqrt {\frac {a}{b x} - 1}}{- 105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} - 105 a^{3} b^{6} x^{5}} + \frac {34 a^{3} b^{\frac {13}{2}} x^{2} \sqrt {\frac {a}{b x} - 1}}{- 105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} - 105 a^{3} b^{6} x^{5}} - \frac {6 a^{2} b^{\frac {15}{2}} x^{3} \sqrt {\frac {a}{b x} - 1}}{- 105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} - 105 a^{3} b^{6} x^{5}} + \frac {24 a b^{\frac {17}{2}} x^{4} \sqrt {\frac {a}{b x} - 1}}{- 105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} - 105 a^{3} b^{6} x^{5}} - \frac {16 b^{\frac {19}{2}} x^{5} \sqrt {\frac {a}{b x} - 1}}{- 105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} - 105 a^{3} b^{6} x^{5}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\\frac {30 i a^{5} b^{\frac {9}{2}} \sqrt {- \frac {a}{b x} + 1}}{- 105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} - 105 a^{3} b^{6} x^{5}} - \frac {66 i a^{4} b^{\frac {11}{2}} x \sqrt {- \frac {a}{b x} + 1}}{- 105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} - 105 a^{3} b^{6} x^{5}} + \frac {34 i a^{3} b^{\frac {13}{2}} x^{2} \sqrt {- \frac {a}{b x} + 1}}{- 105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} - 105 a^{3} b^{6} x^{5}} - \frac {6 i a^{2} b^{\frac {15}{2}} x^{3} \sqrt {- \frac {a}{b x} + 1}}{- 105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} - 105 a^{3} b^{6} x^{5}} + \frac {24 i a b^{\frac {17}{2}} x^{4} \sqrt {- \frac {a}{b x} + 1}}{- 105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} - 105 a^{3} b^{6} x^{5}} - \frac {16 i b^{\frac {19}{2}} x^{5} \sqrt {- \frac {a}{b x} + 1}}{- 105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} - 105 a^{3} b^{6} x^{5}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((30*a**5*b**(9/2)*sqrt(a/(b*x) - 1)/(-105*a**5*b**4*x**3 + 210*a**4*b**5*x**4 - 105*a**3*b**6*x**5)
- 66*a**4*b**(11/2)*x*sqrt(a/(b*x) - 1)/(-105*a**5*b**4*x**3 + 210*a**4*b**5*x**4 - 105*a**3*b**6*x**5) + 34*a
**3*b**(13/2)*x**2*sqrt(a/(b*x) - 1)/(-105*a**5*b**4*x**3 + 210*a**4*b**5*x**4 - 105*a**3*b**6*x**5) - 6*a**2*
b**(15/2)*x**3*sqrt(a/(b*x) - 1)/(-105*a**5*b**4*x**3 + 210*a**4*b**5*x**4 - 105*a**3*b**6*x**5) + 24*a*b**(17
/2)*x**4*sqrt(a/(b*x) - 1)/(-105*a**5*b**4*x**3 + 210*a**4*b**5*x**4 - 105*a**3*b**6*x**5) - 16*b**(19/2)*x**5
*sqrt(a/(b*x) - 1)/(-105*a**5*b**4*x**3 + 210*a**4*b**5*x**4 - 105*a**3*b**6*x**5), Abs(a/(b*x)) > 1), (30*I*a
**5*b**(9/2)*sqrt(-a/(b*x) + 1)/(-105*a**5*b**4*x**3 + 210*a**4*b**5*x**4 - 105*a**3*b**6*x**5) - 66*I*a**4*b*
*(11/2)*x*sqrt(-a/(b*x) + 1)/(-105*a**5*b**4*x**3 + 210*a**4*b**5*x**4 - 105*a**3*b**6*x**5) + 34*I*a**3*b**(1
3/2)*x**2*sqrt(-a/(b*x) + 1)/(-105*a**5*b**4*x**3 + 210*a**4*b**5*x**4 - 105*a**3*b**6*x**5) - 6*I*a**2*b**(15
/2)*x**3*sqrt(-a/(b*x) + 1)/(-105*a**5*b**4*x**3 + 210*a**4*b**5*x**4 - 105*a**3*b**6*x**5) + 24*I*a*b**(17/2)
*x**4*sqrt(-a/(b*x) + 1)/(-105*a**5*b**4*x**3 + 210*a**4*b**5*x**4 - 105*a**3*b**6*x**5) - 16*I*b**(19/2)*x**5
*sqrt(-a/(b*x) + 1)/(-105*a**5*b**4*x**3 + 210*a**4*b**5*x**4 - 105*a**3*b**6*x**5), True))

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