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

Optimal. Leaf size=68 \[ -\frac{16 b^2 (a+b x)^{3/2}}{105 a^3 x^{3/2}}+\frac{8 b (a+b x)^{3/2}}{35 a^2 x^{5/2}}-\frac{2 (a+b x)^{3/2}}{7 a x^{7/2}} \]

[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))

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Rubi [A]  time = 0.0096716, antiderivative size = 68, 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} \[ -\frac{16 b^2 (a+b x)^{3/2}}{105 a^3 x^{3/2}}+\frac{8 b (a+b x)^{3/2}}{35 a^2 x^{5/2}}-\frac{2 (a+b x)^{3/2}}{7 a x^{7/2}} \]

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 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{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*}

Mathematica [A]  time = 0.0107707, size = 40, normalized size = 0.59 \[ -\frac{2 (a+b x)^{3/2} \left (15 a^2-12 a b x+8 b^2 x^2\right )}{105 a^3 x^{7/2}} \]

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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Maple [A]  time = 0.005, size = 35, normalized size = 0.5 \begin{align*} -{\frac{16\,{b}^{2}{x}^{2}-24\,abx+30\,{a}^{2}}{105\,{a}^{3}} \left ( bx+a \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{7}{2}}}} \end{align*}

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 = 1.01434, size = 62, normalized size = 0.91 \begin{align*} -\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{align*}

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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Fricas [A]  time = 1.55729, size = 112, normalized size = 1.65 \begin{align*} -\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{align*}

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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Sympy [B]  time = 120.955, size = 347, normalized size = 5.1 \begin{align*} - \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)

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Giac [A]  time = 1.4484, size = 89, normalized size = 1.31 \begin{align*} \frac{{\left (b x + a\right )}^{\frac{3}{2}}{\left (4 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (b x + a\right )}}{a^{4} b^{5}} - \frac{7}{a^{3} b^{5}}\right )} + \frac{35}{a^{2} b^{5}}\right )} b}{40320 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{7}{2}}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/40320*(b*x + a)^(3/2)*(4*(b*x + a)*(2*(b*x + a)/(a^4*b^5) - 7/(a^3*b^5)) + 35/(a^2*b^5))*b/(((b*x + a)*b - a
*b)^(7/2)*abs(b))