3.576 \(\int \frac{1}{x^{9/2} \sqrt{a+b x}} \, dx\)

Optimal. Leaf size=92 \[ -\frac{16 b^2 \sqrt{a+b x}}{35 a^3 x^{3/2}}+\frac{32 b^3 \sqrt{a+b x}}{35 a^4 \sqrt{x}}+\frac{12 b \sqrt{a+b x}}{35 a^2 x^{5/2}}-\frac{2 \sqrt{a+b x}}{7 a x^{7/2}} \]

[Out]

(-2*Sqrt[a + b*x])/(7*a*x^(7/2)) + (12*b*Sqrt[a + b*x])/(35*a^2*x^(5/2)) - (16*b^2*Sqrt[a + b*x])/(35*a^3*x^(3
/2)) + (32*b^3*Sqrt[a + b*x])/(35*a^4*Sqrt[x])

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Rubi [A]  time = 0.0163696, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {45, 37} \[ -\frac{16 b^2 \sqrt{a+b x}}{35 a^3 x^{3/2}}+\frac{32 b^3 \sqrt{a+b x}}{35 a^4 \sqrt{x}}+\frac{12 b \sqrt{a+b x}}{35 a^2 x^{5/2}}-\frac{2 \sqrt{a+b x}}{7 a x^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(9/2)*Sqrt[a + b*x]),x]

[Out]

(-2*Sqrt[a + b*x])/(7*a*x^(7/2)) + (12*b*Sqrt[a + b*x])/(35*a^2*x^(5/2)) - (16*b^2*Sqrt[a + b*x])/(35*a^3*x^(3
/2)) + (32*b^3*Sqrt[a + b*x])/(35*a^4*Sqrt[x])

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{1}{x^{9/2} \sqrt{a+b x}} \, dx &=-\frac{2 \sqrt{a+b x}}{7 a x^{7/2}}-\frac{(6 b) \int \frac{1}{x^{7/2} \sqrt{a+b x}} \, dx}{7 a}\\ &=-\frac{2 \sqrt{a+b x}}{7 a x^{7/2}}+\frac{12 b \sqrt{a+b x}}{35 a^2 x^{5/2}}+\frac{\left (24 b^2\right ) \int \frac{1}{x^{5/2} \sqrt{a+b x}} \, dx}{35 a^2}\\ &=-\frac{2 \sqrt{a+b x}}{7 a x^{7/2}}+\frac{12 b \sqrt{a+b x}}{35 a^2 x^{5/2}}-\frac{16 b^2 \sqrt{a+b x}}{35 a^3 x^{3/2}}-\frac{\left (16 b^3\right ) \int \frac{1}{x^{3/2} \sqrt{a+b x}} \, dx}{35 a^3}\\ &=-\frac{2 \sqrt{a+b x}}{7 a x^{7/2}}+\frac{12 b \sqrt{a+b x}}{35 a^2 x^{5/2}}-\frac{16 b^2 \sqrt{a+b x}}{35 a^3 x^{3/2}}+\frac{32 b^3 \sqrt{a+b x}}{35 a^4 \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.011458, size = 51, normalized size = 0.55 \[ -\frac{2 \sqrt{a+b x} \left (-6 a^2 b x+5 a^3+8 a b^2 x^2-16 b^3 x^3\right )}{35 a^4 x^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(9/2)*Sqrt[a + b*x]),x]

[Out]

(-2*Sqrt[a + b*x]*(5*a^3 - 6*a^2*b*x + 8*a*b^2*x^2 - 16*b^3*x^3))/(35*a^4*x^(7/2))

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Maple [A]  time = 0.004, size = 46, normalized size = 0.5 \begin{align*} -{\frac{-32\,{b}^{3}{x}^{3}+16\,a{b}^{2}{x}^{2}-12\,{a}^{2}bx+10\,{a}^{3}}{35\,{a}^{4}}\sqrt{bx+a}{x}^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/35*(b*x+a)^(1/2)*(-16*b^3*x^3+8*a*b^2*x^2-6*a^2*b*x+5*a^3)/x^(7/2)/a^4

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Maxima [A]  time = 1.3045, size = 82, normalized size = 0.89 \begin{align*} \frac{2 \,{\left (\frac{35 \, \sqrt{b x + a} b^{3}}{\sqrt{x}} - \frac{35 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{2}}{x^{\frac{3}{2}}} + \frac{21 \,{\left (b x + a\right )}^{\frac{5}{2}} b}{x^{\frac{5}{2}}} - \frac{5 \,{\left (b x + a\right )}^{\frac{7}{2}}}{x^{\frac{7}{2}}}\right )}}{35 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(35*sqrt(b*x + a)*b^3/sqrt(x) - 35*(b*x + a)^(3/2)*b^2/x^(3/2) + 21*(b*x + a)^(5/2)*b/x^(5/2) - 5*(b*x +
a)^(7/2)/x^(7/2))/a^4

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Fricas [A]  time = 1.77953, size = 109, normalized size = 1.18 \begin{align*} \frac{2 \,{\left (16 \, b^{3} x^{3} - 8 \, a b^{2} x^{2} + 6 \, a^{2} b x - 5 \, a^{3}\right )} \sqrt{b x + a}}{35 \, a^{4} x^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(16*b^3*x^3 - 8*a*b^2*x^2 + 6*a^2*b*x - 5*a^3)*sqrt(b*x + a)/(a^4*x^(7/2))

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Sympy [B]  time = 174.652, size = 488, normalized size = 5.3 \begin{align*} - \frac{10 a^{6} b^{\frac{19}{2}} \sqrt{\frac{a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} - \frac{18 a^{5} b^{\frac{21}{2}} x \sqrt{\frac{a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} - \frac{10 a^{4} b^{\frac{23}{2}} x^{2} \sqrt{\frac{a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} + \frac{10 a^{3} b^{\frac{25}{2}} x^{3} \sqrt{\frac{a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} + \frac{60 a^{2} b^{\frac{27}{2}} x^{4} \sqrt{\frac{a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} + \frac{80 a b^{\frac{29}{2}} x^{5} \sqrt{\frac{a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} + \frac{32 b^{\frac{31}{2}} x^{6} \sqrt{\frac{a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-10*a**6*b**(19/2)*sqrt(a/(b*x) + 1)/(35*a**7*b**9*x**3 + 105*a**6*b**10*x**4 + 105*a**5*b**11*x**5 + 35*a**4*
b**12*x**6) - 18*a**5*b**(21/2)*x*sqrt(a/(b*x) + 1)/(35*a**7*b**9*x**3 + 105*a**6*b**10*x**4 + 105*a**5*b**11*
x**5 + 35*a**4*b**12*x**6) - 10*a**4*b**(23/2)*x**2*sqrt(a/(b*x) + 1)/(35*a**7*b**9*x**3 + 105*a**6*b**10*x**4
+ 105*a**5*b**11*x**5 + 35*a**4*b**12*x**6) + 10*a**3*b**(25/2)*x**3*sqrt(a/(b*x) + 1)/(35*a**7*b**9*x**3 + 1
05*a**6*b**10*x**4 + 105*a**5*b**11*x**5 + 35*a**4*b**12*x**6) + 60*a**2*b**(27/2)*x**4*sqrt(a/(b*x) + 1)/(35*
a**7*b**9*x**3 + 105*a**6*b**10*x**4 + 105*a**5*b**11*x**5 + 35*a**4*b**12*x**6) + 80*a*b**(29/2)*x**5*sqrt(a/
(b*x) + 1)/(35*a**7*b**9*x**3 + 105*a**6*b**10*x**4 + 105*a**5*b**11*x**5 + 35*a**4*b**12*x**6) + 32*b**(31/2)
*x**6*sqrt(a/(b*x) + 1)/(35*a**7*b**9*x**3 + 105*a**6*b**10*x**4 + 105*a**5*b**11*x**5 + 35*a**4*b**12*x**6)

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Giac [A]  time = 1.0962, size = 111, normalized size = 1.21 \begin{align*} -\frac{{\left (2 \,{\left (b x + a\right )}{\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 )} - \frac{35}{a b^{5}}\right )} \sqrt{b x + a} b}{13440 \,{\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(1/x^(9/2)/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

-1/13440*(2*(b*x + a)*(4*(b*x + a)*(2*(b*x + a)/(a^4*b^5) - 7/(a^3*b^5)) + 35/(a^2*b^5)) - 35/(a*b^5))*sqrt(b*
x + a)*b/(((b*x + a)*b - a*b)^(7/2)*abs(b))