3.1752 \(\int \sqrt{a+\frac{b}{x}} x^{7/2} \, dx\)

Optimal. Leaf size=100 \[ \frac{16 b^2 x^{5/2} \left (a+\frac{b}{x}\right )^{3/2}}{105 a^3}-\frac{32 b^3 x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}{315 a^4}-\frac{4 b x^{7/2} \left (a+\frac{b}{x}\right )^{3/2}}{21 a^2}+\frac{2 x^{9/2} \left (a+\frac{b}{x}\right )^{3/2}}{9 a} \]

[Out]

(-32*b^3*(a + b/x)^(3/2)*x^(3/2))/(315*a^4) + (16*b^2*(a + b/x)^(3/2)*x^(5/2))/(105*a^3) - (4*b*(a + b/x)^(3/2
)*x^(7/2))/(21*a^2) + (2*(a + b/x)^(3/2)*x^(9/2))/(9*a)

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Rubi [A]  time = 0.031134, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {271, 264} \[ \frac{16 b^2 x^{5/2} \left (a+\frac{b}{x}\right )^{3/2}}{105 a^3}-\frac{32 b^3 x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}{315 a^4}-\frac{4 b x^{7/2} \left (a+\frac{b}{x}\right )^{3/2}}{21 a^2}+\frac{2 x^{9/2} \left (a+\frac{b}{x}\right )^{3/2}}{9 a} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/x]*x^(7/2),x]

[Out]

(-32*b^3*(a + b/x)^(3/2)*x^(3/2))/(315*a^4) + (16*b^2*(a + b/x)^(3/2)*x^(5/2))/(105*a^3) - (4*b*(a + b/x)^(3/2
)*x^(7/2))/(21*a^2) + (2*(a + b/x)^(3/2)*x^(9/2))/(9*a)

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \sqrt{a+\frac{b}{x}} x^{7/2} \, dx &=\frac{2 \left (a+\frac{b}{x}\right )^{3/2} x^{9/2}}{9 a}-\frac{(2 b) \int \sqrt{a+\frac{b}{x}} x^{5/2} \, dx}{3 a}\\ &=-\frac{4 b \left (a+\frac{b}{x}\right )^{3/2} x^{7/2}}{21 a^2}+\frac{2 \left (a+\frac{b}{x}\right )^{3/2} x^{9/2}}{9 a}+\frac{\left (8 b^2\right ) \int \sqrt{a+\frac{b}{x}} x^{3/2} \, dx}{21 a^2}\\ &=\frac{16 b^2 \left (a+\frac{b}{x}\right )^{3/2} x^{5/2}}{105 a^3}-\frac{4 b \left (a+\frac{b}{x}\right )^{3/2} x^{7/2}}{21 a^2}+\frac{2 \left (a+\frac{b}{x}\right )^{3/2} x^{9/2}}{9 a}-\frac{\left (16 b^3\right ) \int \sqrt{a+\frac{b}{x}} \sqrt{x} \, dx}{105 a^3}\\ &=-\frac{32 b^3 \left (a+\frac{b}{x}\right )^{3/2} x^{3/2}}{315 a^4}+\frac{16 b^2 \left (a+\frac{b}{x}\right )^{3/2} x^{5/2}}{105 a^3}-\frac{4 b \left (a+\frac{b}{x}\right )^{3/2} x^{7/2}}{21 a^2}+\frac{2 \left (a+\frac{b}{x}\right )^{3/2} x^{9/2}}{9 a}\\ \end{align*}

Mathematica [A]  time = 0.015364, size = 58, normalized size = 0.58 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} (a x+b) \left (-30 a^2 b x^2+35 a^3 x^3+24 a b^2 x-16 b^3\right )}{315 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/x]*x^(7/2),x]

[Out]

(2*Sqrt[a + b/x]*Sqrt[x]*(b + a*x)*(-16*b^3 + 24*a*b^2*x - 30*a^2*b*x^2 + 35*a^3*x^3))/(315*a^4)

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Maple [A]  time = 0.005, size = 55, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 35\,{a}^{3}{x}^{3}-30\,{a}^{2}b{x}^{2}+24\,xa{b}^{2}-16\,{b}^{3} \right ) }{315\,{a}^{4}}\sqrt{x}\sqrt{{\frac{ax+b}{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(a*x+b)*(35*a^3*x^3-30*a^2*b*x^2+24*a*b^2*x-16*b^3)*x^(1/2)*((a*x+b)/x)^(1/2)/a^4

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Maxima [A]  time = 0.959088, size = 93, normalized size = 0.93 \begin{align*} \frac{2 \,{\left (35 \,{\left (a + \frac{b}{x}\right )}^{\frac{9}{2}} x^{\frac{9}{2}} - 135 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} b x^{\frac{7}{2}} + 189 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} b^{2} x^{\frac{5}{2}} - 105 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b^{3} x^{\frac{3}{2}}\right )}}{315 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*(a + b/x)^(9/2)*x^(9/2) - 135*(a + b/x)^(7/2)*b*x^(7/2) + 189*(a + b/x)^(5/2)*b^2*x^(5/2) - 105*(a +
b/x)^(3/2)*b^3*x^(3/2))/a^4

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Fricas [A]  time = 1.43864, size = 136, normalized size = 1.36 \begin{align*} \frac{2 \,{\left (35 \, a^{4} x^{4} + 5 \, a^{3} b x^{3} - 6 \, a^{2} b^{2} x^{2} + 8 \, a b^{3} x - 16 \, b^{4}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{315 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*a^4*x^4 + 5*a^3*b*x^3 - 6*a^2*b^2*x^2 + 8*a*b^3*x - 16*b^4)*sqrt(x)*sqrt((a*x + b)/x)/a^4

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.17217, size = 82, normalized size = 0.82 \begin{align*} \frac{2}{315} \,{\left (\frac{16 \, b^{\frac{9}{2}}}{a^{4}} + \frac{35 \,{\left (a x + b\right )}^{\frac{9}{2}} - 135 \,{\left (a x + b\right )}^{\frac{7}{2}} b + 189 \,{\left (a x + b\right )}^{\frac{5}{2}} b^{2} - 105 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{3}}{a^{4}}\right )} \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(16*b^(9/2)/a^4 + (35*(a*x + b)^(9/2) - 135*(a*x + b)^(7/2)*b + 189*(a*x + b)^(5/2)*b^2 - 105*(a*x + b)^
(3/2)*b^3)/a^4)*sgn(x)