3.1768 \(\int (a+\frac{b}{x})^{5/2} x^{11/2} \, dx\)

Optimal. Leaf size=100 \[ \frac{16 b^2 x^{9/2} \left (a+\frac{b}{x}\right )^{7/2}}{429 a^3}-\frac{32 b^3 x^{7/2} \left (a+\frac{b}{x}\right )^{7/2}}{3003 a^4}-\frac{12 b x^{11/2} \left (a+\frac{b}{x}\right )^{7/2}}{143 a^2}+\frac{2 x^{13/2} \left (a+\frac{b}{x}\right )^{7/2}}{13 a} \]

[Out]

(-32*b^3*(a + b/x)^(7/2)*x^(7/2))/(3003*a^4) + (16*b^2*(a + b/x)^(7/2)*x^(9/2))/(429*a^3) - (12*b*(a + b/x)^(7
/2)*x^(11/2))/(143*a^2) + (2*(a + b/x)^(7/2)*x^(13/2))/(13*a)

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Rubi [A]  time = 0.0345881, 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} \[ \frac{16 b^2 x^{9/2} \left (a+\frac{b}{x}\right )^{7/2}}{429 a^3}-\frac{32 b^3 x^{7/2} \left (a+\frac{b}{x}\right )^{7/2}}{3003 a^4}-\frac{12 b x^{11/2} \left (a+\frac{b}{x}\right )^{7/2}}{143 a^2}+\frac{2 x^{13/2} \left (a+\frac{b}{x}\right )^{7/2}}{13 a} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^(5/2)*x^(11/2),x]

[Out]

(-32*b^3*(a + b/x)^(7/2)*x^(7/2))/(3003*a^4) + (16*b^2*(a + b/x)^(7/2)*x^(9/2))/(429*a^3) - (12*b*(a + b/x)^(7
/2)*x^(11/2))/(143*a^2) + (2*(a + b/x)^(7/2)*x^(13/2))/(13*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 \left (a+\frac{b}{x}\right )^{5/2} x^{11/2} \, dx &=\frac{2 \left (a+\frac{b}{x}\right )^{7/2} x^{13/2}}{13 a}-\frac{(6 b) \int \left (a+\frac{b}{x}\right )^{5/2} x^{9/2} \, dx}{13 a}\\ &=-\frac{12 b \left (a+\frac{b}{x}\right )^{7/2} x^{11/2}}{143 a^2}+\frac{2 \left (a+\frac{b}{x}\right )^{7/2} x^{13/2}}{13 a}+\frac{\left (24 b^2\right ) \int \left (a+\frac{b}{x}\right )^{5/2} x^{7/2} \, dx}{143 a^2}\\ &=\frac{16 b^2 \left (a+\frac{b}{x}\right )^{7/2} x^{9/2}}{429 a^3}-\frac{12 b \left (a+\frac{b}{x}\right )^{7/2} x^{11/2}}{143 a^2}+\frac{2 \left (a+\frac{b}{x}\right )^{7/2} x^{13/2}}{13 a}-\frac{\left (16 b^3\right ) \int \left (a+\frac{b}{x}\right )^{5/2} x^{5/2} \, dx}{429 a^3}\\ &=-\frac{32 b^3 \left (a+\frac{b}{x}\right )^{7/2} x^{7/2}}{3003 a^4}+\frac{16 b^2 \left (a+\frac{b}{x}\right )^{7/2} x^{9/2}}{429 a^3}-\frac{12 b \left (a+\frac{b}{x}\right )^{7/2} x^{11/2}}{143 a^2}+\frac{2 \left (a+\frac{b}{x}\right )^{7/2} x^{13/2}}{13 a}\\ \end{align*}

Mathematica [A]  time = 0.01934, size = 60, normalized size = 0.6 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} (a x+b)^3 \left (-126 a^2 b x^2+231 a^3 x^3+56 a b^2 x-16 b^3\right )}{3003 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^(5/2)*x^(11/2),x]

[Out]

(2*Sqrt[a + b/x]*Sqrt[x]*(b + a*x)^3*(-16*b^3 + 56*a*b^2*x - 126*a^2*b*x^2 + 231*a^3*x^3))/(3003*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 ( 231\,{a}^{3}{x}^{3}-126\,{a}^{2}b{x}^{2}+56\,xa{b}^{2}-16\,{b}^{3} \right ) }{3003\,{a}^{4}}{x}^{{\frac{5}{2}}} \left ({\frac{ax+b}{x}} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^(5/2)*x^(11/2),x)

[Out]

2/3003*(a*x+b)*(231*a^3*x^3-126*a^2*b*x^2+56*a*b^2*x-16*b^3)*x^(5/2)*((a*x+b)/x)^(5/2)/a^4

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Maxima [A]  time = 0.987642, size = 93, normalized size = 0.93 \begin{align*} \frac{2 \,{\left (231 \,{\left (a + \frac{b}{x}\right )}^{\frac{13}{2}} x^{\frac{13}{2}} - 819 \,{\left (a + \frac{b}{x}\right )}^{\frac{11}{2}} b x^{\frac{11}{2}} + 1001 \,{\left (a + \frac{b}{x}\right )}^{\frac{9}{2}} b^{2} x^{\frac{9}{2}} - 429 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} b^{3} x^{\frac{7}{2}}\right )}}{3003 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(5/2)*x^(11/2),x, algorithm="maxima")

[Out]

2/3003*(231*(a + b/x)^(13/2)*x^(13/2) - 819*(a + b/x)^(11/2)*b*x^(11/2) + 1001*(a + b/x)^(9/2)*b^2*x^(9/2) - 4
29*(a + b/x)^(7/2)*b^3*x^(7/2))/a^4

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Fricas [A]  time = 1.47645, size = 188, normalized size = 1.88 \begin{align*} \frac{2 \,{\left (231 \, a^{6} x^{6} + 567 \, a^{5} b x^{5} + 371 \, a^{4} b^{2} x^{4} + 5 \, a^{3} b^{3} x^{3} - 6 \, a^{2} b^{4} x^{2} + 8 \, a b^{5} x - 16 \, b^{6}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{3003 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(5/2)*x^(11/2),x, algorithm="fricas")

[Out]

2/3003*(231*a^6*x^6 + 567*a^5*b*x^5 + 371*a^4*b^2*x^4 + 5*a^3*b^3*x^3 - 6*a^2*b^4*x^2 + 8*a*b^5*x - 16*b^6)*sq
rt(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)**(5/2)*x**(11/2),x)

[Out]

Timed out

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Giac [B]  time = 1.24083, size = 309, normalized size = 3.09 \begin{align*} \frac{2}{315} \, b^{2}{\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 ) - \frac{4}{3465} \, a b{\left (\frac{128 \, b^{\frac{11}{2}}}{a^{5}} - \frac{315 \,{\left (a x + b\right )}^{\frac{11}{2}} - 1540 \,{\left (a x + b\right )}^{\frac{9}{2}} b + 2970 \,{\left (a x + b\right )}^{\frac{7}{2}} b^{2} - 2772 \,{\left (a x + b\right )}^{\frac{5}{2}} b^{3} + 1155 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{4}}{a^{5}}\right )} \mathrm{sgn}\left (x\right ) + \frac{2}{9009} \, a^{2}{\left (\frac{256 \, b^{\frac{13}{2}}}{a^{6}} + \frac{693 \,{\left (a x + b\right )}^{\frac{13}{2}} - 4095 \,{\left (a x + b\right )}^{\frac{11}{2}} b + 10010 \,{\left (a x + b\right )}^{\frac{9}{2}} b^{2} - 12870 \,{\left (a x + b\right )}^{\frac{7}{2}} b^{3} + 9009 \,{\left (a x + b\right )}^{\frac{5}{2}} b^{4} - 3003 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{5}}{a^{6}}\right )} \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(5/2)*x^(11/2),x, algorithm="giac")

[Out]

2/315*b^2*(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) - 4/3465*a*b*(128*b^(11/2)/a^5 - (315*(a*x + b)^(11/2) - 1540*(a*x + b)^(9/2)*b + 2
970*(a*x + b)^(7/2)*b^2 - 2772*(a*x + b)^(5/2)*b^3 + 1155*(a*x + b)^(3/2)*b^4)/a^5)*sgn(x) + 2/9009*a^2*(256*b
^(13/2)/a^6 + (693*(a*x + b)^(13/2) - 4095*(a*x + b)^(11/2)*b + 10010*(a*x + b)^(9/2)*b^2 - 12870*(a*x + b)^(7
/2)*b^3 + 9009*(a*x + b)^(5/2)*b^4 - 3003*(a*x + b)^(3/2)*b^5)/a^6)*sgn(x)