Optimal. Leaf size=59 \[ -\frac {2 a^2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b^3}-\frac {2 \left (a+\frac {b}{x}\right )^{11/2}}{11 b^3}+\frac {4 a \left (a+\frac {b}{x}\right )^{9/2}}{9 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 59, normalized size of antiderivative = 1.00, 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 = {266, 43} \[ -\frac {2 a^2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b^3}-\frac {2 \left (a+\frac {b}{x}\right )^{11/2}}{11 b^3}+\frac {4 a \left (a+\frac {b}{x}\right )^{9/2}}{9 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^4} \, dx &=-\operatorname {Subst}\left (\int x^2 (a+b x)^{5/2} \, dx,x,\frac {1}{x}\right )\\ &=-\operatorname {Subst}\left (\int \left (\frac {a^2 (a+b x)^{5/2}}{b^2}-\frac {2 a (a+b x)^{7/2}}{b^2}+\frac {(a+b x)^{9/2}}{b^2}\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {2 a^2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b^3}+\frac {4 a \left (a+\frac {b}{x}\right )^{9/2}}{9 b^3}-\frac {2 \left (a+\frac {b}{x}\right )^{11/2}}{11 b^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 47, normalized size = 0.80 \[ -\frac {2 \sqrt {a+\frac {b}{x}} (a x+b)^3 \left (8 a^2 x^2-28 a b x+63 b^2\right )}{693 b^3 x^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.91, size = 71, normalized size = 1.20 \[ -\frac {2 \, {\left (8 \, a^{5} x^{5} - 4 \, a^{4} b x^{4} + 3 \, a^{3} b^{2} x^{3} + 113 \, a^{2} b^{3} x^{2} + 161 \, a b^{4} x + 63 \, b^{5}\right )} \sqrt {\frac {a x + b}{x}}}{693 \, b^{3} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.29, size = 270, normalized size = 4.58 \[ \frac {2 \, {\left (924 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{8} a^{4} \mathrm {sgn}\relax (x) + 4851 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{7} a^{\frac {7}{2}} b \mathrm {sgn}\relax (x) + 11781 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{6} a^{3} b^{2} \mathrm {sgn}\relax (x) + 16863 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{5} a^{\frac {5}{2}} b^{3} \mathrm {sgn}\relax (x) + 15345 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{4} a^{2} b^{4} \mathrm {sgn}\relax (x) + 9009 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} a^{\frac {3}{2}} b^{5} \mathrm {sgn}\relax (x) + 3311 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a b^{6} \mathrm {sgn}\relax (x) + 693 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{7} \mathrm {sgn}\relax (x) + 63 \, b^{8} \mathrm {sgn}\relax (x)\right )}}{693 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 44, normalized size = 0.75 \[ -\frac {2 \left (a x +b \right ) \left (8 a^{2} x^{2}-28 a b x +63 b^{2}\right ) \left (\frac {a x +b}{x}\right )^{\frac {5}{2}}}{693 b^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.99, size = 47, normalized size = 0.80 \[ -\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {11}{2}}}{11 \, b^{3}} + \frac {4 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{2}} a}{9 \, b^{3}} - \frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} a^{2}}{7 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.41, size = 108, normalized size = 1.83 \[ \frac {8\,a^4\,\sqrt {a+\frac {b}{x}}}{693\,b^2\,x}-\frac {226\,a^2\,\sqrt {a+\frac {b}{x}}}{693\,x^3}-\frac {2\,b^2\,\sqrt {a+\frac {b}{x}}}{11\,x^5}-\frac {2\,a^3\,\sqrt {a+\frac {b}{x}}}{231\,b\,x^2}-\frac {16\,a^5\,\sqrt {a+\frac {b}{x}}}{693\,b^3}-\frac {46\,a\,b\,\sqrt {a+\frac {b}{x}}}{99\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 2.76, size = 1073, normalized size = 18.19 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________