Optimal. Leaf size=129 \[ \frac {63}{8} a^2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )+\frac {21}{4} b^3 x \left (a+b x^2\right )^{3/2}+\frac {63}{8} a b^3 x \sqrt {a+b x^2}-\frac {21 b^2 \left (a+b x^2\right )^{5/2}}{5 x}-\frac {\left (a+b x^2\right )^{9/2}}{5 x^5}-\frac {3 b \left (a+b x^2\right )^{7/2}}{5 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {277, 195, 217, 206} \[ \frac {63}{8} a^2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {21 b^2 \left (a+b x^2\right )^{5/2}}{5 x}+\frac {21}{4} b^3 x \left (a+b x^2\right )^{3/2}+\frac {63}{8} a b^3 x \sqrt {a+b x^2}-\frac {\left (a+b x^2\right )^{9/2}}{5 x^5}-\frac {3 b \left (a+b x^2\right )^{7/2}}{5 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 195
Rule 206
Rule 217
Rule 277
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{9/2}}{x^6} \, dx &=-\frac {\left (a+b x^2\right )^{9/2}}{5 x^5}+\frac {1}{5} (9 b) \int \frac {\left (a+b x^2\right )^{7/2}}{x^4} \, dx\\ &=-\frac {3 b \left (a+b x^2\right )^{7/2}}{5 x^3}-\frac {\left (a+b x^2\right )^{9/2}}{5 x^5}+\frac {1}{5} \left (21 b^2\right ) \int \frac {\left (a+b x^2\right )^{5/2}}{x^2} \, dx\\ &=-\frac {21 b^2 \left (a+b x^2\right )^{5/2}}{5 x}-\frac {3 b \left (a+b x^2\right )^{7/2}}{5 x^3}-\frac {\left (a+b x^2\right )^{9/2}}{5 x^5}+\left (21 b^3\right ) \int \left (a+b x^2\right )^{3/2} \, dx\\ &=\frac {21}{4} b^3 x \left (a+b x^2\right )^{3/2}-\frac {21 b^2 \left (a+b x^2\right )^{5/2}}{5 x}-\frac {3 b \left (a+b x^2\right )^{7/2}}{5 x^3}-\frac {\left (a+b x^2\right )^{9/2}}{5 x^5}+\frac {1}{4} \left (63 a b^3\right ) \int \sqrt {a+b x^2} \, dx\\ &=\frac {63}{8} a b^3 x \sqrt {a+b x^2}+\frac {21}{4} b^3 x \left (a+b x^2\right )^{3/2}-\frac {21 b^2 \left (a+b x^2\right )^{5/2}}{5 x}-\frac {3 b \left (a+b x^2\right )^{7/2}}{5 x^3}-\frac {\left (a+b x^2\right )^{9/2}}{5 x^5}+\frac {1}{8} \left (63 a^2 b^3\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx\\ &=\frac {63}{8} a b^3 x \sqrt {a+b x^2}+\frac {21}{4} b^3 x \left (a+b x^2\right )^{3/2}-\frac {21 b^2 \left (a+b x^2\right )^{5/2}}{5 x}-\frac {3 b \left (a+b x^2\right )^{7/2}}{5 x^3}-\frac {\left (a+b x^2\right )^{9/2}}{5 x^5}+\frac {1}{8} \left (63 a^2 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )\\ &=\frac {63}{8} a b^3 x \sqrt {a+b x^2}+\frac {21}{4} b^3 x \left (a+b x^2\right )^{3/2}-\frac {21 b^2 \left (a+b x^2\right )^{5/2}}{5 x}-\frac {3 b \left (a+b x^2\right )^{7/2}}{5 x^3}-\frac {\left (a+b x^2\right )^{9/2}}{5 x^5}+\frac {63}{8} a^2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 54, normalized size = 0.42 \[ -\frac {a^4 \sqrt {a+b x^2} \, _2F_1\left (-\frac {9}{2},-\frac {5}{2};-\frac {3}{2};-\frac {b x^2}{a}\right )}{5 x^5 \sqrt {\frac {b x^2}{a}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.04, size = 191, normalized size = 1.48 \[ \left [\frac {315 \, a^{2} b^{\frac {5}{2}} x^{5} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (10 \, b^{4} x^{8} + 85 \, a b^{3} x^{6} - 288 \, a^{2} b^{2} x^{4} - 56 \, a^{3} b x^{2} - 8 \, a^{4}\right )} \sqrt {b x^{2} + a}}{80 \, x^{5}}, -\frac {315 \, a^{2} \sqrt {-b} b^{2} x^{5} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (10 \, b^{4} x^{8} + 85 \, a b^{3} x^{6} - 288 \, a^{2} b^{2} x^{4} - 56 \, a^{3} b x^{2} - 8 \, a^{4}\right )} \sqrt {b x^{2} + a}}{40 \, x^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.08, size = 200, normalized size = 1.55 \[ -\frac {63}{16} \, a^{2} b^{\frac {5}{2}} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right ) + \frac {1}{8} \, {\left (2 \, b^{4} x^{2} + 17 \, a b^{3}\right )} \sqrt {b x^{2} + a} x + \frac {4 \, {\left (25 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{3} b^{\frac {5}{2}} - 75 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{4} b^{\frac {5}{2}} + 105 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{5} b^{\frac {5}{2}} - 65 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{6} b^{\frac {5}{2}} + 18 \, a^{7} b^{\frac {5}{2}}\right )}}{5 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 166, normalized size = 1.29 \[ \frac {63 a^{2} b^{\frac {5}{2}} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{8}+\frac {63 \sqrt {b \,x^{2}+a}\, a \,b^{3} x}{8}+\frac {21 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{3} x}{4}+\frac {21 \left (b \,x^{2}+a \right )^{\frac {5}{2}} b^{3} x}{5 a}+\frac {18 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{3} x}{5 a^{2}}+\frac {16 \left (b \,x^{2}+a \right )^{\frac {9}{2}} b^{3} x}{5 a^{3}}-\frac {16 \left (b \,x^{2}+a \right )^{\frac {11}{2}} b^{2}}{5 a^{3} x}-\frac {2 \left (b \,x^{2}+a \right )^{\frac {11}{2}} b}{5 a^{2} x^{3}}-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{5 a \,x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.45, size = 140, normalized size = 1.09 \[ \frac {21}{4} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3} x + \frac {18 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3} x}{5 \, a^{2}} + \frac {21 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3} x}{5 \, a} + \frac {63}{8} \, \sqrt {b x^{2} + a} a b^{3} x + \frac {63}{8} \, a^{2} b^{\frac {5}{2}} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} b^{2}}{5 \, a^{2} x} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b}{5 \, a^{2} x^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}}}{5 \, a x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (b\,x^2+a\right )}^{9/2}}{x^6} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 7.82, size = 175, normalized size = 1.36 \[ - \frac {a^{\frac {9}{2}}}{5 x^{5} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {8 a^{\frac {7}{2}} b}{5 x^{3} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {43 a^{\frac {5}{2}} b^{2}}{5 x \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {203 a^{\frac {3}{2}} b^{3} x}{40 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {19 \sqrt {a} b^{4} x^{3}}{8 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {63 a^{2} b^{\frac {5}{2}} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{8} + \frac {b^{5} x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________