Optimal. Leaf size=86 \[ \frac {5}{2} a b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )+\frac {5}{2} b^2 x \sqrt {a+b x^2}-\frac {5 b \left (a+b x^2\right )^{3/2}}{3 x}-\frac {\left (a+b x^2\right )^{5/2}}{3 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {277, 195, 217, 206} \[ \frac {5}{2} b^2 x \sqrt {a+b x^2}+\frac {5}{2} a b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {\left (a+b x^2\right )^{5/2}}{3 x^3}-\frac {5 b \left (a+b x^2\right )^{3/2}}{3 x} \]
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 )^{5/2}}{x^4} \, dx &=-\frac {\left (a+b x^2\right )^{5/2}}{3 x^3}+\frac {1}{3} (5 b) \int \frac {\left (a+b x^2\right )^{3/2}}{x^2} \, dx\\ &=-\frac {5 b \left (a+b x^2\right )^{3/2}}{3 x}-\frac {\left (a+b x^2\right )^{5/2}}{3 x^3}+\left (5 b^2\right ) \int \sqrt {a+b x^2} \, dx\\ &=\frac {5}{2} b^2 x \sqrt {a+b x^2}-\frac {5 b \left (a+b x^2\right )^{3/2}}{3 x}-\frac {\left (a+b x^2\right )^{5/2}}{3 x^3}+\frac {1}{2} \left (5 a b^2\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx\\ &=\frac {5}{2} b^2 x \sqrt {a+b x^2}-\frac {5 b \left (a+b x^2\right )^{3/2}}{3 x}-\frac {\left (a+b x^2\right )^{5/2}}{3 x^3}+\frac {1}{2} \left (5 a b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )\\ &=\frac {5}{2} b^2 x \sqrt {a+b x^2}-\frac {5 b \left (a+b x^2\right )^{3/2}}{3 x}-\frac {\left (a+b x^2\right )^{5/2}}{3 x^3}+\frac {5}{2} a b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 54, normalized size = 0.63 \[ -\frac {a^2 \sqrt {a+b x^2} \, _2F_1\left (-\frac {5}{2},-\frac {3}{2};-\frac {1}{2};-\frac {b x^2}{a}\right )}{3 x^3 \sqrt {\frac {b x^2}{a}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.93, size = 141, normalized size = 1.64 \[ \left [\frac {15 \, a b^{\frac {3}{2}} x^{3} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (3 \, b^{2} x^{4} - 14 \, a b x^{2} - 2 \, a^{2}\right )} \sqrt {b x^{2} + a}}{12 \, x^{3}}, -\frac {15 \, a \sqrt {-b} b x^{3} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (3 \, b^{2} x^{4} - 14 \, a b x^{2} - 2 \, a^{2}\right )} \sqrt {b x^{2} + a}}{6 \, x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.12, size = 132, normalized size = 1.53 \[ \frac {1}{2} \, \sqrt {b x^{2} + a} b^{2} x - \frac {5}{4} \, a b^{\frac {3}{2}} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right ) + \frac {2 \, {\left (9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b^{\frac {3}{2}} - 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{3} b^{\frac {3}{2}} + 7 \, a^{4} b^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 110, normalized size = 1.28 \[ \frac {5 a \,b^{\frac {3}{2}} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2}+\frac {5 \sqrt {b \,x^{2}+a}\, b^{2} x}{2}+\frac {5 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{2} x}{3 a}+\frac {4 \left (b \,x^{2}+a \right )^{\frac {5}{2}} b^{2} x}{3 a^{2}}-\frac {4 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b}{3 a^{2} x}-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{3 a \,x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.32, size = 84, normalized size = 0.98 \[ \frac {5}{2} \, \sqrt {b x^{2} + a} b^{2} x + \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2} x}{3 \, a} + \frac {5}{2} \, a b^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b}{3 \, a x} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}}}{3 \, a x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (b\,x^2+a\right )}^{5/2}}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 3.16, size = 112, normalized size = 1.30 \[ - \frac {a^{2} \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{3 x^{2}} - \frac {7 a b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3} - \frac {5 a b^{\frac {3}{2}} \log {\left (\frac {a}{b x^{2}} \right )}}{4} + \frac {5 a b^{\frac {3}{2}} \log {\left (\sqrt {\frac {a}{b x^{2}} + 1} + 1 \right )}}{2} + \frac {b^{\frac {5}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________