Optimal. Leaf size=116 \[ \frac{5 a^4 \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{128 b^{3/2}}-\frac{5 a^3 \sqrt{a+\frac{b}{x^2}}}{128 b x}-\frac{5 a^2 \sqrt{a+\frac{b}{x^2}}}{64 x^3}-\frac{5 a \left (a+\frac{b}{x^2}\right )^{3/2}}{48 x^3}-\frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{8 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0617103, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {335, 279, 321, 217, 206} \[ \frac{5 a^4 \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{128 b^{3/2}}-\frac{5 a^3 \sqrt{a+\frac{b}{x^2}}}{128 b x}-\frac{5 a^2 \sqrt{a+\frac{b}{x^2}}}{64 x^3}-\frac{5 a \left (a+\frac{b}{x^2}\right )^{3/2}}{48 x^3}-\frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{8 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 335
Rule 279
Rule 321
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{x^4} \, dx &=-\operatorname{Subst}\left (\int x^2 \left (a+b x^2\right )^{5/2} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{8 x^3}-\frac{1}{8} (5 a) \operatorname{Subst}\left (\int x^2 \left (a+b x^2\right )^{3/2} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{5 a \left (a+\frac{b}{x^2}\right )^{3/2}}{48 x^3}-\frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{8 x^3}-\frac{1}{16} \left (5 a^2\right ) \operatorname{Subst}\left (\int x^2 \sqrt{a+b x^2} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{5 a^2 \sqrt{a+\frac{b}{x^2}}}{64 x^3}-\frac{5 a \left (a+\frac{b}{x^2}\right )^{3/2}}{48 x^3}-\frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{8 x^3}-\frac{1}{64} \left (5 a^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{5 a^2 \sqrt{a+\frac{b}{x^2}}}{64 x^3}-\frac{5 a \left (a+\frac{b}{x^2}\right )^{3/2}}{48 x^3}-\frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{8 x^3}-\frac{5 a^3 \sqrt{a+\frac{b}{x^2}}}{128 b x}+\frac{\left (5 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{x}\right )}{128 b}\\ &=-\frac{5 a^2 \sqrt{a+\frac{b}{x^2}}}{64 x^3}-\frac{5 a \left (a+\frac{b}{x^2}\right )^{3/2}}{48 x^3}-\frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{8 x^3}-\frac{5 a^3 \sqrt{a+\frac{b}{x^2}}}{128 b x}+\frac{\left (5 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{1}{\sqrt{a+\frac{b}{x^2}} x}\right )}{128 b}\\ &=-\frac{5 a^2 \sqrt{a+\frac{b}{x^2}}}{64 x^3}-\frac{5 a \left (a+\frac{b}{x^2}\right )^{3/2}}{48 x^3}-\frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{8 x^3}-\frac{5 a^3 \sqrt{a+\frac{b}{x^2}}}{128 b x}+\frac{5 a^4 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a+\frac{b}{x^2}} x}\right )}{128 b^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0146198, size = 49, normalized size = 0.42 \[ -\frac{a^4 x^5 \left (a+\frac{b}{x^2}\right )^{5/2} \left (a x^2+b\right ) \, _2F_1\left (\frac{7}{2},5;\frac{9}{2};\frac{a x^2}{b}+1\right )}{7 b^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.017, size = 186, normalized size = 1.6 \begin{align*}{\frac{1}{384\,{x}^{3}{b}^{4}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{{\frac{5}{2}}} \left ( -3\, \left ( a{x}^{2}+b \right ) ^{5/2}{x}^{8}{a}^{4}+3\, \left ( a{x}^{2}+b \right ) ^{7/2}{x}^{6}{a}^{3}-5\, \left ( a{x}^{2}+b \right ) ^{3/2}{x}^{8}{a}^{4}b+15\,{b}^{5/2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{2}+b}+b}{x}} \right ){x}^{8}{a}^{4}-15\,\sqrt{a{x}^{2}+b}{x}^{8}{a}^{4}{b}^{2}+2\, \left ( a{x}^{2}+b \right ) ^{7/2}{x}^{4}{a}^{2}b+8\, \left ( a{x}^{2}+b \right ) ^{7/2}{x}^{2}a{b}^{2}-48\, \left ( a{x}^{2}+b \right ) ^{7/2}{b}^{3} \right ) \left ( a{x}^{2}+b \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.60664, size = 487, normalized size = 4.2 \begin{align*} \left [\frac{15 \, a^{4} \sqrt{b} x^{7} \log \left (-\frac{a x^{2} + 2 \, \sqrt{b} x \sqrt{\frac{a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right ) - 2 \,{\left (15 \, a^{3} b x^{6} + 118 \, a^{2} b^{2} x^{4} + 136 \, a b^{3} x^{2} + 48 \, b^{4}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{768 \, b^{2} x^{7}}, -\frac{15 \, a^{4} \sqrt{-b} x^{7} \arctan \left (\frac{\sqrt{-b} x \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) +{\left (15 \, a^{3} b x^{6} + 118 \, a^{2} b^{2} x^{4} + 136 \, a b^{3} x^{2} + 48 \, b^{4}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{384 \, b^{2} x^{7}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 10.5566, size = 150, normalized size = 1.29 \begin{align*} - \frac{5 a^{\frac{7}{2}}}{128 b x \sqrt{1 + \frac{b}{a x^{2}}}} - \frac{133 a^{\frac{5}{2}}}{384 x^{3} \sqrt{1 + \frac{b}{a x^{2}}}} - \frac{127 a^{\frac{3}{2}} b}{192 x^{5} \sqrt{1 + \frac{b}{a x^{2}}}} - \frac{23 \sqrt{a} b^{2}}{48 x^{7} \sqrt{1 + \frac{b}{a x^{2}}}} + \frac{5 a^{4} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x} \right )}}{128 b^{\frac{3}{2}}} - \frac{b^{3}}{8 \sqrt{a} x^{9} \sqrt{1 + \frac{b}{a x^{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.28517, size = 130, normalized size = 1.12 \begin{align*} -\frac{1}{384} \, a^{4}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{a x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} + \frac{15 \,{\left (a x^{2} + b\right )}^{\frac{7}{2}} + 73 \,{\left (a x^{2} + b\right )}^{\frac{5}{2}} b - 55 \,{\left (a x^{2} + b\right )}^{\frac{3}{2}} b^{2} + 15 \, \sqrt{a x^{2} + b} b^{3}}{a^{4} b x^{8}}\right )} \mathrm{sgn}\left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]