Optimal. Leaf size=77 \[ -\frac {b^{3/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{7/2}}+\frac {b x (b B-A c)}{c^3}-\frac {x^3 (b B-A c)}{3 c^2}+\frac {B x^5}{5 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1584, 459, 302, 205} \[ -\frac {b^{3/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{7/2}}-\frac {x^3 (b B-A c)}{3 c^2}+\frac {b x (b B-A c)}{c^3}+\frac {B x^5}{5 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 302
Rule 459
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^6 \left (A+B x^2\right )}{b x^2+c x^4} \, dx &=\int \frac {x^4 \left (A+B x^2\right )}{b+c x^2} \, dx\\ &=\frac {B x^5}{5 c}-\frac {(5 b B-5 A c) \int \frac {x^4}{b+c x^2} \, dx}{5 c}\\ &=\frac {B x^5}{5 c}-\frac {(5 b B-5 A c) \int \left (-\frac {b}{c^2}+\frac {x^2}{c}+\frac {b^2}{c^2 \left (b+c x^2\right )}\right ) \, dx}{5 c}\\ &=\frac {b (b B-A c) x}{c^3}-\frac {(b B-A c) x^3}{3 c^2}+\frac {B x^5}{5 c}-\frac {\left (b^2 (b B-A c)\right ) \int \frac {1}{b+c x^2} \, dx}{c^3}\\ &=\frac {b (b B-A c) x}{c^3}-\frac {(b B-A c) x^3}{3 c^2}+\frac {B x^5}{5 c}-\frac {b^{3/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 77, normalized size = 1.00 \[ -\frac {b^{3/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{7/2}}+\frac {b x (b B-A c)}{c^3}+\frac {x^3 (A c-b B)}{3 c^2}+\frac {B x^5}{5 c} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.96, size = 178, normalized size = 2.31 \[ \left [\frac {6 \, B c^{2} x^{5} - 10 \, {\left (B b c - A c^{2}\right )} x^{3} - 15 \, {\left (B b^{2} - A b c\right )} \sqrt {-\frac {b}{c}} \log \left (\frac {c x^{2} + 2 \, c x \sqrt {-\frac {b}{c}} - b}{c x^{2} + b}\right ) + 30 \, {\left (B b^{2} - A b c\right )} x}{30 \, c^{3}}, \frac {3 \, B c^{2} x^{5} - 5 \, {\left (B b c - A c^{2}\right )} x^{3} - 15 \, {\left (B b^{2} - A b c\right )} \sqrt {\frac {b}{c}} \arctan \left (\frac {c x \sqrt {\frac {b}{c}}}{b}\right ) + 15 \, {\left (B b^{2} - A b c\right )} x}{15 \, c^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 85, normalized size = 1.10 \[ -\frac {{\left (B b^{3} - A b^{2} c\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} c^{3}} + \frac {3 \, B c^{4} x^{5} - 5 \, B b c^{3} x^{3} + 5 \, A c^{4} x^{3} + 15 \, B b^{2} c^{2} x - 15 \, A b c^{3} x}{15 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 92, normalized size = 1.19 \[ \frac {B \,x^{5}}{5 c}+\frac {A \,x^{3}}{3 c}-\frac {B b \,x^{3}}{3 c^{2}}+\frac {A \,b^{2} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c}\, c^{2}}-\frac {B \,b^{3} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c}\, c^{3}}-\frac {A b x}{c^{2}}+\frac {B \,b^{2} x}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.86, size = 78, normalized size = 1.01 \[ -\frac {{\left (B b^{3} - A b^{2} c\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} c^{3}} + \frac {3 \, B c^{2} x^{5} - 5 \, {\left (B b c - A c^{2}\right )} x^{3} + 15 \, {\left (B b^{2} - A b c\right )} x}{15 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.07, size = 96, normalized size = 1.25 \[ x^3\,\left (\frac {A}{3\,c}-\frac {B\,b}{3\,c^2}\right )+\frac {B\,x^5}{5\,c}-\frac {b^{3/2}\,\mathrm {atan}\left (\frac {b^{3/2}\,\sqrt {c}\,x\,\left (A\,c-B\,b\right )}{B\,b^3-A\,b^2\,c}\right )\,\left (A\,c-B\,b\right )}{c^{7/2}}-\frac {b\,x\,\left (\frac {A}{c}-\frac {B\,b}{c^2}\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.37, size = 153, normalized size = 1.99 \[ \frac {B x^{5}}{5 c} + x^{3} \left (\frac {A}{3 c} - \frac {B b}{3 c^{2}}\right ) + x \left (- \frac {A b}{c^{2}} + \frac {B b^{2}}{c^{3}}\right ) + \frac {\sqrt {- \frac {b^{3}}{c^{7}}} \left (- A c + B b\right ) \log {\left (- \frac {c^{3} \sqrt {- \frac {b^{3}}{c^{7}}} \left (- A c + B b\right )}{- A b c + B b^{2}} + x \right )}}{2} - \frac {\sqrt {- \frac {b^{3}}{c^{7}}} \left (- A c + B b\right ) \log {\left (\frac {c^{3} \sqrt {- \frac {b^{3}}{c^{7}}} \left (- A c + B b\right )}{- A b c + B b^{2}} + x \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________