Optimal. Leaf size=63 \[ -\frac {A \left (b+c x^2\right )^4}{8 b x^8}-\frac {b^3 B}{6 x^6}-\frac {3 b^2 B c}{4 x^4}-\frac {3 b B c^2}{2 x^2}+B c^3 \log (x) \]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 63, 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, 446, 78, 43} \begin {gather*} -\frac {A \left (b+c x^2\right )^4}{8 b x^8}-\frac {3 b^2 B c}{4 x^4}-\frac {b^3 B}{6 x^6}-\frac {3 b B c^2}{2 x^2}+B c^3 \log (x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 78
Rule 446
Rule 1584
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^3}{x^{15}} \, dx &=\int \frac {\left (A+B x^2\right ) \left (b+c x^2\right )^3}{x^9} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(A+B x) (b+c x)^3}{x^5} \, dx,x,x^2\right )\\ &=-\frac {A \left (b+c x^2\right )^4}{8 b x^8}+\frac {1}{2} B \operatorname {Subst}\left (\int \frac {(b+c x)^3}{x^4} \, dx,x,x^2\right )\\ &=-\frac {A \left (b+c x^2\right )^4}{8 b x^8}+\frac {1}{2} B \operatorname {Subst}\left (\int \left (\frac {b^3}{x^4}+\frac {3 b^2 c}{x^3}+\frac {3 b c^2}{x^2}+\frac {c^3}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac {b^3 B}{6 x^6}-\frac {3 b^2 B c}{4 x^4}-\frac {3 b B c^2}{2 x^2}-\frac {A \left (b+c x^2\right )^4}{8 b x^8}+B c^3 \log (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 77, normalized size = 1.22 \begin {gather*} B c^3 \log (x)-\frac {3 A \left (b^3+4 b^2 c x^2+6 b c^2 x^4+4 c^3 x^6\right )+2 b B x^2 \left (2 b^2+9 b c x^2+18 c^2 x^4\right )}{24 x^8} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^3}{x^{15}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.40, size = 77, normalized size = 1.22 \begin {gather*} \frac {24 \, B c^{3} x^{8} \log \relax (x) - 12 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} - 18 \, {\left (B b^{2} c + A b c^{2}\right )} x^{4} - 3 \, A b^{3} - 4 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2}}{24 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 90, normalized size = 1.43 \begin {gather*} \frac {1}{2} \, B c^{3} \log \left (x^{2}\right ) - \frac {25 \, B c^{3} x^{8} + 36 \, B b c^{2} x^{6} + 12 \, A c^{3} x^{6} + 18 \, B b^{2} c x^{4} + 18 \, A b c^{2} x^{4} + 4 \, B b^{3} x^{2} + 12 \, A b^{2} c x^{2} + 3 \, A b^{3}}{24 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 76, normalized size = 1.21 \begin {gather*} B \,c^{3} \ln \relax (x )-\frac {A \,c^{3}}{2 x^{2}}-\frac {3 B b \,c^{2}}{2 x^{2}}-\frac {3 A b \,c^{2}}{4 x^{4}}-\frac {3 B \,b^{2} c}{4 x^{4}}-\frac {A \,b^{2} c}{2 x^{6}}-\frac {B \,b^{3}}{6 x^{6}}-\frac {A \,b^{3}}{8 x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.32, size = 77, normalized size = 1.22 \begin {gather*} \frac {1}{2} \, B c^{3} \log \left (x^{2}\right ) - \frac {12 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 18 \, {\left (B b^{2} c + A b c^{2}\right )} x^{4} + 3 \, A b^{3} + 4 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2}}{24 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.10, size = 75, normalized size = 1.19 \begin {gather*} B\,c^3\,\ln \relax (x)-\frac {x^4\,\left (\frac {3\,B\,b^2\,c}{4}+\frac {3\,A\,b\,c^2}{4}\right )+\frac {A\,b^3}{8}+x^2\,\left (\frac {B\,b^3}{6}+\frac {A\,c\,b^2}{2}\right )+x^6\,\left (\frac {A\,c^3}{2}+\frac {3\,B\,b\,c^2}{2}\right )}{x^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 2.37, size = 82, normalized size = 1.30 \begin {gather*} B c^{3} \log {\relax (x )} + \frac {- 3 A b^{3} + x^{6} \left (- 12 A c^{3} - 36 B b c^{2}\right ) + x^{4} \left (- 18 A b c^{2} - 18 B b^{2} c\right ) + x^{2} \left (- 12 A b^{2} c - 4 B b^{3}\right )}{24 x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________