Optimal. Leaf size=71 \[ -\frac {A b^3}{6 x^6}-\frac {b^2 (b B+3 A c)}{4 x^4}-\frac {3 b c (b B+A c)}{2 x^2}+\frac {1}{2} B c^3 x^2+c^2 (3 b B+A c) \log (x) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1598, 457, 77}
\begin {gather*} -\frac {A b^3}{6 x^6}-\frac {b^2 (3 A c+b B)}{4 x^4}+c^2 \log (x) (A c+3 b B)-\frac {3 b c (A c+b B)}{2 x^2}+\frac {1}{2} B c^3 x^2 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 77
Rule 457
Rule 1598
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^3}{x^{13}} \, dx &=\int \frac {\left (A+B x^2\right ) \left (b+c x^2\right )^3}{x^7} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {(A+B x) (b+c x)^3}{x^4} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (B c^3+\frac {A b^3}{x^4}+\frac {b^2 (b B+3 A c)}{x^3}+\frac {3 b c (b B+A c)}{x^2}+\frac {c^2 (3 b B+A c)}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac {A b^3}{6 x^6}-\frac {b^2 (b B+3 A c)}{4 x^4}-\frac {3 b c (b B+A c)}{2 x^2}+\frac {1}{2} B c^3 x^2+c^2 (3 b B+A c) \log (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 71, normalized size = 1.00 \begin {gather*} -\frac {A b^3}{6 x^6}-\frac {b^2 (b B+3 A c)}{4 x^4}-\frac {3 b c (b B+A c)}{2 x^2}+\frac {1}{2} B c^3 x^2+c^2 (3 b B+A c) \log (x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.38, size = 64, normalized size = 0.90
method | result | size |
default | \(-\frac {A \,b^{3}}{6 x^{6}}-\frac {b^{2} \left (3 A c +B b \right )}{4 x^{4}}-\frac {3 b c \left (A c +B b \right )}{2 x^{2}}+\frac {B \,c^{3} x^{2}}{2}+c^{2} \left (A c +3 B b \right ) \ln \left (x \right )\) | \(64\) |
risch | \(\frac {B \,c^{3} x^{2}}{2}+\frac {\left (-\frac {3}{2} A b \,c^{2}-\frac {3}{2} B \,b^{2} c \right ) x^{4}+\left (-\frac {3}{4} A \,b^{2} c -\frac {1}{4} B \,b^{3}\right ) x^{2}-\frac {A \,b^{3}}{6}}{x^{6}}+A \ln \left (x \right ) c^{3}+3 B \ln \left (x \right ) b \,c^{2}\) | \(75\) |
norman | \(\frac {\left (-\frac {3}{2} A b \,c^{2}-\frac {3}{2} B \,b^{2} c \right ) x^{10}+\left (-\frac {3}{4} A \,b^{2} c -\frac {1}{4} B \,b^{3}\right ) x^{8}-\frac {A \,b^{3} x^{6}}{6}+\frac {B \,c^{3} x^{14}}{2}}{x^{12}}+\left (A \,c^{3}+3 B b \,c^{2}\right ) \ln \left (x \right )\) | \(78\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 77, normalized size = 1.08 \begin {gather*} \frac {1}{2} \, B c^{3} x^{2} + \frac {1}{2} \, {\left (3 \, B b c^{2} + A c^{3}\right )} \log \left (x^{2}\right ) - \frac {18 \, {\left (B b^{2} c + A b c^{2}\right )} x^{4} + 2 \, A b^{3} + 3 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2}}{12 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.70, size = 77, normalized size = 1.08 \begin {gather*} \frac {6 \, B c^{3} x^{8} + 12 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} \log \left (x\right ) - 18 \, {\left (B b^{2} c + A b c^{2}\right )} x^{4} - 2 \, A b^{3} - 3 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2}}{12 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.68, size = 78, normalized size = 1.10 \begin {gather*} \frac {B c^{3} x^{2}}{2} + c^{2} \left (A c + 3 B b\right ) \log {\left (x \right )} + \frac {- 2 A b^{3} + x^{4} \left (- 18 A b c^{2} - 18 B b^{2} c\right ) + x^{2} \left (- 9 A b^{2} c - 3 B b^{3}\right )}{12 x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.69, size = 99, normalized size = 1.39 \begin {gather*} \frac {1}{2} \, B c^{3} x^{2} + \frac {1}{2} \, {\left (3 \, B b c^{2} + A c^{3}\right )} \log \left (x^{2}\right ) - \frac {33 \, B b c^{2} x^{6} + 11 \, A c^{3} x^{6} + 18 \, B b^{2} c x^{4} + 18 \, A b c^{2} x^{4} + 3 \, B b^{3} x^{2} + 9 \, A b^{2} c x^{2} + 2 \, A b^{3}}{12 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.09, size = 75, normalized size = 1.06 \begin {gather*} \ln \left (x\right )\,\left (A\,c^3+3\,B\,b\,c^2\right )-\frac {x^4\,\left (\frac {3\,B\,b^2\,c}{2}+\frac {3\,A\,b\,c^2}{2}\right )+\frac {A\,b^3}{6}+x^2\,\left (\frac {B\,b^3}{4}+\frac {3\,A\,c\,b^2}{4}\right )}{x^6}+\frac {B\,c^3\,x^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________