Optimal. Leaf size=71 \[ -\frac {A b^3}{2 x^2}+\frac {3}{2} b c (b B+A c) x^2+\frac {1}{4} c^2 (3 b B+A c) x^4+\frac {1}{6} B c^3 x^6+b^2 (b B+3 A c) \log (x) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, 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}{2 x^2}+b^2 \log (x) (3 A c+b B)+\frac {1}{4} c^2 x^4 (A c+3 b B)+\frac {3}{2} b c x^2 (A c+b B)+\frac {1}{6} B c^3 x^6 \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^9} \, dx &=\int \frac {\left (A+B x^2\right ) \left (b+c x^2\right )^3}{x^3} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {(A+B x) (b+c x)^3}{x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (3 b c (b B+A c)+\frac {A b^3}{x^2}+\frac {b^2 (b B+3 A c)}{x}+c^2 (3 b B+A c) x+B c^3 x^2\right ) \, dx,x,x^2\right )\\ &=-\frac {A b^3}{2 x^2}+\frac {3}{2} b c (b B+A c) x^2+\frac {1}{4} c^2 (3 b B+A c) x^4+\frac {1}{6} B c^3 x^6+b^2 (b B+3 A c) \log (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 73, normalized size = 1.03 \begin {gather*} -\frac {A b^3}{2 x^2}+\frac {3}{2} b c (b B+A c) x^2+\frac {1}{4} c^2 (3 b B+A c) x^4+\frac {1}{6} B c^3 x^6+\left (b^3 B+3 A b^2 c\right ) \log (x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.37, size = 73, normalized size = 1.03
method | result | size |
default | \(\frac {B \,c^{3} x^{6}}{6}+\frac {A \,c^{3} x^{4}}{4}+\frac {3 B b \,c^{2} x^{4}}{4}+\frac {3 A b \,c^{2} x^{2}}{2}+\frac {3 B \,b^{2} c \,x^{2}}{2}-\frac {A \,b^{3}}{2 x^{2}}+b^{2} \left (3 A c +B b \right ) \ln \left (x \right )\) | \(73\) |
risch | \(\frac {B \,c^{3} x^{6}}{6}+\frac {A \,c^{3} x^{4}}{4}+\frac {3 B b \,c^{2} x^{4}}{4}+\frac {3 A b \,c^{2} x^{2}}{2}+\frac {3 B \,b^{2} c \,x^{2}}{2}-\frac {A \,b^{3}}{2 x^{2}}+3 A \ln \left (x \right ) b^{2} c +B \ln \left (x \right ) b^{3}\) | \(75\) |
norman | \(\frac {\left (\frac {1}{4} A \,c^{3}+\frac {3}{4} B b \,c^{2}\right ) x^{12}+\left (\frac {3}{2} A b \,c^{2}+\frac {3}{2} B \,b^{2} c \right ) x^{10}-\frac {A \,b^{3} x^{6}}{2}+\frac {B \,c^{3} x^{14}}{6}}{x^{8}}+\left (3 A \,b^{2} c +B \,b^{3}\right ) \ln \left (x \right )\) | \(78\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 74, normalized size = 1.04 \begin {gather*} \frac {1}{6} \, B c^{3} x^{6} + \frac {1}{4} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{4} + \frac {3}{2} \, {\left (B b^{2} c + A b c^{2}\right )} x^{2} - \frac {A b^{3}}{2 \, x^{2}} + \frac {1}{2} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} \log \left (x^{2}\right ) \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 {2 \, B c^{3} x^{8} + 3 \, {\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} - 6 \, A b^{3} + 12 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2} \log \left (x\right )}{12 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.12, size = 78, normalized size = 1.10 \begin {gather*} - \frac {A b^{3}}{2 x^{2}} + \frac {B c^{3} x^{6}}{6} + b^{2} \cdot \left (3 A c + B b\right ) \log {\left (x \right )} + x^{4} \left (\frac {A c^{3}}{4} + \frac {3 B b c^{2}}{4}\right ) + x^{2} \cdot \left (\frac {3 A b c^{2}}{2} + \frac {3 B b^{2} c}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.68, size = 97, normalized size = 1.37 \begin {gather*} \frac {1}{6} \, B c^{3} x^{6} + \frac {3}{4} \, B b c^{2} x^{4} + \frac {1}{4} \, A c^{3} x^{4} + \frac {3}{2} \, B b^{2} c x^{2} + \frac {3}{2} \, A b c^{2} x^{2} + \frac {1}{2} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} \log \left (x^{2}\right ) - \frac {B b^{3} x^{2} + 3 \, A b^{2} c x^{2} + A b^{3}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.04, size = 67, normalized size = 0.94 \begin {gather*} x^4\,\left (\frac {A\,c^3}{4}+\frac {3\,B\,b\,c^2}{4}\right )+\ln \left (x\right )\,\left (B\,b^3+3\,A\,c\,b^2\right )-\frac {A\,b^3}{2\,x^2}+\frac {B\,c^3\,x^6}{6}+\frac {3\,b\,c\,x^2\,\left (A\,c+B\,b\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________