Optimal. Leaf size=54 \[ -\frac {a A}{x}+(A b+a C) x+\frac {1}{2} (b B+a D) x^2+\frac {1}{3} b C x^3+\frac {1}{4} b D x^4+a B \log (x) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {1816}
\begin {gather*} x (a C+A b)-\frac {a A}{x}+\frac {1}{2} x^2 (a D+b B)+a B \log (x)+\frac {1}{3} b C x^3+\frac {1}{4} b D x^4 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 1816
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right )}{x^2} \, dx &=\int \left (A b \left (1+\frac {a C}{A b}\right )+\frac {a A}{x^2}+\frac {a B}{x}+(b B+a D) x+b C x^2+b D x^3\right ) \, dx\\ &=-\frac {a A}{x}+(A b+a C) x+\frac {1}{2} (b B+a D) x^2+\frac {1}{3} b C x^3+\frac {1}{4} b D x^4+a B \log (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 54, normalized size = 1.00 \begin {gather*} -\frac {a A}{x}+(A b+a C) x+\frac {1}{2} (b B+a D) x^2+\frac {1}{3} b C x^3+\frac {1}{4} b D x^4+a B \log (x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.02, size = 50, normalized size = 0.93
method | result | size |
default | \(\frac {b D x^{4}}{4}+\frac {b C \,x^{3}}{3}+\frac {b B \,x^{2}}{2}+\frac {D a \,x^{2}}{2}+A b x +a C x +B \ln \left (x \right ) a -\frac {a A}{x}\) | \(50\) |
norman | \(\frac {\left (\frac {B b}{2}+\frac {a D}{2}\right ) x^{3}+\left (A b +a C \right ) x^{2}-A a +\frac {b C \,x^{4}}{3}+\frac {b D x^{5}}{4}}{x}+B \ln \left (x \right ) a\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 48, normalized size = 0.89 \begin {gather*} \frac {1}{4} \, D b x^{4} + \frac {1}{3} \, C b x^{3} + \frac {1}{2} \, {\left (D a + B b\right )} x^{2} + B a \log \left (x\right ) + {\left (C a + A b\right )} x - \frac {A a}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 3.62, size = 55, normalized size = 1.02 \begin {gather*} \frac {3 \, D b x^{5} + 4 \, C b x^{4} + 6 \, {\left (D a + B b\right )} x^{3} + 12 \, B a x \log \left (x\right ) + 12 \, {\left (C a + A b\right )} x^{2} - 12 \, A a}{12 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.06, size = 49, normalized size = 0.91 \begin {gather*} - \frac {A a}{x} + B a \log {\left (x \right )} + \frac {C b x^{3}}{3} + \frac {D b x^{4}}{4} + x^{2} \left (\frac {B b}{2} + \frac {D a}{2}\right ) + x \left (A b + C a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.96, size = 50, normalized size = 0.93 \begin {gather*} \frac {1}{4} \, D b x^{4} + \frac {1}{3} \, C b x^{3} + \frac {1}{2} \, D a x^{2} + \frac {1}{2} \, B b x^{2} + C a x + A b x + B a \log \left ({\left | x \right |}\right ) - \frac {A a}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.14, size = 49, normalized size = 0.91 \begin {gather*} \frac {a\,x^2\,D}{2}+\frac {b\,x^4\,D}{4}+A\,b\,x+C\,a\,x-\frac {A\,a}{x}+\frac {B\,b\,x^2}{2}+\frac {C\,b\,x^3}{3}+B\,a\,\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________