Optimal. Leaf size=31 \[ -\frac {a A}{3 x^3}-\frac {A b+a B}{2 x^2}-\frac {b B}{x} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {77}
\begin {gather*} -\frac {a B+A b}{2 x^2}-\frac {a A}{3 x^3}-\frac {b B}{x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 77
Rubi steps
\begin {align*} \int \frac {(a+b x) (A+B x)}{x^4} \, dx &=\int \left (\frac {a A}{x^4}+\frac {A b+a B}{x^3}+\frac {b B}{x^2}\right ) \, dx\\ &=-\frac {a A}{3 x^3}-\frac {A b+a B}{2 x^2}-\frac {b B}{x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 28, normalized size = 0.90 \begin {gather*} -\frac {3 b x (A+2 B x)+a (2 A+3 B x)}{6 x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.01, size = 28, normalized size = 0.90
method | result | size |
gosper | \(-\frac {6 b B \,x^{2}+3 A b x +3 B a x +2 A a}{6 x^{3}}\) | \(28\) |
default | \(-\frac {b B}{x}-\frac {A b +B a}{2 x^{2}}-\frac {a A}{3 x^{3}}\) | \(28\) |
norman | \(\frac {-b B \,x^{2}+\left (-\frac {A b}{2}-\frac {B a}{2}\right ) x -\frac {A a}{3}}{x^{3}}\) | \(28\) |
risch | \(\frac {-b B \,x^{2}+\left (-\frac {A b}{2}-\frac {B a}{2}\right ) x -\frac {A a}{3}}{x^{3}}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.31, size = 27, normalized size = 0.87 \begin {gather*} -\frac {6 \, B b x^{2} + 2 \, A a + 3 \, {\left (B a + A b\right )} x}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.26, size = 27, normalized size = 0.87 \begin {gather*} -\frac {6 \, B b x^{2} + 2 \, A a + 3 \, {\left (B a + A b\right )} x}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.13, size = 31, normalized size = 1.00 \begin {gather*} \frac {- 2 A a - 6 B b x^{2} + x \left (- 3 A b - 3 B a\right )}{6 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.99, size = 27, normalized size = 0.87 \begin {gather*} -\frac {6 \, B b x^{2} + 3 \, B a x + 3 \, A b x + 2 \, A a}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.05, size = 27, normalized size = 0.87 \begin {gather*} -\frac {B\,b\,x^2+\left (\frac {A\,b}{2}+\frac {B\,a}{2}\right )\,x+\frac {A\,a}{3}}{x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________