Optimal. Leaf size=86 \[ -\frac {b^2 \log (x) (b c-a d)}{a^4}+\frac {b^2 (b c-a d) \log (a+b x)}{a^4}-\frac {b (b c-a d)}{a^3 x}+\frac {b c-a d}{2 a^2 x^2}-\frac {c}{3 a x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77} \[ -\frac {b^2 \log (x) (b c-a d)}{a^4}+\frac {b^2 (b c-a d) \log (a+b x)}{a^4}+\frac {b c-a d}{2 a^2 x^2}-\frac {b (b c-a d)}{a^3 x}-\frac {c}{3 a x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 77
Rubi steps
\begin {align*} \int \frac {c+d x}{x^4 (a+b x)} \, dx &=\int \left (\frac {c}{a x^4}+\frac {-b c+a d}{a^2 x^3}-\frac {b (-b c+a d)}{a^3 x^2}+\frac {b^2 (-b c+a d)}{a^4 x}-\frac {b^3 (-b c+a d)}{a^4 (a+b x)}\right ) \, dx\\ &=-\frac {c}{3 a x^3}+\frac {b c-a d}{2 a^2 x^2}-\frac {b (b c-a d)}{a^3 x}-\frac {b^2 (b c-a d) \log (x)}{a^4}+\frac {b^2 (b c-a d) \log (a+b x)}{a^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 81, normalized size = 0.94 \[ \frac {\frac {a \left (-\left (a^2 (2 c+3 d x)\right )+3 a b x (c+2 d x)-6 b^2 c x^2\right )}{x^3}+6 b^2 \log (x) (a d-b c)+6 b^2 (b c-a d) \log (a+b x)}{6 a^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.90, size = 94, normalized size = 1.09 \[ \frac {6 \, {\left (b^{3} c - a b^{2} d\right )} x^{3} \log \left (b x + a\right ) - 6 \, {\left (b^{3} c - a b^{2} d\right )} x^{3} \log \relax (x) - 2 \, a^{3} c - 6 \, {\left (a b^{2} c - a^{2} b d\right )} x^{2} + 3 \, {\left (a^{2} b c - a^{3} d\right )} x}{6 \, a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.09, size = 99, normalized size = 1.15 \[ -\frac {{\left (b^{3} c - a b^{2} d\right )} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac {{\left (b^{4} c - a b^{3} d\right )} \log \left ({\left | b x + a \right |}\right )}{a^{4} b} - \frac {2 \, a^{3} c + 6 \, {\left (a b^{2} c - a^{2} b d\right )} x^{2} - 3 \, {\left (a^{2} b c - a^{3} d\right )} x}{6 \, a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 101, normalized size = 1.17 \[ \frac {b^{2} d \ln \relax (x )}{a^{3}}-\frac {b^{2} d \ln \left (b x +a \right )}{a^{3}}-\frac {b^{3} c \ln \relax (x )}{a^{4}}+\frac {b^{3} c \ln \left (b x +a \right )}{a^{4}}+\frac {b d}{a^{2} x}-\frac {b^{2} c}{a^{3} x}-\frac {d}{2 a \,x^{2}}+\frac {b c}{2 a^{2} x^{2}}-\frac {c}{3 a \,x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.08, size = 89, normalized size = 1.03 \[ \frac {{\left (b^{3} c - a b^{2} d\right )} \log \left (b x + a\right )}{a^{4}} - \frac {{\left (b^{3} c - a b^{2} d\right )} \log \relax (x)}{a^{4}} - \frac {2 \, a^{2} c + 6 \, {\left (b^{2} c - a b d\right )} x^{2} - 3 \, {\left (a b c - a^{2} d\right )} x}{6 \, a^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.35, size = 97, normalized size = 1.13 \[ \frac {2\,b^2\,\mathrm {atanh}\left (\frac {b^2\,\left (a\,d-b\,c\right )\,\left (a+2\,b\,x\right )}{a\,\left (b^3\,c-a\,b^2\,d\right )}\right )\,\left (a\,d-b\,c\right )}{a^4}-\frac {\frac {c}{3\,a}+\frac {x\,\left (a\,d-b\,c\right )}{2\,a^2}-\frac {b\,x^2\,\left (a\,d-b\,c\right )}{a^3}}{x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.60, size = 165, normalized size = 1.92 \[ \frac {- 2 a^{2} c + x^{2} \left (6 a b d - 6 b^{2} c\right ) + x \left (- 3 a^{2} d + 3 a b c\right )}{6 a^{3} x^{3}} + \frac {b^{2} \left (a d - b c\right ) \log {\left (x + \frac {a^{2} b^{2} d - a b^{3} c - a b^{2} \left (a d - b c\right )}{2 a b^{3} d - 2 b^{4} c} \right )}}{a^{4}} - \frac {b^{2} \left (a d - b c\right ) \log {\left (x + \frac {a^{2} b^{2} d - a b^{3} c + a b^{2} \left (a d - b c\right )}{2 a b^{3} d - 2 b^{4} c} \right )}}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________