Optimal. Leaf size=38 \[ -\frac {b c-a d}{4 b^2 (a+b x)^4}-\frac {d}{3 b^2 (a+b x)^3} \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {43} \begin {gather*} -\frac {b c-a d}{4 b^2 (a+b x)^4}-\frac {d}{3 b^2 (a+b x)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rubi steps
\begin {align*} \int \frac {c+d x}{(a+b x)^5} \, dx &=\int \left (\frac {b c-a d}{b (a+b x)^5}+\frac {d}{b (a+b x)^4}\right ) \, dx\\ &=-\frac {b c-a d}{4 b^2 (a+b x)^4}-\frac {d}{3 b^2 (a+b x)^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 27, normalized size = 0.71 \begin {gather*} -\frac {a d+3 b c+4 b d x}{12 b^2 (a+b x)^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c+d x}{(a+b x)^5} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.20, size = 61, normalized size = 1.61 \begin {gather*} -\frac {4 \, b d x + 3 \, b c + a d}{12 \, {\left (b^{6} x^{4} + 4 \, a b^{5} x^{3} + 6 \, a^{2} b^{4} x^{2} + 4 \, a^{3} b^{3} x + a^{4} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.92, size = 41, normalized size = 1.08 \begin {gather*} -\frac {c}{4 \, {\left (b x + a\right )}^{4} b} - \frac {d}{3 \, {\left (b x + a\right )}^{3} b^{2}} + \frac {a d}{4 \, {\left (b x + a\right )}^{4} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 35, normalized size = 0.92 \begin {gather*} -\frac {d}{3 \left (b x +a \right )^{3} b^{2}}-\frac {-a d +b c}{4 \left (b x +a \right )^{4} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.39, size = 61, normalized size = 1.61 \begin {gather*} -\frac {4 \, b d x + 3 \, b c + a d}{12 \, {\left (b^{6} x^{4} + 4 \, a b^{5} x^{3} + 6 \, a^{2} b^{4} x^{2} + 4 \, a^{3} b^{3} x + a^{4} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 63, normalized size = 1.66 \begin {gather*} -\frac {\frac {a\,d+3\,b\,c}{12\,b^2}+\frac {d\,x}{3\,b}}{a^4+4\,a^3\,b\,x+6\,a^2\,b^2\,x^2+4\,a\,b^3\,x^3+b^4\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.43, size = 65, normalized size = 1.71 \begin {gather*} \frac {- a d - 3 b c - 4 b d x}{12 a^{4} b^{2} + 48 a^{3} b^{3} x + 72 a^{2} b^{4} x^{2} + 48 a b^{5} x^{3} + 12 b^{6} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________