Optimal. Leaf size=58 \[ \frac {a^3}{3 b^4 (a+b x)^3}-\frac {3 a^2}{2 b^4 (a+b x)^2}+\frac {3 a}{b^4 (a+b x)}+\frac {\log (a+b x)}{b^4} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45}
\begin {gather*} \frac {a^3}{3 b^4 (a+b x)^3}-\frac {3 a^2}{2 b^4 (a+b x)^2}+\frac {3 a}{b^4 (a+b x)}+\frac {\log (a+b x)}{b^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rubi steps
\begin {align*} \int \frac {x^3}{(a+b x)^4} \, dx &=\int \left (-\frac {a^3}{b^3 (a+b x)^4}+\frac {3 a^2}{b^3 (a+b x)^3}-\frac {3 a}{b^3 (a+b x)^2}+\frac {1}{b^3 (a+b x)}\right ) \, dx\\ &=\frac {a^3}{3 b^4 (a+b x)^3}-\frac {3 a^2}{2 b^4 (a+b x)^2}+\frac {3 a}{b^4 (a+b x)}+\frac {\log (a+b x)}{b^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 44, normalized size = 0.76 \begin {gather*} \frac {\frac {a \left (11 a^2+27 a b x+18 b^2 x^2\right )}{(a+b x)^3}+6 \log (a+b x)}{6 b^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Mathics [A]
time = 2.32, size = 90, normalized size = 1.55 \begin {gather*} \frac {\frac {a \left (11 a^2+27 a b x+18 b^2 x^2\right )}{6}+\text {Log}\left [a+b x\right ] \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right )}{b^4 \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.08, size = 55, normalized size = 0.95
method | result | size |
norman | \(\frac {\frac {11 a^{3}}{6 b^{4}}+\frac {3 a \,x^{2}}{b^{2}}+\frac {9 a^{2} x}{2 b^{3}}}{\left (b x +a \right )^{3}}+\frac {\ln \left (b x +a \right )}{b^{4}}\) | \(47\) |
risch | \(\frac {\frac {11 a^{3}}{6 b^{4}}+\frac {3 a \,x^{2}}{b^{2}}+\frac {9 a^{2} x}{2 b^{3}}}{\left (b x +a \right )^{3}}+\frac {\ln \left (b x +a \right )}{b^{4}}\) | \(47\) |
default | \(\frac {a^{3}}{3 b^{4} \left (b x +a \right )^{3}}-\frac {3 a^{2}}{2 b^{4} \left (b x +a \right )^{2}}+\frac {3 a}{b^{4} \left (b x +a \right )}+\frac {\ln \left (b x +a \right )}{b^{4}}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 70, normalized size = 1.21 \begin {gather*} \frac {18 \, a b^{2} x^{2} + 27 \, a^{2} b x + 11 \, a^{3}}{6 \, {\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} + \frac {\log \left (b x + a\right )}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.29, size = 94, normalized size = 1.62 \begin {gather*} \frac {18 \, a b^{2} x^{2} + 27 \, a^{2} b x + 11 \, a^{3} + 6 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \log \left (b x + a\right )}{6 \, {\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.15, size = 70, normalized size = 1.21 \begin {gather*} \frac {11 a^{3} + 27 a^{2} b x + 18 a b^{2} x^{2}}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} + \frac {\log {\left (a + b x \right )}}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.00, size = 50, normalized size = 0.86 \begin {gather*} \frac {\frac {1}{6} \left (18 b a x^{2}+27 a^{2} x+\frac {11 a^{3}}{b}\right )}{b^{3} \left (x b+a\right )^{3}}+\frac {\ln \left |x b+a\right |}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.07, size = 45, normalized size = 0.78 \begin {gather*} \frac {\ln \left (a+b\,x\right )+\frac {3\,a}{a+b\,x}-\frac {3\,a^2}{2\,{\left (a+b\,x\right )}^2}+\frac {a^3}{3\,{\left (a+b\,x\right )}^3}}{b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________