Optimal. Leaf size=17 \[ \frac {x^3}{3 a (a+b x)^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {37}
\begin {gather*} \frac {x^3}{3 a (a+b x)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rubi steps
\begin {align*} \int \frac {x^2}{(a+b x)^4} \, dx &=\frac {x^3}{3 a (a+b x)^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 31, normalized size = 1.82 \begin {gather*} -\frac {a^2+3 a b x+3 b^2 x^2}{3 b^3 (a+b x)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(40\) vs.
\(2(15)=30\).
time = 0.09, size = 41, normalized size = 2.41
method | result | size |
gosper | \(-\frac {3 x^{2} b^{2}+3 a b x +a^{2}}{3 \left (b x +a \right )^{3} b^{3}}\) | \(30\) |
norman | \(\frac {-\frac {x^{2}}{b}-\frac {a x}{b^{2}}-\frac {a^{2}}{3 b^{3}}}{\left (b x +a \right )^{3}}\) | \(33\) |
risch | \(\frac {-\frac {x^{2}}{b}-\frac {a x}{b^{2}}-\frac {a^{2}}{3 b^{3}}}{\left (b x +a \right )^{3}}\) | \(33\) |
default | \(-\frac {1}{b^{3} \left (b x +a \right )}+\frac {a}{b^{3} \left (b x +a \right )^{2}}-\frac {a^{2}}{3 b^{3} \left (b x +a \right )^{3}}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs.
\(2 (15) = 30\).
time = 0.27, size = 54, normalized size = 3.18 \begin {gather*} -\frac {3 \, b^{2} x^{2} + 3 \, a b x + a^{2}}{3 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs.
\(2 (15) = 30\).
time = 0.78, size = 54, normalized size = 3.18 \begin {gather*} -\frac {3 \, b^{2} x^{2} + 3 \, a b x + a^{2}}{3 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs.
\(2 (12) = 24\).
time = 0.10, size = 56, normalized size = 3.29 \begin {gather*} \frac {- a^{2} - 3 a b x - 3 b^{2} x^{2}}{3 a^{3} b^{3} + 9 a^{2} b^{4} x + 9 a b^{5} x^{2} + 3 b^{6} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.54, size = 29, normalized size = 1.71 \begin {gather*} -\frac {3 \, b^{2} x^{2} + 3 \, a b x + a^{2}}{3 \, {\left (b x + a\right )}^{3} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.09, size = 56, normalized size = 3.29 \begin {gather*} -\frac {a^2+3\,a\,b\,x+3\,b^2\,x^2}{3\,a^3\,b^3+9\,a^2\,b^4\,x+9\,a\,b^5\,x^2+3\,b^6\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________