Optimal. Leaf size=17 \[ \frac {1}{3} d \left (a+b x+c x^2\right )^3 \]
[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 = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {643}
\begin {gather*} \frac {1}{3} d \left (a+b x+c x^2\right )^3 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 643
Rubi steps
\begin {align*} \int (b d+2 c d x) \left (a+b x+c x^2\right )^2 \, dx &=\frac {1}{3} d \left (a+b x+c x^2\right )^3\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(37\) vs. \(2(17)=34\).
time = 0.01, size = 37, normalized size = 2.18 \begin {gather*} \frac {1}{3} d x (b+c x) \left (3 a^2+3 a x (b+c x)+x^2 (b+c x)^2\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.60, size = 16, normalized size = 0.94
method | result | size |
default | \(\frac {d \left (c \,x^{2}+b x +a \right )^{3}}{3}\) | \(16\) |
gosper | \(\frac {d x \left (c^{3} x^{5}+3 x^{4} b \,c^{2}+3 a \,c^{2} x^{3}+3 b^{2} c \,x^{3}+6 a b c \,x^{2}+b^{3} x^{2}+3 a^{2} c x +3 a \,b^{2} x +3 a^{2} b \right )}{3}\) | \(75\) |
norman | \(\left (2 a b c d +\frac {1}{3} b^{3} d \right ) x^{3}+\left (a \,c^{2} d +b^{2} c d \right ) x^{4}+\left (a^{2} c d +a \,b^{2} d \right ) x^{2}+a^{2} b d x +b \,c^{2} d \,x^{5}+\frac {c^{3} d \,x^{6}}{3}\) | \(78\) |
risch | \(\frac {1}{3} c^{3} d \,x^{6}+b \,c^{2} d \,x^{5}+x^{4} a \,c^{2} d +b^{2} c d \,x^{4}+2 a b c d \,x^{3}+\frac {1}{3} b^{3} d \,x^{3}+a^{2} c d \,x^{2}+a \,b^{2} d \,x^{2}+a^{2} b d x +\frac {1}{3} a^{3} d\) | \(87\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 15, normalized size = 0.88 \begin {gather*} \frac {1}{3} \, {\left (c x^{2} + b x + a\right )}^{3} d \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. 73 vs.
\(2 (15) = 30\).
time = 2.28, size = 73, normalized size = 4.29 \begin {gather*} \frac {1}{3} \, c^{3} d x^{6} + b c^{2} d x^{5} + {\left (b^{2} c + a c^{2}\right )} d x^{4} + a^{2} b d x + \frac {1}{3} \, {\left (b^{3} + 6 \, a b c\right )} d x^{3} + {\left (a b^{2} + a^{2} c\right )} d x^{2} \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. 80 vs.
\(2 (14) = 28\).
time = 0.02, size = 80, normalized size = 4.71 \begin {gather*} a^{2} b d x + b c^{2} d x^{5} + \frac {c^{3} d x^{6}}{3} + x^{4} \left (a c^{2} d + b^{2} c d\right ) + x^{3} \cdot \left (2 a b c d + \frac {b^{3} d}{3}\right ) + x^{2} \left (a^{2} c d + a b^{2} d\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 52 vs.
\(2 (15) = 30\).
time = 0.85, size = 52, normalized size = 3.06 \begin {gather*} {\left (c d x^{2} + b d x\right )} a^{2} + \frac {3 \, {\left (c d x^{2} + b d x\right )}^{2} a d + {\left (c d x^{2} + b d x\right )}^{3}}{3 \, d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.41, size = 67, normalized size = 3.94 \begin {gather*} \frac {c^3\,d\,x^6}{3}+a\,d\,x^2\,\left (b^2+a\,c\right )+\frac {b\,d\,x^3\,\left (b^2+6\,a\,c\right )}{3}+c\,d\,x^4\,\left (b^2+a\,c\right )+a^2\,b\,d\,x+b\,c^2\,d\,x^5 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________