Optimal. Leaf size=34 \[ \frac {b (c x)^{m+3}}{c^3 (m+3)}+\frac {(c x)^{m+5}}{c^4 (m+5)} \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {14} \begin {gather*} \frac {b (c x)^{m+3}}{c^3 (m+3)}+\frac {(c x)^{m+5}}{c^4 (m+5)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rubi steps
\begin {align*} \int (c x)^m \left (b x^2+c x^4\right ) \, dx &=\int \left (\frac {b (c x)^{2+m}}{c^2}+\frac {(c x)^{4+m}}{c^3}\right ) \, dx\\ &=\frac {b (c x)^{3+m}}{c^3 (3+m)}+\frac {(c x)^{5+m}}{c^4 (5+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 27, normalized size = 0.79 \begin {gather*} x^3 (c x)^m \left (\frac {b}{m+3}+\frac {c x^2}{m+5}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.05, size = 0, normalized size = 0.00 \begin {gather*} \int (c x)^m \left (b x^2+c x^4\right ) \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.80, size = 39, normalized size = 1.15 \begin {gather*} \frac {{\left ({\left (c m + 3 \, c\right )} x^{5} + {\left (b m + 5 \, b\right )} x^{3}\right )} \left (c x\right )^{m}}{m^{2} + 8 \, m + 15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.20, size = 56, normalized size = 1.65 \begin {gather*} \frac {\left (c x\right )^{m} c m x^{5} + 3 \, \left (c x\right )^{m} c x^{5} + \left (c x\right )^{m} b m x^{3} + 5 \, \left (c x\right )^{m} b x^{3}}{m^{2} + 8 \, m + 15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 39, normalized size = 1.15 \begin {gather*} \frac {\left (c m \,x^{2}+3 c \,x^{2}+b m +5 b \right ) x^{3} \left (c x \right )^{m}}{\left (m +5\right ) \left (m +3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.43, size = 34, normalized size = 1.00 \begin {gather*} \frac {c^{m + 1} x^{5} x^{m}}{m + 5} + \frac {b c^{m} x^{3} x^{m}}{m + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.15, size = 38, normalized size = 1.12 \begin {gather*} \frac {x^3\,{\left (c\,x\right )}^m\,\left (5\,b+b\,m+3\,c\,x^2+c\,m\,x^2\right )}{m^2+8\,m+15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.76, size = 119, normalized size = 3.50 \begin {gather*} \begin {cases} \frac {- \frac {b}{2 x^{2}} + c \log {\relax (x )}}{c^{5}} & \text {for}\: m = -5 \\\frac {b \log {\relax (x )} + \frac {c x^{2}}{2}}{c^{3}} & \text {for}\: m = -3 \\\frac {b c^{m} m x^{3} x^{m}}{m^{2} + 8 m + 15} + \frac {5 b c^{m} x^{3} x^{m}}{m^{2} + 8 m + 15} + \frac {c c^{m} m x^{5} x^{m}}{m^{2} + 8 m + 15} + \frac {3 c c^{m} x^{5} x^{m}}{m^{2} + 8 m + 15} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________