Optimal. Leaf size=19 \[ \frac {\left (b x+d x^3\right )^{1+n}}{1+n} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1602}
\begin {gather*} \frac {\left (b x+d x^3\right )^{n+1}}{n+1} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 1602
Rubi steps
\begin {align*} \int \left (b+3 d x^2\right ) \left (b x+d x^3\right )^n \, dx &=\frac {\left (b x+d x^3\right )^{1+n}}{1+n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 19, normalized size = 1.00 \begin {gather*} \frac {\left (x \left (b+d x^2\right )\right )^{1+n}}{1+n} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.16, size = 20, normalized size = 1.05
method | result | size |
derivativedivides | \(\frac {\left (d \,x^{3}+b x \right )^{1+n}}{1+n}\) | \(20\) |
default | \(\frac {\left (d \,x^{3}+b x \right )^{1+n}}{1+n}\) | \(20\) |
gosper | \(\frac {x \left (d \,x^{2}+b \right ) \left (d \,x^{3}+b x \right )^{n}}{1+n}\) | \(26\) |
risch | \(\frac {x \left (d \,x^{2}+b \right ) \left (x \left (d \,x^{2}+b \right )\right )^{n}}{1+n}\) | \(26\) |
norman | \(\frac {b x \,{\mathrm e}^{n \ln \left (d \,x^{3}+b x \right )}}{1+n}+\frac {d \,x^{3} {\mathrm e}^{n \ln \left (d \,x^{3}+b x \right )}}{1+n}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 19, normalized size = 1.00 \begin {gather*} \frac {{\left (d x^{3} + b x\right )}^{n + 1}}{n + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.40, size = 26, normalized size = 1.37 \begin {gather*} \frac {{\left (d x^{3} + b x\right )} {\left (d x^{3} + b x\right )}^{n}}{n + 1} \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. 63 vs.
\(2 (14) = 28\).
time = 5.35, size = 63, normalized size = 3.32 \begin {gather*} \begin {cases} \frac {b x \left (b x + d x^{3}\right )^{n}}{n + 1} + \frac {d x^{3} \left (b x + d x^{3}\right )^{n}}{n + 1} & \text {for}\: n \neq -1 \\\log {\left (x \right )} + \log {\left (x - \sqrt {- \frac {b}{d}} \right )} + \log {\left (x + \sqrt {- \frac {b}{d}} \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 3.95, size = 19, normalized size = 1.00 \begin {gather*} \frac {{\left (d x^{3} + b x\right )}^{n + 1}}{n + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 2.13, size = 25, normalized size = 1.32 \begin {gather*} \frac {x\,{\left (d\,x^3+b\,x\right )}^n\,\left (d\,x^2+b\right )}{n+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________