Optimal. Leaf size=27 \[ \frac {x^{1+p} \left (b x+c x^3\right )^{1+p}}{2 (1+p)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {1604}
\begin {gather*} \frac {x^{p+1} \left (b x+c x^3\right )^{p+1}}{2 (p+1)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 1604
Rubi steps
\begin {align*} \int x^{1+p} \left (b+2 c x^2\right ) \left (b x+c x^3\right )^p \, dx &=\frac {x^{1+p} \left (b x+c x^3\right )^{1+p}}{2 (1+p)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.05, size = 27, normalized size = 1.00 \begin {gather*} \frac {x^{1+p} \left (x \left (b+c x^2\right )\right )^{1+p}}{2 (1+p)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.24, size = 31, normalized size = 1.15
method | result | size |
gosper | \(\frac {x^{2+p} \left (c \,x^{2}+b \right ) \left (c \,x^{3}+b x \right )^{p}}{2+2 p}\) | \(31\) |
risch | \(\frac {\left (c \,x^{2}+b \right ) x \,x^{1+p} {\mathrm e}^{\frac {p \left (-i \pi \mathrm {csgn}\left (i x \left (c \,x^{2}+b \right )\right )^{3}+i \pi \mathrm {csgn}\left (i x \left (c \,x^{2}+b \right )\right )^{2} \mathrm {csgn}\left (i x \right )+i \pi \mathrm {csgn}\left (i x \left (c \,x^{2}+b \right )\right )^{2} \mathrm {csgn}\left (i \left (c \,x^{2}+b \right )\right )-i \pi \,\mathrm {csgn}\left (i x \left (c \,x^{2}+b \right )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (c \,x^{2}+b \right )\right )+2 \ln \left (c \,x^{2}+b \right )+2 \ln \left (x \right )\right )}{2}}}{2+2 p}\) | \(142\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.31, size = 35, normalized size = 1.30 \begin {gather*} \frac {{\left (c x^{4} + b x^{2}\right )} e^{\left (p \log \left (c x^{2} + b\right ) + 2 \, p \log \left (x\right )\right )}}{2 \, {\left (p + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.38, size = 32, normalized size = 1.19 \begin {gather*} \frac {{\left (c x^{3} + b x\right )} {\left (c x^{3} + b x\right )}^{p} x^{p + 1}}{2 \, {\left (p + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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. 54 vs.
\(2 (25) = 50\).
time = 2.96, size = 54, normalized size = 2.00 \begin {gather*} \frac {c x^{3} e^{\left (p \log \left (c x^{2} + b\right ) + 2 \, p \log \left (x\right ) + \log \left (x\right )\right )} + b x e^{\left (p \log \left (c x^{2} + b\right ) + 2 \, p \log \left (x\right ) + \log \left (x\right )\right )}}{2 \, {\left (p + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 2.21, size = 45, normalized size = 1.67 \begin {gather*} {\left (c\,x^3+b\,x\right )}^p\,\left (\frac {b\,x\,x^{p+1}}{2\,p+2}+\frac {c\,x^{p+1}\,x^3}{2\,p+2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________