Optimal. Leaf size=22 \[ \frac {x^{n+1} \left (b+d x^2\right )^{n+1}}{n+1} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {449} \[ \frac {x^{n+1} \left (b+d x^2\right )^{n+1}}{n+1} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 449
Rubi steps
\begin {align*} \int x^n \left (b+d x^2\right )^n \left (b+3 d x^2\right ) \, dx &=\frac {x^{1+n} \left (b+d x^2\right )^{1+n}}{1+n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.04, size = 108, normalized size = 4.91 \[ \frac {x^{n+1} \left (b+d x^2\right )^n \left (\frac {d x^2}{b}+1\right )^{-n} \left (3 d (n+1) x^2 \, _2F_1\left (-n,\frac {n+3}{2};\frac {n+5}{2};-\frac {d x^2}{b}\right )+b (n+3) \, _2F_1\left (-n,\frac {n+1}{2};\frac {n+3}{2};-\frac {d x^2}{b}\right )\right )}{(n+1) (n+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.07, size = 27, normalized size = 1.23 \[ \frac {{\left (d x^{3} + b x\right )} {\left (d x^{2} + b\right )}^{n} x^{n}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.34, size = 39, normalized size = 1.77 \[ \frac {{\left (d x^{2} + b\right )}^{n} d x^{3} x^{n} + {\left (d x^{2} + b\right )}^{n} b x x^{n}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 23, normalized size = 1.05 \[ \frac {x^{n +1} \left (d \,x^{2}+b \right )^{n +1}}{n +1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.21, size = 31, normalized size = 1.41 \[ \frac {{\left (d x^{3} + b x\right )} e^{\left (n \log \left (d x^{2} + b\right ) + n \log \relax (x)\right )}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.16, size = 26, normalized size = 1.18 \[ \frac {x\,x^n\,{\left (d\,x^2+b\right )}^n\,\left (d\,x^2+b\right )}{n+1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 52.84, size = 76, normalized size = 3.45 \[ \begin {cases} \frac {b x x^{n} \left (b + d x^{2}\right )^{n}}{n + 1} + \frac {d x^{3} x^{n} \left (b + d x^{2}\right )^{n}}{n + 1} & \text {for}\: n \neq -1 \\\log {\relax (x )} + \log {\left (- i \sqrt {b} \sqrt {\frac {1}{d}} + x \right )} + \log {\left (i \sqrt {b} \sqrt {\frac {1}{d}} + x \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________