Optimal. Leaf size=49 \[ \frac {d x \left (\frac {a+b}{d}+\frac {c x^2}{d}\right )^{1+m} \, _2F_1\left (1,\frac {3}{2}+m;\frac {3}{2};-\frac {c x^2}{a+b}\right )}{a+b} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 57, normalized size of antiderivative = 1.16, number of steps
used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1997, 252, 251}
\begin {gather*} x \left (\frac {c x^2}{a+b}+1\right )^{-m} \left (\frac {a+b}{d}+\frac {c x^2}{d}\right )^m \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};-\frac {c x^2}{a+b}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 251
Rule 252
Rule 1997
Rubi steps
\begin {align*} \int \left (\frac {a+b+c x^2}{d}\right )^m \, dx &=\int \left (\frac {a+b}{d}+\frac {c x^2}{d}\right )^m \, dx\\ &=\left (\left (1+\frac {c x^2}{a+b}\right )^{-m} \left (\frac {a+b}{d}+\frac {c x^2}{d}\right )^m\right ) \int \left (1+\frac {c x^2}{a+b}\right )^m \, dx\\ &=x \left (1+\frac {c x^2}{a+b}\right )^{-m} \left (\frac {a+b}{d}+\frac {c x^2}{d}\right )^m \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};-\frac {c x^2}{a+b}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 53, normalized size = 1.08 \begin {gather*} x \left (\frac {a+b+c x^2}{d}\right )^m \left (1+\frac {c x^2}{a+b}\right )^{-m} \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};-\frac {c x^2}{a+b}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \left (\frac {c \,x^{2}+a +b}{d}\right )^{m}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\frac {a + b + c x^{2}}{d}\right )^{m}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 4.67, size = 54, normalized size = 1.10 \begin {gather*} \frac {x\,{\left (\frac {a+b}{d}+\frac {c\,x^2}{d}\right )}^m\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},-m;\ \frac {3}{2};\ -\frac {c\,x^2}{a+b}\right )}{{\left (\frac {c\,x^2}{a+b}+1\right )}^m} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________