Optimal. Leaf size=49 \[ \frac {d x \left (\frac {a+b}{d}+\frac {c x^2}{d}\right )^{m+1} \, _2F_1\left (1,m+\frac {3}{2};\frac {3}{2};-\frac {c x^2}{a+b}\right )}{a+b} \]
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Rubi [A] time = 0.02, 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 = {1972, 246, 245} \[ x \left (\frac {c x^2}{a+b}+1\right )^{-m} \left (\frac {a+b}{d}+\frac {c x^2}{d}\right )^m \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};-\frac {c x^2}{a+b}\right ) \]
Antiderivative was successfully verified.
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Rule 245
Rule 246
Rule 1972
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*}
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Mathematica [A] time = 0.01, size = 53, normalized size = 1.08 \[ x \left (\frac {c x^2}{a+b}+1\right )^{-m} \left (\frac {a+b+c x^2}{d}\right )^m \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};-\frac {c x^2}{a+b}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (\frac {c x^{2} + a + b}{d}\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\frac {c x^{2} + a + b}{d}\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, 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.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\frac {c x^{2} + a + b}{d}\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.67, size = 54, normalized size = 1.10 \[ \frac {x\,{\left (\frac {a+b}{d}+\frac {c\,x^2}{d}\right )}^m\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},-m;\ \frac {3}{2};\ -\frac {c\,x^2}{a+b}\right )}{{\left (\frac {c\,x^2}{a+b}+1\right )}^m} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\frac {a + b + c x^{2}}{d}\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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