Optimal. Leaf size=158 \[ -\frac {a^2 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {b (e+f x)}{b e-a f}\right )}{b (n+1) (b c-a d) (b e-a f)}+\frac {c^2 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {d (e+f x)}{d e-c f}\right )}{d (n+1) (b c-a d) (d e-c f)}+\frac {(e+f x)^{n+1}}{b d f (n+1)} \]
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Rubi [A] time = 0.11, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {180, 68} \[ -\frac {a^2 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {b (e+f x)}{b e-a f}\right )}{b (n+1) (b c-a d) (b e-a f)}+\frac {c^2 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {d (e+f x)}{d e-c f}\right )}{d (n+1) (b c-a d) (d e-c f)}+\frac {(e+f x)^{n+1}}{b d f (n+1)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 180
Rubi steps
\begin {align*} \int \frac {x^2 (e+f x)^n}{(a+b x) (c+d x)} \, dx &=\int \left (\frac {(e+f x)^n}{b d}+\frac {a^2 (e+f x)^n}{b (b c-a d) (a+b x)}+\frac {c^2 (e+f x)^n}{d (-b c+a d) (c+d x)}\right ) \, dx\\ &=\frac {(e+f x)^{1+n}}{b d f (1+n)}+\frac {a^2 \int \frac {(e+f x)^n}{a+b x} \, dx}{b (b c-a d)}-\frac {c^2 \int \frac {(e+f x)^n}{c+d x} \, dx}{d (b c-a d)}\\ &=\frac {(e+f x)^{1+n}}{b d f (1+n)}-\frac {a^2 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {b (e+f x)}{b e-a f}\right )}{b (b c-a d) (b e-a f) (1+n)}+\frac {c^2 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {d (e+f x)}{d e-c f}\right )}{d (b c-a d) (d e-c f) (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 153, normalized size = 0.97 \[ \frac {(e+f x)^{n+1} \left (a^2 d f (c f-d e) \, _2F_1\left (1,n+1;n+2;\frac {b (e+f x)}{b e-a f}\right )+(b e-a f) \left (b c^2 f \, _2F_1\left (1,n+1;n+2;\frac {d (e+f x)}{d e-c f}\right )-(b c-a d) (c f-d e)\right )\right )}{b d f (n+1) (b c-a d) (b e-a f) (d e-c f)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (f x + e\right )}^{n} x^{2}}{b d x^{2} + a c + {\left (b c + a d\right )} x}, x\right ) \]
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 \[ \int \frac {{\left (f x + e\right )}^{n} x^{2}}{{\left (b x + a\right )} {\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.25, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \left (f x +e \right )^{n}}{\left (b x +a \right ) \left (d x +c \right )}\, dx \]
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 \[ \int \frac {{\left (f x + e\right )}^{n} x^{2}}{{\left (b x + a\right )} {\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,{\left (e+f\,x\right )}^n}{\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]
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 \[ \int \frac {x^{2} \left (e + f x\right )^{n}}{\left (a + b x\right ) \left (c + d x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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