Optimal. Leaf size=222 \[ -\frac {b^3 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {b (e+f x)}{b e-a f}\right )}{a^2 (n+1) (b c-a d) (b e-a f)}+\frac {(a d+b c) (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {f x}{e}+1\right )}{a^2 c^2 e (n+1)}+\frac {d^3 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {d (e+f x)}{d e-c f}\right )}{c^2 (n+1) (b c-a d) (d e-c f)}+\frac {f (e+f x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac {f x}{e}+1\right )}{a c e^2 (n+1)} \]
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Rubi [A] time = 0.15, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {180, 65, 68} \[ -\frac {b^3 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {b (e+f x)}{b e-a f}\right )}{a^2 (n+1) (b c-a d) (b e-a f)}+\frac {(a d+b c) (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {f x}{e}+1\right )}{a^2 c^2 e (n+1)}+\frac {d^3 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {d (e+f x)}{d e-c f}\right )}{c^2 (n+1) (b c-a d) (d e-c f)}+\frac {f (e+f x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac {f x}{e}+1\right )}{a c e^2 (n+1)} \]
Antiderivative was successfully verified.
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Rule 65
Rule 68
Rule 180
Rubi steps
\begin {align*} \int \frac {(e+f x)^n}{x^2 (a+b x) (c+d x)} \, dx &=\int \left (\frac {(e+f x)^n}{a c x^2}+\frac {(-b c-a d) (e+f x)^n}{a^2 c^2 x}-\frac {b^3 (e+f x)^n}{a^2 (-b c+a d) (a+b x)}-\frac {d^3 (e+f x)^n}{c^2 (b c-a d) (c+d x)}\right ) \, dx\\ &=\frac {\int \frac {(e+f x)^n}{x^2} \, dx}{a c}+\frac {b^3 \int \frac {(e+f x)^n}{a+b x} \, dx}{a^2 (b c-a d)}-\frac {d^3 \int \frac {(e+f x)^n}{c+d x} \, dx}{c^2 (b c-a d)}-\frac {(b c+a d) \int \frac {(e+f x)^n}{x} \, dx}{a^2 c^2}\\ &=-\frac {b^3 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {b (e+f x)}{b e-a f}\right )}{a^2 (b c-a d) (b e-a f) (1+n)}+\frac {d^3 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {d (e+f x)}{d e-c f}\right )}{c^2 (b c-a d) (d e-c f) (1+n)}+\frac {(b c+a d) (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac {f x}{e}\right )}{a^2 c^2 e (1+n)}+\frac {f (e+f x)^{1+n} \, _2F_1\left (2,1+n;2+n;1+\frac {f x}{e}\right )}{a c e^2 (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 177, normalized size = 0.80 \[ \frac {(e+f x)^{n+1} \left (\frac {\frac {e (a d+b c) \, _2F_1\left (1,n+1;n+2;\frac {f x}{e}+1\right )+a c f \, _2F_1\left (2,n+1;n+2;\frac {f x}{e}+1\right )}{a^2 e^2}-\frac {d^3 \, _2F_1\left (1,n+1;n+2;\frac {d (e+f x)}{d e-c f}\right )}{(b c-a d) (c f-d e)}}{c^2}-\frac {b^3 \, _2F_1\left (1,n+1;n+2;\frac {b (e+f x)}{b e-a f}\right )}{a^2 (b c-a d) (b e-a f)}\right )}{n+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (f x + e\right )}^{n}}{b d x^{4} + a c x^{2} + {\left (b c + a d\right )} x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (f x + e\right )}^{n}}{{\left (b x + a\right )} {\left (d x + c\right )} x^{2}}\,{d x} \]
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 \[ \int \frac {\left (f x +e \right )^{n}}{\left (b x +a \right ) \left (d x +c \right ) x^{2}}\, dx \]
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 \[ \int \frac {{\left (f x + e\right )}^{n}}{{\left (b x + a\right )} {\left (d x + c\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e+f\,x\right )}^n}{x^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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