Optimal. Leaf size=175 \[ \frac {b^2 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {b (e+f x)}{b e-a f}\right )}{a (n+1) (b c-a d) (b e-a f)}-\frac {d^2 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {d (e+f x)}{d e-c f}\right )}{c (n+1) (b c-a d) (d e-c f)}-\frac {(e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {f x}{e}+1\right )}{a c e (n+1)} \]
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Rubi [A] time = 0.12, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {180, 65, 68} \[ \frac {b^2 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {b (e+f x)}{b e-a f}\right )}{a (n+1) (b c-a d) (b e-a f)}-\frac {d^2 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {d (e+f x)}{d e-c f}\right )}{c (n+1) (b c-a d) (d e-c f)}-\frac {(e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {f x}{e}+1\right )}{a c e (n+1)} \]
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 (a+b x) (c+d x)} \, dx &=\int \left (\frac {(e+f x)^n}{a c x}+\frac {b^2 (e+f x)^n}{a (-b c+a d) (a+b x)}+\frac {d^2 (e+f x)^n}{c (b c-a d) (c+d x)}\right ) \, dx\\ &=\frac {\int \frac {(e+f x)^n}{x} \, dx}{a c}-\frac {b^2 \int \frac {(e+f x)^n}{a+b x} \, dx}{a (b c-a d)}+\frac {d^2 \int \frac {(e+f x)^n}{c+d x} \, dx}{c (b c-a d)}\\ &=\frac {b^2 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {b (e+f x)}{b e-a f}\right )}{a (b c-a d) (b e-a f) (1+n)}-\frac {d^2 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {d (e+f x)}{d e-c f}\right )}{c (b c-a d) (d e-c f) (1+n)}-\frac {(e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac {f x}{e}\right )}{a c e (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 170, normalized size = 0.97 \[ -\frac {(e+f x)^{n+1} \left (b^2 c e (d e-c f) \, _2F_1\left (1,n+1;n+2;\frac {b (e+f x)}{b e-a f}\right )+(a f-b e) \left (a d^2 e \, _2F_1\left (1,n+1;n+2;\frac {d (e+f x)}{d e-c f}\right )-(b c-a d) (c f-d e) \, _2F_1\left (1,n+1;n+2;\frac {f x}{e}+1\right )\right )\right )}{a c e (n+1) (a d-b c) (a f-b e) (c f-d e)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (f x + e\right )}^{n}}{b d x^{3} + a c x + {\left (b c + a d\right )} x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (f x + e\right )}^{n}}{{\left (b x + a\right )} {\left (d x + c\right )} x}\,{d x} \]
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 \[ \int \frac {\left (f x +e \right )^{n}}{\left (b x +a \right ) \left (d x +c \right ) x}\, dx \]
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 \[ \int \frac {{\left (f x + e\right )}^{n}}{{\left (b x + a\right )} {\left (d x + c\right )} x}\,{d x} \]
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 \[ \int \frac {{\left (e+f\,x\right )}^n}{x\,\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]
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 \[ \int \frac {\left (e + f x\right )^{n}}{x \left (a + b x\right ) \left (c + d x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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