Optimal. Leaf size=187 \[ \frac {d^2 (a+b x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac {d (a+b x)}{b c-a d}\right )}{(m+1) (b c-a d) (d e-c f)^2}+\frac {f (a+b x)^{m+1} (a d f-b (c f m+d e (1-m))) \, _2F_1\left (1,m+1;m+2;-\frac {f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f)^2 (d e-c f)^2}-\frac {f (a+b x)^{m+1}}{(e+f x) (b e-a f) (d e-c f)} \]
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Rubi [A] time = 0.18, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {103, 156, 68} \[ \frac {d^2 (a+b x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac {d (a+b x)}{b c-a d}\right )}{(m+1) (b c-a d) (d e-c f)^2}+\frac {f (a+b x)^{m+1} (a d f-b c f m-b d e (1-m)) \, _2F_1\left (1,m+1;m+2;-\frac {f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f)^2 (d e-c f)^2}-\frac {f (a+b x)^{m+1}}{(e+f x) (b e-a f) (d e-c f)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 103
Rule 156
Rubi steps
\begin {align*} \int \frac {(a+b x)^m}{(c+d x) (e+f x)^2} \, dx &=-\frac {f (a+b x)^{1+m}}{(b e-a f) (d e-c f) (e+f x)}-\frac {\int \frac {(a+b x)^m (a d f-b (d e+c f m)-b d f m x)}{(c+d x) (e+f x)} \, dx}{(b e-a f) (d e-c f)}\\ &=-\frac {f (a+b x)^{1+m}}{(b e-a f) (d e-c f) (e+f x)}+\frac {d^2 \int \frac {(a+b x)^m}{c+d x} \, dx}{(d e-c f)^2}+\frac {(f (a d f-b d e (1-m)-b c f m)) \int \frac {(a+b x)^m}{e+f x} \, dx}{(b e-a f) (d e-c f)^2}\\ &=-\frac {f (a+b x)^{1+m}}{(b e-a f) (d e-c f) (e+f x)}+\frac {d^2 (a+b x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d) (d e-c f)^2 (1+m)}+\frac {f (a d f-b d e (1-m)-b c f m) (a+b x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {f (a+b x)}{b e-a f}\right )}{(b e-a f)^2 (d e-c f)^2 (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 174, normalized size = 0.93 \[ \frac {(a+b x)^{m+1} \left (-\frac {d^2 (b e-a f) \, _2F_1\left (1,m+1;m+2;\frac {d (a+b x)}{a d-b c}\right )}{(m+1) (b c-a d) (c f-d e)}+\frac {f (a d f-b c f m+b d e (m-1)) \, _2F_1\left (1,m+1;m+2;\frac {f (a+b x)}{a f-b e}\right )}{(m+1) (b e-a f) (d e-c f)}-\frac {f}{e+f x}\right )}{(b e-a f) (d e-c f)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{m}}{d f^{2} x^{3} + c e^{2} + {\left (2 \, d e f + c f^{2}\right )} x^{2} + {\left (d e^{2} + 2 \, c e f\right )} x}, x\right ) \]
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 \[ \int \frac {{\left (b x + a\right )}^{m}}{{\left (d x + c\right )} {\left (f x + e\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.24, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right )^{m}}{\left (d x +c \right ) \left (f x +e \right )^{2}}\, dx \]
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 \[ \int \frac {{\left (b x + a\right )}^{m}}{{\left (d x + c\right )} {\left (f x + e\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,x\right )}^m}{{\left (e+f\,x\right )}^2\,\left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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