Optimal. Leaf size=187 \[ -\frac {f (a+b x)^{1+m}}{(b e-a f) (d e-c f) (e+f x)}+\frac {d^2 (a+b x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d) (d e-c f)^2 (1+m)}+\frac {f (a d f-b (d e (1-m)+c f m)) (a+b x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {f (a+b x)}{b e-a f}\right )}{(b e-a f)^2 (d e-c f)^2 (1+m)} \]
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Rubi [A]
time = 0.12, 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 = {105, 162, 70}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 105
Rule 162
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 = 0.22, size = 174, normalized size = 0.93 \begin {gather*} \frac {(a+b x)^{1+m} \left (-\frac {f}{e+f x}-\frac {d^2 (b e-a f) \, _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) (1+m)}+\frac {f (a d f+b d e (-1+m)-b c f m) \, _2F_1\left (1,1+m;2+m;\frac {f (a+b x)}{-b e+a f}\right )}{(b e-a f) (d e-c f) (1+m)}\right )}{(b e-a f) (d e-c f)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^m}{{\left (e+f\,x\right )}^2\,\left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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