Optimal. Leaf size=281 \[ \frac {f (b d e+b c f-2 a d f) (a+b x)^{1+m}}{(b c-a d) (b e-a f) (d e-c f)^2 (e+f x)}+\frac {d (a+b x)^{1+m}}{(b c-a d) (d e-c f) (c+d x) (e+f x)}+\frac {d^2 (2 a d f-b (c f (2-m)+d e m)) (a+b x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d)^2 (d e-c f)^3 (1+m)}-\frac {f^2 (2 a d f-b (d e (2-m)+c f m)) (a+b x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {f (a+b x)}{b e-a f}\right )}{(b e-a f)^2 (d e-c f)^3 (1+m)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.28, antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {105, 156, 162,
70} \begin {gather*} \frac {d^2 (a+b x)^{m+1} (2 a d f-b c f (2-m)-b d e m) \, _2F_1\left (1,m+1;m+2;-\frac {d (a+b x)}{b c-a d}\right )}{(m+1) (b c-a d)^2 (d e-c f)^3}-\frac {f^2 (a+b x)^{m+1} (2 a d f-b c f m-b d e (2-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)^3}+\frac {f (a+b x)^{m+1} (-2 a d f+b c f+b d e)}{(e+f x) (b c-a d) (b e-a f) (d e-c f)^2}+\frac {d (a+b x)^{m+1}}{(c+d x) (e+f x) (b c-a d) (d e-c f)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 70
Rule 105
Rule 156
Rule 162
Rubi steps
\begin {align*} \int \frac {(a+b x)^m}{(c+d x)^2 (e+f x)^2} \, dx &=\frac {d (a+b x)^{1+m}}{(b c-a d) (d e-c f) (c+d x) (e+f x)}+\frac {\int \frac {(a+b x)^m (2 a d f-b (c f+d e m)+b d f (1-m) x)}{(c+d x) (e+f x)^2} \, dx}{(b c-a d) (d e-c f)}\\ &=\frac {f (b d e+b c f-2 a d f) (a+b x)^{1+m}}{(b c-a d) (b e-a f) (d e-c f)^2 (e+f x)}+\frac {d (a+b x)^{1+m}}{(b c-a d) (d e-c f) (c+d x) (e+f x)}-\frac {\int \frac {(a+b x)^m \left (2 a^2 d^2 f^2-a b d f (d e+c f) (2+m)+b^2 \left (2 c d e f+d^2 e^2 m+c^2 f^2 m\right )+b d f (b d e+b c f-2 a d f) m x\right )}{(c+d x) (e+f x)} \, dx}{(b c-a d) (b e-a f) (d e-c f)^2}\\ &=\frac {f (b d e+b c f-2 a d f) (a+b x)^{1+m}}{(b c-a d) (b e-a f) (d e-c f)^2 (e+f x)}+\frac {d (a+b x)^{1+m}}{(b c-a d) (d e-c f) (c+d x) (e+f x)}+\frac {\left (d^2 (2 a d f-b c f (2-m)-b d e m)\right ) \int \frac {(a+b x)^m}{c+d x} \, dx}{(b c-a d) (d e-c f)^3}-\frac {\left (f^2 (2 a d f-b d e (2-m)-b c f m)\right ) \int \frac {(a+b x)^m}{e+f x} \, dx}{(b e-a f) (d e-c f)^3}\\ &=\frac {f (b d e+b c f-2 a d f) (a+b x)^{1+m}}{(b c-a d) (b e-a f) (d e-c f)^2 (e+f x)}+\frac {d (a+b x)^{1+m}}{(b c-a d) (d e-c f) (c+d x) (e+f x)}+\frac {d^2 (2 a d f-b c f (2-m)-b d e m) (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)^2 (d e-c f)^3 (1+m)}-\frac {f^2 (2 a d f-b d e (2-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)^3 (1+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.56, size = 247, normalized size = 0.88 \begin {gather*} \frac {(a+b x)^{1+m} \left (-\frac {f (b d e+b c f-2 a d f)}{(b e-a f) (d e-c f) (e+f x)}-\frac {d}{(c+d x) (e+f x)}-\frac {-d^2 (b e-a f)^2 (-2 a d f-b c f (-2+m)+b d e m) \, _2F_1\left (1,1+m;2+m;\frac {d (a+b x)}{-b c+a d}\right )+(b c-a d)^2 f^2 (-2 a d f-b d e (-2+m)+b c f m) \, _2F_1\left (1,1+m;2+m;\frac {f (a+b x)}{-b e+a f}\right )}{(b c-a d) (b e-a f)^2 (d e-c f)^2 (1+m)}\right )}{(b c-a d) (-d e+c f)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{m}}{\left (d x +c \right )^{2} \left (f x +e \right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^m}{{\left (e+f\,x\right )}^2\,{\left (c+d\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________