Optimal. Leaf size=542 \[ -\frac {f (a+b x)^{1+m} (c+d x)^{2-m}}{5 (b e-a f) (d e-c f) (e+f x)^5}-\frac {f (b (7 d e-c f (4-m))-a d f (3+m)) (a+b x)^{1+m} (c+d x)^{2-m}}{20 (b e-a f)^2 (d e-c f)^2 (e+f x)^4}-\frac {f \left (a^2 d^2 f^2 \left (6+5 m+m^2\right )-a b d f \left (3 d e (7+4 m)-c f \left (9+2 m-2 m^2\right )\right )+b^2 \left (27 d^2 e^2-3 c d e f (11-4 m)+c^2 f^2 \left (12-7 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{2-m}}{60 (b e-a f)^3 (d e-c f)^3 (e+f x)^3}+\frac {(b c-a d)^2 \left (3 a^2 b d^2 f^2 (5 d e-c f (2-m)) \left (2+3 m+m^2\right )-a^3 d^3 f^3 \left (6+11 m+6 m^2+m^3\right )-3 a b^2 d f (1+m) \left (20 d^2 e^2-10 c d e f (2-m)+c^2 f^2 \left (6-5 m+m^2\right )\right )+b^3 \left (60 d^3 e^3-60 c d^2 e^2 f (2-m)+15 c^2 d e f^2 \left (6-5 m+m^2\right )-c^3 f^3 \left (24-26 m+9 m^2-m^3\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m} \, _2F_1\left (3,1+m;2+m;\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{60 (b e-a f)^6 (d e-c f)^3 (1+m)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.64, antiderivative size = 541, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {105, 156, 12,
133} \begin {gather*} -\frac {f (a+b x)^{m+1} (c+d x)^{2-m} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )-a b d f \left (3 d e (4 m+7)-c f \left (-2 m^2+2 m+9\right )\right )+b^2 \left (c^2 f^2 \left (m^2-7 m+12\right )-3 c d e f (11-4 m)+27 d^2 e^2\right )\right )}{60 (e+f x)^3 (b e-a f)^3 (d e-c f)^3}+\frac {(b c-a d)^2 (a+b x)^{m+1} (c+d x)^{-m-1} \left (-a^3 d^3 f^3 \left (m^3+6 m^2+11 m+6\right )+3 a^2 b d^2 f^2 \left (m^2+3 m+2\right ) (5 d e-c f (2-m))-3 a b^2 d f (m+1) \left (c^2 f^2 \left (m^2-5 m+6\right )-10 c d e f (2-m)+20 d^2 e^2\right )+b^3 \left (-c^3 f^3 \left (-m^3+9 m^2-26 m+24\right )+15 c^2 d e f^2 \left (m^2-5 m+6\right )-60 c d^2 e^2 f (2-m)+60 d^3 e^3\right )\right ) \, _2F_1\left (3,m+1;m+2;\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{60 (m+1) (b e-a f)^6 (d e-c f)^3}-\frac {f (a+b x)^{m+1} (c+d x)^{2-m} (-a d f (m+3)-b c f (4-m)+7 b d e)}{20 (e+f x)^4 (b e-a f)^2 (d e-c f)^2}-\frac {f (a+b x)^{m+1} (c+d x)^{2-m}}{5 (e+f x)^5 (b e-a f) (d e-c f)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 105
Rule 133
Rule 156
Rubi steps
\begin {align*} \int \frac {(a+b x)^m (c+d x)^{1-m}}{(e+f x)^6} \, dx &=-\frac {f (a+b x)^{1+m} (c+d x)^{2-m}}{5 (b e-a f) (d e-c f) (e+f x)^5}-\frac {\int \frac {(a+b x)^m (c+d x)^{1-m} (-b (5 d e-c f (4-m))+a d f (3+m)+2 b d f x)}{(e+f x)^5} \, dx}{5 (b e-a f) (d e-c f)}\\ &=-\frac {f (a+b x)^{1+m} (c+d x)^{2-m}}{5 (b e-a f) (d e-c f) (e+f x)^5}-\frac {f (7 b d e-b c f (4-m)-a d f (3+m)) (a+b x)^{1+m} (c+d x)^{2-m}}{20 (b e-a f)^2 (d e-c f)^2 (e+f x)^4}+\frac {\int \frac {(a+b x)^m (c+d x)^{1-m} \left (a^2 d^2 f^2 \left (6+5 m+m^2\right )-a b d f \left (d e (18+11 m)-c f \left (9+2 m-2 m^2\right )\right )+b^2 \left (20 d^2 e^2-c d e f (29-11 m)+c^2 f^2 \left (12-7 m+m^2\right )\right )-b d f (7 b d e-b c f (4-m)-a d f (3+m)) x\right )}{(e+f x)^4} \, dx}{20 (b e-a f)^2 (d e-c f)^2}\\ &=-\frac {f (a+b x)^{1+m} (c+d x)^{2-m}}{5 (b e-a f) (d e-c f) (e+f x)^5}-\frac {f (7 b d e-b c f (4-m)-a d f (3+m)) (a+b x)^{1+m} (c+d x)^{2-m}}{20 (b e-a f)^2 (d e-c f)^2 (e+f x)^4}-\frac {f \left (a^2 d^2 f^2 \left (6+5 m+m^2\right )-a b d f \left (3 d e (7+4 m)-c f \left (9+2 m-2 m^2\right )\right )+b^2 \left (27 d^2 e^2-3 c d e f (11-4 m)+c^2 f^2 \left (12-7 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{2-m}}{60 (b e-a f)^3 (d e-c f)^3 (e+f x)^3}-\frac {\int \frac {\left (-3 a^2 b d^2 f^2 (5 d e-c f (2-m)) \left (2+3 m+m^2\right )+a^3 d^3 f^3 \left (6+11 m+6 m^2+m^3\right )+3 a b^2 d f (1+m) \left (20 d^2 e^2-10 c d e f (2-m)+c^2 f^2 \left (6-5 m+m^2\right )\right )-b^3 \left (60 d^3 e^3-60 c d^2 e^2 f (2-m)+15 c^2 d e f^2 \left (6-5 m+m^2\right )-c^3 f^3 \left (24-26 m+9 m^2-m^3\right )\right )\right ) (a+b x)^m (c+d x)^{1-m}}{(e+f x)^3} \, dx}{60 (b e-a f)^3 (d e-c f)^3}\\ &=-\frac {f (a+b x)^{1+m} (c+d x)^{2-m}}{5 (b e-a f) (d e-c f) (e+f x)^5}-\frac {f (7 b d e-b c f (4-m)-a d f (3+m)) (a+b x)^{1+m} (c+d x)^{2-m}}{20 (b e-a f)^2 (d e-c f)^2 (e+f x)^4}-\frac {f \left (a^2 d^2 f^2 \left (6+5 m+m^2\right )-a b d f \left (3 d e (7+4 m)-c f \left (9+2 m-2 m^2\right )\right )+b^2 \left (27 d^2 e^2-3 c d e f (11-4 m)+c^2 f^2 \left (12-7 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{2-m}}{60 (b e-a f)^3 (d e-c f)^3 (e+f x)^3}+\frac {\left (3 a^2 b d^2 f^2 (5 d e-c f (2-m)) \left (2+3 m+m^2\right )-a^3 d^3 f^3 \left (6+11 m+6 m^2+m^3\right )-3 a b^2 d f (1+m) \left (20 d^2 e^2-10 c d e f (2-m)+c^2 f^2 \left (6-5 m+m^2\right )\right )+b^3 \left (60 d^3 e^3-60 c d^2 e^2 f (2-m)+15 c^2 d e f^2 \left (6-5 m+m^2\right )-c^3 f^3 \left (24-26 m+9 m^2-m^3\right )\right )\right ) \int \frac {(a+b x)^m (c+d x)^{1-m}}{(e+f x)^3} \, dx}{60 (b e-a f)^3 (d e-c f)^3}\\ &=-\frac {f (a+b x)^{1+m} (c+d x)^{2-m}}{5 (b e-a f) (d e-c f) (e+f x)^5}-\frac {f (7 b d e-b c f (4-m)-a d f (3+m)) (a+b x)^{1+m} (c+d x)^{2-m}}{20 (b e-a f)^2 (d e-c f)^2 (e+f x)^4}-\frac {f \left (a^2 d^2 f^2 \left (6+5 m+m^2\right )-a b d f \left (3 d e (7+4 m)-c f \left (9+2 m-2 m^2\right )\right )+b^2 \left (27 d^2 e^2-3 c d e f (11-4 m)+c^2 f^2 \left (12-7 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{2-m}}{60 (b e-a f)^3 (d e-c f)^3 (e+f x)^3}+\frac {(b c-a d)^2 \left (3 a^2 b d^2 f^2 (5 d e-c f (2-m)) \left (2+3 m+m^2\right )-a^3 d^3 f^3 \left (6+11 m+6 m^2+m^3\right )-3 a b^2 d f (1+m) \left (20 d^2 e^2-10 c d e f (2-m)+c^2 f^2 \left (6-5 m+m^2\right )\right )+b^3 \left (60 d^3 e^3-60 c d^2 e^2 f (2-m)+15 c^2 d e f^2 \left (6-5 m+m^2\right )-c^3 f^3 \left (24-26 m+9 m^2-m^3\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m} \, _2F_1\left (3,1+m;2+m;\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{60 (b e-a f)^6 (d e-c f)^3 (1+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 2.37, size = 484, normalized size = 0.89 \begin {gather*} \frac {(a+b x)^{1+m} (c+d x)^{-1-m} \left (-12 f (c+d x)^3+\frac {3 f (-b (7 d e+c f (-4+m))+a d f (3+m)) (c+d x)^3 (e+f x)}{(b e-a f) (d e-c f)}-\frac {(e+f x)^2 \left (f (b e-a f)^3 (1+m) \left (a^2 d^2 f^2 \left (6+5 m+m^2\right )+a b d f \left (-3 d e (7+4 m)+c f \left (9+2 m-2 m^2\right )\right )+b^2 \left (27 d^2 e^2+3 c d e f (-11+4 m)+c^2 f^2 \left (12-7 m+m^2\right )\right )\right ) (c+d x)^3-(b c-a d)^2 \left (3 a^2 b d^2 f^2 (5 d e+c f (-2+m)) \left (2+3 m+m^2\right )-a^3 d^3 f^3 \left (6+11 m+6 m^2+m^3\right )-3 a b^2 d f (1+m) \left (20 d^2 e^2+10 c d e f (-2+m)+c^2 f^2 \left (6-5 m+m^2\right )\right )+b^3 \left (60 d^3 e^3+60 c d^2 e^2 f (-2+m)+15 c^2 d e f^2 \left (6-5 m+m^2\right )+c^3 f^3 \left (-24+26 m-9 m^2+m^3\right )\right )\right ) (e+f x)^3 \, _2F_1\left (3,1+m;2+m;\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )\right )}{(b e-a f)^5 (d e-c f)^2 (1+m)}\right )}{60 (b e-a f) (d e-c f) (e+f x)^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{m} \left (d x +c \right )^{1-m}}{\left (f x +e \right )^{6}}\, dx\]
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 (c+d\,x\right )}^{1-m}}{{\left (e+f\,x\right )}^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________