Optimal. Leaf size=299 \[ \frac {(b c-a d) (c+d x)^{-4+n} (e+f x)^{1-n}}{d (d e-c f) (4-n)}+\frac {(3 a d f+b (c f (1-n)-d e (4-n))) (c+d x)^{-3+n} (e+f x)^{1-n}}{d (d e-c f)^2 (3-n) (4-n)}-\frac {2 f (3 a d f+b (c f (1-n)-d e (4-n))) (c+d x)^{-2+n} (e+f x)^{1-n}}{d (d e-c f)^3 (2-n) (3-n) (4-n)}+\frac {2 f^2 (3 a d f+b (c f (1-n)-d e (4-n))) (c+d x)^{-1+n} (e+f x)^{1-n}}{d (d e-c f)^4 (1-n) (2-n) (3-n) (4-n)} \]
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Rubi [A]
time = 0.14, antiderivative size = 296, normalized size of antiderivative = 0.99, number of steps
used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {80, 47, 37}
\begin {gather*} \frac {2 f^2 (c+d x)^{n-1} (e+f x)^{1-n} (3 a d f+b c f (1-n)-b d e (4-n))}{d (1-n) (2-n) (3-n) (4-n) (d e-c f)^4}+\frac {(b c-a d) (c+d x)^{n-4} (e+f x)^{1-n}}{d (4-n) (d e-c f)}+\frac {(c+d x)^{n-3} (e+f x)^{1-n} (3 a d f+b c f (1-n)-b d e (4-n))}{d (3-n) (4-n) (d e-c f)^2}-\frac {2 f (c+d x)^{n-2} (e+f x)^{1-n} (3 a d f+b c f (1-n)-b d e (4-n))}{d (2-n) (3-n) (4-n) (d e-c f)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rule 80
Rubi steps
\begin {align*} \int (a+b x) (c+d x)^{-5+n} (e+f x)^{-n} \, dx &=\frac {(b c-a d) (c+d x)^{-4+n} (e+f x)^{1-n}}{d (d e-c f) (4-n)}-\frac {(3 a d f+b c f (1-n)-b d e (4-n)) \int (c+d x)^{-4+n} (e+f x)^{-n} \, dx}{d (d e-c f) (4-n)}\\ &=\frac {(b c-a d) (c+d x)^{-4+n} (e+f x)^{1-n}}{d (d e-c f) (4-n)}+\frac {(3 a d f+b c f (1-n)-b d e (4-n)) (c+d x)^{-3+n} (e+f x)^{1-n}}{d (d e-c f)^2 (3-n) (4-n)}+\frac {(2 f (3 a d f+b c f (1-n)-b d e (4-n))) \int (c+d x)^{-3+n} (e+f x)^{-n} \, dx}{d (d e-c f)^2 (3-n) (4-n)}\\ &=\frac {(b c-a d) (c+d x)^{-4+n} (e+f x)^{1-n}}{d (d e-c f) (4-n)}+\frac {(3 a d f+b c f (1-n)-b d e (4-n)) (c+d x)^{-3+n} (e+f x)^{1-n}}{d (d e-c f)^2 (3-n) (4-n)}-\frac {2 f (3 a d f+b c f (1-n)-b d e (4-n)) (c+d x)^{-2+n} (e+f x)^{1-n}}{d (d e-c f)^3 (2-n) (3-n) (4-n)}-\frac {\left (2 f^2 (3 a d f+b c f (1-n)-b d e (4-n))\right ) \int (c+d x)^{-2+n} (e+f x)^{-n} \, dx}{d (d e-c f)^3 (2-n) (3-n) (4-n)}\\ &=\frac {(b c-a d) (c+d x)^{-4+n} (e+f x)^{1-n}}{d (d e-c f) (4-n)}+\frac {(3 a d f+b c f (1-n)-b d e (4-n)) (c+d x)^{-3+n} (e+f x)^{1-n}}{d (d e-c f)^2 (3-n) (4-n)}-\frac {2 f (3 a d f+b c f (1-n)-b d e (4-n)) (c+d x)^{-2+n} (e+f x)^{1-n}}{d (d e-c f)^3 (2-n) (3-n) (4-n)}+\frac {2 f^2 (3 a d f+b c f (1-n)-b d e (4-n)) (c+d x)^{-1+n} (e+f x)^{1-n}}{d (d e-c f)^4 (1-n) (2-n) (3-n) (4-n)}\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 165, normalized size = 0.55 \begin {gather*} \frac {(c+d x)^{-4+n} (e+f x)^{1-n} \left (-b c+a d+\frac {(3 a d f+b d e (-4+n)-b c f (-1+n)) (c+d x) \left (c^2 f^2 \left (6-5 n+n^2\right )-2 c d f (-3+n) (e (-1+n)+f x)+d^2 \left (e^2 \left (2-3 n+n^2\right )+2 e f (-1+n) x+2 f^2 x^2\right )\right )}{(d e-c f)^3 (-3+n) (-2+n) (-1+n)}\right )}{d (d e-c f) (-4+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1186\) vs.
\(2(299)=598\).
time = 0.10, size = 1187, normalized size = 3.97
method | result | size |
gosper | \(-\frac {\left (d x +c \right )^{-4+n} \left (f x +e \right ) \left (b \,c^{3} f^{3} n^{3} x -3 b \,c^{2} d e \,f^{2} n^{3} x -2 b \,c^{2} d \,f^{3} n^{2} x^{2}+3 b c \,d^{2} e^{2} f \,n^{3} x +4 b c \,d^{2} e \,f^{2} n^{2} x^{2}+2 b c \,d^{2} f^{3} n \,x^{3}-b \,d^{3} e^{3} n^{3} x -2 b \,d^{3} e^{2} f \,n^{2} x^{2}-2 b \,d^{3} e \,f^{2} n \,x^{3}+a \,c^{3} f^{3} n^{3}-3 a \,c^{2} d e \,f^{2} n^{3}-3 a \,c^{2} d \,f^{3} n^{2} x +3 a c \,d^{2} e^{2} f \,n^{3}+6 a c \,d^{2} e \,f^{2} n^{2} x +6 a c \,d^{2} f^{3} n \,x^{2}-a \,d^{3} e^{3} n^{3}-3 a \,d^{3} e^{2} f \,n^{2} x -6 a \,d^{3} e \,f^{2} n \,x^{2}-6 a \,d^{3} f^{3} x^{3}-8 b \,c^{3} f^{3} n^{2} x +23 b \,c^{2} d e \,f^{2} n^{2} x +10 b \,c^{2} d \,f^{3} n \,x^{2}-22 b c \,d^{2} e^{2} f \,n^{2} x -20 b c \,d^{2} e \,f^{2} n \,x^{2}-2 b c \,d^{2} f^{3} x^{3}+7 b \,d^{3} e^{3} n^{2} x +10 b \,d^{3} e^{2} f n \,x^{2}+8 b \,d^{3} e \,f^{2} x^{3}-9 a \,c^{3} f^{3} n^{2}+24 a \,c^{2} d e \,f^{2} n^{2}+21 a \,c^{2} d \,f^{3} n x -21 a c \,d^{2} e^{2} f \,n^{2}-30 a c \,d^{2} e \,f^{2} n x -24 a c \,d^{2} f^{3} x^{2}+6 a \,d^{3} e^{3} n^{2}+9 a \,d^{3} e^{2} f n x +6 a \,d^{3} e \,f^{2} x^{2}+b \,c^{3} e \,f^{2} n^{2}+19 b \,c^{3} f^{3} n x -2 b \,c^{2} d \,e^{2} f \,n^{2}-58 b \,c^{2} d e \,f^{2} n x -8 b \,c^{2} d \,f^{3} x^{2}+b c \,d^{2} e^{3} n^{2}+53 b c \,d^{2} e^{2} f n x +34 b c \,d^{2} e \,f^{2} x^{2}-14 b \,d^{3} e^{3} n x -8 b \,d^{3} e^{2} f \,x^{2}+26 a \,c^{3} f^{3} n -57 a \,c^{2} d e \,f^{2} n -36 a \,c^{2} d \,f^{3} x +42 a c \,d^{2} e^{2} f n +24 a c \,d^{2} e \,f^{2} x -11 a \,d^{3} e^{3} n -6 a \,d^{3} e^{2} f x -7 b \,c^{3} e \,f^{2} n -12 b \,c^{3} f^{3} x +10 b \,c^{2} d \,e^{2} f n +56 b \,c^{2} d e \,f^{2} x -3 b c \,d^{2} e^{3} n -34 b c \,d^{2} e^{2} f x +8 b \,d^{3} e^{3} x -24 a \,c^{3} f^{3}+36 a \,c^{2} d e \,f^{2}-24 a c \,d^{2} e^{2} f +6 a \,d^{3} e^{3}+12 b \,c^{3} e \,f^{2}-8 b \,c^{2} d \,e^{2} f +2 b c \,d^{2} e^{3}\right ) \left (f x +e \right )^{-n}}{c^{4} f^{4} n^{4}-4 c^{3} d e \,f^{3} n^{4}+6 c^{2} d^{2} e^{2} f^{2} n^{4}-4 c \,d^{3} e^{3} f \,n^{4}+d^{4} e^{4} n^{4}-10 c^{4} f^{4} n^{3}+40 c^{3} d e \,f^{3} n^{3}-60 c^{2} d^{2} e^{2} f^{2} n^{3}+40 c \,d^{3} e^{3} f \,n^{3}-10 d^{4} e^{4} n^{3}+35 c^{4} f^{4} n^{2}-140 c^{3} d e \,f^{3} n^{2}+210 c^{2} d^{2} e^{2} f^{2} n^{2}-140 c \,d^{3} e^{3} f \,n^{2}+35 d^{4} e^{4} n^{2}-50 c^{4} f^{4} n +200 c^{3} d e \,f^{3} n -300 c^{2} d^{2} e^{2} f^{2} n +200 c \,d^{3} e^{3} f n -50 d^{4} e^{4} n +24 c^{4} f^{4}-96 c^{3} d e \,f^{3}+144 c^{2} d^{2} e^{2} f^{2}-96 c \,d^{3} e^{3} f +24 d^{4} e^{4}}\) | \(1187\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 1734 vs.
\(2 (281) = 562\).
time = 2.05, size = 1734, normalized size = 5.80 \begin {gather*} -\frac {{\left (2 \, {\left (b c d^{3} f^{4} n - {\left (b c d^{3} + 3 \, a d^{4}\right )} f^{4}\right )} x^{5} - 2 \, {\left (b c^{2} d^{2} f^{4} n^{2} - 3 \, {\left (2 \, b c^{2} d^{2} + a c d^{3}\right )} f^{4} n + 5 \, {\left (b c^{2} d^{2} + 3 \, a c d^{3}\right )} f^{4}\right )} x^{4} + {\left (b c^{3} d f^{4} n^{3} - {\left (10 \, b c^{3} d + 3 \, a c^{2} d^{2}\right )} f^{4} n^{2} + {\left (29 \, b c^{3} d + 27 \, a c^{2} d^{2}\right )} f^{4} n - 20 \, {\left (b c^{3} d + 3 \, a c^{2} d^{2}\right )} f^{4}\right )} x^{3} + {\left ({\left (b c^{4} + a c^{3} d\right )} f^{4} n^{3} - 4 \, {\left (2 \, b c^{4} + 3 \, a c^{3} d\right )} f^{4} n^{2} + {\left (19 \, b c^{4} + 47 \, a c^{3} d\right )} f^{4} n - 12 \, {\left (b c^{4} + 5 \, a c^{3} d\right )} f^{4}\right )} x^{2} + {\left (a c^{4} f^{4} n^{3} - 9 \, a c^{4} f^{4} n^{2} + 26 \, a c^{4} f^{4} n - 24 \, a c^{4} f^{4}\right )} x - {\left (a c d^{3} n^{3} - 2 \, b c^{2} d^{2} - 6 \, a c d^{3} - {\left (b c^{2} d^{2} + 6 \, a c d^{3}\right )} n^{2} + {\left (b d^{4} n^{3} - 7 \, b d^{4} n^{2} + 14 \, b d^{4} n - 8 \, b d^{4}\right )} x^{2} + {\left (3 \, b c^{2} d^{2} + 11 \, a c d^{3}\right )} n - {\left (10 \, b c d^{3} + 6 \, a d^{4} - {\left (b c d^{3} + a d^{4}\right )} n^{3} + 2 \, {\left (4 \, b c d^{3} + 3 \, a d^{4}\right )} n^{2} - {\left (17 \, b c d^{3} + 11 \, a d^{4}\right )} n\right )} x\right )} e^{4} + {\left (3 \, a c^{2} d^{2} f n^{3} - {\left (2 \, b c^{3} d + 21 \, a c^{2} d^{2}\right )} f n^{2} - {\left (b d^{4} f n^{3} - 5 \, b d^{4} f n^{2} + 4 \, b d^{4} f n\right )} x^{3} + 2 \, {\left (5 \, b c^{3} d + 21 \, a c^{2} d^{2}\right )} f n - {\left (32 \, b c d^{3} f - {\left (2 \, b c d^{3} - a d^{4}\right )} f n^{3} + {\left (16 \, b c d^{3} - 3 \, a d^{4}\right )} f n^{2} - 2 \, {\left (23 \, b c d^{3} - a d^{4}\right )} f n\right )} x^{2} - 8 \, {\left (b c^{3} d + 3 \, a c^{2} d^{2}\right )} f + {\left ({\left (3 \, b c^{2} d^{2} + 2 \, a c d^{3}\right )} f n^{3} - {\left (23 \, b c^{2} d^{2} + 18 \, a c d^{3}\right )} f n^{2} + 20 \, {\left (3 \, b c^{2} d^{2} + 2 \, a c d^{3}\right )} f n - 8 \, {\left (5 \, b c^{2} d^{2} + 3 \, a c d^{3}\right )} f\right )} x\right )} e^{3} - {\left (3 \, a c^{3} d f^{2} n^{3} - {\left (b c^{4} + 24 \, a c^{3} d\right )} f^{2} n^{2} + 2 \, {\left (b d^{4} f^{2} n^{2} - 4 \, b d^{4} f^{2} n\right )} x^{4} + {\left (7 \, b c^{4} + 57 \, a c^{3} d\right )} f^{2} n - {\left (3 \, b c d^{3} f^{2} n^{3} - {\left (20 \, b c d^{3} + 3 \, a d^{4}\right )} f^{2} n^{2} + {\left (41 \, b c d^{3} + 3 \, a d^{4}\right )} f^{2} n\right )} x^{3} - 12 \, {\left (b c^{4} + 3 \, a c^{3} d\right )} f^{2} - 3 \, {\left (a c d^{3} f^{2} n^{3} + 16 \, b c^{2} d^{2} f^{2} + {\left (b c^{2} d^{2} - 6 \, a c d^{3}\right )} f^{2} n^{2} - 5 \, {\left (b c^{2} d^{2} - a c d^{3}\right )} f^{2} n\right )} x^{2} + {\left (3 \, b c^{3} d f^{2} n^{3} - {\left (22 \, b c^{3} d + 9 \, a c^{2} d^{2}\right )} f^{2} n^{2} + 5 \, {\left (11 \, b c^{3} d + 9 \, a c^{2} d^{2}\right )} f^{2} n - 12 \, {\left (5 \, b c^{3} d + 3 \, a c^{2} d^{2}\right )} f^{2}\right )} x\right )} e^{2} + {\left (a c^{4} f^{3} n^{3} - 9 \, a c^{4} f^{3} n^{2} + 26 \, a c^{4} f^{3} n - 24 \, a c^{4} f^{3} - 2 \, {\left (b d^{4} f^{3} n - 4 \, b d^{4} f^{3}\right )} x^{5} + 2 \, {\left (2 \, b c d^{3} f^{3} n^{2} + 20 \, b c d^{3} f^{3} - {\left (10 \, b c d^{3} + 3 \, a d^{4}\right )} f^{3} n\right )} x^{4} - {\left (3 \, b c^{2} d^{2} f^{3} n^{3} - 80 \, b c^{2} d^{2} f^{3} - {\left (25 \, b c^{2} d^{2} + 6 \, a c d^{3}\right )} f^{3} n^{2} + 6 \, {\left (11 \, b c^{2} d^{2} + 5 \, a c d^{3}\right )} f^{3} n\right )} x^{3} + {\left (48 \, b c^{3} d f^{3} - {\left (2 \, b c^{3} d + 3 \, a c^{2} d^{2}\right )} f^{3} n^{3} + {\left (14 \, b c^{3} d + 27 \, a c^{2} d^{2}\right )} f^{3} n^{2} - 12 \, {\left (3 \, b c^{3} d + 5 \, a c^{2} d^{2}\right )} f^{3} n\right )} x^{2} - {\left (24 \, a c^{3} d f^{3} - {\left (b c^{4} - 2 \, a c^{3} d\right )} f^{3} n^{3} + {\left (7 \, b c^{4} - 12 \, a c^{3} d\right )} f^{3} n^{2} - 2 \, {\left (6 \, b c^{4} - 5 \, a c^{3} d\right )} f^{3} n\right )} x\right )} e\right )} {\left (d x + c\right )}^{n - 5}}{{\left (c^{4} f^{4} n^{4} - 10 \, c^{4} f^{4} n^{3} + 35 \, c^{4} f^{4} n^{2} - 50 \, c^{4} f^{4} n + 24 \, c^{4} f^{4} + {\left (d^{4} n^{4} - 10 \, d^{4} n^{3} + 35 \, d^{4} n^{2} - 50 \, d^{4} n + 24 \, d^{4}\right )} e^{4} - 4 \, {\left (c d^{3} f n^{4} - 10 \, c d^{3} f n^{3} + 35 \, c d^{3} f n^{2} - 50 \, c d^{3} f n + 24 \, c d^{3} f\right )} e^{3} + 6 \, {\left (c^{2} d^{2} f^{2} n^{4} - 10 \, c^{2} d^{2} f^{2} n^{3} + 35 \, c^{2} d^{2} f^{2} n^{2} - 50 \, c^{2} d^{2} f^{2} n + 24 \, c^{2} d^{2} f^{2}\right )} e^{2} - 4 \, {\left (c^{3} d f^{3} n^{4} - 10 \, c^{3} d f^{3} n^{3} + 35 \, c^{3} d f^{3} n^{2} - 50 \, c^{3} d f^{3} n + 24 \, c^{3} d f^{3}\right )} e\right )} {\left (f x + e\right )}^{n}} \end {gather*}
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 \begin {gather*} \text {Exception raised: SystemError} \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 [B]
time = 4.67, size = 1657, normalized size = 5.54 \begin {gather*} \frac {x\,{\left (c+d\,x\right )}^{n-5}\,\left (-b\,c^4\,e\,f^3\,n^3+7\,b\,c^4\,e\,f^3\,n^2-12\,b\,c^4\,e\,f^3\,n-a\,c^4\,f^4\,n^3+9\,a\,c^4\,f^4\,n^2-26\,a\,c^4\,f^4\,n+24\,a\,c^4\,f^4+3\,b\,c^3\,d\,e^2\,f^2\,n^3-22\,b\,c^3\,d\,e^2\,f^2\,n^2+55\,b\,c^3\,d\,e^2\,f^2\,n-60\,b\,c^3\,d\,e^2\,f^2+2\,a\,c^3\,d\,e\,f^3\,n^3-12\,a\,c^3\,d\,e\,f^3\,n^2+10\,a\,c^3\,d\,e\,f^3\,n+24\,a\,c^3\,d\,e\,f^3-3\,b\,c^2\,d^2\,e^3\,f\,n^3+23\,b\,c^2\,d^2\,e^3\,f\,n^2-60\,b\,c^2\,d^2\,e^3\,f\,n+40\,b\,c^2\,d^2\,e^3\,f-9\,a\,c^2\,d^2\,e^2\,f^2\,n^2+45\,a\,c^2\,d^2\,e^2\,f^2\,n-36\,a\,c^2\,d^2\,e^2\,f^2+b\,c\,d^3\,e^4\,n^3-8\,b\,c\,d^3\,e^4\,n^2+17\,b\,c\,d^3\,e^4\,n-10\,b\,c\,d^3\,e^4-2\,a\,c\,d^3\,e^3\,f\,n^3+18\,a\,c\,d^3\,e^3\,f\,n^2-40\,a\,c\,d^3\,e^3\,f\,n+24\,a\,c\,d^3\,e^3\,f+a\,d^4\,e^4\,n^3-6\,a\,d^4\,e^4\,n^2+11\,a\,d^4\,e^4\,n-6\,a\,d^4\,e^4\right )}{{\left (e+f\,x\right )}^n\,{\left (c\,f-d\,e\right )}^4\,\left (n^4-10\,n^3+35\,n^2-50\,n+24\right )}-\frac {{\left (c+d\,x\right )}^{n-5}\,\left (b\,c^4\,e^2\,f^2\,n^2-7\,b\,c^4\,e^2\,f^2\,n+12\,b\,c^4\,e^2\,f^2+a\,c^4\,e\,f^3\,n^3-9\,a\,c^4\,e\,f^3\,n^2+26\,a\,c^4\,e\,f^3\,n-24\,a\,c^4\,e\,f^3-2\,b\,c^3\,d\,e^3\,f\,n^2+10\,b\,c^3\,d\,e^3\,f\,n-8\,b\,c^3\,d\,e^3\,f-3\,a\,c^3\,d\,e^2\,f^2\,n^3+24\,a\,c^3\,d\,e^2\,f^2\,n^2-57\,a\,c^3\,d\,e^2\,f^2\,n+36\,a\,c^3\,d\,e^2\,f^2+b\,c^2\,d^2\,e^4\,n^2-3\,b\,c^2\,d^2\,e^4\,n+2\,b\,c^2\,d^2\,e^4+3\,a\,c^2\,d^2\,e^3\,f\,n^3-21\,a\,c^2\,d^2\,e^3\,f\,n^2+42\,a\,c^2\,d^2\,e^3\,f\,n-24\,a\,c^2\,d^2\,e^3\,f-a\,c\,d^3\,e^4\,n^3+6\,a\,c\,d^3\,e^4\,n^2-11\,a\,c\,d^3\,e^4\,n+6\,a\,c\,d^3\,e^4\right )}{{\left (e+f\,x\right )}^n\,{\left (c\,f-d\,e\right )}^4\,\left (n^4-10\,n^3+35\,n^2-50\,n+24\right )}+\frac {x^2\,{\left (c+d\,x\right )}^{n-5}\,\left (-b\,c^4\,f^4\,n^3+8\,b\,c^4\,f^4\,n^2-19\,b\,c^4\,f^4\,n+12\,b\,c^4\,f^4+2\,b\,c^3\,d\,e\,f^3\,n^3-14\,b\,c^3\,d\,e\,f^3\,n^2+36\,b\,c^3\,d\,e\,f^3\,n-48\,b\,c^3\,d\,e\,f^3-a\,c^3\,d\,f^4\,n^3+12\,a\,c^3\,d\,f^4\,n^2-47\,a\,c^3\,d\,f^4\,n+60\,a\,c^3\,d\,f^4-3\,b\,c^2\,d^2\,e^2\,f^2\,n^2+15\,b\,c^2\,d^2\,e^2\,f^2\,n-48\,b\,c^2\,d^2\,e^2\,f^2+3\,a\,c^2\,d^2\,e\,f^3\,n^3-27\,a\,c^2\,d^2\,e\,f^3\,n^2+60\,a\,c^2\,d^2\,e\,f^3\,n-2\,b\,c\,d^3\,e^3\,f\,n^3+16\,b\,c\,d^3\,e^3\,f\,n^2-46\,b\,c\,d^3\,e^3\,f\,n+32\,b\,c\,d^3\,e^3\,f-3\,a\,c\,d^3\,e^2\,f^2\,n^3+18\,a\,c\,d^3\,e^2\,f^2\,n^2-15\,a\,c\,d^3\,e^2\,f^2\,n+b\,d^4\,e^4\,n^3-7\,b\,d^4\,e^4\,n^2+14\,b\,d^4\,e^4\,n-8\,b\,d^4\,e^4+a\,d^4\,e^3\,f\,n^3-3\,a\,d^4\,e^3\,f\,n^2+2\,a\,d^4\,e^3\,f\,n\right )}{{\left (e+f\,x\right )}^n\,{\left (c\,f-d\,e\right )}^4\,\left (n^4-10\,n^3+35\,n^2-50\,n+24\right )}+\frac {2\,d^3\,f^3\,x^5\,{\left (c+d\,x\right )}^{n-5}\,\left (3\,a\,d\,f+b\,c\,f-4\,b\,d\,e-b\,c\,f\,n+b\,d\,e\,n\right )}{{\left (e+f\,x\right )}^n\,{\left (c\,f-d\,e\right )}^4\,\left (n^4-10\,n^3+35\,n^2-50\,n+24\right )}+\frac {2\,d^2\,f^2\,x^4\,{\left (c+d\,x\right )}^{n-5}\,\left (5\,c\,f-c\,f\,n+d\,e\,n\right )\,\left (3\,a\,d\,f+b\,c\,f-4\,b\,d\,e-b\,c\,f\,n+b\,d\,e\,n\right )}{{\left (e+f\,x\right )}^n\,{\left (c\,f-d\,e\right )}^4\,\left (n^4-10\,n^3+35\,n^2-50\,n+24\right )}+\frac {d\,f\,x^3\,{\left (c+d\,x\right )}^{n-5}\,\left (3\,a\,d\,f+b\,c\,f-4\,b\,d\,e-b\,c\,f\,n+b\,d\,e\,n\right )\,\left (c^2\,f^2\,n^2-9\,c^2\,f^2\,n+20\,c^2\,f^2-2\,c\,d\,e\,f\,n^2+10\,c\,d\,e\,f\,n+d^2\,e^2\,n^2-d^2\,e^2\,n\right )}{{\left (e+f\,x\right )}^n\,{\left (c\,f-d\,e\right )}^4\,\left (n^4-10\,n^3+35\,n^2-50\,n+24\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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