Optimal. Leaf size=123 \[ \frac {(b c-a d) (c+d x)^{n-2} (e+f x)^{1-n}}{d (2-n) (d e-c f)}+\frac {(c+d x)^{n-1} (e+f x)^{1-n} (a d f+b (c f (1-n)-d e (2-n)))}{d (1-n) (2-n) (d e-c f)^2} \]
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Rubi [A] time = 0.06, antiderivative size = 122, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {79, 37} \[ \frac {(b c-a d) (c+d x)^{n-2} (e+f x)^{1-n}}{d (2-n) (d e-c f)}+\frac {(c+d x)^{n-1} (e+f x)^{1-n} (a d f+b c f (1-n)-b d e (2-n))}{d (1-n) (2-n) (d e-c f)^2} \]
Antiderivative was successfully verified.
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Rule 37
Rule 79
Rubi steps
\begin {align*} \int (a+b x) (c+d x)^{-3+n} (e+f x)^{-n} \, dx &=\frac {(b c-a d) (c+d x)^{-2+n} (e+f x)^{1-n}}{d (d e-c f) (2-n)}-\frac {(a d f+b c f (1-n)-b d e (2-n)) \int (c+d x)^{-2+n} (e+f x)^{-n} \, dx}{d (d e-c f) (2-n)}\\ &=\frac {(b c-a d) (c+d x)^{-2+n} (e+f x)^{1-n}}{d (d e-c f) (2-n)}+\frac {(a d f+b c f (1-n)-b d e (2-n)) (c+d x)^{-1+n} (e+f x)^{1-n}}{d (d e-c f)^2 (1-n) (2-n)}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 82, normalized size = 0.67 \[ \frac {(c+d x)^{n-2} (e+f x)^{1-n} (-a c f (n-2)+a d e (n-1)+a d f x-b c (e+f (n-1) x)+b d e (n-2) x)}{(n-2) (n-1) (d e-c f)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.60, size = 324, normalized size = 2.63 \[ \frac {{\left (2 \, a c^{2} e f - {\left (2 \, b d^{2} e f - {\left (b c d + a d^{2}\right )} f^{2} - {\left (b d^{2} e f - b c d f^{2}\right )} n\right )} x^{3} - {\left (b c^{2} + a c d\right )} e^{2} - {\left (2 \, b d^{2} e^{2} + 2 \, b c d e f - {\left (b c^{2} + 3 \, a c d\right )} f^{2} - {\left (b d^{2} e^{2} + a d^{2} e f - {\left (b c^{2} + a c d\right )} f^{2}\right )} n\right )} x^{2} + {\left (a c d e^{2} - a c^{2} e f\right )} n + {\left (2 \, a c d e f + 2 \, a c^{2} f^{2} - {\left (3 \, b c d + a d^{2}\right )} e^{2} - {\left (b c^{2} e f + a c^{2} f^{2} - {\left (b c d + a d^{2}\right )} e^{2}\right )} n\right )} x\right )} {\left (d x + c\right )}^{n - 3}}{{\left (2 \, d^{2} e^{2} - 4 \, c d e f + 2 \, c^{2} f^{2} + {\left (d^{2} e^{2} - 2 \, c d e f + c^{2} f^{2}\right )} n^{2} - 3 \, {\left (d^{2} e^{2} - 2 \, c d e f + c^{2} f^{2}\right )} n\right )} {\left (f x + e\right )}^{n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.18, size = 1058, normalized size = 8.60 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 161, normalized size = 1.31 \[ -\frac {\left (f x +e \right ) \left (b c f n x -b d e n x +a c f n -a d e n -a d f x -b c f x +2 b d e x -2 a c f +a d e +b c e \right ) \left (d x +c \right )^{n -2} \left (f x +e \right )^{-n}}{c^{2} f^{2} n^{2}-2 c d e f \,n^{2}+d^{2} e^{2} n^{2}-3 c^{2} f^{2} n +6 c d e f n -3 d^{2} e^{2} n +2 c^{2} f^{2}-4 c d e f +2 d^{2} e^{2}} \]
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 \[ \int \frac {{\left (b x + a\right )} {\left (d x + c\right )}^{n - 3}}{{\left (f x + e\right )}^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.85, size = 358, normalized size = 2.91 \[ \frac {x\,{\left (c+d\,x\right )}^{n-3}\,\left (2\,a\,c^2\,f^2-a\,d^2\,e^2-3\,b\,c\,d\,e^2-a\,c^2\,f^2\,n+a\,d^2\,e^2\,n+2\,a\,c\,d\,e\,f+b\,c\,d\,e^2\,n-b\,c^2\,e\,f\,n\right )}{{\left (e+f\,x\right )}^n\,{\left (c\,f-d\,e\right )}^2\,\left (n^2-3\,n+2\right )}+\frac {x^2\,{\left (c+d\,x\right )}^{n-3}\,\left (b\,c^2\,f^2-2\,b\,d^2\,e^2+3\,a\,c\,d\,f^2-b\,c^2\,f^2\,n+b\,d^2\,e^2\,n-2\,b\,c\,d\,e\,f-a\,c\,d\,f^2\,n+a\,d^2\,e\,f\,n\right )}{{\left (e+f\,x\right )}^n\,{\left (c\,f-d\,e\right )}^2\,\left (n^2-3\,n+2\right )}-\frac {c\,e\,{\left (c+d\,x\right )}^{n-3}\,\left (a\,d\,e-2\,a\,c\,f+b\,c\,e+a\,c\,f\,n-a\,d\,e\,n\right )}{{\left (e+f\,x\right )}^n\,{\left (c\,f-d\,e\right )}^2\,\left (n^2-3\,n+2\right )}+\frac {d\,f\,x^3\,{\left (c+d\,x\right )}^{n-3}\,\left (a\,d\,f+b\,c\,f-2\,b\,d\,e-b\,c\,f\,n+b\,d\,e\,n\right )}{{\left (e+f\,x\right )}^n\,{\left (c\,f-d\,e\right )}^2\,\left (n^2-3\,n+2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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