Optimal. Leaf size=185 \[ \frac {6 b^3 (a+b x)^{m+1} (c+d x)^{-m-1}}{(m+1) (m+2) (m+3) (m+4) (b c-a d)^4}+\frac {6 b^2 (a+b x)^{m+1} (c+d x)^{-m-2}}{(m+2) (m+3) (m+4) (b c-a d)^3}+\frac {(a+b x)^{m+1} (c+d x)^{-m-4}}{(m+4) (b c-a d)}+\frac {3 b (a+b x)^{m+1} (c+d x)^{-m-3}}{(m+3) (m+4) (b c-a d)^2} \]
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Rubi [A] time = 0.06, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {45, 37} \begin {gather*} \frac {6 b^2 (a+b x)^{m+1} (c+d x)^{-m-2}}{(m+2) (m+3) (m+4) (b c-a d)^3}+\frac {6 b^3 (a+b x)^{m+1} (c+d x)^{-m-1}}{(m+1) (m+2) (m+3) (m+4) (b c-a d)^4}+\frac {(a+b x)^{m+1} (c+d x)^{-m-4}}{(m+4) (b c-a d)}+\frac {3 b (a+b x)^{m+1} (c+d x)^{-m-3}}{(m+3) (m+4) (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int (a+b x)^m (c+d x)^{-5-m} \, dx &=\frac {(a+b x)^{1+m} (c+d x)^{-4-m}}{(b c-a d) (4+m)}+\frac {(3 b) \int (a+b x)^m (c+d x)^{-4-m} \, dx}{(b c-a d) (4+m)}\\ &=\frac {(a+b x)^{1+m} (c+d x)^{-4-m}}{(b c-a d) (4+m)}+\frac {3 b (a+b x)^{1+m} (c+d x)^{-3-m}}{(b c-a d)^2 (3+m) (4+m)}+\frac {\left (6 b^2\right ) \int (a+b x)^m (c+d x)^{-3-m} \, dx}{(b c-a d)^2 (3+m) (4+m)}\\ &=\frac {(a+b x)^{1+m} (c+d x)^{-4-m}}{(b c-a d) (4+m)}+\frac {3 b (a+b x)^{1+m} (c+d x)^{-3-m}}{(b c-a d)^2 (3+m) (4+m)}+\frac {6 b^2 (a+b x)^{1+m} (c+d x)^{-2-m}}{(b c-a d)^3 (2+m) (3+m) (4+m)}+\frac {\left (6 b^3\right ) \int (a+b x)^m (c+d x)^{-2-m} \, dx}{(b c-a d)^3 (2+m) (3+m) (4+m)}\\ &=\frac {(a+b x)^{1+m} (c+d x)^{-4-m}}{(b c-a d) (4+m)}+\frac {3 b (a+b x)^{1+m} (c+d x)^{-3-m}}{(b c-a d)^2 (3+m) (4+m)}+\frac {6 b^2 (a+b x)^{1+m} (c+d x)^{-2-m}}{(b c-a d)^3 (2+m) (3+m) (4+m)}+\frac {6 b^3 (a+b x)^{1+m} (c+d x)^{-1-m}}{(b c-a d)^4 (1+m) (2+m) (3+m) (4+m)}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 195, normalized size = 1.05 \begin {gather*} \frac {(a+b x)^{m+1} (c+d x)^{-m-4} \left (-a^3 d^3 \left (m^3+6 m^2+11 m+6\right )+3 a^2 b d^2 \left (m^2+3 m+2\right ) (c (m+4)+d x)-3 a b^2 d (m+1) \left (c^2 \left (m^2+7 m+12\right )+2 c d (m+4) x+2 d^2 x^2\right )+b^3 \left (c^3 \left (m^3+9 m^2+26 m+24\right )+3 c^2 d \left (m^2+7 m+12\right ) x+6 c d^2 (m+4) x^2+6 d^3 x^3\right )\right )}{(m+1) (m+2) (m+3) (m+4) (b c-a d)^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.06, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x)^m (c+d x)^{-5-m} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.24, size = 954, normalized size = 5.16 \begin {gather*} \frac {{\left (6 \, b^{4} d^{4} x^{5} + 24 \, a b^{3} c^{4} - 36 \, a^{2} b^{2} c^{3} d + 24 \, a^{3} b c^{2} d^{2} - 6 \, a^{4} c d^{3} + 6 \, {\left (5 \, b^{4} c d^{3} + {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} m\right )} x^{4} + {\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3}\right )} m^{3} + 3 \, {\left (20 \, b^{4} c^{2} d^{2} + {\left (b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} m^{2} + {\left (9 \, b^{4} c^{2} d^{2} - 10 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} m\right )} x^{3} + 3 \, {\left (3 \, a b^{3} c^{4} - 8 \, a^{2} b^{2} c^{3} d + 7 \, a^{3} b c^{2} d^{2} - 2 \, a^{4} c d^{3}\right )} m^{2} + {\left (60 \, b^{4} c^{3} d + {\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} m^{3} + 3 \, {\left (4 \, b^{4} c^{3} d - 9 \, a b^{3} c^{2} d^{2} + 6 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} m^{2} + {\left (47 \, b^{4} c^{3} d - 60 \, a b^{3} c^{2} d^{2} + 15 \, a^{2} b^{2} c d^{3} - 2 \, a^{3} b d^{4}\right )} m\right )} x^{2} + {\left (26 \, a b^{3} c^{4} - 57 \, a^{2} b^{2} c^{3} d + 42 \, a^{3} b c^{2} d^{2} - 11 \, a^{4} c d^{3}\right )} m + {\left (24 \, b^{4} c^{4} + 24 \, a b^{3} c^{3} d - 36 \, a^{2} b^{2} c^{2} d^{2} + 24 \, a^{3} b c d^{3} - 6 \, a^{4} d^{4} + {\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} m^{3} + 3 \, {\left (3 \, b^{4} c^{4} - 4 \, a b^{3} c^{3} d - 3 \, a^{2} b^{2} c^{2} d^{2} + 6 \, a^{3} b c d^{3} - 2 \, a^{4} d^{4}\right )} m^{2} + {\left (26 \, b^{4} c^{4} - 10 \, a b^{3} c^{3} d - 45 \, a^{2} b^{2} c^{2} d^{2} + 40 \, a^{3} b c d^{3} - 11 \, a^{4} d^{4}\right )} m\right )} x\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 5}}{24 \, b^{4} c^{4} - 96 \, a b^{3} c^{3} d + 144 \, a^{2} b^{2} c^{2} d^{2} - 96 \, a^{3} b c d^{3} + 24 \, a^{4} d^{4} + {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} m^{4} + 10 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} m^{3} + 35 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} m^{2} + 50 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} m} \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*} \int {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 5}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 662, normalized size = 3.58 \begin {gather*} -\frac {\left (a^{3} d^{3} m^{3}-3 a^{2} b c \,d^{2} m^{3}-3 a^{2} b \,d^{3} m^{2} x +3 a \,b^{2} c^{2} d \,m^{3}+6 a \,b^{2} c \,d^{2} m^{2} x +6 a \,b^{2} d^{3} m \,x^{2}-b^{3} c^{3} m^{3}-3 b^{3} c^{2} d \,m^{2} x -6 b^{3} c \,d^{2} m \,x^{2}-6 b^{3} d^{3} x^{3}+6 a^{3} d^{3} m^{2}-21 a^{2} b c \,d^{2} m^{2}-9 a^{2} b \,d^{3} m x +24 a \,b^{2} c^{2} d \,m^{2}+30 a \,b^{2} c \,d^{2} m x +6 a \,b^{2} d^{3} x^{2}-9 b^{3} c^{3} m^{2}-21 b^{3} c^{2} d m x -24 b^{3} c \,d^{2} x^{2}+11 a^{3} d^{3} m -42 a^{2} b c \,d^{2} m -6 a^{2} b \,d^{3} x +57 a \,b^{2} c^{2} d m +24 a \,b^{2} c \,d^{2} x -26 b^{3} c^{3} m -36 b^{3} c^{2} d x +6 a^{3} d^{3}-24 a^{2} b c \,d^{2}+36 a \,b^{2} c^{2} d -24 b^{3} c^{3}\right ) \left (b x +a \right )^{m +1} \left (d x +c \right )^{-m -4}}{a^{4} d^{4} m^{4}-4 a^{3} b c \,d^{3} m^{4}+6 a^{2} b^{2} c^{2} d^{2} m^{4}-4 a \,b^{3} c^{3} d \,m^{4}+b^{4} c^{4} m^{4}+10 a^{4} d^{4} m^{3}-40 a^{3} b c \,d^{3} m^{3}+60 a^{2} b^{2} c^{2} d^{2} m^{3}-40 a \,b^{3} c^{3} d \,m^{3}+10 b^{4} c^{4} m^{3}+35 a^{4} d^{4} m^{2}-140 a^{3} b c \,d^{3} m^{2}+210 a^{2} b^{2} c^{2} d^{2} m^{2}-140 a \,b^{3} c^{3} d \,m^{2}+35 b^{4} c^{4} m^{2}+50 a^{4} d^{4} m -200 a^{3} b c \,d^{3} m +300 a^{2} b^{2} c^{2} d^{2} m -200 a \,b^{3} c^{3} d m +50 b^{4} c^{4} m +24 a^{4} d^{4}-96 a^{3} b c \,d^{3}+144 a^{2} b^{2} c^{2} d^{2}-96 a \,b^{3} c^{3} d +24 b^{4} c^{4}} \end {gather*}
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*} \int {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 5}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.54, size = 945, normalized size = 5.11 \begin {gather*} \frac {6\,b^4\,d^4\,x^5\,{\left (a+b\,x\right )}^m}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{m+5}\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}-\frac {x\,{\left (a+b\,x\right )}^m\,\left (a^4\,d^4\,m^3+6\,a^4\,d^4\,m^2+11\,a^4\,d^4\,m+6\,a^4\,d^4-2\,a^3\,b\,c\,d^3\,m^3-18\,a^3\,b\,c\,d^3\,m^2-40\,a^3\,b\,c\,d^3\,m-24\,a^3\,b\,c\,d^3+9\,a^2\,b^2\,c^2\,d^2\,m^2+45\,a^2\,b^2\,c^2\,d^2\,m+36\,a^2\,b^2\,c^2\,d^2+2\,a\,b^3\,c^3\,d\,m^3+12\,a\,b^3\,c^3\,d\,m^2+10\,a\,b^3\,c^3\,d\,m-24\,a\,b^3\,c^3\,d-b^4\,c^4\,m^3-9\,b^4\,c^4\,m^2-26\,b^4\,c^4\,m-24\,b^4\,c^4\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{m+5}\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}-\frac {a\,c\,{\left (a+b\,x\right )}^m\,\left (a^3\,d^3\,m^3+6\,a^3\,d^3\,m^2+11\,a^3\,d^3\,m+6\,a^3\,d^3-3\,a^2\,b\,c\,d^2\,m^3-21\,a^2\,b\,c\,d^2\,m^2-42\,a^2\,b\,c\,d^2\,m-24\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d\,m^3+24\,a\,b^2\,c^2\,d\,m^2+57\,a\,b^2\,c^2\,d\,m+36\,a\,b^2\,c^2\,d-b^3\,c^3\,m^3-9\,b^3\,c^3\,m^2-26\,b^3\,c^3\,m-24\,b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{m+5}\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {3\,b^2\,d^2\,x^3\,{\left (a+b\,x\right )}^m\,\left (a^2\,d^2\,m^2+a^2\,d^2\,m-2\,a\,b\,c\,d\,m^2-10\,a\,b\,c\,d\,m+b^2\,c^2\,m^2+9\,b^2\,c^2\,m+20\,b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{m+5}\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {6\,b^3\,d^3\,x^4\,{\left (a+b\,x\right )}^m\,\left (5\,b\,c-a\,d\,m+b\,c\,m\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{m+5}\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {b\,d\,x^2\,{\left (a+b\,x\right )}^m\,\left (-a^3\,d^3\,m^3-3\,a^3\,d^3\,m^2-2\,a^3\,d^3\,m+3\,a^2\,b\,c\,d^2\,m^3+18\,a^2\,b\,c\,d^2\,m^2+15\,a^2\,b\,c\,d^2\,m-3\,a\,b^2\,c^2\,d\,m^3-27\,a\,b^2\,c^2\,d\,m^2-60\,a\,b^2\,c^2\,d\,m+b^3\,c^3\,m^3+12\,b^3\,c^3\,m^2+47\,b^3\,c^3\,m+60\,b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{m+5}\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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