Optimal. Leaf size=64 \[ -\frac {c^5 (A b-a B) (a c+b c x)^{m-5}}{b^2 (5-m)}-\frac {B c^4 (a c+b c x)^{m-4}}{b^2 (4-m)} \]
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Rubi [A] time = 0.05, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {27, 21, 43} \begin {gather*} -\frac {c^5 (A b-a B) (a c+b c x)^{m-5}}{b^2 (5-m)}-\frac {B c^4 (a c+b c x)^{m-4}}{b^2 (4-m)} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 27
Rule 43
Rubi steps
\begin {align*} \int \frac {(A+B x) (a c+b c x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {(A+B x) (a c+b c x)^m}{(a+b x)^6} \, dx\\ &=c^6 \int (A+B x) (a c+b c x)^{-6+m} \, dx\\ &=c^6 \int \left (\frac {(A b-a B) (a c+b c x)^{-6+m}}{b}+\frac {B (a c+b c x)^{-5+m}}{b c}\right ) \, dx\\ &=-\frac {(A b-a B) c^5 (a c+b c x)^{-5+m}}{b^2 (5-m)}-\frac {B c^4 (a c+b c x)^{-4+m}}{b^2 (4-m)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 48, normalized size = 0.75 \begin {gather*} \frac {(c (a+b x))^m (-a B+A b (m-4)+b B (m-5) x)}{b^2 (m-5) (m-4) (a+b x)^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.22, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) (a c+b c x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.45, size = 212, normalized size = 3.31 \begin {gather*} \frac {{\left (A b m - B a - 4 \, A b + {\left (B b m - 5 \, B b\right )} x\right )} {\left (b c x + a c\right )}^{m}}{a^{5} b^{2} m^{2} - 9 \, a^{5} b^{2} m + 20 \, a^{5} b^{2} + {\left (b^{7} m^{2} - 9 \, b^{7} m + 20 \, b^{7}\right )} x^{5} + 5 \, {\left (a b^{6} m^{2} - 9 \, a b^{6} m + 20 \, a b^{6}\right )} x^{4} + 10 \, {\left (a^{2} b^{5} m^{2} - 9 \, a^{2} b^{5} m + 20 \, a^{2} b^{5}\right )} x^{3} + 10 \, {\left (a^{3} b^{4} m^{2} - 9 \, a^{3} b^{4} m + 20 \, a^{3} b^{4}\right )} x^{2} + 5 \, {\left (a^{4} b^{3} m^{2} - 9 \, a^{4} b^{3} m + 20 \, a^{4} b^{3}\right )} x} \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 \frac {{\left (B x + A\right )} {\left (b c x + a c\right )}^{m}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 73, normalized size = 1.14 \begin {gather*} \frac {\left (B b m x +A b m -5 B b x -4 A b -B a \right ) \left (b c x +a c \right )^{m}}{\left (b x +a \right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2} \left (m^{2}-9 m +20\right ) b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.74, size = 216, normalized size = 3.38 \begin {gather*} \frac {{\left (b c^{m} {\left (m - 5\right )} x - a c^{m}\right )} {\left (b x + a\right )}^{m} B}{{\left (m^{2} - 9 \, m + 20\right )} b^{7} x^{5} + 5 \, {\left (m^{2} - 9 \, m + 20\right )} a b^{6} x^{4} + 10 \, {\left (m^{2} - 9 \, m + 20\right )} a^{2} b^{5} x^{3} + 10 \, {\left (m^{2} - 9 \, m + 20\right )} a^{3} b^{4} x^{2} + 5 \, {\left (m^{2} - 9 \, m + 20\right )} a^{4} b^{3} x + {\left (m^{2} - 9 \, m + 20\right )} a^{5} b^{2}} + \frac {{\left (b x + a\right )}^{m} A c^{m}}{b^{6} {\left (m - 5\right )} x^{5} + 5 \, a b^{5} {\left (m - 5\right )} x^{4} + 10 \, a^{2} b^{4} {\left (m - 5\right )} x^{3} + 10 \, a^{3} b^{3} {\left (m - 5\right )} x^{2} + 5 \, a^{4} b^{2} {\left (m - 5\right )} x + a^{5} b {\left (m - 5\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.20, size = 113, normalized size = 1.77 \begin {gather*} -\frac {{\left (a\,c+b\,c\,x\right )}^m\,\left (\frac {4\,A\,b+B\,a-A\,b\,m}{b^7\,\left (m^2-9\,m+20\right )}-\frac {B\,x\,\left (m-5\right )}{b^6\,\left (m^2-9\,m+20\right )}\right )}{x^5+\frac {a^5}{b^5}+\frac {5\,a\,x^4}{b}+\frac {5\,a^4\,x}{b^4}+\frac {10\,a^2\,x^3}{b^2}+\frac {10\,a^3\,x^2}{b^3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.73, size = 1268, normalized size = 19.81 \begin {gather*} \begin {cases} \frac {\left (a c\right )^{m} \left (A x + \frac {B x^{2}}{2}\right )}{a^{6}} & \text {for}\: b = 0 \\- \frac {A b c^{4}}{a b^{2} + b^{3} x} + \frac {B a c^{4} \log {\left (\frac {a}{b} + x \right )}}{a b^{2} + b^{3} x} + \frac {B a c^{4}}{a b^{2} + b^{3} x} + \frac {B b c^{4} x \log {\left (\frac {a}{b} + x \right )}}{a b^{2} + b^{3} x} & \text {for}\: m = 4 \\\frac {A c^{5} \log {\left (\frac {a}{b} + x \right )}}{b} - \frac {B a c^{5} \log {\left (\frac {a}{b} + x \right )}}{b^{2}} + \frac {B c^{5} x}{b} & \text {for}\: m = 5 \\\frac {A b m \left (a c + b c x\right )^{m}}{a^{5} b^{2} m^{2} - 9 a^{5} b^{2} m + 20 a^{5} b^{2} + 5 a^{4} b^{3} m^{2} x - 45 a^{4} b^{3} m x + 100 a^{4} b^{3} x + 10 a^{3} b^{4} m^{2} x^{2} - 90 a^{3} b^{4} m x^{2} + 200 a^{3} b^{4} x^{2} + 10 a^{2} b^{5} m^{2} x^{3} - 90 a^{2} b^{5} m x^{3} + 200 a^{2} b^{5} x^{3} + 5 a b^{6} m^{2} x^{4} - 45 a b^{6} m x^{4} + 100 a b^{6} x^{4} + b^{7} m^{2} x^{5} - 9 b^{7} m x^{5} + 20 b^{7} x^{5}} - \frac {4 A b \left (a c + b c x\right )^{m}}{a^{5} b^{2} m^{2} - 9 a^{5} b^{2} m + 20 a^{5} b^{2} + 5 a^{4} b^{3} m^{2} x - 45 a^{4} b^{3} m x + 100 a^{4} b^{3} x + 10 a^{3} b^{4} m^{2} x^{2} - 90 a^{3} b^{4} m x^{2} + 200 a^{3} b^{4} x^{2} + 10 a^{2} b^{5} m^{2} x^{3} - 90 a^{2} b^{5} m x^{3} + 200 a^{2} b^{5} x^{3} + 5 a b^{6} m^{2} x^{4} - 45 a b^{6} m x^{4} + 100 a b^{6} x^{4} + b^{7} m^{2} x^{5} - 9 b^{7} m x^{5} + 20 b^{7} x^{5}} - \frac {B a \left (a c + b c x\right )^{m}}{a^{5} b^{2} m^{2} - 9 a^{5} b^{2} m + 20 a^{5} b^{2} + 5 a^{4} b^{3} m^{2} x - 45 a^{4} b^{3} m x + 100 a^{4} b^{3} x + 10 a^{3} b^{4} m^{2} x^{2} - 90 a^{3} b^{4} m x^{2} + 200 a^{3} b^{4} x^{2} + 10 a^{2} b^{5} m^{2} x^{3} - 90 a^{2} b^{5} m x^{3} + 200 a^{2} b^{5} x^{3} + 5 a b^{6} m^{2} x^{4} - 45 a b^{6} m x^{4} + 100 a b^{6} x^{4} + b^{7} m^{2} x^{5} - 9 b^{7} m x^{5} + 20 b^{7} x^{5}} + \frac {B b m x \left (a c + b c x\right )^{m}}{a^{5} b^{2} m^{2} - 9 a^{5} b^{2} m + 20 a^{5} b^{2} + 5 a^{4} b^{3} m^{2} x - 45 a^{4} b^{3} m x + 100 a^{4} b^{3} x + 10 a^{3} b^{4} m^{2} x^{2} - 90 a^{3} b^{4} m x^{2} + 200 a^{3} b^{4} x^{2} + 10 a^{2} b^{5} m^{2} x^{3} - 90 a^{2} b^{5} m x^{3} + 200 a^{2} b^{5} x^{3} + 5 a b^{6} m^{2} x^{4} - 45 a b^{6} m x^{4} + 100 a b^{6} x^{4} + b^{7} m^{2} x^{5} - 9 b^{7} m x^{5} + 20 b^{7} x^{5}} - \frac {5 B b x \left (a c + b c x\right )^{m}}{a^{5} b^{2} m^{2} - 9 a^{5} b^{2} m + 20 a^{5} b^{2} + 5 a^{4} b^{3} m^{2} x - 45 a^{4} b^{3} m x + 100 a^{4} b^{3} x + 10 a^{3} b^{4} m^{2} x^{2} - 90 a^{3} b^{4} m x^{2} + 200 a^{3} b^{4} x^{2} + 10 a^{2} b^{5} m^{2} x^{3} - 90 a^{2} b^{5} m x^{3} + 200 a^{2} b^{5} x^{3} + 5 a b^{6} m^{2} x^{4} - 45 a b^{6} m x^{4} + 100 a b^{6} x^{4} + b^{7} m^{2} x^{5} - 9 b^{7} m x^{5} + 20 b^{7} x^{5}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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