Optimal. Leaf size=162 \[ -\frac {a^3 A}{8 x^8}-\frac {a^2 (a B+3 A b)}{7 x^7}-\frac {a \left (A \left (a c+b^2\right )+a b B\right )}{2 x^6}-\frac {c \left (a B c+A b c+b^2 B\right )}{x^3}-\frac {3 a A c^2+6 a b B c+3 A b^2 c+b^3 B}{4 x^4}-\frac {A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )}{5 x^5}-\frac {c^2 (A c+3 b B)}{2 x^2}-\frac {B c^3}{x} \]
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Rubi [A] time = 0.10, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {765} \begin {gather*} -\frac {a^2 (a B+3 A b)}{7 x^7}-\frac {a^3 A}{8 x^8}-\frac {3 a A c^2+6 a b B c+3 A b^2 c+b^3 B}{4 x^4}-\frac {a \left (A \left (a c+b^2\right )+a b B\right )}{2 x^6}-\frac {c \left (a B c+A b c+b^2 B\right )}{x^3}-\frac {A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )}{5 x^5}-\frac {c^2 (A c+3 b B)}{2 x^2}-\frac {B c^3}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^9} \, dx &=\int \left (\frac {a^3 A}{x^9}+\frac {a^2 (3 A b+a B)}{x^8}+\frac {3 a \left (a b B+A \left (b^2+a c\right )\right )}{x^7}+\frac {3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )}{x^6}+\frac {b^3 B+3 A b^2 c+6 a b B c+3 a A c^2}{x^5}+\frac {3 c \left (b^2 B+A b c+a B c\right )}{x^4}+\frac {c^2 (3 b B+A c)}{x^3}+\frac {B c^3}{x^2}\right ) \, dx\\ &=-\frac {a^3 A}{8 x^8}-\frac {a^2 (3 A b+a B)}{7 x^7}-\frac {a \left (a b B+A \left (b^2+a c\right )\right )}{2 x^6}-\frac {3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )}{5 x^5}-\frac {b^3 B+3 A b^2 c+6 a b B c+3 a A c^2}{4 x^4}-\frac {c \left (b^2 B+A b c+a B c\right )}{x^3}-\frac {c^2 (3 b B+A c)}{2 x^2}-\frac {B c^3}{x}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 172, normalized size = 1.06 \begin {gather*} -\frac {5 a^3 (7 A+8 B x)+4 a^2 x (5 A (6 b+7 c x)+7 B x (5 b+6 c x))+14 a x^2 \left (A \left (10 b^2+24 b c x+15 c^2 x^2\right )+2 B x \left (6 b^2+15 b c x+10 c^2 x^2\right )\right )+14 x^3 \left (A \left (4 b^3+15 b^2 c x+20 b c^2 x^2+10 c^3 x^3\right )+5 B x \left (b^3+4 b^2 c x+6 b c^2 x^2+4 c^3 x^3\right )\right )}{280 x^8} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^9} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 166, normalized size = 1.02 \begin {gather*} -\frac {280 \, B c^{3} x^{7} + 140 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 280 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{5} + 70 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 35 \, A a^{3} + 56 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 140 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 40 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{280 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 191, normalized size = 1.18 \begin {gather*} -\frac {280 \, B c^{3} x^{7} + 420 \, B b c^{2} x^{6} + 140 \, A c^{3} x^{6} + 280 \, B b^{2} c x^{5} + 280 \, B a c^{2} x^{5} + 280 \, A b c^{2} x^{5} + 70 \, B b^{3} x^{4} + 420 \, B a b c x^{4} + 210 \, A b^{2} c x^{4} + 210 \, A a c^{2} x^{4} + 168 \, B a b^{2} x^{3} + 56 \, A b^{3} x^{3} + 168 \, B a^{2} c x^{3} + 336 \, A a b c x^{3} + 140 \, B a^{2} b x^{2} + 140 \, A a b^{2} x^{2} + 140 \, A a^{2} c x^{2} + 40 \, B a^{3} x + 120 \, A a^{2} b x + 35 \, A a^{3}}{280 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 154, normalized size = 0.95 \begin {gather*} -\frac {B \,c^{3}}{x}-\frac {\left (A c +3 b B \right ) c^{2}}{2 x^{2}}-\frac {\left (A b c +a B c +b^{2} B \right ) c}{x^{3}}-\frac {A \,a^{3}}{8 x^{8}}-\frac {3 A a \,c^{2}+3 A \,b^{2} c +6 a b B c +b^{3} B}{4 x^{4}}-\frac {\left (3 A b +B a \right ) a^{2}}{7 x^{7}}-\frac {\left (A a c +A \,b^{2}+B a b \right ) a}{2 x^{6}}-\frac {6 A a b c +A \,b^{3}+3 B \,a^{2} c +3 B a \,b^{2}}{5 x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 166, normalized size = 1.02 \begin {gather*} -\frac {280 \, B c^{3} x^{7} + 140 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 280 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{5} + 70 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 35 \, A a^{3} + 56 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 140 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 40 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{280 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 165, normalized size = 1.02 \begin {gather*} -\frac {x^3\,\left (\frac {3\,B\,c\,a^2}{5}+\frac {3\,B\,a\,b^2}{5}+\frac {6\,A\,c\,a\,b}{5}+\frac {A\,b^3}{5}\right )+x^4\,\left (\frac {B\,b^3}{4}+\frac {3\,A\,b^2\,c}{4}+\frac {3\,B\,a\,b\,c}{2}+\frac {3\,A\,a\,c^2}{4}\right )+x\,\left (\frac {B\,a^3}{7}+\frac {3\,A\,b\,a^2}{7}\right )+\frac {A\,a^3}{8}+x^6\,\left (\frac {A\,c^3}{2}+\frac {3\,B\,b\,c^2}{2}\right )+x^2\,\left (\frac {B\,a^2\,b}{2}+\frac {A\,c\,a^2}{2}+\frac {A\,a\,b^2}{2}\right )+x^5\,\left (B\,b^2\,c+A\,b\,c^2+B\,a\,c^2\right )+B\,c^3\,x^7}{x^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 119.40, size = 196, normalized size = 1.21 \begin {gather*} \frac {- 35 A a^{3} - 280 B c^{3} x^{7} + x^{6} \left (- 140 A c^{3} - 420 B b c^{2}\right ) + x^{5} \left (- 280 A b c^{2} - 280 B a c^{2} - 280 B b^{2} c\right ) + x^{4} \left (- 210 A a c^{2} - 210 A b^{2} c - 420 B a b c - 70 B b^{3}\right ) + x^{3} \left (- 336 A a b c - 56 A b^{3} - 168 B a^{2} c - 168 B a b^{2}\right ) + x^{2} \left (- 140 A a^{2} c - 140 A a b^{2} - 140 B a^{2} b\right ) + x \left (- 120 A a^{2} b - 40 B a^{3}\right )}{280 x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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