Optimal. Leaf size=92 \[ -\frac {d^2 (b c-a d)}{2 b^4 (a+b x)^6}-\frac {3 d (b c-a d)^2}{7 b^4 (a+b x)^7}-\frac {(b c-a d)^3}{8 b^4 (a+b x)^8}-\frac {d^3}{5 b^4 (a+b x)^5} \]
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Rubi [A] time = 0.05, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {43} \[ -\frac {d^2 (b c-a d)}{2 b^4 (a+b x)^6}-\frac {3 d (b c-a d)^2}{7 b^4 (a+b x)^7}-\frac {(b c-a d)^3}{8 b^4 (a+b x)^8}-\frac {d^3}{5 b^4 (a+b x)^5} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {(c+d x)^3}{(a+b x)^9} \, dx &=\int \left (\frac {(b c-a d)^3}{b^3 (a+b x)^9}+\frac {3 d (b c-a d)^2}{b^3 (a+b x)^8}+\frac {3 d^2 (b c-a d)}{b^3 (a+b x)^7}+\frac {d^3}{b^3 (a+b x)^6}\right ) \, dx\\ &=-\frac {(b c-a d)^3}{8 b^4 (a+b x)^8}-\frac {3 d (b c-a d)^2}{7 b^4 (a+b x)^7}-\frac {d^2 (b c-a d)}{2 b^4 (a+b x)^6}-\frac {d^3}{5 b^4 (a+b x)^5}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 97, normalized size = 1.05 \[ -\frac {a^3 d^3+a^2 b d^2 (5 c+8 d x)+a b^2 d \left (15 c^2+40 c d x+28 d^2 x^2\right )+b^3 \left (35 c^3+120 c^2 d x+140 c d^2 x^2+56 d^3 x^3\right )}{280 b^4 (a+b x)^8} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 193, normalized size = 2.10 \[ -\frac {56 \, b^{3} d^{3} x^{3} + 35 \, b^{3} c^{3} + 15 \, a b^{2} c^{2} d + 5 \, a^{2} b c d^{2} + a^{3} d^{3} + 28 \, {\left (5 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 8 \, {\left (15 \, b^{3} c^{2} d + 5 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{280 \, {\left (b^{12} x^{8} + 8 \, a b^{11} x^{7} + 28 \, a^{2} b^{10} x^{6} + 56 \, a^{3} b^{9} x^{5} + 70 \, a^{4} b^{8} x^{4} + 56 \, a^{5} b^{7} x^{3} + 28 \, a^{6} b^{6} x^{2} + 8 \, a^{7} b^{5} x + a^{8} b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.86, size = 114, normalized size = 1.24 \[ -\frac {56 \, b^{3} d^{3} x^{3} + 140 \, b^{3} c d^{2} x^{2} + 28 \, a b^{2} d^{3} x^{2} + 120 \, b^{3} c^{2} d x + 40 \, a b^{2} c d^{2} x + 8 \, a^{2} b d^{3} x + 35 \, b^{3} c^{3} + 15 \, a b^{2} c^{2} d + 5 \, a^{2} b c d^{2} + a^{3} d^{3}}{280 \, {\left (b x + a\right )}^{8} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 122, normalized size = 1.33 \[ -\frac {d^{3}}{5 \left (b x +a \right )^{5} b^{4}}+\frac {\left (a d -b c \right ) d^{2}}{2 \left (b x +a \right )^{6} b^{4}}-\frac {3 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) d}{7 \left (b x +a \right )^{7} b^{4}}-\frac {-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}}{8 \left (b x +a \right )^{8} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.52, size = 193, normalized size = 2.10 \[ -\frac {56 \, b^{3} d^{3} x^{3} + 35 \, b^{3} c^{3} + 15 \, a b^{2} c^{2} d + 5 \, a^{2} b c d^{2} + a^{3} d^{3} + 28 \, {\left (5 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 8 \, {\left (15 \, b^{3} c^{2} d + 5 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{280 \, {\left (b^{12} x^{8} + 8 \, a b^{11} x^{7} + 28 \, a^{2} b^{10} x^{6} + 56 \, a^{3} b^{9} x^{5} + 70 \, a^{4} b^{8} x^{4} + 56 \, a^{5} b^{7} x^{3} + 28 \, a^{6} b^{6} x^{2} + 8 \, a^{7} b^{5} x + a^{8} b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 187, normalized size = 2.03 \[ -\frac {\frac {a^3\,d^3+5\,a^2\,b\,c\,d^2+15\,a\,b^2\,c^2\,d+35\,b^3\,c^3}{280\,b^4}+\frac {d^3\,x^3}{5\,b}+\frac {d\,x\,\left (a^2\,d^2+5\,a\,b\,c\,d+15\,b^2\,c^2\right )}{35\,b^3}+\frac {d^2\,x^2\,\left (a\,d+5\,b\,c\right )}{10\,b^2}}{a^8+8\,a^7\,b\,x+28\,a^6\,b^2\,x^2+56\,a^5\,b^3\,x^3+70\,a^4\,b^4\,x^4+56\,a^3\,b^5\,x^5+28\,a^2\,b^6\,x^6+8\,a\,b^7\,x^7+b^8\,x^8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.95, size = 207, normalized size = 2.25 \[ \frac {- a^{3} d^{3} - 5 a^{2} b c d^{2} - 15 a b^{2} c^{2} d - 35 b^{3} c^{3} - 56 b^{3} d^{3} x^{3} + x^{2} \left (- 28 a b^{2} d^{3} - 140 b^{3} c d^{2}\right ) + x \left (- 8 a^{2} b d^{3} - 40 a b^{2} c d^{2} - 120 b^{3} c^{2} d\right )}{280 a^{8} b^{4} + 2240 a^{7} b^{5} x + 7840 a^{6} b^{6} x^{2} + 15680 a^{5} b^{7} x^{3} + 19600 a^{4} b^{8} x^{4} + 15680 a^{3} b^{9} x^{5} + 7840 a^{2} b^{10} x^{6} + 2240 a b^{11} x^{7} + 280 b^{12} x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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