Optimal. Leaf size=304 \[ \frac {2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )}{5 e^6 (d+e x)^5}+\frac {B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{6 e^6 (d+e x)^6}+\frac {\left (a e^2-b d e+c d^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{7 e^6 (d+e x)^7}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{8 e^6 (d+e x)^8}+\frac {c (-A c e-2 b B e+5 B c d)}{4 e^6 (d+e x)^4}-\frac {B c^2}{3 e^6 (d+e x)^3} \]
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Rubi [A] time = 0.32, antiderivative size = 302, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {771} \[ \frac {2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )}{5 e^6 (d+e x)^5}+\frac {B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{6 e^6 (d+e x)^6}-\frac {\left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{7 e^6 (d+e x)^7}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{8 e^6 (d+e x)^8}+\frac {c (-A c e-2 b B e+5 B c d)}{4 e^6 (d+e x)^4}-\frac {B c^2}{3 e^6 (d+e x)^3} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^9} \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2-b d e+a e^2\right )^2}{e^5 (d+e x)^9}+\frac {\left (c d^2-b d e+a e^2\right ) \left (5 B c d^2-B e (3 b d-a e)-2 A e (2 c d-b e)\right )}{e^5 (d+e x)^8}+\frac {-B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )+A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )}{e^5 (d+e x)^7}+\frac {-2 A c e (2 c d-b e)+B \left (10 c^2 d^2+b^2 e^2-2 c e (4 b d-a e)\right )}{e^5 (d+e x)^6}+\frac {c (-5 B c d+2 b B e+A c e)}{e^5 (d+e x)^5}+\frac {B c^2}{e^5 (d+e x)^4}\right ) \, dx\\ &=\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^2}{8 e^6 (d+e x)^8}-\frac {\left (c d^2-b d e+a e^2\right ) \left (5 B c d^2-B e (3 b d-a e)-2 A e (2 c d-b e)\right )}{7 e^6 (d+e x)^7}+\frac {B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )-A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )}{6 e^6 (d+e x)^6}+\frac {2 A c e (2 c d-b e)-B \left (10 c^2 d^2+b^2 e^2-2 c e (4 b d-a e)\right )}{5 e^6 (d+e x)^5}+\frac {c (5 B c d-2 b B e-A c e)}{4 e^6 (d+e x)^4}-\frac {B c^2}{3 e^6 (d+e x)^3}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 375, normalized size = 1.23 \[ -\frac {A e \left (5 e^2 \left (21 a^2 e^2+6 a b e (d+8 e x)+b^2 \left (d^2+8 d e x+28 e^2 x^2\right )\right )+2 c e \left (5 a e \left (d^2+8 d e x+28 e^2 x^2\right )+3 b \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )\right )+3 c^2 \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )\right )+B \left (e^2 \left (15 a^2 e^2 (d+8 e x)+10 a b e \left (d^2+8 d e x+28 e^2 x^2\right )+3 b^2 \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )\right )+6 c e \left (a e \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+b \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )\right )+5 c^2 \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )\right )}{840 e^6 (d+e x)^8} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 468, normalized size = 1.54 \[ -\frac {280 \, B c^{2} e^{5} x^{5} + 5 \, B c^{2} d^{5} + 105 \, A a^{2} e^{5} + 3 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e^{2} + 5 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} + 15 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{4} + 70 \, {\left (5 \, B c^{2} d e^{4} + 3 \, {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 56 \, {\left (5 \, B c^{2} d^{2} e^{3} + 3 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} + 28 \, {\left (5 \, B c^{2} d^{3} e^{2} + 3 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{4} + 5 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 8 \, {\left (5 \, B c^{2} d^{4} e + 3 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{3} + 5 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} + 15 \, {\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x}{840 \, {\left (e^{14} x^{8} + 8 \, d e^{13} x^{7} + 28 \, d^{2} e^{12} x^{6} + 56 \, d^{3} e^{11} x^{5} + 70 \, d^{4} e^{10} x^{4} + 56 \, d^{5} e^{9} x^{3} + 28 \, d^{6} e^{8} x^{2} + 8 \, d^{7} e^{7} x + d^{8} e^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 462, normalized size = 1.52 \[ -\frac {{\left (280 \, B c^{2} x^{5} e^{5} + 350 \, B c^{2} d x^{4} e^{4} + 280 \, B c^{2} d^{2} x^{3} e^{3} + 140 \, B c^{2} d^{3} x^{2} e^{2} + 40 \, B c^{2} d^{4} x e + 5 \, B c^{2} d^{5} + 420 \, B b c x^{4} e^{5} + 210 \, A c^{2} x^{4} e^{5} + 336 \, B b c d x^{3} e^{4} + 168 \, A c^{2} d x^{3} e^{4} + 168 \, B b c d^{2} x^{2} e^{3} + 84 \, A c^{2} d^{2} x^{2} e^{3} + 48 \, B b c d^{3} x e^{2} + 24 \, A c^{2} d^{3} x e^{2} + 6 \, B b c d^{4} e + 3 \, A c^{2} d^{4} e + 168 \, B b^{2} x^{3} e^{5} + 336 \, B a c x^{3} e^{5} + 336 \, A b c x^{3} e^{5} + 84 \, B b^{2} d x^{2} e^{4} + 168 \, B a c d x^{2} e^{4} + 168 \, A b c d x^{2} e^{4} + 24 \, B b^{2} d^{2} x e^{3} + 48 \, B a c d^{2} x e^{3} + 48 \, A b c d^{2} x e^{3} + 3 \, B b^{2} d^{3} e^{2} + 6 \, B a c d^{3} e^{2} + 6 \, A b c d^{3} e^{2} + 280 \, B a b x^{2} e^{5} + 140 \, A b^{2} x^{2} e^{5} + 280 \, A a c x^{2} e^{5} + 80 \, B a b d x e^{4} + 40 \, A b^{2} d x e^{4} + 80 \, A a c d x e^{4} + 10 \, B a b d^{2} e^{3} + 5 \, A b^{2} d^{2} e^{3} + 10 \, A a c d^{2} e^{3} + 120 \, B a^{2} x e^{5} + 240 \, A a b x e^{5} + 15 \, B a^{2} d e^{4} + 30 \, A a b d e^{4} + 105 \, A a^{2} e^{5}\right )} e^{\left (-6\right )}}{840 \, {\left (x e + d\right )}^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 453, normalized size = 1.49 \[ -\frac {B \,c^{2}}{3 \left (e x +d \right )^{3} e^{6}}-\frac {\left (A c e +2 B b e -5 B c d \right ) c}{4 \left (e x +d \right )^{4} e^{6}}-\frac {A \,a^{2} e^{5}-2 A a b d \,e^{4}+2 A \,d^{2} a c \,e^{3}+A \,d^{2} b^{2} e^{3}-2 A b c \,d^{3} e^{2}+A \,c^{2} d^{4} e -B \,a^{2} d \,e^{4}+2 B \,d^{2} a b \,e^{3}-2 B \,d^{3} a c \,e^{2}-B \,d^{3} b^{2} e^{2}+2 B \,d^{4} b c e -B \,c^{2} d^{5}}{8 \left (e x +d \right )^{8} e^{6}}-\frac {2 A a c \,e^{3}+A \,b^{2} e^{3}-6 A b c d \,e^{2}+6 A \,c^{2} d^{2} e +2 B a b \,e^{3}-6 B a c d \,e^{2}-3 B \,b^{2} d \,e^{2}+12 B b c \,d^{2} e -10 B \,c^{2} d^{3}}{6 \left (e x +d \right )^{6} e^{6}}-\frac {2 A b c \,e^{2}-4 A \,c^{2} d e +2 a B c \,e^{2}+b^{2} B \,e^{2}-8 B b c d e +10 B \,c^{2} d^{2}}{5 \left (e x +d \right )^{5} e^{6}}-\frac {2 A a b \,e^{4}-4 A a c d \,e^{3}-2 A d \,b^{2} e^{3}+6 A b c \,d^{2} e^{2}-4 A \,c^{2} d^{3} e +B \,a^{2} e^{4}-4 B d a b \,e^{3}+6 B a c \,d^{2} e^{2}+3 B \,b^{2} d^{2} e^{2}-8 B \,d^{3} b c e +5 B \,c^{2} d^{4}}{7 \left (e x +d \right )^{7} e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 468, normalized size = 1.54 \[ -\frac {280 \, B c^{2} e^{5} x^{5} + 5 \, B c^{2} d^{5} + 105 \, A a^{2} e^{5} + 3 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e^{2} + 5 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} + 15 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{4} + 70 \, {\left (5 \, B c^{2} d e^{4} + 3 \, {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 56 \, {\left (5 \, B c^{2} d^{2} e^{3} + 3 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} + 28 \, {\left (5 \, B c^{2} d^{3} e^{2} + 3 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{4} + 5 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 8 \, {\left (5 \, B c^{2} d^{4} e + 3 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{3} + 5 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} + 15 \, {\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x}{840 \, {\left (e^{14} x^{8} + 8 \, d e^{13} x^{7} + 28 \, d^{2} e^{12} x^{6} + 56 \, d^{3} e^{11} x^{5} + 70 \, d^{4} e^{10} x^{4} + 56 \, d^{5} e^{9} x^{3} + 28 \, d^{6} e^{8} x^{2} + 8 \, d^{7} e^{7} x + d^{8} e^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.43, size = 516, normalized size = 1.70 \[ -\frac {\frac {15\,B\,a^2\,d\,e^4+105\,A\,a^2\,e^5+10\,B\,a\,b\,d^2\,e^3+30\,A\,a\,b\,d\,e^4+6\,B\,a\,c\,d^3\,e^2+10\,A\,a\,c\,d^2\,e^3+3\,B\,b^2\,d^3\,e^2+5\,A\,b^2\,d^2\,e^3+6\,B\,b\,c\,d^4\,e+6\,A\,b\,c\,d^3\,e^2+5\,B\,c^2\,d^5+3\,A\,c^2\,d^4\,e}{840\,e^6}+\frac {x^3\,\left (3\,B\,b^2\,e^2+6\,B\,b\,c\,d\,e+6\,A\,b\,c\,e^2+5\,B\,c^2\,d^2+3\,A\,c^2\,d\,e+6\,B\,a\,c\,e^2\right )}{15\,e^3}+\frac {x^2\,\left (3\,B\,b^2\,d\,e^2+5\,A\,b^2\,e^3+6\,B\,b\,c\,d^2\,e+6\,A\,b\,c\,d\,e^2+10\,B\,a\,b\,e^3+5\,B\,c^2\,d^3+3\,A\,c^2\,d^2\,e+6\,B\,a\,c\,d\,e^2+10\,A\,a\,c\,e^3\right )}{30\,e^4}+\frac {x\,\left (15\,B\,a^2\,e^4+10\,B\,a\,b\,d\,e^3+30\,A\,a\,b\,e^4+6\,B\,a\,c\,d^2\,e^2+10\,A\,a\,c\,d\,e^3+3\,B\,b^2\,d^2\,e^2+5\,A\,b^2\,d\,e^3+6\,B\,b\,c\,d^3\,e+6\,A\,b\,c\,d^2\,e^2+5\,B\,c^2\,d^4+3\,A\,c^2\,d^3\,e\right )}{105\,e^5}+\frac {c\,x^4\,\left (3\,A\,c\,e+6\,B\,b\,e+5\,B\,c\,d\right )}{12\,e^2}+\frac {B\,c^2\,x^5}{3\,e}}{d^8+8\,d^7\,e\,x+28\,d^6\,e^2\,x^2+56\,d^5\,e^3\,x^3+70\,d^4\,e^4\,x^4+56\,d^3\,e^5\,x^5+28\,d^2\,e^6\,x^6+8\,d\,e^7\,x^7+e^8\,x^8} \]
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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