Optimal. Leaf size=304 \[ \frac {2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )}{4 e^6 (d+e x)^4}+\frac {B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{5 e^6 (d+e x)^5}+\frac {\left (a e^2-b d e+c d^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{6 e^6 (d+e x)^6}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{7 e^6 (d+e x)^7}+\frac {c (-A c e-2 b B e+5 B c d)}{3 e^6 (d+e x)^3}-\frac {B c^2}{2 e^6 (d+e x)^2} \]
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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} \[ \frac {2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )}{4 e^6 (d+e x)^4}+\frac {B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{5 e^6 (d+e x)^5}-\frac {\left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{6 e^6 (d+e x)^6}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{7 e^6 (d+e x)^7}+\frac {c (-A c e-2 b B e+5 B c d)}{3 e^6 (d+e x)^3}-\frac {B c^2}{2 e^6 (d+e x)^2} \]
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)^8} \, 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)^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 )}{e^5 (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 )}{e^5 (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 )}{e^5 (d+e x)^5}+\frac {c (-5 B c d+2 b B e+A c e)}{e^5 (d+e x)^4}+\frac {B c^2}{e^5 (d+e x)^3}\right ) \, dx\\ &=\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^2}{7 e^6 (d+e x)^7}-\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 )}{6 e^6 (d+e x)^6}+\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 )}{5 e^6 (d+e x)^5}+\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 )}{4 e^6 (d+e x)^4}+\frac {c (5 B c d-2 b B e-A c e)}{3 e^6 (d+e x)^3}-\frac {B c^2}{2 e^6 (d+e x)^2}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 377, normalized size = 1.24 \[ -\frac {2 A e \left (2 e^2 \left (15 a^2 e^2+5 a b e (d+7 e x)+b^2 \left (d^2+7 d e x+21 e^2 x^2\right )\right )+c e \left (4 a e \left (d^2+7 d e x+21 e^2 x^2\right )+3 b \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )\right )+2 c^2 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )\right )+B \left (e^2 \left (10 a^2 e^2 (d+7 e x)+8 a b e \left (d^2+7 d e x+21 e^2 x^2\right )+3 b^2 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )\right )+2 c e \left (3 a e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+4 b \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )\right )+10 c^2 \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )\right )}{420 e^6 (d+e x)^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 457, normalized size = 1.50 \[ -\frac {210 \, B c^{2} e^{5} x^{5} + 10 \, B c^{2} d^{5} + 60 \, A a^{2} e^{5} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e^{2} + 4 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} + 10 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{4} + 70 \, {\left (5 \, B c^{2} d e^{4} + 2 \, {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 35 \, {\left (10 \, B c^{2} d^{2} e^{3} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} + 21 \, {\left (10 \, B c^{2} d^{3} e^{2} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{4} + 4 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 7 \, {\left (10 \, B c^{2} d^{4} e + 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{3} + 4 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} + 10 \, {\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x}{420 \, {\left (e^{13} x^{7} + 7 \, d e^{12} x^{6} + 21 \, d^{2} e^{11} x^{5} + 35 \, d^{3} e^{10} x^{4} + 35 \, d^{4} e^{9} x^{3} + 21 \, d^{5} e^{8} x^{2} + 7 \, d^{6} e^{7} x + d^{7} e^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 462, normalized size = 1.52 \[ -\frac {{\left (210 \, B c^{2} x^{5} e^{5} + 350 \, B c^{2} d x^{4} e^{4} + 350 \, B c^{2} d^{2} x^{3} e^{3} + 210 \, B c^{2} d^{3} x^{2} e^{2} + 70 \, B c^{2} d^{4} x e + 10 \, B c^{2} d^{5} + 280 \, B b c x^{4} e^{5} + 140 \, A c^{2} x^{4} e^{5} + 280 \, B b c d x^{3} e^{4} + 140 \, A c^{2} d x^{3} e^{4} + 168 \, B b c d^{2} x^{2} e^{3} + 84 \, A c^{2} d^{2} x^{2} e^{3} + 56 \, B b c d^{3} x e^{2} + 28 \, A c^{2} d^{3} x e^{2} + 8 \, B b c d^{4} e + 4 \, A c^{2} d^{4} e + 105 \, B b^{2} x^{3} e^{5} + 210 \, B a c x^{3} e^{5} + 210 \, A b c x^{3} e^{5} + 63 \, B b^{2} d x^{2} e^{4} + 126 \, B a c d x^{2} e^{4} + 126 \, A b c d x^{2} e^{4} + 21 \, B b^{2} d^{2} x e^{3} + 42 \, B a c d^{2} x e^{3} + 42 \, A b c d^{2} x e^{3} + 3 \, B b^{2} d^{3} e^{2} + 6 \, B a c d^{3} e^{2} + 6 \, A b c d^{3} e^{2} + 168 \, B a b x^{2} e^{5} + 84 \, A b^{2} x^{2} e^{5} + 168 \, A a c x^{2} e^{5} + 56 \, B a b d x e^{4} + 28 \, A b^{2} d x e^{4} + 56 \, A a c d x e^{4} + 8 \, B a b d^{2} e^{3} + 4 \, A b^{2} d^{2} e^{3} + 8 \, A a c d^{2} e^{3} + 70 \, B a^{2} x e^{5} + 140 \, A a b x e^{5} + 10 \, B a^{2} d e^{4} + 20 \, A a b d e^{4} + 60 \, A a^{2} e^{5}\right )} e^{\left (-6\right )}}{420 \, {\left (x e + d\right )}^{7}} \]
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 \[ -\frac {B \,c^{2}}{2 \left (e x +d \right )^{2} e^{6}}-\frac {\left (A c e +2 B b e -5 B c d \right ) c}{3 \left (e x +d \right )^{3} e^{6}}-\frac {2 A a b \,e^{4}-4 A a c d \,e^{3}-2 A d \,b^{2} e^{3}+6 A b c \,d^{2} e^{2}-4 A \,c^{2} d^{3} e +B \,a^{2} e^{4}-4 B d a b \,e^{3}+6 B a c \,d^{2} e^{2}+3 B \,b^{2} d^{2} e^{2}-8 B \,d^{3} b c e +5 B \,c^{2} d^{4}}{6 \left (e x +d \right )^{6} e^{6}}-\frac {2 A b c \,e^{2}-4 A \,c^{2} d e +2 a B c \,e^{2}+b^{2} B \,e^{2}-8 B b c d e +10 B \,c^{2} d^{2}}{4 \left (e x +d \right )^{4} e^{6}}-\frac {2 A a c \,e^{3}+A \,b^{2} e^{3}-6 A b c d \,e^{2}+6 A \,c^{2} d^{2} e +2 B a b \,e^{3}-6 B a c d \,e^{2}-3 B \,b^{2} d \,e^{2}+12 B b c \,d^{2} e -10 B \,c^{2} d^{3}}{5 \left (e x +d \right )^{5} e^{6}}-\frac {A \,a^{2} e^{5}-2 A a b d \,e^{4}+2 A \,d^{2} a c \,e^{3}+A \,d^{2} b^{2} e^{3}-2 A b c \,d^{3} e^{2}+A \,c^{2} d^{4} e -B \,a^{2} d \,e^{4}+2 B \,d^{2} a b \,e^{3}-2 B \,d^{3} a c \,e^{2}-B \,d^{3} b^{2} e^{2}+2 B \,d^{4} b c e -B \,c^{2} d^{5}}{7 \left (e x +d \right )^{7} e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 457, normalized size = 1.50 \[ -\frac {210 \, B c^{2} e^{5} x^{5} + 10 \, B c^{2} d^{5} + 60 \, A a^{2} e^{5} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e^{2} + 4 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} + 10 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{4} + 70 \, {\left (5 \, B c^{2} d e^{4} + 2 \, {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 35 \, {\left (10 \, B c^{2} d^{2} e^{3} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} + 21 \, {\left (10 \, B c^{2} d^{3} e^{2} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{4} + 4 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 7 \, {\left (10 \, B c^{2} d^{4} e + 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{3} + 4 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} + 10 \, {\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x}{420 \, {\left (e^{13} x^{7} + 7 \, d e^{12} x^{6} + 21 \, d^{2} e^{11} x^{5} + 35 \, d^{3} e^{10} x^{4} + 35 \, d^{4} e^{9} x^{3} + 21 \, d^{5} e^{8} x^{2} + 7 \, d^{6} e^{7} x + d^{7} e^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 505, normalized size = 1.66 \[ -\frac {\frac {10\,B\,a^2\,d\,e^4+60\,A\,a^2\,e^5+8\,B\,a\,b\,d^2\,e^3+20\,A\,a\,b\,d\,e^4+6\,B\,a\,c\,d^3\,e^2+8\,A\,a\,c\,d^2\,e^3+3\,B\,b^2\,d^3\,e^2+4\,A\,b^2\,d^2\,e^3+8\,B\,b\,c\,d^4\,e+6\,A\,b\,c\,d^3\,e^2+10\,B\,c^2\,d^5+4\,A\,c^2\,d^4\,e}{420\,e^6}+\frac {x^3\,\left (3\,B\,b^2\,e^2+8\,B\,b\,c\,d\,e+6\,A\,b\,c\,e^2+10\,B\,c^2\,d^2+4\,A\,c^2\,d\,e+6\,B\,a\,c\,e^2\right )}{12\,e^3}+\frac {x^2\,\left (3\,B\,b^2\,d\,e^2+4\,A\,b^2\,e^3+8\,B\,b\,c\,d^2\,e+6\,A\,b\,c\,d\,e^2+8\,B\,a\,b\,e^3+10\,B\,c^2\,d^3+4\,A\,c^2\,d^2\,e+6\,B\,a\,c\,d\,e^2+8\,A\,a\,c\,e^3\right )}{20\,e^4}+\frac {x\,\left (10\,B\,a^2\,e^4+8\,B\,a\,b\,d\,e^3+20\,A\,a\,b\,e^4+6\,B\,a\,c\,d^2\,e^2+8\,A\,a\,c\,d\,e^3+3\,B\,b^2\,d^2\,e^2+4\,A\,b^2\,d\,e^3+8\,B\,b\,c\,d^3\,e+6\,A\,b\,c\,d^2\,e^2+10\,B\,c^2\,d^4+4\,A\,c^2\,d^3\,e\right )}{60\,e^5}+\frac {c\,x^4\,\left (2\,A\,c\,e+4\,B\,b\,e+5\,B\,c\,d\right )}{6\,e^2}+\frac {B\,c^2\,x^5}{2\,e}}{d^7+7\,d^6\,e\,x+21\,d^5\,e^2\,x^2+35\,d^4\,e^3\,x^3+35\,d^3\,e^4\,x^4+21\,d^2\,e^5\,x^5+7\,d\,e^6\,x^6+e^7\,x^7} \]
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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