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 = 0.316739, antiderivative size = 302, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, 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*}
Mathematica [A] time = 0.211291, size = 375, normalized size = 1.23 \[ -\frac{A e \left (5 e^2 \left (21 a^2 e^2+6 a b e (d+8 e x)+b^2 \left (d^2+8 d e x+28 e^2 x^2\right )\right )+2 c e \left (5 a e \left (d^2+8 d e x+28 e^2 x^2\right )+3 b \left (8 d^2 e x+d^3+28 d e^2 x^2+56 e^3 x^3\right )\right )+3 c^2 \left (28 d^2 e^2 x^2+8 d^3 e x+d^4+56 d e^3 x^3+70 e^4 x^4\right )\right )+B \left (e^2 \left (15 a^2 e^2 (d+8 e x)+10 a b e \left (d^2+8 d e x+28 e^2 x^2\right )+3 b^2 \left (8 d^2 e x+d^3+28 d e^2 x^2+56 e^3 x^3\right )\right )+6 c e \left (a e \left (8 d^2 e x+d^3+28 d e^2 x^2+56 e^3 x^3\right )+b \left (28 d^2 e^2 x^2+8 d^3 e x+d^4+56 d e^3 x^3+70 e^4 x^4\right )\right )+5 c^2 \left (28 d^3 e^2 x^2+56 d^2 e^3 x^3+8 d^4 e x+d^5+70 d e^4 x^4+56 e^5 x^5\right )\right )}{840 e^6 (d+e x)^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 453, normalized size = 1.5 \begin{align*} -{\frac{2\,aA{e}^{3}c+A{b}^{2}{e}^{3}-6\,Abcd{e}^{2}+6\,A{c}^{2}{d}^{2}e+2\,B{e}^{3}ab-6\,aBd{e}^{2}c-3\,B{b}^{2}d{e}^{2}+12\,Bbc{d}^{2}e-10\,B{c}^{2}{d}^{3}}{6\,{e}^{6} \left ( ex+d \right ) ^{6}}}-{\frac{2\,Abc{e}^{2}-4\,A{c}^{2}de+2\,aBc{e}^{2}+B{e}^{2}{b}^{2}-8\,Bbcde+10\,B{c}^{2}{d}^{2}}{5\,{e}^{6} \left ( ex+d \right ) ^{5}}}-{\frac{2\,Aab{e}^{4}-4\,Aacd{e}^{3}-2\,Ad{b}^{2}{e}^{3}+6\,A{d}^{2}bc{e}^{2}-4\,A{c}^{2}{d}^{3}e+B{a}^{2}{e}^{4}-4\,Bdab{e}^{3}+6\,Bac{d}^{2}{e}^{2}+3\,B{b}^{2}{d}^{2}{e}^{2}-8\,B{d}^{3}bce+5\,B{c}^{2}{d}^{4}}{7\,{e}^{6} \left ( ex+d \right ) ^{7}}}-{\frac{B{c}^{2}}{3\,{e}^{6} \left ( ex+d \right ) ^{3}}}-{\frac{c \left ( Ace+2\,bBe-5\,Bcd \right ) }{4\,{e}^{6} \left ( ex+d \right ) ^{4}}}-{\frac{A{a}^{2}{e}^{5}-2\,Adab{e}^{4}+2\,A{d}^{2}ac{e}^{3}+A{d}^{2}{b}^{2}{e}^{3}-2\,A{d}^{3}bc{e}^{2}+A{d}^{4}{c}^{2}e-B{a}^{2}d{e}^{4}+2\,B{d}^{2}ab{e}^{3}-2\,aBc{d}^{3}{e}^{2}-B{d}^{3}{b}^{2}{e}^{2}+2\,B{d}^{4}bce-B{c}^{2}{d}^{5}}{8\,{e}^{6} \left ( ex+d \right ) ^{8}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10529, size = 632, normalized size = 2.08 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.01948, size = 1019, normalized size = 3.35 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12075, size = 624, normalized size = 2.05 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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