Optimal. Leaf size=330 \[ \frac {c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{5 e^8 (d+e x)^5}-\frac {c^2 \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8 (d+e x)^3}+\frac {c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{4 e^8 (d+e x)^4}-\frac {\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{7 e^8 (d+e x)^7}+\frac {\left (a e^2+c d^2\right )^3 (B d-A e)}{8 e^8 (d+e x)^8}+\frac {c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{2 e^8 (d+e x)^6}+\frac {c^3 (7 B d-A e)}{2 e^8 (d+e x)^2}-\frac {B c^3}{e^8 (d+e x)} \]
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Rubi [A] time = 0.27, antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {772} \begin {gather*} \frac {c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{5 e^8 (d+e x)^5}-\frac {c^2 \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8 (d+e x)^3}+\frac {c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{4 e^8 (d+e x)^4}+\frac {c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{2 e^8 (d+e x)^6}-\frac {\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{7 e^8 (d+e x)^7}+\frac {\left (a e^2+c d^2\right )^3 (B d-A e)}{8 e^8 (d+e x)^8}+\frac {c^3 (7 B d-A e)}{2 e^8 (d+e x)^2}-\frac {B c^3}{e^8 (d+e x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 772
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^9} \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2+a e^2\right )^3}{e^7 (d+e x)^9}+\frac {\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right )}{e^7 (d+e x)^8}+\frac {3 c \left (c d^2+a e^2\right ) \left (-7 B c d^3+5 A c d^2 e-3 a B d e^2+a A e^3\right )}{e^7 (d+e x)^7}-\frac {c \left (-35 B c^2 d^4+20 A c^2 d^3 e-30 a B c d^2 e^2+12 a A c d e^3-3 a^2 B e^4\right )}{e^7 (d+e x)^6}+\frac {c^2 \left (-35 B c d^3+15 A c d^2 e-15 a B d e^2+3 a A e^3\right )}{e^7 (d+e x)^5}-\frac {3 c^2 \left (-7 B c d^2+2 A c d e-a B e^2\right )}{e^7 (d+e x)^4}+\frac {c^3 (-7 B d+A e)}{e^7 (d+e x)^3}+\frac {B c^3}{e^7 (d+e x)^2}\right ) \, dx\\ &=\frac {(B d-A e) \left (c d^2+a e^2\right )^3}{8 e^8 (d+e x)^8}-\frac {\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right )}{7 e^8 (d+e x)^7}+\frac {c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right )}{2 e^8 (d+e x)^6}+\frac {c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right )}{5 e^8 (d+e x)^5}+\frac {c^2 \left (35 B c d^3-15 A c d^2 e+15 a B d e^2-3 a A e^3\right )}{4 e^8 (d+e x)^4}-\frac {c^2 \left (7 B c d^2-2 A c d e+a B e^2\right )}{e^8 (d+e x)^3}+\frac {c^3 (7 B d-A e)}{2 e^8 (d+e x)^2}-\frac {B c^3}{e^8 (d+e x)}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 357, normalized size = 1.08 \begin {gather*} -\frac {A e \left (35 a^3 e^6+5 a^2 c e^4 \left (d^2+8 d e x+28 e^2 x^2\right )+3 a c^2 e^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 )+5 c^3 \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )\right )+B \left (5 a^3 e^6 (d+8 e x)+3 a^2 c e^4 \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+5 a c^2 e^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 )+35 c^3 \left (d^7+8 d^6 e x+28 d^5 e^2 x^2+56 d^4 e^3 x^3+70 d^3 e^4 x^4+56 d^2 e^5 x^5+28 d e^6 x^6+8 e^7 x^7\right )\right )}{280 e^8 (d+e 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+c x^2\right )^3}{(d+e x)^9} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 532, normalized size = 1.61 \begin {gather*} -\frac {280 \, B c^{3} e^{7} x^{7} + 35 \, B c^{3} d^{7} + 5 \, A c^{3} d^{6} e + 5 \, B a c^{2} d^{5} e^{2} + 3 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} + 5 \, A a^{2} c d^{2} e^{5} + 5 \, B a^{3} d e^{6} + 35 \, A a^{3} e^{7} + 140 \, {\left (7 \, B c^{3} d e^{6} + A c^{3} e^{7}\right )} x^{6} + 280 \, {\left (7 \, B c^{3} d^{2} e^{5} + A c^{3} d e^{6} + B a c^{2} e^{7}\right )} x^{5} + 70 \, {\left (35 \, B c^{3} d^{3} e^{4} + 5 \, A c^{3} d^{2} e^{5} + 5 \, B a c^{2} d e^{6} + 3 \, A a c^{2} e^{7}\right )} x^{4} + 56 \, {\left (35 \, B c^{3} d^{4} e^{3} + 5 \, A c^{3} d^{3} e^{4} + 5 \, B a c^{2} d^{2} e^{5} + 3 \, A a c^{2} d e^{6} + 3 \, B a^{2} c e^{7}\right )} x^{3} + 28 \, {\left (35 \, B c^{3} d^{5} e^{2} + 5 \, A c^{3} d^{4} e^{3} + 5 \, B a c^{2} d^{3} e^{4} + 3 \, A a c^{2} d^{2} e^{5} + 3 \, B a^{2} c d e^{6} + 5 \, A a^{2} c e^{7}\right )} x^{2} + 8 \, {\left (35 \, B c^{3} d^{6} e + 5 \, A c^{3} d^{5} e^{2} + 5 \, B a c^{2} d^{4} e^{3} + 3 \, A a c^{2} d^{3} e^{4} + 3 \, B a^{2} c d^{2} e^{5} + 5 \, A a^{2} c d e^{6} + 5 \, B a^{3} e^{7}\right )} x}{280 \, {\left (e^{16} x^{8} + 8 \, d e^{15} x^{7} + 28 \, d^{2} e^{14} x^{6} + 56 \, d^{3} e^{13} x^{5} + 70 \, d^{4} e^{12} x^{4} + 56 \, d^{5} e^{11} x^{3} + 28 \, d^{6} e^{10} x^{2} + 8 \, d^{7} e^{9} x + d^{8} e^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 457, normalized size = 1.38 \begin {gather*} -\frac {{\left (280 \, B c^{3} x^{7} e^{7} + 980 \, B c^{3} d x^{6} e^{6} + 1960 \, B c^{3} d^{2} x^{5} e^{5} + 2450 \, B c^{3} d^{3} x^{4} e^{4} + 1960 \, B c^{3} d^{4} x^{3} e^{3} + 980 \, B c^{3} d^{5} x^{2} e^{2} + 280 \, B c^{3} d^{6} x e + 35 \, B c^{3} d^{7} + 140 \, A c^{3} x^{6} e^{7} + 280 \, A c^{3} d x^{5} e^{6} + 350 \, A c^{3} d^{2} x^{4} e^{5} + 280 \, A c^{3} d^{3} x^{3} e^{4} + 140 \, A c^{3} d^{4} x^{2} e^{3} + 40 \, A c^{3} d^{5} x e^{2} + 5 \, A c^{3} d^{6} e + 280 \, B a c^{2} x^{5} e^{7} + 350 \, B a c^{2} d x^{4} e^{6} + 280 \, B a c^{2} d^{2} x^{3} e^{5} + 140 \, B a c^{2} d^{3} x^{2} e^{4} + 40 \, B a c^{2} d^{4} x e^{3} + 5 \, B a c^{2} d^{5} e^{2} + 210 \, A a c^{2} x^{4} e^{7} + 168 \, A a c^{2} d x^{3} e^{6} + 84 \, A a c^{2} d^{2} x^{2} e^{5} + 24 \, A a c^{2} d^{3} x e^{4} + 3 \, A a c^{2} d^{4} e^{3} + 168 \, B a^{2} c x^{3} e^{7} + 84 \, B a^{2} c d x^{2} e^{6} + 24 \, B a^{2} c d^{2} x e^{5} + 3 \, B a^{2} c d^{3} e^{4} + 140 \, A a^{2} c x^{2} e^{7} + 40 \, A a^{2} c d x e^{6} + 5 \, A a^{2} c d^{2} e^{5} + 40 \, B a^{3} x e^{7} + 5 \, B a^{3} d e^{6} + 35 \, A a^{3} e^{7}\right )} e^{\left (-8\right )}}{280 \, {\left (x e + d\right )}^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 448, normalized size = 1.36 \begin {gather*} -\frac {B \,c^{3}}{\left (e x +d \right ) e^{8}}-\frac {\left (A e -7 B d \right ) c^{3}}{2 \left (e x +d \right )^{2} e^{8}}-\frac {\left (3 a A \,e^{3}+15 A c \,d^{2} e -15 a B d \,e^{2}-35 B c \,d^{3}\right ) c^{2}}{4 \left (e x +d \right )^{4} e^{8}}+\frac {\left (2 A c d e -B a \,e^{2}-7 B c \,d^{2}\right ) c^{2}}{\left (e x +d \right )^{3} e^{8}}-\frac {\left (A \,a^{2} e^{5}+6 A \,d^{2} a c \,e^{3}+5 A \,c^{2} d^{4} e -3 B \,a^{2} d \,e^{4}-10 B \,d^{3} a c \,e^{2}-7 B \,c^{2} d^{5}\right ) c}{2 \left (e x +d \right )^{6} e^{8}}+\frac {\left (12 A d a c \,e^{3}+20 A \,c^{2} d^{3} e -3 B \,a^{2} e^{4}-30 B \,d^{2} a c \,e^{2}-35 B \,c^{2} d^{4}\right ) c}{5 \left (e x +d \right )^{5} e^{8}}-\frac {A \,a^{3} e^{7}+3 A \,d^{2} a^{2} c \,e^{5}+3 A a \,c^{2} d^{4} e^{3}+A \,d^{6} c^{3} e -B d \,a^{3} e^{6}-3 B \,d^{3} a^{2} c \,e^{4}-3 B \,d^{5} a \,c^{2} e^{2}-B \,d^{7} c^{3}}{8 \left (e x +d \right )^{8} e^{8}}-\frac {-6 A d \,a^{2} c \,e^{5}-12 A \,d^{3} a \,c^{2} e^{3}-6 A \,c^{3} d^{5} e +B \,a^{3} e^{6}+9 B \,d^{2} a^{2} c \,e^{4}+15 B \,d^{4} a \,c^{2} e^{2}+7 B \,d^{6} c^{3}}{7 \left (e x +d \right )^{7} e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 532, normalized size = 1.61 \begin {gather*} -\frac {280 \, B c^{3} e^{7} x^{7} + 35 \, B c^{3} d^{7} + 5 \, A c^{3} d^{6} e + 5 \, B a c^{2} d^{5} e^{2} + 3 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} + 5 \, A a^{2} c d^{2} e^{5} + 5 \, B a^{3} d e^{6} + 35 \, A a^{3} e^{7} + 140 \, {\left (7 \, B c^{3} d e^{6} + A c^{3} e^{7}\right )} x^{6} + 280 \, {\left (7 \, B c^{3} d^{2} e^{5} + A c^{3} d e^{6} + B a c^{2} e^{7}\right )} x^{5} + 70 \, {\left (35 \, B c^{3} d^{3} e^{4} + 5 \, A c^{3} d^{2} e^{5} + 5 \, B a c^{2} d e^{6} + 3 \, A a c^{2} e^{7}\right )} x^{4} + 56 \, {\left (35 \, B c^{3} d^{4} e^{3} + 5 \, A c^{3} d^{3} e^{4} + 5 \, B a c^{2} d^{2} e^{5} + 3 \, A a c^{2} d e^{6} + 3 \, B a^{2} c e^{7}\right )} x^{3} + 28 \, {\left (35 \, B c^{3} d^{5} e^{2} + 5 \, A c^{3} d^{4} e^{3} + 5 \, B a c^{2} d^{3} e^{4} + 3 \, A a c^{2} d^{2} e^{5} + 3 \, B a^{2} c d e^{6} + 5 \, A a^{2} c e^{7}\right )} x^{2} + 8 \, {\left (35 \, B c^{3} d^{6} e + 5 \, A c^{3} d^{5} e^{2} + 5 \, B a c^{2} d^{4} e^{3} + 3 \, A a c^{2} d^{3} e^{4} + 3 \, B a^{2} c d^{2} e^{5} + 5 \, A a^{2} c d e^{6} + 5 \, B a^{3} e^{7}\right )} x}{280 \, {\left (e^{16} x^{8} + 8 \, d e^{15} x^{7} + 28 \, d^{2} e^{14} x^{6} + 56 \, d^{3} e^{13} x^{5} + 70 \, d^{4} e^{12} x^{4} + 56 \, d^{5} e^{11} x^{3} + 28 \, d^{6} e^{10} x^{2} + 8 \, d^{7} e^{9} x + d^{8} e^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 570, normalized size = 1.73 \begin {gather*} -\frac {5\,B\,a^3\,d\,e^6+40\,B\,a^3\,e^7\,x+35\,A\,a^3\,e^7+3\,B\,a^2\,c\,d^3\,e^4+24\,B\,a^2\,c\,d^2\,e^5\,x+5\,A\,a^2\,c\,d^2\,e^5+84\,B\,a^2\,c\,d\,e^6\,x^2+40\,A\,a^2\,c\,d\,e^6\,x+168\,B\,a^2\,c\,e^7\,x^3+140\,A\,a^2\,c\,e^7\,x^2+5\,B\,a\,c^2\,d^5\,e^2+40\,B\,a\,c^2\,d^4\,e^3\,x+3\,A\,a\,c^2\,d^4\,e^3+140\,B\,a\,c^2\,d^3\,e^4\,x^2+24\,A\,a\,c^2\,d^3\,e^4\,x+280\,B\,a\,c^2\,d^2\,e^5\,x^3+84\,A\,a\,c^2\,d^2\,e^5\,x^2+350\,B\,a\,c^2\,d\,e^6\,x^4+168\,A\,a\,c^2\,d\,e^6\,x^3+280\,B\,a\,c^2\,e^7\,x^5+210\,A\,a\,c^2\,e^7\,x^4+35\,B\,c^3\,d^7+280\,B\,c^3\,d^6\,e\,x+5\,A\,c^3\,d^6\,e+980\,B\,c^3\,d^5\,e^2\,x^2+40\,A\,c^3\,d^5\,e^2\,x+1960\,B\,c^3\,d^4\,e^3\,x^3+140\,A\,c^3\,d^4\,e^3\,x^2+2450\,B\,c^3\,d^3\,e^4\,x^4+280\,A\,c^3\,d^3\,e^4\,x^3+1960\,B\,c^3\,d^2\,e^5\,x^5+350\,A\,c^3\,d^2\,e^5\,x^4+980\,B\,c^3\,d\,e^6\,x^6+280\,A\,c^3\,d\,e^6\,x^5+280\,B\,c^3\,e^7\,x^7+140\,A\,c^3\,e^7\,x^6}{280\,d^8\,e^8+2240\,d^7\,e^9\,x+7840\,d^6\,e^{10}\,x^2+15680\,d^5\,e^{11}\,x^3+19600\,d^4\,e^{12}\,x^4+15680\,d^3\,e^{13}\,x^5+7840\,d^2\,e^{14}\,x^6+2240\,d\,e^{15}\,x^7+280\,e^{16}\,x^8} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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