Optimal. Leaf size=334 \[ \frac {c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{7 e^8 (d+e x)^7}-\frac {3 c^2 \left (a B e^2-2 A c d e+7 B c d^2\right )}{5 e^8 (d+e x)^5}+\frac {c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{6 e^8 (d+e x)^6}-\frac {\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{9 e^8 (d+e x)^9}+\frac {\left (a e^2+c d^2\right )^3 (B d-A e)}{10 e^8 (d+e x)^{10}}+\frac {3 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{8 e^8 (d+e x)^8}+\frac {c^3 (7 B d-A e)}{4 e^8 (d+e x)^4}-\frac {B c^3}{3 e^8 (d+e x)^3} \]
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Rubi [A] time = 0.26, antiderivative size = 334, 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 )}{7 e^8 (d+e x)^7}-\frac {3 c^2 \left (a B e^2-2 A c d e+7 B c d^2\right )}{5 e^8 (d+e x)^5}+\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 )}{6 e^8 (d+e x)^6}+\frac {3 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 )}{8 e^8 (d+e x)^8}-\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 )}{9 e^8 (d+e x)^9}+\frac {\left (a e^2+c d^2\right )^3 (B d-A e)}{10 e^8 (d+e x)^{10}}+\frac {c^3 (7 B d-A e)}{4 e^8 (d+e x)^4}-\frac {B c^3}{3 e^8 (d+e x)^3} \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)^{11}} \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2+a e^2\right )^3}{e^7 (d+e x)^{11}}+\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)^{10}}+\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)^9}-\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)^8}+\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)^7}-\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)^6}+\frac {c^3 (-7 B d+A e)}{e^7 (d+e x)^5}+\frac {B c^3}{e^7 (d+e x)^4}\right ) \, dx\\ &=\frac {(B d-A e) \left (c d^2+a e^2\right )^3}{10 e^8 (d+e x)^{10}}-\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 )}{9 e^8 (d+e x)^9}+\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 )}{8 e^8 (d+e x)^8}+\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 )}{7 e^8 (d+e x)^7}+\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 )}{6 e^8 (d+e x)^6}-\frac {3 c^2 \left (7 B c d^2-2 A c d e+a B e^2\right )}{5 e^8 (d+e x)^5}+\frac {c^3 (7 B d-A e)}{4 e^8 (d+e x)^4}-\frac {B c^3}{3 e^8 (d+e x)^3}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 357, normalized size = 1.07 \begin {gather*} -\frac {3 A e \left (84 a^3 e^6+7 a^2 c e^4 \left (d^2+10 d e x+45 e^2 x^2\right )+2 a c^2 e^2 \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )+c^3 \left (d^6+10 d^5 e x+45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+252 d e^5 x^5+210 e^6 x^6\right )\right )+B \left (28 a^3 e^6 (d+10 e x)+9 a^2 c e^4 \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+6 a c^2 e^2 \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )+7 c^3 \left (d^7+10 d^6 e x+45 d^5 e^2 x^2+120 d^4 e^3 x^3+210 d^3 e^4 x^4+252 d^2 e^5 x^5+210 d e^6 x^6+120 e^7 x^7\right )\right )}{2520 e^8 (d+e x)^{10}} \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)^{11}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 557, normalized size = 1.67 \begin {gather*} -\frac {840 \, B c^{3} e^{7} x^{7} + 7 \, B c^{3} d^{7} + 3 \, A c^{3} d^{6} e + 6 \, B a c^{2} d^{5} e^{2} + 6 \, A a c^{2} d^{4} e^{3} + 9 \, B a^{2} c d^{3} e^{4} + 21 \, A a^{2} c d^{2} e^{5} + 28 \, B a^{3} d e^{6} + 252 \, A a^{3} e^{7} + 210 \, {\left (7 \, B c^{3} d e^{6} + 3 \, A c^{3} e^{7}\right )} x^{6} + 252 \, {\left (7 \, B c^{3} d^{2} e^{5} + 3 \, A c^{3} d e^{6} + 6 \, B a c^{2} e^{7}\right )} x^{5} + 210 \, {\left (7 \, B c^{3} d^{3} e^{4} + 3 \, A c^{3} d^{2} e^{5} + 6 \, B a c^{2} d e^{6} + 6 \, A a c^{2} e^{7}\right )} x^{4} + 120 \, {\left (7 \, B c^{3} d^{4} e^{3} + 3 \, A c^{3} d^{3} e^{4} + 6 \, B a c^{2} d^{2} e^{5} + 6 \, A a c^{2} d e^{6} + 9 \, B a^{2} c e^{7}\right )} x^{3} + 45 \, {\left (7 \, B c^{3} d^{5} e^{2} + 3 \, A c^{3} d^{4} e^{3} + 6 \, B a c^{2} d^{3} e^{4} + 6 \, A a c^{2} d^{2} e^{5} + 9 \, B a^{2} c d e^{6} + 21 \, A a^{2} c e^{7}\right )} x^{2} + 10 \, {\left (7 \, B c^{3} d^{6} e + 3 \, A c^{3} d^{5} e^{2} + 6 \, B a c^{2} d^{4} e^{3} + 6 \, A a c^{2} d^{3} e^{4} + 9 \, B a^{2} c d^{2} e^{5} + 21 \, A a^{2} c d e^{6} + 28 \, B a^{3} e^{7}\right )} x}{2520 \, {\left (e^{18} x^{10} + 10 \, d e^{17} x^{9} + 45 \, d^{2} e^{16} x^{8} + 120 \, d^{3} e^{15} x^{7} + 210 \, d^{4} e^{14} x^{6} + 252 \, d^{5} e^{13} x^{5} + 210 \, d^{6} e^{12} x^{4} + 120 \, d^{7} e^{11} x^{3} + 45 \, d^{8} e^{10} x^{2} + 10 \, d^{9} e^{9} x + d^{10} e^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 457, normalized size = 1.37 \begin {gather*} -\frac {{\left (840 \, B c^{3} x^{7} e^{7} + 1470 \, B c^{3} d x^{6} e^{6} + 1764 \, B c^{3} d^{2} x^{5} e^{5} + 1470 \, B c^{3} d^{3} x^{4} e^{4} + 840 \, B c^{3} d^{4} x^{3} e^{3} + 315 \, B c^{3} d^{5} x^{2} e^{2} + 70 \, B c^{3} d^{6} x e + 7 \, B c^{3} d^{7} + 630 \, A c^{3} x^{6} e^{7} + 756 \, A c^{3} d x^{5} e^{6} + 630 \, A c^{3} d^{2} x^{4} e^{5} + 360 \, A c^{3} d^{3} x^{3} e^{4} + 135 \, A c^{3} d^{4} x^{2} e^{3} + 30 \, A c^{3} d^{5} x e^{2} + 3 \, A c^{3} d^{6} e + 1512 \, B a c^{2} x^{5} e^{7} + 1260 \, B a c^{2} d x^{4} e^{6} + 720 \, B a c^{2} d^{2} x^{3} e^{5} + 270 \, B a c^{2} d^{3} x^{2} e^{4} + 60 \, B a c^{2} d^{4} x e^{3} + 6 \, B a c^{2} d^{5} e^{2} + 1260 \, A a c^{2} x^{4} e^{7} + 720 \, A a c^{2} d x^{3} e^{6} + 270 \, A a c^{2} d^{2} x^{2} e^{5} + 60 \, A a c^{2} d^{3} x e^{4} + 6 \, A a c^{2} d^{4} e^{3} + 1080 \, B a^{2} c x^{3} e^{7} + 405 \, B a^{2} c d x^{2} e^{6} + 90 \, B a^{2} c d^{2} x e^{5} + 9 \, B a^{2} c d^{3} e^{4} + 945 \, A a^{2} c x^{2} e^{7} + 210 \, A a^{2} c d x e^{6} + 21 \, A a^{2} c d^{2} e^{5} + 280 \, B a^{3} x e^{7} + 28 \, B a^{3} d e^{6} + 252 \, A a^{3} e^{7}\right )} e^{\left (-8\right )}}{2520 \, {\left (x e + d\right )}^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 449, normalized size = 1.34 \begin {gather*} -\frac {B \,c^{3}}{3 \left (e x +d \right )^{3} e^{8}}-\frac {\left (A e -7 B d \right ) c^{3}}{4 \left (e x +d \right )^{4} 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}}{6 \left (e x +d \right )^{6} e^{8}}+\frac {3 \left (2 A c d e -B a \,e^{2}-7 B c \,d^{2}\right ) c^{2}}{5 \left (e x +d \right )^{5} e^{8}}-\frac {3 \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}{8 \left (e x +d \right )^{8} 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}{7 \left (e x +d \right )^{7} 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}}{9 \left (e x +d \right )^{9} 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}}{10 \left (e x +d \right )^{10} e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 557, normalized size = 1.67 \begin {gather*} -\frac {840 \, B c^{3} e^{7} x^{7} + 7 \, B c^{3} d^{7} + 3 \, A c^{3} d^{6} e + 6 \, B a c^{2} d^{5} e^{2} + 6 \, A a c^{2} d^{4} e^{3} + 9 \, B a^{2} c d^{3} e^{4} + 21 \, A a^{2} c d^{2} e^{5} + 28 \, B a^{3} d e^{6} + 252 \, A a^{3} e^{7} + 210 \, {\left (7 \, B c^{3} d e^{6} + 3 \, A c^{3} e^{7}\right )} x^{6} + 252 \, {\left (7 \, B c^{3} d^{2} e^{5} + 3 \, A c^{3} d e^{6} + 6 \, B a c^{2} e^{7}\right )} x^{5} + 210 \, {\left (7 \, B c^{3} d^{3} e^{4} + 3 \, A c^{3} d^{2} e^{5} + 6 \, B a c^{2} d e^{6} + 6 \, A a c^{2} e^{7}\right )} x^{4} + 120 \, {\left (7 \, B c^{3} d^{4} e^{3} + 3 \, A c^{3} d^{3} e^{4} + 6 \, B a c^{2} d^{2} e^{5} + 6 \, A a c^{2} d e^{6} + 9 \, B a^{2} c e^{7}\right )} x^{3} + 45 \, {\left (7 \, B c^{3} d^{5} e^{2} + 3 \, A c^{3} d^{4} e^{3} + 6 \, B a c^{2} d^{3} e^{4} + 6 \, A a c^{2} d^{2} e^{5} + 9 \, B a^{2} c d e^{6} + 21 \, A a^{2} c e^{7}\right )} x^{2} + 10 \, {\left (7 \, B c^{3} d^{6} e + 3 \, A c^{3} d^{5} e^{2} + 6 \, B a c^{2} d^{4} e^{3} + 6 \, A a c^{2} d^{3} e^{4} + 9 \, B a^{2} c d^{2} e^{5} + 21 \, A a^{2} c d e^{6} + 28 \, B a^{3} e^{7}\right )} x}{2520 \, {\left (e^{18} x^{10} + 10 \, d e^{17} x^{9} + 45 \, d^{2} e^{16} x^{8} + 120 \, d^{3} e^{15} x^{7} + 210 \, d^{4} e^{14} x^{6} + 252 \, d^{5} e^{13} x^{5} + 210 \, d^{6} e^{12} x^{4} + 120 \, d^{7} e^{11} x^{3} + 45 \, d^{8} e^{10} x^{2} + 10 \, d^{9} e^{9} x + d^{10} e^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.91, size = 524, normalized size = 1.57 \begin {gather*} -\frac {\frac {28\,B\,a^3\,d\,e^6+252\,A\,a^3\,e^7+9\,B\,a^2\,c\,d^3\,e^4+21\,A\,a^2\,c\,d^2\,e^5+6\,B\,a\,c^2\,d^5\,e^2+6\,A\,a\,c^2\,d^4\,e^3+7\,B\,c^3\,d^7+3\,A\,c^3\,d^6\,e}{2520\,e^8}+\frac {x\,\left (28\,B\,a^3\,e^6+9\,B\,a^2\,c\,d^2\,e^4+21\,A\,a^2\,c\,d\,e^5+6\,B\,a\,c^2\,d^4\,e^2+6\,A\,a\,c^2\,d^3\,e^3+7\,B\,c^3\,d^6+3\,A\,c^3\,d^5\,e\right )}{252\,e^7}+\frac {c^2\,x^4\,\left (7\,B\,c\,d^3+3\,A\,c\,d^2\,e+6\,B\,a\,d\,e^2+6\,A\,a\,e^3\right )}{12\,e^4}+\frac {c\,x^3\,\left (9\,B\,a^2\,e^4+6\,B\,a\,c\,d^2\,e^2+6\,A\,a\,c\,d\,e^3+7\,B\,c^2\,d^4+3\,A\,c^2\,d^3\,e\right )}{21\,e^5}+\frac {c^3\,x^6\,\left (3\,A\,e+7\,B\,d\right )}{12\,e^2}+\frac {c^2\,x^5\,\left (7\,B\,c\,d^2+3\,A\,c\,d\,e+6\,B\,a\,e^2\right )}{10\,e^3}+\frac {c\,x^2\,\left (9\,B\,a^2\,d\,e^4+21\,A\,a^2\,e^5+6\,B\,a\,c\,d^3\,e^2+6\,A\,a\,c\,d^2\,e^3+7\,B\,c^2\,d^5+3\,A\,c^2\,d^4\,e\right )}{56\,e^6}+\frac {B\,c^3\,x^7}{3\,e}}{d^{10}+10\,d^9\,e\,x+45\,d^8\,e^2\,x^2+120\,d^7\,e^3\,x^3+210\,d^6\,e^4\,x^4+252\,d^5\,e^5\,x^5+210\,d^4\,e^6\,x^6+120\,d^3\,e^7\,x^7+45\,d^2\,e^8\,x^8+10\,d\,e^9\,x^9+e^{10}\,x^{10}} \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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