Optimal. Leaf size=204 \[ -\frac {2 c \left (a B e^2-2 A c d e+5 B c d^2\right )}{3 e^6 (d+e x)^3}-\frac {\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{5 e^6 (d+e x)^5}+\frac {\left (a e^2+c d^2\right )^2 (B d-A e)}{6 e^6 (d+e x)^6}+\frac {c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{2 e^6 (d+e x)^4}+\frac {c^2 (5 B d-A e)}{2 e^6 (d+e x)^2}-\frac {B c^2}{e^6 (d+e x)} \]
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Rubi [A] time = 0.14, antiderivative size = 204, 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 {2 c \left (a B e^2-2 A c d e+5 B c d^2\right )}{3 e^6 (d+e x)^3}+\frac {c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{2 e^6 (d+e x)^4}-\frac {\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{5 e^6 (d+e x)^5}+\frac {\left (a e^2+c d^2\right )^2 (B d-A e)}{6 e^6 (d+e x)^6}+\frac {c^2 (5 B d-A e)}{2 e^6 (d+e x)^2}-\frac {B c^2}{e^6 (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 )^2}{(d+e x)^7} \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2+a e^2\right )^2}{e^5 (d+e x)^7}+\frac {\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{e^5 (d+e x)^6}+\frac {2 c \left (-5 B c d^3+3 A c d^2 e-3 a B d e^2+a A e^3\right )}{e^5 (d+e x)^5}-\frac {2 c \left (-5 B c d^2+2 A c d e-a B e^2\right )}{e^5 (d+e x)^4}+\frac {c^2 (-5 B d+A e)}{e^5 (d+e x)^3}+\frac {B c^2}{e^5 (d+e x)^2}\right ) \, dx\\ &=\frac {(B d-A e) \left (c d^2+a e^2\right )^2}{6 e^6 (d+e x)^6}-\frac {\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{5 e^6 (d+e x)^5}+\frac {c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right )}{2 e^6 (d+e x)^4}-\frac {2 c \left (5 B c d^2-2 A c d e+a B e^2\right )}{3 e^6 (d+e x)^3}+\frac {c^2 (5 B d-A e)}{2 e^6 (d+e x)^2}-\frac {B c^2}{e^6 (d+e x)}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 198, normalized size = 0.97 \begin {gather*} -\frac {A e \left (5 a^2 e^4+a c e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+c^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )\right )+B \left (a^2 e^4 (d+6 e x)+a c e^2 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+5 c^2 \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )\right )}{30 e^6 (d+e x)^6} \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 )^2}{(d+e x)^7} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 292, normalized size = 1.43 \begin {gather*} -\frac {30 \, B c^{2} e^{5} x^{5} + 5 \, B c^{2} d^{5} + A c^{2} d^{4} e + B a c d^{3} e^{2} + A a c d^{2} e^{3} + B a^{2} d e^{4} + 5 \, A a^{2} e^{5} + 15 \, {\left (5 \, B c^{2} d e^{4} + A c^{2} e^{5}\right )} x^{4} + 20 \, {\left (5 \, B c^{2} d^{2} e^{3} + A c^{2} d e^{4} + B a c e^{5}\right )} x^{3} + 15 \, {\left (5 \, B c^{2} d^{3} e^{2} + A c^{2} d^{2} e^{3} + B a c d e^{4} + A a c e^{5}\right )} x^{2} + 6 \, {\left (5 \, B c^{2} d^{4} e + A c^{2} d^{3} e^{2} + B a c d^{2} e^{3} + A a c d e^{4} + B a^{2} e^{5}\right )} x}{30 \, {\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 238, normalized size = 1.17 \begin {gather*} -\frac {{\left (30 \, B c^{2} x^{5} e^{5} + 75 \, B c^{2} d x^{4} e^{4} + 100 \, B c^{2} d^{2} x^{3} e^{3} + 75 \, B c^{2} d^{3} x^{2} e^{2} + 30 \, B c^{2} d^{4} x e + 5 \, B c^{2} d^{5} + 15 \, A c^{2} x^{4} e^{5} + 20 \, A c^{2} d x^{3} e^{4} + 15 \, A c^{2} d^{2} x^{2} e^{3} + 6 \, A c^{2} d^{3} x e^{2} + A c^{2} d^{4} e + 20 \, B a c x^{3} e^{5} + 15 \, B a c d x^{2} e^{4} + 6 \, B a c d^{2} x e^{3} + B a c d^{3} e^{2} + 15 \, A a c x^{2} e^{5} + 6 \, A a c d x e^{4} + A a c d^{2} e^{3} + 6 \, B a^{2} x e^{5} + B a^{2} d e^{4} + 5 \, A a^{2} e^{5}\right )} e^{\left (-6\right )}}{30 \, {\left (x e + d\right )}^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 249, normalized size = 1.22 \begin {gather*} -\frac {B \,c^{2}}{\left (e x +d \right ) e^{6}}-\frac {\left (A e -5 B d \right ) c^{2}}{2 \left (e x +d \right )^{2} e^{6}}-\frac {\left (a A \,e^{3}+3 A c \,d^{2} e -3 a B d \,e^{2}-5 B c \,d^{3}\right ) c}{2 \left (e x +d \right )^{4} e^{6}}+\frac {2 \left (2 A c d e -B a \,e^{2}-5 B c \,d^{2}\right ) c}{3 \left (e x +d \right )^{3} e^{6}}-\frac {A \,a^{2} e^{5}+2 A \,d^{2} a c \,e^{3}+A \,c^{2} d^{4} e -B d \,a^{2} e^{4}-2 B \,d^{3} a c \,e^{2}-B \,d^{5} c^{2}}{6 \left (e x +d \right )^{6} e^{6}}-\frac {-4 A d a c \,e^{3}-4 A \,c^{2} d^{3} e +B \,a^{2} e^{4}+6 B \,d^{2} a c \,e^{2}+5 B \,d^{4} c^{2}}{5 \left (e x +d \right )^{5} e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 292, normalized size = 1.43 \begin {gather*} -\frac {30 \, B c^{2} e^{5} x^{5} + 5 \, B c^{2} d^{5} + A c^{2} d^{4} e + B a c d^{3} e^{2} + A a c d^{2} e^{3} + B a^{2} d e^{4} + 5 \, A a^{2} e^{5} + 15 \, {\left (5 \, B c^{2} d e^{4} + A c^{2} e^{5}\right )} x^{4} + 20 \, {\left (5 \, B c^{2} d^{2} e^{3} + A c^{2} d e^{4} + B a c e^{5}\right )} x^{3} + 15 \, {\left (5 \, B c^{2} d^{3} e^{2} + A c^{2} d^{2} e^{3} + B a c d e^{4} + A a c e^{5}\right )} x^{2} + 6 \, {\left (5 \, B c^{2} d^{4} e + A c^{2} d^{3} e^{2} + B a c d^{2} e^{3} + A a c d e^{4} + B a^{2} e^{5}\right )} x}{30 \, {\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 273, normalized size = 1.34 \begin {gather*} -\frac {\frac {B\,a^2\,d\,e^4+5\,A\,a^2\,e^5+B\,a\,c\,d^3\,e^2+A\,a\,c\,d^2\,e^3+5\,B\,c^2\,d^5+A\,c^2\,d^4\,e}{30\,e^6}+\frac {x\,\left (B\,a^2\,e^4+B\,a\,c\,d^2\,e^2+A\,a\,c\,d\,e^3+5\,B\,c^2\,d^4+A\,c^2\,d^3\,e\right )}{5\,e^5}+\frac {2\,c\,x^3\,\left (5\,B\,c\,d^2+A\,c\,d\,e+B\,a\,e^2\right )}{3\,e^3}+\frac {c^2\,x^4\,\left (A\,e+5\,B\,d\right )}{2\,e^2}+\frac {c\,x^2\,\left (5\,B\,c\,d^3+A\,c\,d^2\,e+B\,a\,d\,e^2+A\,a\,e^3\right )}{2\,e^4}+\frac {B\,c^2\,x^5}{e}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 125.89, size = 337, normalized size = 1.65 \begin {gather*} \frac {- 5 A a^{2} e^{5} - A a c d^{2} e^{3} - A c^{2} d^{4} e - B a^{2} d e^{4} - B a c d^{3} e^{2} - 5 B c^{2} d^{5} - 30 B c^{2} e^{5} x^{5} + x^{4} \left (- 15 A c^{2} e^{5} - 75 B c^{2} d e^{4}\right ) + x^{3} \left (- 20 A c^{2} d e^{4} - 20 B a c e^{5} - 100 B c^{2} d^{2} e^{3}\right ) + x^{2} \left (- 15 A a c e^{5} - 15 A c^{2} d^{2} e^{3} - 15 B a c d e^{4} - 75 B c^{2} d^{3} e^{2}\right ) + x \left (- 6 A a c d e^{4} - 6 A c^{2} d^{3} e^{2} - 6 B a^{2} e^{5} - 6 B a c d^{2} e^{3} - 30 B c^{2} d^{4} e\right )}{30 d^{6} e^{6} + 180 d^{5} e^{7} x + 450 d^{4} e^{8} x^{2} + 600 d^{3} e^{9} x^{3} + 450 d^{2} e^{10} x^{4} + 180 d e^{11} x^{5} + 30 e^{12} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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