Optimal. Leaf size=206 \[ \frac {2 c^2 d^2 (d+e x)^{-2 (p+1)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+1) (p+2) (p+3) \left (c d^2-a e^2\right )^3}+\frac {2 c d (d+e x)^{-2 p-3} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+2) (p+3) \left (c d^2-a e^2\right )^2}+\frac {(d+e x)^{-2 (p+2)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+3) \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.09, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {658, 650} \[ \frac {2 c^2 d^2 (d+e x)^{-2 (p+1)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+1) (p+2) (p+3) \left (c d^2-a e^2\right )^3}+\frac {2 c d (d+e x)^{-2 p-3} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+2) (p+3) \left (c d^2-a e^2\right )^2}+\frac {(d+e x)^{-2 (p+2)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+3) \left (c d^2-a e^2\right )} \]
Antiderivative was successfully verified.
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Rule 650
Rule 658
Rubi steps
\begin {align*} \int (d+e x)^{-4-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx &=\frac {(d+e x)^{-2 (2+p)} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p}}{\left (c d^2-a e^2\right ) (3+p)}+\frac {(2 c d) \int (d+e x)^{-3-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx}{\left (c d^2-a e^2\right ) (3+p)}\\ &=\frac {2 c d (d+e x)^{-3-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p}}{\left (c d^2-a e^2\right )^2 (2+p) (3+p)}+\frac {(d+e x)^{-2 (2+p)} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p}}{\left (c d^2-a e^2\right ) (3+p)}+\frac {\left (2 c^2 d^2\right ) \int (d+e x)^{-2-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx}{\left (c d^2-a e^2\right )^2 (2+p) (3+p)}\\ &=\frac {2 c d (d+e x)^{-3-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p}}{\left (c d^2-a e^2\right )^2 (2+p) (3+p)}+\frac {2 c^2 d^2 (d+e x)^{-2 (1+p)} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p}}{\left (c d^2-a e^2\right )^3 (1+p) (2+p) (3+p)}+\frac {(d+e x)^{-2 (2+p)} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p}}{\left (c d^2-a e^2\right ) (3+p)}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 131, normalized size = 0.64 \[ \frac {(d+e x)^{-2 (p+2)} ((d+e x) (a e+c d x))^{p+1} \left (a^2 e^4 \left (p^2+3 p+2\right )-2 a c d e^2 (p+1) (d (p+3)+e x)+c^2 d^2 \left (d^2 \left (p^2+5 p+6\right )+2 d e (p+3) x+2 e^2 x^2\right )\right )}{(p+1) (p+2) (p+3) \left (c d^2-a e^2\right )^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.88, size = 580, normalized size = 2.82 \[ \frac {{\left (2 \, c^{3} d^{3} e^{3} x^{4} + 6 \, a c^{2} d^{5} e - 6 \, a^{2} c d^{3} e^{3} + 2 \, a^{3} d e^{5} + 2 \, {\left (4 \, c^{3} d^{4} e^{2} + {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} p\right )} x^{3} + {\left (a c^{2} d^{5} e - 2 \, a^{2} c d^{3} e^{3} + a^{3} d e^{5}\right )} p^{2} + {\left (12 \, c^{3} d^{5} e + {\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} p^{2} + {\left (7 \, c^{3} d^{5} e - 8 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} p\right )} x^{2} + {\left (5 \, a c^{2} d^{5} e - 8 \, a^{2} c d^{3} e^{3} + 3 \, a^{3} d e^{5}\right )} p + {\left (6 \, c^{3} d^{6} + 6 \, a c^{2} d^{4} e^{2} - 6 \, a^{2} c d^{2} e^{4} + 2 \, a^{3} e^{6} + {\left (c^{3} d^{6} - a c^{2} d^{4} e^{2} - a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} p^{2} + {\left (5 \, c^{3} d^{6} - a c^{2} d^{4} e^{2} - 7 \, a^{2} c d^{2} e^{4} + 3 \, a^{3} e^{6}\right )} p\right )} x\right )} {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 4}}{6 \, c^{3} d^{6} - 18 \, a c^{2} d^{4} e^{2} + 18 \, a^{2} c d^{2} e^{4} - 6 \, a^{3} e^{6} + {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} p^{3} + 6 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} p^{2} + 11 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} p} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 381, normalized size = 1.85 \[ -\frac {\left (c d x +a e \right ) \left (a^{2} e^{4} p^{2}-2 a c \,d^{2} e^{2} p^{2}-2 a c d \,e^{3} p x +c^{2} d^{4} p^{2}+2 c^{2} d^{3} e p x +2 c^{2} d^{2} e^{2} x^{2}+3 a^{2} e^{4} p -8 a c \,d^{2} e^{2} p -2 a c d \,e^{3} x +5 c^{2} d^{4} p +6 c^{2} d^{3} e x +2 a^{2} e^{4}-6 a c \,d^{2} e^{2}+6 c^{2} d^{4}\right ) \left (e x +d \right )^{-2 p -3} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{p}}{a^{3} e^{6} p^{3}-3 a^{2} c \,d^{2} e^{4} p^{3}+3 a \,c^{2} d^{4} e^{2} p^{3}-c^{3} d^{6} p^{3}+6 a^{3} e^{6} p^{2}-18 a^{2} c \,d^{2} e^{4} p^{2}+18 a \,c^{2} d^{4} e^{2} p^{2}-6 c^{3} d^{6} p^{2}+11 a^{3} e^{6} p -33 a^{2} c \,d^{2} e^{4} p +33 a \,c^{2} d^{4} e^{2} p -11 c^{3} d^{6} p +6 a^{3} e^{6}-18 a^{2} c \,d^{2} e^{4}+18 a \,c^{2} d^{4} e^{2}-6 c^{3} d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.36, size = 595, normalized size = 2.89 \[ -{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^p\,\left (\frac {x\,\left (a^3\,e^6\,p^2+3\,a^3\,e^6\,p+2\,a^3\,e^6-a^2\,c\,d^2\,e^4\,p^2-7\,a^2\,c\,d^2\,e^4\,p-6\,a^2\,c\,d^2\,e^4-a\,c^2\,d^4\,e^2\,p^2-a\,c^2\,d^4\,e^2\,p+6\,a\,c^2\,d^4\,e^2+c^3\,d^6\,p^2+5\,c^3\,d^6\,p+6\,c^3\,d^6\right )}{{\left (a\,e^2-c\,d^2\right )}^3\,{\left (d+e\,x\right )}^{2\,p+4}\,\left (p^3+6\,p^2+11\,p+6\right )}+\frac {2\,c^3\,d^3\,e^3\,x^4}{{\left (a\,e^2-c\,d^2\right )}^3\,{\left (d+e\,x\right )}^{2\,p+4}\,\left (p^3+6\,p^2+11\,p+6\right )}+\frac {a\,d\,e\,\left (a^2\,e^4\,p^2+3\,a^2\,e^4\,p+2\,a^2\,e^4-2\,a\,c\,d^2\,e^2\,p^2-8\,a\,c\,d^2\,e^2\,p-6\,a\,c\,d^2\,e^2+c^2\,d^4\,p^2+5\,c^2\,d^4\,p+6\,c^2\,d^4\right )}{{\left (a\,e^2-c\,d^2\right )}^3\,{\left (d+e\,x\right )}^{2\,p+4}\,\left (p^3+6\,p^2+11\,p+6\right )}+\frac {c\,d\,e\,x^2\,\left (a^2\,e^4\,p^2+a^2\,e^4\,p-2\,a\,c\,d^2\,e^2\,p^2-8\,a\,c\,d^2\,e^2\,p+c^2\,d^4\,p^2+7\,c^2\,d^4\,p+12\,c^2\,d^4\right )}{{\left (a\,e^2-c\,d^2\right )}^3\,{\left (d+e\,x\right )}^{2\,p+4}\,\left (p^3+6\,p^2+11\,p+6\right )}+\frac {2\,c^2\,d^2\,e^2\,x^3\,\left (4\,c\,d^2-a\,e^2\,p+c\,d^2\,p\right )}{{\left (a\,e^2-c\,d^2\right )}^3\,{\left (d+e\,x\right )}^{2\,p+4}\,\left (p^3+6\,p^2+11\,p+6\right )}\right ) \]
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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