Optimal. Leaf size=128 \[ \frac {(d+e x)^{-2 p-3} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+2) \left (c d^2-a e^2\right )}+\frac {c d (d+e x)^{-2 (p+1)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+1) (p+2) \left (c d^2-a e^2\right )^2} \]
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Rubi [A] time = 0.05, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {658, 650} \begin {gather*} \frac {(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) \left (c d^2-a e^2\right )}+\frac {c d (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) \left (c d^2-a e^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 650
Rule 658
Rubi steps
\begin {align*} \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 &=\frac {(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+p)}+\frac {(c d) \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+p)}\\ &=\frac {(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+p)}+\frac {c d (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 )^2 (1+p) (2+p)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 76, normalized size = 0.59 \begin {gather*} \frac {(d+e x)^{-2 p-3} ((d+e x) (a e+c d x))^{p+1} \left (c d (d (p+2)+e x)-a e^2 (p+1)\right )}{(p+1) (p+2) \left (c d^2-a e^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.18, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.44, size = 255, normalized size = 1.99 \begin {gather*} \frac {{\left (c^{2} d^{2} e^{2} x^{3} + 2 \, a c d^{3} e - a^{2} d e^{3} + {\left (3 \, c^{2} d^{3} e + {\left (c^{2} d^{3} e - a c d e^{3}\right )} p\right )} x^{2} + {\left (a c d^{3} e - a^{2} d e^{3}\right )} p + {\left (2 \, c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4} + {\left (c^{2} d^{4} - a^{2} e^{4}\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 - 3}}{2 \, c^{2} d^{4} - 4 \, a c d^{2} e^{2} + 2 \, a^{2} e^{4} + {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} p^{2} + 3 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} p} \end {gather*}
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 \begin {gather*} \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 - 3}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 170, normalized size = 1.33 \begin {gather*} -\frac {\left (c d x +a e \right ) \left (a \,e^{2} p -c \,d^{2} p -c d e x +a \,e^{2}-2 c \,d^{2}\right ) \left (e x +d \right )^{-2 p -2} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{p}}{a^{2} e^{4} p^{2}-2 a c \,d^{2} e^{2} p^{2}+c^{2} d^{4} p^{2}+3 a^{2} e^{4} p -6 a c \,d^{2} e^{2} p +3 c^{2} d^{4} p +2 a^{2} e^{4}-4 a c \,d^{2} e^{2}+2 c^{2} d^{4}} \end {gather*}
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 \begin {gather*} \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 - 3}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.98, size = 293, normalized size = 2.29 \begin {gather*} {\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^p\,\left (\frac {x\,\left (2\,c^2\,d^4-a^2\,e^4-a^2\,e^4\,p+c^2\,d^4\,p+2\,a\,c\,d^2\,e^2\right )}{{\left (a\,e^2-c\,d^2\right )}^2\,{\left (d+e\,x\right )}^{2\,p+3}\,\left (p^2+3\,p+2\right )}+\frac {c^2\,d^2\,e^2\,x^3}{{\left (a\,e^2-c\,d^2\right )}^2\,{\left (d+e\,x\right )}^{2\,p+3}\,\left (p^2+3\,p+2\right )}-\frac {a\,d\,e\,\left (a\,e^2-2\,c\,d^2+a\,e^2\,p-c\,d^2\,p\right )}{{\left (a\,e^2-c\,d^2\right )}^2\,{\left (d+e\,x\right )}^{2\,p+3}\,\left (p^2+3\,p+2\right )}+\frac {c\,d\,e\,x^2\,\left (3\,c\,d^2-a\,e^2\,p+c\,d^2\,p\right )}{{\left (a\,e^2-c\,d^2\right )}^2\,{\left (d+e\,x\right )}^{2\,p+3}\,\left (p^2+3\,p+2\right )}\right ) \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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