Optimal. Leaf size=42 \[ -\frac {c (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{p-1}}{e (1-2 p)} \]
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Rubi [A] time = 0.03, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {642, 609} \begin {gather*} -\frac {c (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{p-1}}{e (1-2 p)} \end {gather*}
Antiderivative was successfully verified.
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Rule 609
Rule 642
Rubi steps
\begin {align*} \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^p}{(d+e x)^2} \, dx &=c \int \left (c d^2+2 c d e x+c e^2 x^2\right )^{-1+p} \, dx\\ &=-\frac {c (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{-1+p}}{e (1-2 p)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 30, normalized size = 0.71 \begin {gather*} \frac {c (d+e x) \left (c (d+e x)^2\right )^{p-1}}{e (2 p-1)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.16, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^p}{(d+e x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.42, size = 49, normalized size = 1.17 \begin {gather*} \frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}}{2 \, d e p - d e + {\left (2 \, e^{2} p - e^{2}\right )} x} \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 \frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}}{{\left (e x + d\right )}^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 41, normalized size = 0.98 \begin {gather*} \frac {\left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{p}}{\left (e x +d \right ) \left (2 p -1\right ) e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.49, size = 34, normalized size = 0.81 \begin {gather*} \frac {{\left (e x + d\right )}^{2 \, p} c^{p}}{e^{2} {\left (2 \, p - 1\right )} x + d e {\left (2 \, p - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.46, size = 42, normalized size = 1.00 \begin {gather*} \frac {{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^p}{e^2\,\left (2\,p-1\right )\,\left (x+\frac {d}{e}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} \text {NaN} & \text {for}\: d = 0 \wedge e = 0 \wedge p = \frac {1}{2} \\0^{p} \tilde {\infty } x & \text {for}\: d = - e x \\\frac {x \left (c d^{2}\right )^{p}}{d^{2}} & \text {for}\: e = 0 \\\int \frac {\sqrt {c \left (d + e x\right )^{2}}}{\left (d + e x\right )^{2}}\, dx & \text {for}\: p = \frac {1}{2} \\\frac {\left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 d e p - d e + 2 e^{2} p x - e^{2} x} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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