Optimal. Leaf size=43 \[ \frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{p+1}}{c e (2 p+3)} \]
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Rubi [A] time = 0.03, antiderivative size = 43, 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 {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{p+1}}{c e (2 p+3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 609
Rule 642
Rubi steps
\begin {align*} \int (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^p \, dx &=\frac {\int \left (c d^2+2 c d e x+c e^2 x^2\right )^{1+p} \, dx}{c}\\ &=\frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{1+p}}{c e (3+2 p)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 32, normalized size = 0.74 \begin {gather*} \frac {(d+e x) \left (c (d+e x)^2\right )^{p+1}}{c e (2 p+3)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.13, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^p \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.42, size = 60, normalized size = 1.40 \begin {gather*} \frac {{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} {\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}}{2 \, e p + 3 \, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 128, normalized size = 2.98 \begin {gather*} \frac {{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{p} x^{3} e^{3} + 3 \, {\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{p} d x^{2} e^{2} + 3 \, {\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{p} d^{2} x e + {\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{p} d^{3}}{2 \, p e + 3 \, e} \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 = 0.95 \begin {gather*} \frac {\left (e x +d \right )^{3} \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{p}}{\left (2 p +3\right ) e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.63, size = 182, normalized size = 4.23 \begin {gather*} \frac {{\left (c^{p} e x + c^{p} d\right )} {\left (e x + d\right )}^{2 \, p} d^{2}}{e {\left (2 \, p + 1\right )}} + \frac {{\left (c^{p} e^{2} {\left (2 \, p + 1\right )} x^{2} + 2 \, c^{p} d e p x - c^{p} d^{2}\right )} {\left (e x + d\right )}^{2 \, p} d}{{\left (2 \, p^{2} + 3 \, p + 1\right )} e} + \frac {{\left ({\left (2 \, p^{2} + 3 \, p + 1\right )} c^{p} e^{3} x^{3} + {\left (2 \, p^{2} + p\right )} c^{p} d e^{2} x^{2} - 2 \, c^{p} d^{2} e p x + c^{p} d^{3}\right )} {\left (e x + d\right )}^{2 \, p}}{{\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 79, normalized size = 1.84 \begin {gather*} {\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^p\,\left (\frac {3\,d^2\,x}{2\,p+3}+\frac {d^3}{e\,\left (2\,p+3\right )}+\frac {e^2\,x^3}{2\,p+3}+\frac {3\,d\,e\,x^2}{2\,p+3}\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} \frac {d^{2} x}{\left (c d^{2}\right )^{\frac {3}{2}}} & \text {for}\: e = 0 \wedge p = - \frac {3}{2} \\d^{2} x \left (c d^{2}\right )^{p} & \text {for}\: e = 0 \\\int \frac {\left (d + e x\right )^{2}}{\left (c \left (d + e x\right )^{2}\right )^{\frac {3}{2}}}\, dx & \text {for}\: p = - \frac {3}{2} \\\frac {d^{3} \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p + 3 e} + \frac {3 d^{2} e x \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p + 3 e} + \frac {3 d e^{2} x^{2} \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p + 3 e} + \frac {e^{3} x^{3} \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p + 3 e} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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