Optimal. Leaf size=42 \[ \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2} (d+e x)^{m+1}}{e (m+2)} \]
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Rubi [A] time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {644, 32} \[ \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2} (d+e x)^{m+1}}{e (m+2)} \]
Antiderivative was successfully verified.
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Rule 32
Rule 644
Rubi steps
\begin {align*} \int (d+e x)^m \sqrt {c d^2+2 c d e x+c e^2 x^2} \, dx &=\frac {\sqrt {c d^2+2 c d e x+c e^2 x^2} \int (d+e x)^{1+m} \, dx}{d+e x}\\ &=\frac {(d+e x)^{1+m} \sqrt {c d^2+2 c d e x+c e^2 x^2}}{e (2+m)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 31, normalized size = 0.74 \[ \frac {\sqrt {c (d+e x)^2} (d+e x)^{m+1}}{e (m+2)} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.30, size = 44, normalized size = 1.05 \[ \frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} {\left (e x + d\right )} {\left (e x + d\right )}^{m}}{e m + 2 \, e} \]
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 \[ \int \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} {\left (e x + d\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 41, normalized size = 0.98 \[ \frac {\sqrt {c \,e^{2} x^{2}+2 c d e x +c \,d^{2}}\, \left (e x +d \right )^{m +1}}{\left (m +2\right ) e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 42, normalized size = 1.00 \[ \frac {{\left (\sqrt {c} e^{2} x^{2} + 2 \, \sqrt {c} d e x + \sqrt {c} d^{2}\right )} {\left (e x + d\right )}^{m}}{e {\left (m + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (d+e\,x\right )}^m\,\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2} \,d x \]
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 \[ \int \sqrt {c \left (d + e x\right )^{2}} \left (d + e x\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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