Optimal. Leaf size=39 \[ \frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{6 c e} \]
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Rubi [A] time = 0.02, antiderivative size = 39, 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 = {642, 609} \[ \frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{6 c e} \]
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 )^{3/2} \, dx &=\frac {\int \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx}{c}\\ &=\frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{6 c e}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 28, normalized size = 0.72 \[ \frac {(d+e x) \left (c (d+e x)^2\right )^{5/2}}{6 c e} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.84, size = 91, normalized size = 2.33 \[ \frac {{\left (c e^{5} x^{6} + 6 \, c d e^{4} x^{5} + 15 \, c d^{2} e^{3} x^{4} + 20 \, c d^{3} e^{2} x^{3} + 15 \, c d^{4} e x^{2} + 6 \, c d^{5} x\right )} \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{6 \, {\left (e x + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 77, normalized size = 1.97 \[ \frac {1}{6} \, {\left (c d^{5} e^{\left (-1\right )} + {\left (5 \, c d^{4} + {\left (10 \, c d^{3} e + {\left (10 \, c d^{2} e^{2} + {\left (c x e^{4} + 5 \, c d e^{3}\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 84, normalized size = 2.15 \[ \frac {\left (e^{5} x^{5}+6 d \,e^{4} x^{4}+15 e^{3} x^{3} d^{2}+20 d^{3} e^{2} x^{2}+15 d^{4} e x +6 d^{5}\right ) \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}} x}{6 \left (e x +d \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.52, size = 60, normalized size = 1.54 \[ \frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {5}{2}} x}{6 \, c} + \frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {5}{2}} d}{6 \, c e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int {\left (d+e\,x\right )}^2\,{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{3/2} \,d x \]
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 \[ \int \left (c \left (d + e x\right )^{2}\right )^{\frac {3}{2}} \left (d + e x\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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