Optimal. Leaf size=34 \[ \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{7 c e} \]
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Rubi [A] time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {629} \begin {gather*} \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{7 c e} \end {gather*}
Antiderivative was successfully verified.
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Rule 629
Rubi steps
\begin {align*} \int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx &=\frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{7 c e}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 23, normalized size = 0.68 \begin {gather*} \frac {\left (c (d+e x)^2\right )^{7/2}}{7 c e} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 23, normalized size = 0.68 \begin {gather*} \frac {\left (c (d+e x)^2\right )^{7/2}}{7 c e} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 117, normalized size = 3.44 \begin {gather*} \frac {{\left (c^{2} e^{6} x^{7} + 7 \, c^{2} d e^{5} x^{6} + 21 \, c^{2} d^{2} e^{4} x^{5} + 35 \, c^{2} d^{3} e^{3} x^{4} + 35 \, c^{2} d^{4} e^{2} x^{3} + 21 \, c^{2} d^{5} e x^{2} + 7 \, c^{2} d^{6} x\right )} \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{7 \, {\left (e x + d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 102, normalized size = 3.00 \begin {gather*} \frac {1}{7} \, {\left (c^{2} d^{6} e^{\left (-1\right )} + {\left (6 \, c^{2} d^{5} + {\left (15 \, c^{2} d^{4} e + {\left (20 \, c^{2} d^{3} e^{2} + {\left (15 \, c^{2} d^{2} e^{3} + {\left (c^{2} x e^{5} + 6 \, c^{2} d e^{4}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 95, normalized size = 2.79 \begin {gather*} \frac {\left (e^{6} x^{6}+7 e^{5} x^{5} d +21 e^{4} x^{4} d^{2}+35 d^{3} e^{3} x^{3}+35 d^{4} e^{2} x^{2}+21 d^{5} e x +7 d^{6}\right ) \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}} x}{7 \left (e x +d \right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.41, size = 30, normalized size = 0.88 \begin {gather*} \frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {7}{2}}}{7 \, c e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.65, size = 19, normalized size = 0.56 \begin {gather*} \frac {{\left (c\,{\left (d+e\,x\right )}^2\right )}^{7/2}}{7\,c\,e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.71, size = 287, normalized size = 8.44 \begin {gather*} \begin {cases} \frac {c^{2} d^{6} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7 e} + \frac {6 c^{2} d^{5} x \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac {15 c^{2} d^{4} e x^{2} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac {20 c^{2} d^{3} e^{2} x^{3} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac {15 c^{2} d^{2} e^{3} x^{4} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac {6 c^{2} d e^{4} x^{5} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac {c^{2} e^{5} x^{6} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} & \text {for}\: e \neq 0 \\d x \left (c d^{2}\right )^{\frac {5}{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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