Optimal. Leaf size=39 \[ \frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{8 c e} \]
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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} \begin {gather*} \frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{8 c e} \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 )^{5/2} \, dx &=\frac {\int \left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2} \, dx}{c}\\ &=\frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{8 c e}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 28, normalized size = 0.72 \begin {gather*} \frac {(d+e x) \left (c (d+e x)^2\right )^{7/2}}{8 c e} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.99, 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 )^{5/2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.39, size = 131, normalized size = 3.36 \begin {gather*} \frac {{\left (c^{2} e^{7} x^{8} + 8 \, c^{2} d e^{6} x^{7} + 28 \, c^{2} d^{2} e^{5} x^{6} + 56 \, c^{2} d^{3} e^{4} x^{5} + 70 \, c^{2} d^{4} e^{3} x^{4} + 56 \, c^{2} d^{5} e^{2} x^{3} + 28 \, c^{2} d^{6} e x^{2} + 8 \, c^{2} d^{7} x\right )} \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{8 \, {\left (e x + d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.31, size = 115, normalized size = 2.95 \begin {gather*} \frac {1}{8} \, {\left (c^{2} d^{7} e^{\left (-1\right )} + {\left (7 \, c^{2} d^{6} + {\left (21 \, c^{2} d^{5} e + {\left (35 \, c^{2} d^{4} e^{2} + {\left (35 \, c^{2} d^{3} e^{3} + {\left (21 \, c^{2} d^{2} e^{4} + {\left (c^{2} x e^{6} + 7 \, c^{2} d e^{5}\right )} x\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 = 0.04, size = 106, normalized size = 2.72 \begin {gather*} \frac {\left (e^{7} x^{7}+8 d \,e^{6} x^{6}+28 e^{5} x^{5} d^{2}+56 e^{4} x^{4} d^{3}+70 d^{4} e^{3} x^{3}+56 d^{5} e^{2} x^{2}+28 d^{6} e x +8 d^{7}\right ) \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}} x}{8 \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 = 1.48, size = 60, normalized size = 1.54 \begin {gather*} \frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {7}{2}} x}{8 \, c} + \frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {7}{2}} d}{8 \, c e} \end {gather*}
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 \begin {gather*} \int {\left (d+e\,x\right )}^2\,{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{5/2} \,d x \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*} \int \left (c \left (d + e x\right )^{2}\right )^{\frac {5}{2}} \left (d + e x\right )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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