Optimal. Leaf size=39 \[ \frac {(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 c^3 e} \]
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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) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 c^3 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 609
Rule 642
Rubi steps
\begin {align*} \int \frac {(d+e x)^6}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx &=\frac {\int \sqrt {c d^2+2 c d e x+c e^2 x^2} \, dx}{c^3}\\ &=\frac {(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 c^3 e}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 33, normalized size = 0.85 \begin {gather*} \frac {x (d+e x) (2 d+e x)}{2 c^2 \sqrt {c (d+e x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.47, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^6}{\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 [A] time = 0.41, size = 48, normalized size = 1.23 \begin {gather*} \frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} {\left (e x^{2} + 2 \, d x\right )}}{2 \, {\left (c^{3} e x + c^{3} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.66, size = 108, normalized size = 2.77 \begin {gather*} -\frac {\frac {9 \, d^{5} e^{\left (-1\right )}}{c} - 4 \, C_{0} d^{3} e^{\left (-3\right )} - {\left (12 \, C_{0} d^{2} e^{\left (-2\right )} - \frac {25 \, d^{4}}{c} - {\left (\frac {20 \, d^{3} e}{c} - 12 \, C_{0} d e^{\left (-1\right )} - {\left (x {\left (\frac {x e^{4}}{c} + \frac {5 \, d e^{3}}{c}\right )} + 4 \, C_{0}\right )} x\right )} x\right )} x}{2 \, {\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 40, normalized size = 1.03 \begin {gather*} \frac {\left (e x +2 d \right ) \left (e x +d \right )^{5} x}{2 \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.73, size = 175, normalized size = 4.49 \begin {gather*} \frac {e^{4} x^{5}}{2 \, {\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {3}{2}} c} + \frac {5 \, d e^{3} x^{4}}{2 \, {\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {3}{2}} c} - \frac {10 \, d^{3} e x^{2}}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {3}{2}} c} - \frac {26 \, d^{5}}{3 \, {\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {3}{2}} c e} - \frac {25 \, d^{4}}{2 \, c^{\frac {5}{2}} e^{3} {\left (x + \frac {d}{e}\right )}^{2}} + \frac {50 \, d^{5}}{3 \, c^{\frac {5}{2}} e^{4} {\left (x + \frac {d}{e}\right )}^{3}} \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 \frac {{\left (d+e\,x\right )}^6}{{\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 \frac {\left (d + e x\right )^{6}}{\left (c \left (d + e x\right )^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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