Optimal. Leaf size=31 \[ \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{c^2 e} \]
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Rubi [A] time = 0.02, antiderivative size = 31, 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 = {643, 629} \begin {gather*} \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{c^2 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 629
Rule 643
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx &=\frac {\int \frac {d+e x}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx}{c}\\ &=\frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{c^2 e}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 0.71 \begin {gather*} \frac {x (d+e x)^3}{\left (c (d+e x)^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 20, normalized size = 0.65 \begin {gather*} \frac {\sqrt {c (d+e x)^2}}{c^2 e} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 38, normalized size = 1.23 \begin {gather*} \frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} x}{c^{2} e x + c^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 53, normalized size = 1.71 \begin {gather*} \frac {2 \, C_{0} d e^{\left (-1\right )} + {\left (2 \, C_{0} + \frac {x e}{c}\right )} x - \frac {d^{2} e^{\left (-1\right )}}{c}}{\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 [A] time = 0.04, size = 32, normalized size = 1.03 \begin {gather*} \frac {\left (e x +d \right )^{3} x}{\left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.45, size = 64, normalized size = 2.06 \begin {gather*} \frac {e x^{2}}{\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c} - \frac {d^{2}}{\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} 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 \frac {{\left (d+e\,x\right )}^3}{{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.80, size = 42, normalized size = 1.35 \begin {gather*} \begin {cases} \frac {\sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{c^{2} e} & \text {for}\: e \neq 0 \\\frac {d^{3} x}{\left (c d^{2}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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