Optimal. Leaf size=41 \[ -\frac {2 \left (a-\frac {c d^2}{e^2}\right )}{3 (d+e x)^{3/2}}-\frac {2 c d}{e^2 \sqrt {d+e x}} \]
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Rubi [A] time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {24, 43} \begin {gather*} -\frac {2 \left (a-\frac {c d^2}{e^2}\right )}{3 (d+e x)^{3/2}}-\frac {2 c d}{e^2 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 24
Rule 43
Rubi steps
\begin {align*} \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^{7/2}} \, dx &=\frac {\int \frac {a e^3+c d e^2 x}{(d+e x)^{5/2}} \, dx}{e^2}\\ &=\frac {\int \left (\frac {-c d^2 e+a e^3}{(d+e x)^{5/2}}+\frac {c d e}{(d+e x)^{3/2}}\right ) \, dx}{e^2}\\ &=-\frac {2 \left (a-\frac {c d^2}{e^2}\right )}{3 (d+e x)^{3/2}}-\frac {2 c d}{e^2 \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 0.80 \begin {gather*} -\frac {2 \left (a e^2+c d (2 d+3 e x)\right )}{3 e^2 (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 37, normalized size = 0.90 \begin {gather*} -\frac {2 \left (a e^2-c d^2+3 c d (d+e x)\right )}{3 e^2 (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 51, normalized size = 1.24 \begin {gather*} -\frac {2 \, {\left (3 \, c d e x + 2 \, c d^{2} + a e^{2}\right )} \sqrt {e x + d}}{3 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 47, normalized size = 1.15 \begin {gather*} -\frac {2 \, {\left (3 \, {\left (x e + d\right )}^{2} c d - {\left (x e + d\right )} c d^{2} + {\left (x e + d\right )} a e^{2}\right )} e^{\left (-2\right )}}{3 \, {\left (x e + d\right )}^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 31, normalized size = 0.76 \begin {gather*} -\frac {2 \left (3 c d e x +a \,e^{2}+2 c \,d^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.11, size = 33, normalized size = 0.80 \begin {gather*} -\frac {2 \, {\left (3 \, {\left (e x + d\right )} c d - c d^{2} + a e^{2}\right )}}{3 \, {\left (e x + d\right )}^{\frac {3}{2}} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.60, size = 34, normalized size = 0.83 \begin {gather*} -\frac {2\,a\,e^2-2\,c\,d^2+6\,c\,d\,\left (d+e\,x\right )}{3\,e^2\,{\left (d+e\,x\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.18, size = 126, normalized size = 3.07 \begin {gather*} \begin {cases} - \frac {2 a e^{2}}{3 d e^{2} \sqrt {d + e x} + 3 e^{3} x \sqrt {d + e x}} - \frac {4 c d^{2}}{3 d e^{2} \sqrt {d + e x} + 3 e^{3} x \sqrt {d + e x}} - \frac {6 c d e x}{3 d e^{2} \sqrt {d + e x} + 3 e^{3} x \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {c x^{2}}{2 d^{\frac {3}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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