Optimal. Leaf size=43 \[ -\frac {2 \left (a-\frac {c d^2}{e^2}\right )}{7 (d+e x)^{7/2}}-\frac {2 c d}{5 e^2 (d+e x)^{5/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 43, 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} \[ -\frac {2 \left (a-\frac {c d^2}{e^2}\right )}{7 (d+e x)^{7/2}}-\frac {2 c d}{5 e^2 (d+e x)^{5/2}} \]
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)^{11/2}} \, dx &=\frac {\int \frac {a e^3+c d e^2 x}{(d+e x)^{9/2}} \, dx}{e^2}\\ &=\frac {\int \left (\frac {-c d^2 e+a e^3}{(d+e x)^{9/2}}+\frac {c d e}{(d+e x)^{7/2}}\right ) \, dx}{e^2}\\ &=-\frac {2 \left (a-\frac {c d^2}{e^2}\right )}{7 (d+e x)^{7/2}}-\frac {2 c d}{5 e^2 (d+e x)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 34, normalized size = 0.79 \[ -\frac {2 \left (5 a e^2+c d (2 d+7 e x)\right )}{35 e^2 (d+e x)^{7/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.92, size = 74, normalized size = 1.72 \[ -\frac {2 \, {\left (7 \, c d e x + 2 \, c d^{2} + 5 \, a e^{2}\right )} \sqrt {e x + d}}{35 \, {\left (e^{6} x^{4} + 4 \, d e^{5} x^{3} + 6 \, d^{2} e^{4} x^{2} + 4 \, d^{3} e^{3} x + d^{4} e^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 48, normalized size = 1.12 \[ -\frac {2 \, {\left (7 \, {\left (x e + d\right )}^{2} c d - 5 \, {\left (x e + d\right )} c d^{2} + 5 \, {\left (x e + d\right )} a e^{2}\right )} e^{\left (-2\right )}}{35 \, {\left (x e + d\right )}^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 32, normalized size = 0.74 \[ -\frac {2 \left (7 c d e x +5 a \,e^{2}+2 c \,d^{2}\right )}{35 \left (e x +d \right )^{\frac {7}{2}} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.18, size = 34, normalized size = 0.79 \[ -\frac {2 \, {\left (7 \, {\left (e x + d\right )} c d - 5 \, c d^{2} + 5 \, a e^{2}\right )}}{35 \, {\left (e x + d\right )}^{\frac {7}{2}} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.59, size = 31, normalized size = 0.72 \[ -\frac {4\,c\,d^2+14\,c\,x\,d\,e+10\,a\,e^2}{35\,e^2\,{\left (d+e\,x\right )}^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 13.21, size = 248, normalized size = 5.77 \[ \begin {cases} - \frac {10 a e^{2}}{35 d^{3} e^{2} \sqrt {d + e x} + 105 d^{2} e^{3} x \sqrt {d + e x} + 105 d e^{4} x^{2} \sqrt {d + e x} + 35 e^{5} x^{3} \sqrt {d + e x}} - \frac {4 c d^{2}}{35 d^{3} e^{2} \sqrt {d + e x} + 105 d^{2} e^{3} x \sqrt {d + e x} + 105 d e^{4} x^{2} \sqrt {d + e x} + 35 e^{5} x^{3} \sqrt {d + e x}} - \frac {14 c d e x}{35 d^{3} e^{2} \sqrt {d + e x} + 105 d^{2} e^{3} x \sqrt {d + e x} + 105 d e^{4} x^{2} \sqrt {d + e x} + 35 e^{5} x^{3} \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {c x^{2}}{2 d^{\frac {7}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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