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 = 0.01, 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, 45}
\begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Rule 24
Rule 45
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 = 0.03, size = 34, normalized size = 0.79 \begin {gather*} -\frac {2 \left (5 a e^2+c d (2 d+7 e x)\right )}{35 e^2 (d+e x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.46, size = 39, normalized size = 0.91
method | result | size |
gosper | \(-\frac {2 \left (7 c d e x +5 e^{2} a +2 c \,d^{2}\right )}{35 \left (e x +d \right )^{\frac {7}{2}} e^{2}}\) | \(32\) |
trager | \(-\frac {2 \left (7 c d e x +5 e^{2} a +2 c \,d^{2}\right )}{35 \left (e x +d \right )^{\frac {7}{2}} e^{2}}\) | \(32\) |
derivativedivides | \(\frac {-\frac {2 c d}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 \left (e^{2} a -c \,d^{2}\right )}{7 \left (e x +d \right )^{\frac {7}{2}}}}{e^{2}}\) | \(39\) |
default | \(\frac {-\frac {2 c d}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 \left (e^{2} a -c \,d^{2}\right )}{7 \left (e x +d \right )^{\frac {7}{2}}}}{e^{2}}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 34, normalized size = 0.79 \begin {gather*} -\frac {2 \, {\left (7 \, {\left (x e + d\right )} c d - 5 \, c d^{2} + 5 \, a e^{2}\right )} e^{\left (-2\right )}}{35 \, {\left (x e + d\right )}^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.83, size = 70, normalized size = 1.63 \begin {gather*} -\frac {2 \, {\left (7 \, c d x e + 2 \, c d^{2} + 5 \, a e^{2}\right )} \sqrt {x e + d}}{35 \, {\left (x^{4} e^{6} + 4 \, d x^{3} e^{5} + 6 \, d^{2} x^{2} e^{4} + 4 \, d^{3} x e^{3} + d^{4} e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 248 vs.
\(2 (42) = 84\).
time = 1.52, size = 248, normalized size = 5.77 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.27, size = 34, normalized size = 0.79 \begin {gather*} -\frac {2 \, {\left (7 \, {\left (x e + d\right )} c d - 5 \, c d^{2} + 5 \, a e^{2}\right )} e^{\left (-2\right )}}{35 \, {\left (x e + d\right )}^{\frac {7}{2}}} \end {gather*}
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 \begin {gather*} -\frac {4\,c\,d^2+14\,c\,x\,d\,e+10\,a\,e^2}{35\,e^2\,{\left (d+e\,x\right )}^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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