Optimal. Leaf size=79 \[ -\frac {2 \left (c d^2-a e^2\right )^2}{3 e^3 (d+e x)^{3/2}}+\frac {4 c d \left (c d^2-a e^2\right )}{e^3 \sqrt {d+e x}}+\frac {2 c^2 d^2 \sqrt {d+e x}}{e^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {640, 45}
\begin {gather*} \frac {4 c d \left (c d^2-a e^2\right )}{e^3 \sqrt {d+e x}}-\frac {2 \left (c d^2-a e^2\right )^2}{3 e^3 (d+e x)^{3/2}}+\frac {2 c^2 d^2 \sqrt {d+e x}}{e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 640
Rubi steps
\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^{9/2}} \, dx &=\int \frac {(a e+c d x)^2}{(d+e x)^{5/2}} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^2}{e^2 (d+e x)^{5/2}}-\frac {2 c d \left (c d^2-a e^2\right )}{e^2 (d+e x)^{3/2}}+\frac {c^2 d^2}{e^2 \sqrt {d+e x}}\right ) \, dx\\ &=-\frac {2 \left (c d^2-a e^2\right )^2}{3 e^3 (d+e x)^{3/2}}+\frac {4 c d \left (c d^2-a e^2\right )}{e^3 \sqrt {d+e x}}+\frac {2 c^2 d^2 \sqrt {d+e x}}{e^3}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 68, normalized size = 0.86 \begin {gather*} \frac {-2 a^2 e^4-4 a c d e^2 (2 d+3 e x)+2 c^2 d^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )}{3 e^3 (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.72, size = 78, normalized size = 0.99
method | result | size |
risch | \(\frac {2 c^{2} d^{2} \sqrt {e x +d}}{e^{3}}-\frac {2 \left (6 c d e x +e^{2} a +5 c \,d^{2}\right ) \left (e^{2} a -c \,d^{2}\right )}{3 e^{3} \left (e x +d \right )^{\frac {3}{2}}}\) | \(62\) |
gosper | \(-\frac {2 \left (-3 e^{2} x^{2} c^{2} d^{2}+6 a c d \,e^{3} x -12 c^{2} d^{3} e x +a^{2} e^{4}+4 a c \,d^{2} e^{2}-8 c^{2} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{3}}\) | \(72\) |
trager | \(-\frac {2 \left (-3 e^{2} x^{2} c^{2} d^{2}+6 a c d \,e^{3} x -12 c^{2} d^{3} e x +a^{2} e^{4}+4 a c \,d^{2} e^{2}-8 c^{2} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{3}}\) | \(72\) |
derivativedivides | \(\frac {2 c^{2} d^{2} \sqrt {e x +d}-\frac {2 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {4 c d \left (e^{2} a -c \,d^{2}\right )}{\sqrt {e x +d}}}{e^{3}}\) | \(78\) |
default | \(\frac {2 c^{2} d^{2} \sqrt {e x +d}-\frac {2 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {4 c d \left (e^{2} a -c \,d^{2}\right )}{\sqrt {e x +d}}}{e^{3}}\) | \(78\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 81, normalized size = 1.03 \begin {gather*} \frac {2}{3} \, {\left (3 \, \sqrt {x e + d} c^{2} d^{2} e^{\left (-2\right )} - \frac {{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4} - 6 \, {\left (c^{2} d^{3} - a c d e^{2}\right )} {\left (x e + d\right )}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{\frac {3}{2}}}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.34, size = 88, normalized size = 1.11 \begin {gather*} \frac {2 \, {\left (12 \, c^{2} d^{3} x e + 8 \, c^{2} d^{4} - 6 \, a c d x e^{3} - a^{2} e^{4} + {\left (3 \, c^{2} d^{2} x^{2} - 4 \, a c d^{2}\right )} e^{2}\right )} \sqrt {x e + d}}{3 \, {\left (x^{2} e^{5} + 2 \, d x e^{4} + d^{2} e^{3}\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. 264 vs.
\(2 (73) = 146\).
time = 1.05, size = 264, normalized size = 3.34 \begin {gather*} \begin {cases} - \frac {2 a^{2} e^{4}}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} - \frac {8 a c d^{2} e^{2}}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} - \frac {12 a c d e^{3} x}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} + \frac {16 c^{2} d^{4}}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} + \frac {24 c^{2} d^{3} e x}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} + \frac {6 c^{2} d^{2} e^{2} x^{2}}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {c^{2} x^{3}}{3 \sqrt {d}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.26, size = 83, normalized size = 1.05 \begin {gather*} 2 \, \sqrt {x e + d} c^{2} d^{2} e^{\left (-3\right )} + \frac {2 \, {\left (6 \, {\left (x e + d\right )} c^{2} d^{3} - c^{2} d^{4} - 6 \, {\left (x e + d\right )} a c d e^{2} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} e^{\left (-3\right )}}{3 \, {\left (x e + d\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.63, size = 80, normalized size = 1.01 \begin {gather*} -\frac {2\,a^2\,e^4+2\,c^2\,d^4-6\,c^2\,d^2\,{\left (d+e\,x\right )}^2-12\,c^2\,d^3\,\left (d+e\,x\right )-4\,a\,c\,d^2\,e^2+12\,a\,c\,d\,e^2\,\left (d+e\,x\right )}{3\,e^3\,{\left (d+e\,x\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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