Optimal. Leaf size=115 \[ \frac {2 \left (c d^2-a e^2\right )^3}{3 e^4 (d+e x)^{3/2}}-\frac {6 c d \left (c d^2-a e^2\right )^2}{e^4 \sqrt {d+e x}}-\frac {6 c^2 d^2 \left (c d^2-a e^2\right ) \sqrt {d+e x}}{e^4}+\frac {2 c^3 d^3 (d+e x)^{3/2}}{3 e^4} \]
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Rubi [A]
time = 0.04, antiderivative size = 115, 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 {6 c^2 d^2 \sqrt {d+e x} \left (c d^2-a e^2\right )}{e^4}-\frac {6 c d \left (c d^2-a e^2\right )^2}{e^4 \sqrt {d+e x}}+\frac {2 \left (c d^2-a e^2\right )^3}{3 e^4 (d+e x)^{3/2}}+\frac {2 c^3 d^3 (d+e x)^{3/2}}{3 e^4} \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 )^3}{(d+e x)^{11/2}} \, dx &=\int \frac {(a e+c d x)^3}{(d+e x)^{5/2}} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^3}{e^3 (d+e x)^{5/2}}+\frac {3 c d \left (c d^2-a e^2\right )^2}{e^3 (d+e x)^{3/2}}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right )}{e^3 \sqrt {d+e x}}+\frac {c^3 d^3 \sqrt {d+e x}}{e^3}\right ) \, dx\\ &=\frac {2 \left (c d^2-a e^2\right )^3}{3 e^4 (d+e x)^{3/2}}-\frac {6 c d \left (c d^2-a e^2\right )^2}{e^4 \sqrt {d+e x}}-\frac {6 c^2 d^2 \left (c d^2-a e^2\right ) \sqrt {d+e x}}{e^4}+\frac {2 c^3 d^3 (d+e x)^{3/2}}{3 e^4}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 110, normalized size = 0.96 \begin {gather*} -\frac {2 \left (a^3 e^6+3 a^2 c d e^4 (2 d+3 e x)-3 a c^2 d^2 e^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )+c^3 d^3 \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )\right )}{3 e^4 (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.72, size = 141, normalized size = 1.23
method | result | size |
risch | \(\frac {2 c^{2} d^{2} \left (c d e x +9 e^{2} a -8 c \,d^{2}\right ) \sqrt {e x +d}}{3 e^{4}}-\frac {2 \left (9 c d e x +e^{2} a +8 c \,d^{2}\right ) \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{3 e^{4} \left (e x +d \right )^{\frac {3}{2}}}\) | \(93\) |
gosper | \(-\frac {2 \left (-c^{3} d^{3} e^{3} x^{3}-9 a \,c^{2} d^{2} e^{4} x^{2}+6 c^{3} d^{4} e^{2} x^{2}+9 a^{2} c d \,e^{5} x -36 a \,c^{2} d^{3} e^{3} x +24 c^{3} d^{5} e x +e^{6} a^{3}+6 e^{4} d^{2} a^{2} c -24 d^{4} e^{2} c^{2} a +16 d^{6} c^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{4}}\) | \(130\) |
trager | \(-\frac {2 \left (-c^{3} d^{3} e^{3} x^{3}-9 a \,c^{2} d^{2} e^{4} x^{2}+6 c^{3} d^{4} e^{2} x^{2}+9 a^{2} c d \,e^{5} x -36 a \,c^{2} d^{3} e^{3} x +24 c^{3} d^{5} e x +e^{6} a^{3}+6 e^{4} d^{2} a^{2} c -24 d^{4} e^{2} c^{2} a +16 d^{6} c^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{4}}\) | \(130\) |
derivativedivides | \(\frac {\frac {2 c^{3} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+6 a \,c^{2} d^{2} e^{2} \sqrt {e x +d}-6 c^{3} d^{4} \sqrt {e x +d}-\frac {6 c d \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{\sqrt {e x +d}}-\frac {2 \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{4}}\) | \(141\) |
default | \(\frac {\frac {2 c^{3} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+6 a \,c^{2} d^{2} e^{2} \sqrt {e x +d}-6 c^{3} d^{4} \sqrt {e x +d}-\frac {6 c d \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{\sqrt {e x +d}}-\frac {2 \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{4}}\) | \(141\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 136, normalized size = 1.18 \begin {gather*} \frac {2}{3} \, {\left ({\left ({\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{3} - 9 \, {\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} \sqrt {x e + d}\right )} e^{\left (-3\right )} + \frac {{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} - 9 \, {\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} {\left (x e + d\right )}\right )} e^{\left (-3\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 = 2.81, size = 142, normalized size = 1.23 \begin {gather*} -\frac {2 \, {\left (24 \, c^{3} d^{5} x e + 16 \, c^{3} d^{6} + 9 \, a^{2} c d x e^{5} + a^{3} e^{6} - 3 \, {\left (3 \, a c^{2} d^{2} x^{2} - 2 \, a^{2} c d^{2}\right )} e^{4} - {\left (c^{3} d^{3} x^{3} + 36 \, a c^{2} d^{3} x\right )} e^{3} + 6 \, {\left (c^{3} d^{4} x^{2} - 4 \, a c^{2} d^{4}\right )} e^{2}\right )} \sqrt {x e + d}}{3 \, {\left (x^{2} e^{6} + 2 \, d x e^{5} + d^{2} e^{4}\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. 450 vs.
\(2 (107) = 214\).
time = 1.55, size = 450, normalized size = 3.91 \begin {gather*} \begin {cases} - \frac {2 a^{3} e^{6}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {12 a^{2} c d^{2} e^{4}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {18 a^{2} c d e^{5} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {48 a c^{2} d^{4} e^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {72 a c^{2} d^{3} e^{3} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {18 a c^{2} d^{2} e^{4} x^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {32 c^{3} d^{6}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {48 c^{3} d^{5} e x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {12 c^{3} d^{4} e^{2} x^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {2 c^{3} d^{3} e^{3} x^{3}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {c^{3} \sqrt {d} x^{4}}{4} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.94, size = 155, normalized size = 1.35 \begin {gather*} \frac {2}{3} \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{3} e^{8} - 9 \, \sqrt {x e + d} c^{3} d^{4} e^{8} + 9 \, \sqrt {x e + d} a c^{2} d^{2} e^{10}\right )} e^{\left (-12\right )} - \frac {2 \, {\left (9 \, {\left (x e + d\right )} c^{3} d^{5} - c^{3} d^{6} - 18 \, {\left (x e + d\right )} a c^{2} d^{3} e^{2} + 3 \, a c^{2} d^{4} e^{2} + 9 \, {\left (x e + d\right )} a^{2} c d e^{4} - 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} e^{\left (-4\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 = 0.65, size = 147, normalized size = 1.28 \begin {gather*} -\frac {2\,a^3\,e^6-2\,c^3\,d^6-2\,c^3\,d^3\,{\left (d+e\,x\right )}^3+18\,c^3\,d^4\,{\left (d+e\,x\right )}^2+18\,c^3\,d^5\,\left (d+e\,x\right )+6\,a\,c^2\,d^4\,e^2-6\,a^2\,c\,d^2\,e^4-36\,a\,c^2\,d^3\,e^2\,\left (d+e\,x\right )-18\,a\,c^2\,d^2\,e^2\,{\left (d+e\,x\right )}^2+18\,a^2\,c\,d\,e^4\,\left (d+e\,x\right )}{3\,e^4\,{\left (d+e\,x\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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