Optimal. Leaf size=115 \[ -\frac {6 c^2 d^2 \sqrt {d+e x} \left (c d^2-a e^2\right )}{e^4}-\frac {6 c d \left (c d^2-a e^2\right )^2}{e^4 \sqrt {d+e x}}+\frac {2 \left (c d^2-a e^2\right )^3}{3 e^4 (d+e x)^{3/2}}+\frac {2 c^3 d^3 (d+e x)^{3/2}}{3 e^4} \]
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Rubi [A] time = 0.05, 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 = {626, 43} \[ -\frac {6 c^2 d^2 \sqrt {d+e x} \left (c d^2-a e^2\right )}{e^4}-\frac {6 c d \left (c d^2-a e^2\right )^2}{e^4 \sqrt {d+e x}}+\frac {2 \left (c d^2-a e^2\right )^3}{3 e^4 (d+e x)^{3/2}}+\frac {2 c^3 d^3 (d+e x)^{3/2}}{3 e^4} \]
Antiderivative was successfully verified.
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Rule 43
Rule 626
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 = 0.06, size = 110, normalized size = 0.96 \[ -\frac {2 \left (a^3 e^6+3 a^2 c d e^4 (2 d+3 e x)-3 a c^2 d^2 e^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )+c^3 d^3 \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )\right )}{3 e^4 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 151, normalized size = 1.31 \[ \frac {2 \, {\left (c^{3} d^{3} e^{3} x^{3} - 16 \, c^{3} d^{6} + 24 \, a c^{2} d^{4} e^{2} - 6 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} - 3 \, {\left (2 \, c^{3} d^{4} e^{2} - 3 \, a c^{2} d^{2} e^{4}\right )} x^{2} - 3 \, {\left (8 \, c^{3} d^{5} e - 12 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {e x + d}}{3 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 193, normalized size = 1.68 \[ \frac {2}{3} \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{3} e^{8} - 9 \, \sqrt {x e + d} c^{3} d^{4} e^{8} + 9 \, \sqrt {x e + d} a c^{2} d^{2} e^{10}\right )} e^{\left (-12\right )} - \frac {2 \, {\left (9 \, {\left (x e + d\right )}^{4} c^{3} d^{5} - {\left (x e + d\right )}^{3} c^{3} d^{6} - 18 \, {\left (x e + d\right )}^{4} a c^{2} d^{3} e^{2} + 3 \, {\left (x e + d\right )}^{3} a c^{2} d^{4} e^{2} + 9 \, {\left (x e + d\right )}^{4} a^{2} c d e^{4} - 3 \, {\left (x e + d\right )}^{3} a^{2} c d^{2} e^{4} + {\left (x e + d\right )}^{3} a^{3} e^{6}\right )} e^{\left (-4\right )}}{3 \, {\left (x e + d\right )}^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 130, normalized size = 1.13 \[ -\frac {2 \left (-c^{3} d^{3} e^{3} x^{3}-9 a \,c^{2} d^{2} e^{4} x^{2}+6 c^{3} d^{4} e^{2} x^{2}+9 a^{2} c d \,e^{5} x -36 a \,c^{2} d^{3} e^{3} x +24 c^{3} d^{5} e x +a^{3} e^{6}+6 a^{2} c \,d^{2} e^{4}-24 a \,c^{2} d^{4} e^{2}+16 c^{3} d^{6}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.12, size = 141, normalized size = 1.23 \[ \frac {2 \, {\left (\frac {{\left (e x + d\right )}^{\frac {3}{2}} c^{3} d^{3} - 9 \, {\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} \sqrt {e x + d}}{e^{3}} + \frac {c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} - 9 \, {\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{3}}\right )}}{3 \, e} \]
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 \[ -\frac {2\,a^3\,e^6-2\,c^3\,d^6-2\,c^3\,d^3\,{\left (d+e\,x\right )}^3+18\,c^3\,d^4\,{\left (d+e\,x\right )}^2+18\,c^3\,d^5\,\left (d+e\,x\right )+6\,a\,c^2\,d^4\,e^2-6\,a^2\,c\,d^2\,e^4-36\,a\,c^2\,d^3\,e^2\,\left (d+e\,x\right )-18\,a\,c^2\,d^2\,e^2\,{\left (d+e\,x\right )}^2+18\,a^2\,c\,d\,e^4\,\left (d+e\,x\right )}{3\,e^4\,{\left (d+e\,x\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 25.51, size = 450, normalized size = 3.91 \[ \begin {cases} - \frac {2 a^{3} e^{6}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {12 a^{2} c d^{2} e^{4}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {18 a^{2} c d e^{5} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {48 a c^{2} d^{4} e^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {72 a c^{2} d^{3} e^{3} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {18 a c^{2} d^{2} e^{4} x^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {32 c^{3} d^{6}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {48 c^{3} d^{5} e x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {12 c^{3} d^{4} e^{2} x^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {2 c^{3} d^{3} e^{3} x^{3}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {c^{3} \sqrt {d} x^{4}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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