Optimal. Leaf size=81 \[ \frac {4 c d \left (c d^2-a e^2\right )}{3 e^3 (d+e x)^{3/2}}-\frac {2 \left (c d^2-a e^2\right )^2}{5 e^3 (d+e x)^{5/2}}-\frac {2 c^2 d^2}{e^3 \sqrt {d+e x}} \]
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Rubi [A] time = 0.04, antiderivative size = 81, 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} \[ \frac {4 c d \left (c d^2-a e^2\right )}{3 e^3 (d+e x)^{3/2}}-\frac {2 \left (c d^2-a e^2\right )^2}{5 e^3 (d+e x)^{5/2}}-\frac {2 c^2 d^2}{e^3 \sqrt {d+e x}} \]
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 )^2}{(d+e x)^{11/2}} \, dx &=\int \frac {(a e+c d x)^2}{(d+e x)^{7/2}} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^2}{e^2 (d+e x)^{7/2}}-\frac {2 c d \left (c d^2-a e^2\right )}{e^2 (d+e x)^{5/2}}+\frac {c^2 d^2}{e^2 (d+e x)^{3/2}}\right ) \, dx\\ &=-\frac {2 \left (c d^2-a e^2\right )^2}{5 e^3 (d+e x)^{5/2}}+\frac {4 c d \left (c d^2-a e^2\right )}{3 e^3 (d+e x)^{3/2}}-\frac {2 c^2 d^2}{e^3 \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 67, normalized size = 0.83 \[ -\frac {2 \left (3 a^2 e^4+2 a c d e^2 (2 d+5 e x)+c^2 d^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )}{15 e^3 (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 105, normalized size = 1.30 \[ -\frac {2 \, {\left (15 \, c^{2} d^{2} e^{2} x^{2} + 8 \, c^{2} d^{4} + 4 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 10 \, {\left (2 \, c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 108, normalized size = 1.33 \[ -\frac {2 \, {\left (15 \, {\left (x e + d\right )}^{4} c^{2} d^{2} - 10 \, {\left (x e + d\right )}^{3} c^{2} d^{3} + 3 \, {\left (x e + d\right )}^{2} c^{2} d^{4} + 10 \, {\left (x e + d\right )}^{3} a c d e^{2} - 6 \, {\left (x e + d\right )}^{2} a c d^{2} e^{2} + 3 \, {\left (x e + d\right )}^{2} a^{2} e^{4}\right )} e^{\left (-3\right )}}{15 \, {\left (x e + d\right )}^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 73, normalized size = 0.90 \[ -\frac {2 \left (15 c^{2} d^{2} e^{2} x^{2}+10 a c d \,e^{3} x +20 c^{2} d^{3} e x +3 a^{2} e^{4}+4 a c \,d^{2} e^{2}+8 c^{2} d^{4}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 77, normalized size = 0.95 \[ -\frac {2 \, {\left (15 \, {\left (e x + d\right )}^{2} c^{2} d^{2} + 3 \, c^{2} d^{4} - 6 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} - 10 \, {\left (c^{2} d^{3} - a c d e^{2}\right )} {\left (e x + d\right )}\right )}}{15 \, {\left (e x + d\right )}^{\frac {5}{2}} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 78, normalized size = 0.96 \[ -\frac {\frac {2\,a^2\,e^4}{5}+\frac {2\,c^2\,d^4}{5}-\left (\frac {4\,c^2\,d^3}{3}-\frac {4\,a\,c\,d\,e^2}{3}\right )\,\left (d+e\,x\right )+2\,c^2\,d^2\,{\left (d+e\,x\right )}^2-\frac {4\,a\,c\,d^2\,e^2}{5}}{e^3\,{\left (d+e\,x\right )}^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 17.07, size = 388, normalized size = 4.79 \[ \begin {cases} - \frac {6 a^{2} e^{4}}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} - \frac {8 a c d^{2} e^{2}}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} - \frac {20 a c d e^{3} x}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} - \frac {16 c^{2} d^{4}}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} - \frac {40 c^{2} d^{3} e x}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} - \frac {30 c^{2} d^{2} e^{2} x^{2}}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {c^{2} x^{3}}{3 d^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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