Optimal. Leaf size=83 \[ \frac {4 c d \left (c d^2-a e^2\right )}{5 e^3 (d+e x)^{5/2}}-\frac {2 \left (c d^2-a e^2\right )^2}{7 e^3 (d+e x)^{7/2}}-\frac {2 c^2 d^2}{3 e^3 (d+e x)^{3/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 83, 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} \[ \frac {4 c d \left (c d^2-a e^2\right )}{5 e^3 (d+e x)^{5/2}}-\frac {2 \left (c d^2-a e^2\right )^2}{7 e^3 (d+e x)^{7/2}}-\frac {2 c^2 d^2}{3 e^3 (d+e x)^{3/2}} \]
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)^{13/2}} \, dx &=\int \frac {(a e+c d x)^2}{(d+e x)^{9/2}} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^2}{e^2 (d+e x)^{9/2}}-\frac {2 c d \left (c d^2-a e^2\right )}{e^2 (d+e x)^{7/2}}+\frac {c^2 d^2}{e^2 (d+e x)^{5/2}}\right ) \, dx\\ &=-\frac {2 \left (c d^2-a e^2\right )^2}{7 e^3 (d+e x)^{7/2}}+\frac {4 c d \left (c d^2-a e^2\right )}{5 e^3 (d+e x)^{5/2}}-\frac {2 c^2 d^2}{3 e^3 (d+e x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 67, normalized size = 0.81 \[ -\frac {2 \left (15 a^2 e^4+6 a c d e^2 (2 d+7 e x)+c^2 d^2 \left (8 d^2+28 d e x+35 e^2 x^2\right )\right )}{105 e^3 (d+e x)^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 117, normalized size = 1.41 \[ -\frac {2 \, {\left (35 \, c^{2} d^{2} e^{2} x^{2} + 8 \, c^{2} d^{4} + 12 \, a c d^{2} e^{2} + 15 \, a^{2} e^{4} + 14 \, {\left (2 \, c^{2} d^{3} e + 3 \, a c d e^{3}\right )} x\right )} \sqrt {e x + d}}{105 \, {\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 108, normalized size = 1.30 \[ -\frac {2 \, {\left (35 \, {\left (x e + d\right )}^{4} c^{2} d^{2} - 42 \, {\left (x e + d\right )}^{3} c^{2} d^{3} + 15 \, {\left (x e + d\right )}^{2} c^{2} d^{4} + 42 \, {\left (x e + d\right )}^{3} a c d e^{2} - 30 \, {\left (x e + d\right )}^{2} a c d^{2} e^{2} + 15 \, {\left (x e + d\right )}^{2} a^{2} e^{4}\right )} e^{\left (-3\right )}}{105 \, {\left (x e + d\right )}^{\frac {11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 73, normalized size = 0.88 \[ -\frac {2 \left (35 c^{2} d^{2} e^{2} x^{2}+42 a c d \,e^{3} x +28 c^{2} d^{3} e x +15 a^{2} e^{4}+12 a c \,d^{2} e^{2}+8 c^{2} d^{4}\right )}{105 \left (e x +d \right )^{\frac {7}{2}} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 77, normalized size = 0.93 \[ -\frac {2 \, {\left (35 \, {\left (e x + d\right )}^{2} c^{2} d^{2} + 15 \, c^{2} d^{4} - 30 \, a c d^{2} e^{2} + 15 \, a^{2} e^{4} - 42 \, {\left (c^{2} d^{3} - a c d e^{2}\right )} {\left (e x + d\right )}\right )}}{105 \, {\left (e x + d\right )}^{\frac {7}{2}} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.64, size = 78, normalized size = 0.94 \[ -\frac {\frac {2\,a^2\,e^4}{7}+\frac {2\,c^2\,d^4}{7}-\left (\frac {4\,c^2\,d^3}{5}-\frac {4\,a\,c\,d\,e^2}{5}\right )\,\left (d+e\,x\right )+\frac {2\,c^2\,d^2\,{\left (d+e\,x\right )}^2}{3}-\frac {4\,a\,c\,d^2\,e^2}{7}}{e^3\,{\left (d+e\,x\right )}^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 29.81, size = 510, normalized size = 6.14 \[ \begin {cases} - \frac {30 a^{2} e^{4}}{105 d^{3} e^{3} \sqrt {d + e x} + 315 d^{2} e^{4} x \sqrt {d + e x} + 315 d e^{5} x^{2} \sqrt {d + e x} + 105 e^{6} x^{3} \sqrt {d + e x}} - \frac {24 a c d^{2} e^{2}}{105 d^{3} e^{3} \sqrt {d + e x} + 315 d^{2} e^{4} x \sqrt {d + e x} + 315 d e^{5} x^{2} \sqrt {d + e x} + 105 e^{6} x^{3} \sqrt {d + e x}} - \frac {84 a c d e^{3} x}{105 d^{3} e^{3} \sqrt {d + e x} + 315 d^{2} e^{4} x \sqrt {d + e x} + 315 d e^{5} x^{2} \sqrt {d + e x} + 105 e^{6} x^{3} \sqrt {d + e x}} - \frac {16 c^{2} d^{4}}{105 d^{3} e^{3} \sqrt {d + e x} + 315 d^{2} e^{4} x \sqrt {d + e x} + 315 d e^{5} x^{2} \sqrt {d + e x} + 105 e^{6} x^{3} \sqrt {d + e x}} - \frac {56 c^{2} d^{3} e x}{105 d^{3} e^{3} \sqrt {d + e x} + 315 d^{2} e^{4} x \sqrt {d + e x} + 315 d e^{5} x^{2} \sqrt {d + e x} + 105 e^{6} x^{3} \sqrt {d + e x}} - \frac {70 c^{2} d^{2} e^{2} x^{2}}{105 d^{3} e^{3} \sqrt {d + e x} + 315 d^{2} e^{4} x \sqrt {d + e x} + 315 d e^{5} x^{2} \sqrt {d + e x} + 105 e^{6} x^{3} \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {c^{2} x^{3}}{3 d^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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