Optimal. Leaf size=81 \[ -\frac {2 \left (c d^2-a e^2\right )^2}{5 e^3 (d+e x)^{5/2}}+\frac {4 c d \left (c d^2-a e^2\right )}{3 e^3 (d+e x)^{3/2}}-\frac {2 c^2 d^2}{e^3 \sqrt {d+e x}} \]
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Rubi [A]
time = 0.03, 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 = {640, 45}
\begin {gather*} \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}} \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)^{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 = 0.05, size = 67, normalized size = 0.83 \begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.70, size = 79, normalized size = 0.98
method | result | size |
gosper | \(-\frac {2 \left (15 e^{2} x^{2} c^{2} d^{2}+10 a c d \,e^{3} x +20 c^{2} d^{3} e x +3 a^{2} e^{4}+4 a c \,d^{2} e^{2}+8 c^{2} d^{4}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{3}}\) | \(73\) |
trager | \(-\frac {2 \left (15 e^{2} x^{2} c^{2} d^{2}+10 a c d \,e^{3} x +20 c^{2} d^{3} e x +3 a^{2} e^{4}+4 a c \,d^{2} e^{2}+8 c^{2} d^{4}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{3}}\) | \(73\) |
derivativedivides | \(\frac {-\frac {2 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {4 c d \left (e^{2} a -c \,d^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 c^{2} d^{2}}{\sqrt {e x +d}}}{e^{3}}\) | \(79\) |
default | \(\frac {-\frac {2 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {4 c d \left (e^{2} a -c \,d^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 c^{2} d^{2}}{\sqrt {e x +d}}}{e^{3}}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 76, normalized size = 0.94 \begin {gather*} -\frac {2 \, {\left (15 \, {\left (x e + 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 (x e + d\right )}\right )} e^{\left (-3\right )}}{15 \, {\left (x e + d\right )}^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.60, size = 98, normalized size = 1.21 \begin {gather*} -\frac {2 \, {\left (20 \, c^{2} d^{3} x e + 8 \, c^{2} d^{4} + 10 \, a c d x e^{3} + 3 \, a^{2} e^{4} + {\left (15 \, c^{2} d^{2} x^{2} + 4 \, a c d^{2}\right )} e^{2}\right )} \sqrt {x e + d}}{15 \, {\left (x^{3} e^{6} + 3 \, d x^{2} e^{5} + 3 \, d^{2} x e^{4} + d^{3} 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. 388 vs.
\(2 (75) = 150\).
time = 1.52, size = 388, normalized size = 4.79 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.33, size = 80, normalized size = 0.99 \begin {gather*} -\frac {2 \, {\left (15 \, {\left (x e + d\right )}^{2} c^{2} d^{2} - 10 \, {\left (x e + d\right )} c^{2} d^{3} + 3 \, c^{2} d^{4} + 10 \, {\left (x e + d\right )} a c d e^{2} - 6 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4}\right )} e^{\left (-3\right )}}{15 \, {\left (x e + d\right )}^{\frac {5}{2}}} \end {gather*}
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 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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