Optimal. Leaf size=54 \[ \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 \left (c d^2-a e^2\right ) (d+e x)^5} \]
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Rubi [A]
time = 0.01, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {664}
\begin {gather*} \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 (d+e x)^5 \left (c d^2-a e^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 664
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 )^{3/2}}{(d+e x)^5} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 \left (c d^2-a e^2\right ) (d+e x)^5}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 43, normalized size = 0.80 \begin {gather*} \frac {2 ((a e+c d x) (d+e x))^{5/2}}{5 \left (c d^2-a e^2\right ) (d+e x)^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.74, size = 65, normalized size = 1.20
method | result | size |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{5 \left (e x +d \right )^{4} \left (e^{2} a -c \,d^{2}\right )}\) | \(58\) |
default | \(-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5 e^{5} \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{5}}\) | \(65\) |
trager | \(-\frac {2 \left (c^{2} d^{2} x^{2}+2 a c d e x +a^{2} e^{2}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{5 \left (e x +d \right )^{3} \left (e^{2} a -c \,d^{2}\right )}\) | \(75\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 126 vs.
\(2 (51) = 102\).
time = 5.19, size = 126, normalized size = 2.33 \begin {gather*} \frac {2 \, {\left (c^{2} d^{2} x^{2} + 2 \, a c d x e + a^{2} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{5 \, {\left (3 \, c d^{4} x e + c d^{5} - a x^{3} e^{5} - 3 \, a d x^{2} e^{4} + {\left (c d^{2} x^{3} - 3 \, a d^{2} x\right )} e^{3} + {\left (3 \, c d^{3} x^{2} - a d^{3}\right )} e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 855 vs.
\(2 (51) = 102\).
time = 0.90, size = 855, normalized size = 15.83 \begin {gather*} -\frac {2}{15} \, {\left (\frac {3 \, \sqrt {c d} c^{2} d^{2} e^{\frac {1}{2}} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{c d^{2} e^{4} - a e^{6}} - \frac {\frac {{\left (15 \, \sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}} c^{2} d^{2} e^{2} - 10 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {3}{2}} c d e + 3 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {5}{2}}\right )} c^{2} d^{4} e^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{c^{2} d^{4} e^{4} - 2 \, a c d^{2} e^{6} + a^{2} e^{8}} - \frac {10 \, {\left (3 \, \sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}} c d e - {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {3}{2}}\right )} c^{2} d^{3} e \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{c d^{2} e^{2} - a e^{4}} - \frac {2 \, {\left (15 \, \sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}} c^{2} d^{2} e^{2} - 10 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {3}{2}} c d e + 3 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {5}{2}}\right )} a c d^{2} e^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{c^{2} d^{4} e^{4} - 2 \, a c d^{2} e^{6} + a^{2} e^{8}} + 15 \, \sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}} c^{2} d^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + \frac {10 \, {\left (3 \, \sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}} c d e - {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {3}{2}}\right )} a c d e^{3} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{c d^{2} e^{2} - a e^{4}} + \frac {{\left (15 \, \sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}} c^{2} d^{2} e^{2} - 10 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {3}{2}} c d e + 3 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {5}{2}}\right )} a^{2} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{c^{2} d^{4} e^{4} - 2 \, a c d^{2} e^{6} + a^{2} e^{8}}}{c d^{2} e^{4} - a e^{6}}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.93, size = 901, normalized size = 16.69 \begin {gather*} \frac {\left (\frac {d\,\left (\frac {4\,c^3\,d^4}{5\,\left (a\,e^2-c\,d^2\right )\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}-\frac {2\,c^2\,d^2\,\left (5\,a\,e^2-c\,d^2\right )}{5\,\left (a\,e^2-c\,d^2\right )\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}\right )}{e}+\frac {4\,a^2\,c\,d\,e^4+2\,a\,c^2\,d^3\,e^2-2\,c^3\,d^5}{5\,e\,\left (a\,e^2-c\,d^2\right )\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^2}-\frac {\left (\frac {d\,\left (\frac {40\,c^4\,d^5-56\,a\,c^3\,d^3\,e^2}{15\,e\,{\left (a\,e^2-c\,d^2\right )}^3}+\frac {8\,c^4\,d^5}{15\,e\,{\left (a\,e^2-c\,d^2\right )}^3}\right )}{e}+\frac {8\,a\,c^2\,d^2\,\left (6\,a\,e^2-5\,c\,d^2\right )}{15\,{\left (a\,e^2-c\,d^2\right )}^3}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x}-\frac {\left (\frac {2\,a^2\,e^3}{5\,a\,e^3-5\,c\,d^2\,e}+\frac {d\,\left (\frac {2\,c^2\,d^3}{5\,a\,e^3-5\,c\,d^2\,e}-\frac {4\,a\,c\,d\,e^2}{5\,a\,e^3-5\,c\,d^2\,e}\right )}{e}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^3}-\frac {\left (\frac {10\,c^3\,d^4-22\,a\,c^2\,d^2\,e^2}{15\,e^2\,{\left (a\,e^2-c\,d^2\right )}^2}+\frac {4\,c^3\,d^4}{5\,e^2\,{\left (a\,e^2-c\,d^2\right )}^2}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x}-\frac {\left (\frac {d\,\left (\frac {12\,c^3\,d^4-20\,a\,c^2\,d^2\,e^2}{5\,\left (a\,e^2-c\,d^2\right )\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}+\frac {4\,c^3\,d^4}{5\,\left (a\,e^2-c\,d^2\right )\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}\right )}{e}+\frac {4\,a\,c\,d\,e\,\left (4\,a\,e^2-3\,c\,d^2\right )}{5\,\left (a\,e^2-c\,d^2\right )\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^2}+\frac {\left (\frac {d\,\left (\frac {8\,c^4\,d^5}{15\,e\,{\left (a\,e^2-c\,d^2\right )}^3}-\frac {4\,c^3\,d^3\,\left (11\,a\,e^2-7\,c\,d^2\right )}{15\,e\,{\left (a\,e^2-c\,d^2\right )}^3}\right )}{e}+\frac {20\,a^2\,c^2\,d^2\,e^4+4\,a\,c^3\,d^4\,e^2-16\,c^4\,d^6}{15\,e^2\,{\left (a\,e^2-c\,d^2\right )}^3}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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