Optimal. Leaf size=132 \[ -\frac {2 \left (c d^2+a e^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {16 c d e \left (c d^2+a e^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {628, 627}
\begin {gather*} \frac {16 c d e \left (a e^2+c d^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 \left (a e^2+c d^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 627
Rule 628
Rubi steps
\begin {align*} \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=-\frac {2 \left (c d^2+a e^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {(8 c d e) \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 \left (c d^2-a e^2\right )^2}\\ &=-\frac {2 \left (c d^2+a e^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {16 c d e \left (c d^2+a e^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 122, normalized size = 0.92 \begin {gather*} -\frac {2 (a e+c d x)^3 \left (e^3-\frac {9 c d e^2 (d+e x)}{a e+c d x}-\frac {9 c^2 d^2 e (d+e x)^2}{(a e+c d x)^2}+\frac {c^3 d^3 (d+e x)^3}{(a e+c d x)^3}\right )}{3 \left (c d^2-a e^2\right )^4 ((a e+c d x) (d+e x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.72, size = 153, normalized size = 1.16
method | result | size |
default | \(\frac {\frac {4}{3} c d e x +\frac {2}{3} e^{2} a +\frac {2}{3} c \,d^{2}}{\left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}+\frac {16 c d e \left (2 c d e x +e^{2} a +c \,d^{2}\right )}{3 \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right )^{2} \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}\) | \(153\) |
trager | \(-\frac {2 \left (-16 c^{3} d^{3} e^{3} x^{3}-24 a \,c^{2} d^{2} e^{4} x^{2}-24 c^{3} d^{4} e^{2} x^{2}-6 a^{2} c d \,e^{5} x -36 a \,c^{2} d^{3} e^{3} x -6 c^{3} d^{5} e x +e^{6} a^{3}-9 e^{4} d^{2} a^{2} c -9 d^{4} e^{2} c^{2} a +d^{6} c^{3}\right )}{3 \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}} \left (e^{2} a -c \,d^{2}\right )}\) | \(201\) |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (e x +d \right ) \left (-16 c^{3} d^{3} e^{3} x^{3}-24 a \,c^{2} d^{2} e^{4} x^{2}-24 c^{3} d^{4} e^{2} x^{2}-6 a^{2} c d \,e^{5} x -36 a \,c^{2} d^{3} e^{3} x -6 c^{3} d^{5} e x +e^{6} a^{3}-9 e^{4} d^{2} a^{2} c -9 d^{4} e^{2} c^{2} a +d^{6} c^{3}\right )}{3 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}\) | \(213\) |
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. 497 vs.
\(2 (125) = 250\).
time = 21.11, size = 497, normalized size = 3.77 \begin {gather*} \frac {2 \, {\left (6 \, c^{3} d^{5} x e - c^{3} d^{6} + 6 \, a^{2} c d x e^{5} - a^{3} e^{6} + 3 \, {\left (8 \, a c^{2} d^{2} x^{2} + 3 \, a^{2} c d^{2}\right )} e^{4} + 4 \, {\left (4 \, c^{3} d^{3} x^{3} + 9 \, a c^{2} d^{3} x\right )} e^{3} + 3 \, {\left (8 \, c^{3} d^{4} x^{2} + 3 \, a c^{2} d^{4}\right )} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{3 \, {\left (c^{6} d^{12} x^{2} + a^{6} x^{2} e^{12} + 2 \, {\left (a^{5} c d x^{3} + a^{6} d x\right )} e^{11} + {\left (a^{4} c^{2} d^{2} x^{4} + a^{6} d^{2}\right )} e^{10} - 6 \, {\left (a^{4} c^{2} d^{3} x^{3} + a^{5} c d^{3} x\right )} e^{9} - {\left (4 \, a^{3} c^{3} d^{4} x^{4} + 9 \, a^{4} c^{2} d^{4} x^{2} + 4 \, a^{5} c d^{4}\right )} e^{8} + 4 \, {\left (a^{3} c^{3} d^{5} x^{3} + a^{4} c^{2} d^{5} x\right )} e^{7} + 2 \, {\left (3 \, a^{2} c^{4} d^{6} x^{4} + 8 \, a^{3} c^{3} d^{6} x^{2} + 3 \, a^{4} c^{2} d^{6}\right )} e^{6} + 4 \, {\left (a^{2} c^{4} d^{7} x^{3} + a^{3} c^{3} d^{7} x\right )} e^{5} - {\left (4 \, a c^{5} d^{8} x^{4} + 9 \, a^{2} c^{4} d^{8} x^{2} + 4 \, a^{3} c^{3} d^{8}\right )} e^{4} - 6 \, {\left (a c^{5} d^{9} x^{3} + a^{2} c^{4} d^{9} x\right )} e^{3} + {\left (c^{6} d^{10} x^{4} + a^{2} c^{4} d^{10}\right )} e^{2} + 2 \, {\left (c^{6} d^{11} x^{3} + a c^{5} d^{11} x\right )} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac {5}{2}}}\, dx \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. 344 vs.
\(2 (125) = 250\).
time = 2.03, size = 344, normalized size = 2.61 \begin {gather*} \frac {2 \, {\left (2 \, {\left (4 \, {\left (\frac {2 \, c^{3} d^{3} x e^{3}}{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} + \frac {3 \, {\left (c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )}}{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}}\right )} x + \frac {3 \, {\left (c^{3} d^{5} e + 6 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )}}{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}}\right )} x - \frac {c^{3} d^{6} - 9 \, a c^{2} d^{4} e^{2} - 9 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}}{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}}\right )}}{3 \, {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.72, size = 131, normalized size = 0.99 \begin {gather*} \frac {\left (2\,c\,d^2+4\,c\,x\,d\,e+2\,a\,e^2\right )\,\left (8\,c^2\,d^2\,e^2\,x^2-{\left (c\,d^2+a\,e^2\right )}^2+12\,a\,c\,d^2\,e^2+8\,c\,d\,e\,x\,\left (c\,d^2+a\,e^2\right )\right )}{3\,{\left ({\left (c\,d^2+a\,e^2\right )}^2-4\,a\,c\,d^2\,e^2\right )}^2\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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