Optimal. Leaf size=192 \[ \frac {2}{5 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {16 c d \left (c d^2+a e^2+2 c d e x\right )}{15 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {128 c^2 d^2 e \left (c d^2+a e^2+2 c d e x\right )}{15 \left (c d^2-a e^2\right )^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {672, 628, 627}
\begin {gather*} \frac {128 c^2 d^2 e \left (a e^2+c d^2+2 c d e x\right )}{15 \left (c d^2-a e^2\right )^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {16 c d \left (a e^2+c d^2+2 c d e x\right )}{15 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac {2}{5 (d+e x) \left (c d^2-a e^2\right ) \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
Rule 672
Rubi steps
\begin {align*} \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=\frac {2}{5 \left (c d^2-a e^2\right ) (d+e x) \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) \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}{5 \left (c d^2-a e^2\right )}\\ &=\frac {2}{5 \left (c d^2-a e^2\right ) (d+e x) \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 \left (c d^2+a e^2+2 c d e x\right )}{15 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {\left (64 c^2 d^2 e\right ) \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}{15 \left (c d^2-a e^2\right )^3}\\ &=\frac {2}{5 \left (c d^2-a e^2\right ) (d+e x) \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 \left (c d^2+a e^2+2 c d e x\right )}{15 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {128 c^2 d^2 e \left (c d^2+a e^2+2 c d e x\right )}{15 \left (c d^2-a e^2\right )^5 \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 = 0.19, size = 153, normalized size = 0.80 \begin {gather*} -\frac {2 (a e+c d x)^5 \left (-3 e^4+\frac {20 c d e^3 (d+e x)}{a e+c d x}-\frac {90 c^2 d^2 e^2 (d+e x)^2}{(a e+c d x)^2}-\frac {60 c^3 d^3 e (d+e x)^3}{(a e+c d x)^3}+\frac {5 c^4 d^4 (d+e x)^4}{(a e+c d x)^4}\right )}{15 \left (c d^2-a e^2\right )^5 ((a e+c d x) (d+e x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.72, size = 242, normalized size = 1.26
method | result | size |
default | \(\frac {-\frac {2}{5 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {8 c d e \left (-\frac {2 \left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right )}{3 \left (e^{2} a -c \,d^{2}\right )^{2} \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {16 c d e \left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right )}{3 \left (e^{2} a -c \,d^{2}\right )^{4} \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}\right )}{5 \left (e^{2} a -c \,d^{2}\right )}}{e}\) | \(242\) |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (128 c^{4} d^{4} e^{4} x^{4}+192 a \,c^{3} d^{3} e^{5} x^{3}+320 c^{4} d^{5} e^{3} x^{3}+48 a^{2} c^{2} d^{2} e^{6} x^{2}+480 a \,c^{3} d^{4} e^{4} x^{2}+240 c^{4} d^{6} e^{2} x^{2}-8 a^{3} c d \,e^{7} x +120 a^{2} c^{2} d^{3} e^{5} x +360 a \,c^{3} d^{5} e^{3} x +40 c^{4} d^{7} e x +3 a^{4} e^{8}-20 a^{3} c \,d^{2} e^{6}+90 a^{2} c^{2} d^{4} e^{4}+60 a \,c^{3} d^{6} e^{2}-5 c^{4} d^{8}\right )}{15 \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}\) | \(300\) |
trager | \(-\frac {2 \left (128 c^{4} d^{4} e^{4} x^{4}+192 a \,c^{3} d^{3} e^{5} x^{3}+320 c^{4} d^{5} e^{3} x^{3}+48 a^{2} c^{2} d^{2} e^{6} x^{2}+480 a \,c^{3} d^{4} e^{4} x^{2}+240 c^{4} d^{6} e^{2} x^{2}-8 a^{3} c d \,e^{7} x +120 a^{2} c^{2} d^{3} e^{5} x +360 a \,c^{3} d^{5} e^{3} x +40 c^{4} d^{7} e x +3 a^{4} e^{8}-20 a^{3} c \,d^{2} e^{6}+90 a^{2} c^{2} d^{4} e^{4}+60 a \,c^{3} d^{6} e^{2}-5 c^{4} d^{8}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{15 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (c d x +a e \right )^{2} \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{3}}\) | \(308\) |
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. 780 vs.
\(2 (182) = 364\).
time = 61.17, size = 780, normalized size = 4.06 \begin {gather*} \frac {2 \, {\left (40 \, c^{4} d^{7} x e - 5 \, c^{4} d^{8} - 8 \, a^{3} c d x e^{7} + 3 \, a^{4} e^{8} + 4 \, {\left (12 \, a^{2} c^{2} d^{2} x^{2} - 5 \, a^{3} c d^{2}\right )} e^{6} + 24 \, {\left (8 \, a c^{3} d^{3} x^{3} + 5 \, a^{2} c^{2} d^{3} x\right )} e^{5} + 2 \, {\left (64 \, c^{4} d^{4} x^{4} + 240 \, a c^{3} d^{4} x^{2} + 45 \, a^{2} c^{2} d^{4}\right )} e^{4} + 40 \, {\left (8 \, c^{4} d^{5} x^{3} + 9 \, a c^{3} d^{5} x\right )} e^{3} + 60 \, {\left (4 \, c^{4} d^{6} x^{2} + a c^{3} d^{6}\right )} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{15 \, {\left (c^{7} d^{15} x^{2} - a^{7} x^{3} e^{15} - {\left (2 \, a^{6} c d x^{4} + 3 \, a^{7} d x^{2}\right )} e^{14} - {\left (a^{5} c^{2} d^{2} x^{5} + a^{6} c d^{2} x^{3} + 3 \, a^{7} d^{2} x\right )} e^{13} + {\left (7 \, a^{5} c^{2} d^{3} x^{4} + 9 \, a^{6} c d^{3} x^{2} - a^{7} d^{3}\right )} e^{12} + {\left (5 \, a^{4} c^{3} d^{4} x^{5} + 17 \, a^{5} c^{2} d^{4} x^{3} + 13 \, a^{6} c d^{4} x\right )} e^{11} - {\left (5 \, a^{4} c^{3} d^{5} x^{4} + a^{5} c^{2} d^{5} x^{2} - 5 \, a^{6} c d^{5}\right )} e^{10} - 5 \, {\left (2 \, a^{3} c^{4} d^{6} x^{5} + 7 \, a^{4} c^{3} d^{6} x^{3} + 4 \, a^{5} c^{2} d^{6} x\right )} e^{9} - 5 \, {\left (2 \, a^{3} c^{4} d^{7} x^{4} + 5 \, a^{4} c^{3} d^{7} x^{2} + 2 \, a^{5} c^{2} d^{7}\right )} e^{8} + 5 \, {\left (2 \, a^{2} c^{5} d^{8} x^{5} + 5 \, a^{3} c^{4} d^{8} x^{3} + 2 \, a^{4} c^{3} d^{8} x\right )} e^{7} + 5 \, {\left (4 \, a^{2} c^{5} d^{9} x^{4} + 7 \, a^{3} c^{4} d^{9} x^{2} + 2 \, a^{4} c^{3} d^{9}\right )} e^{6} - {\left (5 \, a c^{6} d^{10} x^{5} - a^{2} c^{5} d^{10} x^{3} - 5 \, a^{3} c^{4} d^{10} x\right )} e^{5} - {\left (13 \, a c^{6} d^{11} x^{4} + 17 \, a^{2} c^{5} d^{11} x^{2} + 5 \, a^{3} c^{4} d^{11}\right )} e^{4} + {\left (c^{7} d^{12} x^{5} - 9 \, a c^{6} d^{12} x^{3} - 7 \, a^{2} c^{5} d^{12} x\right )} e^{3} + {\left (3 \, c^{7} d^{13} x^{4} + a c^{6} d^{13} x^{2} + a^{2} c^{5} d^{13}\right )} e^{2} + {\left (3 \, c^{7} d^{14} x^{3} + 2 \, a c^{6} d^{14} x\right )} e\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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.65, size = 253, normalized size = 1.32 \begin {gather*} -\frac {2\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (3\,a^4\,e^8-20\,a^3\,c\,d^2\,e^6-8\,a^3\,c\,d\,e^7\,x+90\,a^2\,c^2\,d^4\,e^4+120\,a^2\,c^2\,d^3\,e^5\,x+48\,a^2\,c^2\,d^2\,e^6\,x^2+60\,a\,c^3\,d^6\,e^2+360\,a\,c^3\,d^5\,e^3\,x+480\,a\,c^3\,d^4\,e^4\,x^2+192\,a\,c^3\,d^3\,e^5\,x^3-5\,c^4\,d^8+40\,c^4\,d^7\,e\,x+240\,c^4\,d^6\,e^2\,x^2+320\,c^4\,d^5\,e^3\,x^3+128\,c^4\,d^4\,e^4\,x^4\right )}{15\,{\left (a\,e+c\,d\,x\right )}^2\,{\left (a\,e^2-c\,d^2\right )}^5\,{\left (d+e\,x\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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