Optimal. Leaf size=301 \[ \frac {2}{9 \left (c d^2-a e^2\right ) (d+e x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {20 c d}{63 \left (c d^2-a e^2\right )^2 (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {32 c^2 d^2}{63 \left (c d^2-a e^2\right )^3 (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {64 c^3 d^3}{63 \left (c d^2-a e^2\right )^4 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {256 c^4 d^4 \left (c d^2+a e^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^6 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]
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Rubi [A]
time = 0.11, antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {672, 627}
\begin {gather*} -\frac {256 c^4 d^4 \left (a e^2+c d^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^6 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {64 c^3 d^3}{63 (d+e x) \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 {32 c^2 d^2}{63 (d+e x)^2 \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {20 c d}{63 (d+e x)^3 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {2}{9 (d+e x)^4 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 627
Rule 672
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=\frac {2}{9 \left (c d^2-a e^2\right ) (d+e x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {(10 c d) \int \frac {1}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{9 \left (c d^2-a e^2\right )}\\ &=\frac {2}{9 \left (c d^2-a e^2\right ) (d+e x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {20 c d}{63 \left (c d^2-a e^2\right )^2 (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (80 c^2 d^2\right ) \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{63 \left (c d^2-a e^2\right )^2}\\ &=\frac {2}{9 \left (c d^2-a e^2\right ) (d+e x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {20 c d}{63 \left (c d^2-a e^2\right )^2 (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {32 c^2 d^2}{63 \left (c d^2-a e^2\right )^3 (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (32 c^3 d^3\right ) \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 )^{3/2}} \, dx}{21 \left (c d^2-a e^2\right )^3}\\ &=\frac {2}{9 \left (c d^2-a e^2\right ) (d+e x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {20 c d}{63 \left (c d^2-a e^2\right )^2 (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {32 c^2 d^2}{63 \left (c d^2-a e^2\right )^3 (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {64 c^3 d^3}{63 \left (c d^2-a e^2\right )^4 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (128 c^4 d^4\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}{63 \left (c d^2-a e^2\right )^4}\\ &=\frac {2}{9 \left (c d^2-a e^2\right ) (d+e x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {20 c d}{63 \left (c d^2-a e^2\right )^2 (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {32 c^2 d^2}{63 \left (c d^2-a e^2\right )^3 (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {64 c^3 d^3}{63 \left (c d^2-a e^2\right )^4 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {256 c^4 d^4 \left (c d^2+a e^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^6 \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 = 0.22, size = 188, normalized size = 0.62 \begin {gather*} -\frac {2 (a e+c d x)^6 \left (7 e^5-\frac {45 c d e^4 (d+e x)}{a e+c d x}+\frac {126 c^2 d^2 e^3 (d+e x)^2}{(a e+c d x)^2}-\frac {210 c^3 d^3 e^2 (d+e x)^3}{(a e+c d x)^3}+\frac {315 c^4 d^4 e (d+e x)^4}{(a e+c d x)^4}+\frac {63 c^5 d^5 (d+e x)^5}{(a e+c d x)^5}\right )}{63 \left (c d^2-a e^2\right )^6 (d+e x)^3 ((a e+c d x) (d+e x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.79, size = 389, normalized size = 1.29
method | result | size |
default | \(\frac {-\frac {2}{9 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{4} \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}-\frac {10 c d e \left (-\frac {2}{7 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{3} \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}-\frac {8 c d e \left (-\frac {2}{5 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2} \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}-\frac {6 c d e \left (-\frac {2}{3 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}+\frac {8 c d e \left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right )}{3 \left (e^{2} a -c \,d^{2}\right )^{3} \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}\right )}{5 \left (e^{2} a -c \,d^{2}\right )}\right )}{7 \left (e^{2} a -c \,d^{2}\right )}\right )}{9 \left (e^{2} a -c \,d^{2}\right )}}{e^{4}}\) | \(389\) |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (256 c^{5} d^{5} e^{5} x^{5}+128 a \,c^{4} d^{4} e^{6} x^{4}+1152 c^{5} d^{6} e^{4} x^{4}-32 a^{2} c^{3} d^{3} e^{7} x^{3}+576 a \,c^{4} d^{5} e^{5} x^{3}+2016 c^{5} d^{7} e^{3} x^{3}+16 a^{3} c^{2} d^{2} e^{8} x^{2}-144 a^{2} c^{3} d^{4} e^{6} x^{2}+1008 a \,c^{4} d^{6} e^{4} x^{2}+1680 c^{5} d^{8} e^{2} x^{2}-10 a^{4} c d \,e^{9} x +72 a^{3} c^{2} d^{3} e^{7} x -252 a^{2} c^{3} d^{5} e^{5} x +840 a \,c^{4} d^{7} e^{3} x +630 c^{5} d^{9} e x +7 a^{5} e^{10}-45 a^{4} c \,d^{2} e^{8}+126 a^{3} c^{2} d^{4} e^{6}-210 a^{2} c^{3} d^{6} e^{4}+315 a \,c^{4} d^{8} e^{2}+63 c^{5} d^{10}\right )}{63 \left (e x +d \right )^{3} \left (a^{6} e^{12}-6 a^{5} c \,d^{2} e^{10}+15 a^{4} c^{2} d^{4} e^{8}-20 a^{3} c^{3} d^{6} e^{6}+15 a^{2} c^{4} d^{8} e^{4}-6 a \,c^{5} d^{10} e^{2}+c^{6} d^{12}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\) | \(412\) |
trager | \(-\frac {2 \left (256 c^{5} d^{5} e^{5} x^{5}+128 a \,c^{4} d^{4} e^{6} x^{4}+1152 c^{5} d^{6} e^{4} x^{4}-32 a^{2} c^{3} d^{3} e^{7} x^{3}+576 a \,c^{4} d^{5} e^{5} x^{3}+2016 c^{5} d^{7} e^{3} x^{3}+16 a^{3} c^{2} d^{2} e^{8} x^{2}-144 a^{2} c^{3} d^{4} e^{6} x^{2}+1008 a \,c^{4} d^{6} e^{4} x^{2}+1680 c^{5} d^{8} e^{2} x^{2}-10 a^{4} c d \,e^{9} x +72 a^{3} c^{2} d^{3} e^{7} x -252 a^{2} c^{3} d^{5} e^{5} x +840 a \,c^{4} d^{7} e^{3} x +630 c^{5} d^{9} e x +7 a^{5} e^{10}-45 a^{4} c \,d^{2} e^{8}+126 a^{3} c^{2} d^{4} e^{6}-210 a^{2} c^{3} d^{6} e^{4}+315 a \,c^{4} d^{8} e^{2}+63 c^{5} d^{10}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{63 \left (c d x +a e \right ) \left (e^{2} a -c \,d^{2}\right ) \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) \left (e x +d \right )^{5}}\) | \(415\) |
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. 1019 vs.
\(2 (285) = 570\).
time = 136.06, size = 1019, normalized size = 3.39 \begin {gather*} -\frac {2 \, {\left (630 \, c^{5} d^{9} x e + 63 \, c^{5} d^{10} - 10 \, a^{4} c d x e^{9} + 7 \, a^{5} e^{10} + {\left (16 \, a^{3} c^{2} d^{2} x^{2} - 45 \, a^{4} c d^{2}\right )} e^{8} - 8 \, {\left (4 \, a^{2} c^{3} d^{3} x^{3} - 9 \, a^{3} c^{2} d^{3} x\right )} e^{7} + 2 \, {\left (64 \, a c^{4} d^{4} x^{4} - 72 \, a^{2} c^{3} d^{4} x^{2} + 63 \, a^{3} c^{2} d^{4}\right )} e^{6} + 4 \, {\left (64 \, c^{5} d^{5} x^{5} + 144 \, a c^{4} d^{5} x^{3} - 63 \, a^{2} c^{3} d^{5} x\right )} e^{5} + 6 \, {\left (192 \, c^{5} d^{6} x^{4} + 168 \, a c^{4} d^{6} x^{2} - 35 \, a^{2} c^{3} d^{6}\right )} e^{4} + 168 \, {\left (12 \, c^{5} d^{7} x^{3} + 5 \, a c^{4} d^{7} x\right )} e^{3} + 105 \, {\left (16 \, c^{5} d^{8} x^{2} + 3 \, a c^{4} d^{8}\right )} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{63 \, {\left (c^{7} d^{18} x + a^{7} x^{5} e^{18} + {\left (a^{6} c d x^{6} + 5 \, a^{7} d x^{4}\right )} e^{17} - {\left (a^{6} c d^{2} x^{5} - 10 \, a^{7} d^{2} x^{3}\right )} e^{16} - 2 \, {\left (3 \, a^{5} c^{2} d^{3} x^{6} + 10 \, a^{6} c d^{3} x^{4} - 5 \, a^{7} d^{3} x^{2}\right )} e^{15} - 5 \, {\left (3 \, a^{5} c^{2} d^{4} x^{5} + 10 \, a^{6} c d^{4} x^{3} - a^{7} d^{4} x\right )} e^{14} + {\left (15 \, a^{4} c^{3} d^{5} x^{6} + 15 \, a^{5} c^{2} d^{5} x^{4} - 55 \, a^{6} c d^{5} x^{2} + a^{7} d^{5}\right )} e^{13} + {\left (55 \, a^{4} c^{3} d^{6} x^{5} + 90 \, a^{5} c^{2} d^{6} x^{3} - 29 \, a^{6} c d^{6} x\right )} e^{12} - 2 \, {\left (10 \, a^{3} c^{4} d^{7} x^{6} - 25 \, a^{4} c^{3} d^{7} x^{4} - 60 \, a^{5} c^{2} d^{7} x^{2} + 3 \, a^{6} c d^{7}\right )} e^{11} - {\left (85 \, a^{3} c^{4} d^{8} x^{5} + 50 \, a^{4} c^{3} d^{8} x^{3} - 69 \, a^{5} c^{2} d^{8} x\right )} e^{10} + 5 \, {\left (3 \, a^{2} c^{5} d^{9} x^{6} - 25 \, a^{3} c^{4} d^{9} x^{4} - 25 \, a^{4} c^{3} d^{9} x^{2} + 3 \, a^{5} c^{2} d^{9}\right )} e^{9} + {\left (69 \, a^{2} c^{5} d^{10} x^{5} - 50 \, a^{3} c^{4} d^{10} x^{3} - 85 \, a^{4} c^{3} d^{10} x\right )} e^{8} - 2 \, {\left (3 \, a c^{6} d^{11} x^{6} - 60 \, a^{2} c^{5} d^{11} x^{4} - 25 \, a^{3} c^{4} d^{11} x^{2} + 10 \, a^{4} c^{3} d^{11}\right )} e^{7} - {\left (29 \, a c^{6} d^{12} x^{5} - 90 \, a^{2} c^{5} d^{12} x^{3} - 55 \, a^{3} c^{4} d^{12} x\right )} e^{6} + {\left (c^{7} d^{13} x^{6} - 55 \, a c^{6} d^{13} x^{4} + 15 \, a^{2} c^{5} d^{13} x^{2} + 15 \, a^{3} c^{4} d^{13}\right )} e^{5} + 5 \, {\left (c^{7} d^{14} x^{5} - 10 \, a c^{6} d^{14} x^{3} - 3 \, a^{2} c^{5} d^{14} x\right )} e^{4} + 2 \, {\left (5 \, c^{7} d^{15} x^{4} - 10 \, a c^{6} d^{15} x^{2} - 3 \, a^{2} c^{5} d^{15}\right )} e^{3} + {\left (10 \, c^{7} d^{16} x^{3} - a c^{6} d^{16} x\right )} e^{2} + {\left (5 \, c^{7} d^{17} x^{2} + a c^{6} d^{17}\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 = 4.36, size = 2500, normalized size = 8.31 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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