Optimal. Leaf size=231 \[ \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{9 \left (c d^2-a e^2\right ) (d+e x)^6}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{21 \left (c d^2-a e^2\right )^2 (d+e x)^5}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 \left (c d^2-a e^2\right )^3 (d+e x)^4}+\frac {32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{315 \left (c d^2-a e^2\right )^4 (d+e x)^3} \]
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Rubi [A]
time = 0.08, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {672, 664}
\begin {gather*} \frac {32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{315 (d+e x)^3 \left (c d^2-a e^2\right )^4}+\frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 (d+e x)^4 \left (c d^2-a e^2\right )^3}+\frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{21 (d+e x)^5 \left (c d^2-a e^2\right )^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 (d+e x)^6 \left (c d^2-a e^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 664
Rule 672
Rubi steps
\begin {align*} \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^6} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{9 \left (c d^2-a e^2\right ) (d+e x)^6}+\frac {(2 c d) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^5} \, dx}{3 \left (c d^2-a e^2\right )}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{9 \left (c d^2-a e^2\right ) (d+e x)^6}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{21 \left (c d^2-a e^2\right )^2 (d+e x)^5}+\frac {\left (8 c^2 d^2\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^4} \, dx}{21 \left (c d^2-a e^2\right )^2}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{9 \left (c d^2-a e^2\right ) (d+e x)^6}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{21 \left (c d^2-a e^2\right )^2 (d+e x)^5}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 \left (c d^2-a e^2\right )^3 (d+e x)^4}+\frac {\left (16 c^3 d^3\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^3} \, dx}{105 \left (c d^2-a e^2\right )^3}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{9 \left (c d^2-a e^2\right ) (d+e x)^6}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{21 \left (c d^2-a e^2\right )^2 (d+e x)^5}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 \left (c d^2-a e^2\right )^3 (d+e x)^4}+\frac {32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{315 \left (c d^2-a e^2\right )^4 (d+e x)^3}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 138, normalized size = 0.60 \begin {gather*} \frac {2 ((a e+c d x) (d+e x))^{3/2} \left (-35 a^3 e^6+15 a^2 c d e^4 (9 d+2 e x)-3 a c^2 d^2 e^2 \left (63 d^2+36 d e x+8 e^2 x^2\right )+c^3 d^3 \left (105 d^3+126 d^2 e x+72 d e^2 x^2+16 e^3 x^3\right )\right )}{315 \left (c d^2-a e^2\right )^4 (d+e x)^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.70, size = 293, normalized size = 1.27
method | result | size |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-16 c^{3} d^{3} e^{3} x^{3}+24 a \,c^{2} d^{2} e^{4} x^{2}-72 c^{3} d^{4} e^{2} x^{2}-30 a^{2} c d \,e^{5} x +108 a \,c^{2} d^{3} e^{3} x -126 c^{3} d^{5} e x +35 e^{6} a^{3}-135 e^{4} d^{2} a^{2} c +189 d^{4} e^{2} c^{2} a -105 d^{6} c^{3}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{315 \left (e x +d \right )^{5} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}\) | \(217\) |
trager | \(-\frac {2 \left (-16 c^{4} d^{4} e^{3} x^{4}+8 a \,c^{3} d^{3} e^{4} x^{3}-72 c^{4} d^{5} e^{2} x^{3}-6 a^{2} c^{2} d^{2} e^{5} x^{2}+36 a \,c^{3} d^{4} e^{3} x^{2}-126 c^{4} d^{6} e \,x^{2}+5 d \,e^{6} c \,a^{3} x -27 d^{3} e^{4} a^{2} c^{2} x +63 d^{5} e^{2} c^{3} a x -105 d^{7} c^{4} x +35 e^{7} a^{4}-135 a^{3} c \,d^{2} e^{5}+189 a^{2} c^{2} d^{4} e^{3}-105 a \,c^{3} d^{6} e \right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{315 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (e x +d \right )^{5}}\) | \(271\) |
default | \(\frac {-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{9 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{6}}-\frac {2 c d e \left (-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{7 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{5}}-\frac {4 c d e \left (-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{5 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{4}}+\frac {4 c d e \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{15 \left (e^{2} a -c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )^{3}}\right )}{7 \left (e^{2} a -c \,d^{2}\right )}\right )}{3 \left (e^{2} a -c \,d^{2}\right )}}{e^{6}}\) | \(293\) |
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. 580 vs.
\(2 (219) = 438\).
time = 30.59, size = 580, normalized size = 2.51 \begin {gather*} \frac {2 \, {\left (105 \, c^{4} d^{7} x - 5 \, a^{3} c d x e^{6} - 35 \, a^{4} e^{7} + 3 \, {\left (2 \, a^{2} c^{2} d^{2} x^{2} + 45 \, a^{3} c d^{2}\right )} e^{5} - {\left (8 \, a c^{3} d^{3} x^{3} - 27 \, a^{2} c^{2} d^{3} x\right )} e^{4} + {\left (16 \, c^{4} d^{4} x^{4} - 36 \, a c^{3} d^{4} x^{2} - 189 \, a^{2} c^{2} d^{4}\right )} e^{3} + 9 \, {\left (8 \, c^{4} d^{5} x^{3} - 7 \, a c^{3} d^{5} x\right )} e^{2} + 21 \, {\left (6 \, c^{4} d^{6} x^{2} + 5 \, a c^{3} d^{6}\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{315 \, {\left (5 \, c^{4} d^{12} x e + c^{4} d^{13} + a^{4} x^{5} e^{13} + 5 \, a^{4} d x^{4} e^{12} - 2 \, {\left (2 \, a^{3} c d^{2} x^{5} - 5 \, a^{4} d^{2} x^{3}\right )} e^{11} - 10 \, {\left (2 \, a^{3} c d^{3} x^{4} - a^{4} d^{3} x^{2}\right )} e^{10} + {\left (6 \, a^{2} c^{2} d^{4} x^{5} - 40 \, a^{3} c d^{4} x^{3} + 5 \, a^{4} d^{4} x\right )} e^{9} + {\left (30 \, a^{2} c^{2} d^{5} x^{4} - 40 \, a^{3} c d^{5} x^{2} + a^{4} d^{5}\right )} e^{8} - 4 \, {\left (a c^{3} d^{6} x^{5} - 15 \, a^{2} c^{2} d^{6} x^{3} + 5 \, a^{3} c d^{6} x\right )} e^{7} - 4 \, {\left (5 \, a c^{3} d^{7} x^{4} - 15 \, a^{2} c^{2} d^{7} x^{2} + a^{3} c d^{7}\right )} e^{6} + {\left (c^{4} d^{8} x^{5} - 40 \, a c^{3} d^{8} x^{3} + 30 \, a^{2} c^{2} d^{8} x\right )} e^{5} + {\left (5 \, c^{4} d^{9} x^{4} - 40 \, a c^{3} d^{9} x^{2} + 6 \, a^{2} c^{2} d^{9}\right )} e^{4} + 10 \, {\left (c^{4} d^{10} x^{3} - 2 \, a c^{3} d^{10} x\right )} e^{3} + 2 \, {\left (5 \, c^{4} d^{11} x^{2} - 2 \, a c^{3} d^{11}\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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.58, size = 1192, normalized size = 5.16 \begin {gather*} \frac {\left (\frac {4\,c^2\,d^3}{9\,\left (a\,e^2-c\,d^2\right )\,\left (7\,a\,e^3-7\,c\,d^2\,e\right )}-\frac {4\,a\,c\,d\,e^2}{9\,\left (a\,e^2-c\,d^2\right )\,\left (7\,a\,e^3-7\,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 )}^4}-\frac {\left (\frac {2\,a\,e^2}{9\,a\,e^3-9\,c\,d^2\,e}-\frac {2\,c\,d^2}{9\,a\,e^3-9\,c\,d^2\,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 )}^5}+\frac {\left (\frac {4\,c^3\,d^4+4\,a\,c^2\,d^2\,e^2}{63\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )}-\frac {8\,c^3\,d^4}{63\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (5\,a\,e^3-5\,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 )}^3}+\frac {\left (\frac {8\,c^4\,d^5+8\,a\,c^3\,d^3\,e^2}{315\,{\left (a\,e^2-c\,d^2\right )}^3\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}-\frac {16\,c^4\,d^5}{315\,{\left (a\,e^2-c\,d^2\right )}^3\,\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 {16\,c^5\,d^6+16\,a\,c^4\,d^4\,e^2}{945\,e\,{\left (a\,e^2-c\,d^2\right )}^5}-\frac {32\,c^5\,d^6}{945\,e\,{\left (a\,e^2-c\,d^2\right )}^5}\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\,c^2\,d^3+2\,a\,c\,d\,e^2}{9\,\left (a\,e^2-c\,d^2\right )\,\left (7\,a\,e^3-7\,c\,d^2\,e\right )}-\frac {4\,c^2\,d^3}{9\,\left (a\,e^2-c\,d^2\right )\,\left (7\,a\,e^3-7\,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 )}^4}+\frac {\left (\frac {8\,c^3\,d^4}{63\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )}-\frac {8\,a\,c^2\,d^2\,e^2}{63\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (5\,a\,e^3-5\,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 )}^3}+\frac {\left (\frac {16\,c^4\,d^5}{315\,{\left (a\,e^2-c\,d^2\right )}^3\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}-\frac {16\,a\,c^3\,d^3\,e^2}{315\,{\left (a\,e^2-c\,d^2\right )}^3\,\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 {32\,c^5\,d^6}{945\,e\,{\left (a\,e^2-c\,d^2\right )}^5}-\frac {32\,a\,c^4\,d^4\,e}{945\,{\left (a\,e^2-c\,d^2\right )}^5}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x}-\frac {8\,c^3\,d^3\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{63\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )\,{\left (d+e\,x\right )}^2}+\frac {16\,c^2\,d^2\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{63\,\left (a\,e^2-c\,d^2\right )\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )\,{\left (d+e\,x\right )}^3}+\frac {16\,c^4\,d^4\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{135\,e\,{\left (a\,e^2-c\,d^2\right )}^4\,\left (d+e\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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