Optimal. Leaf size=231 \[ \frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 \left (c d^2-a e^2\right ) (d+e x)^4}+\frac {12 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 \left (c d^2-a e^2\right )^2 (d+e x)^3}+\frac {16 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 \left (c d^2-a e^2\right )^3 (d+e x)^2}+\frac {32 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 \left (c d^2-a e^2\right )^4 (d+e x)} \]
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Rubi [A]
time = 0.09, 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 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 (d+e x) \left (c d^2-a e^2\right )^4}+\frac {16 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 (d+e x)^2 \left (c d^2-a e^2\right )^3}+\frac {12 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 (d+e x)^3 \left (c d^2-a e^2\right )^2}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 (d+e x)^4 \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 {1}{(d+e x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 \left (c d^2-a e^2\right ) (d+e x)^4}+\frac {(6 c d) \int \frac {1}{(d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{7 \left (c d^2-a e^2\right )}\\ &=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 \left (c d^2-a e^2\right ) (d+e x)^4}+\frac {12 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 \left (c d^2-a e^2\right )^2 (d+e x)^3}+\frac {\left (24 c^2 d^2\right ) \int \frac {1}{(d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{35 \left (c d^2-a e^2\right )^2}\\ &=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 \left (c d^2-a e^2\right ) (d+e x)^4}+\frac {12 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 \left (c d^2-a e^2\right )^2 (d+e x)^3}+\frac {16 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 \left (c d^2-a e^2\right )^3 (d+e x)^2}+\frac {\left (16 c^3 d^3\right ) \int \frac {1}{(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{35 \left (c d^2-a e^2\right )^3}\\ &=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 \left (c d^2-a e^2\right ) (d+e x)^4}+\frac {12 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 \left (c d^2-a e^2\right )^2 (d+e x)^3}+\frac {16 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 \left (c d^2-a e^2\right )^3 (d+e x)^2}+\frac {32 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 \left (c d^2-a e^2\right )^4 (d+e x)}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 138, normalized size = 0.60 \begin {gather*} \frac {2 \sqrt {(a e+c d x) (d+e x)} \left (-5 a^3 e^6+3 a^2 c d e^4 (7 d+2 e x)-a c^2 d^2 e^2 \left (35 d^2+28 d e x+8 e^2 x^2\right )+c^3 d^3 \left (35 d^3+70 d^2 e x+56 d e^2 x^2+16 e^3 x^3\right )\right )}{35 \left (c d^2-a e^2\right )^4 (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.86, size = 293, normalized size = 1.27
method | result | size |
trager | \(-\frac {2 \left (-16 c^{3} d^{3} e^{3} x^{3}+8 a \,c^{2} d^{2} e^{4} x^{2}-56 c^{3} d^{4} e^{2} x^{2}-6 a^{2} c d \,e^{5} x +28 a \,c^{2} d^{3} e^{3} x -70 c^{3} d^{5} e x +5 e^{6} a^{3}-21 e^{4} d^{2} a^{2} c +35 d^{4} e^{2} c^{2} a -35 d^{6} c^{3}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{35 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (e x +d \right )^{4}}\) | \(209\) |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-16 c^{3} d^{3} e^{3} x^{3}+8 a \,c^{2} d^{2} e^{4} x^{2}-56 c^{3} d^{4} e^{2} x^{2}-6 a^{2} c d \,e^{5} x +28 a \,c^{2} d^{3} e^{3} x -70 c^{3} d^{5} e x +5 e^{6} a^{3}-21 e^{4} d^{2} a^{2} c +35 d^{4} e^{2} c^{2} a -35 d^{6} c^{3}\right )}{35 \left (e x +d \right )^{3} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}\) | \(217\) |
default | \(\frac {-\frac {2 \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{7 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{4}}-\frac {6 c d e \left (-\frac {2 \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{5 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{3}}-\frac {4 c d e \left (-\frac {2 \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{3 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {4 c d e \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{3 \left (e^{2} a -c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )}\right )}{5 \left (e^{2} a -c \,d^{2}\right )}\right )}{7 \left (e^{2} a -c \,d^{2}\right )}}{e^{4}}\) | \(293\) |
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. 458 vs.
\(2 (219) = 438\).
time = 9.55, size = 458, normalized size = 1.98 \begin {gather*} \frac {2 \, {\left (70 \, c^{3} d^{5} x e + 35 \, c^{3} d^{6} + 6 \, a^{2} c d x e^{5} - 5 \, a^{3} e^{6} - {\left (8 \, a c^{2} d^{2} x^{2} - 21 \, a^{2} c d^{2}\right )} e^{4} + 4 \, {\left (4 \, c^{3} d^{3} x^{3} - 7 \, a c^{2} d^{3} x\right )} e^{3} + 7 \, {\left (8 \, c^{3} d^{4} x^{2} - 5 \, 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}}{35 \, {\left (4 \, c^{4} d^{11} x e + c^{4} d^{12} + a^{4} x^{4} e^{12} + 4 \, a^{4} d x^{3} e^{11} - 2 \, {\left (2 \, a^{3} c d^{2} x^{4} - 3 \, a^{4} d^{2} x^{2}\right )} e^{10} - 4 \, {\left (4 \, a^{3} c d^{3} x^{3} - a^{4} d^{3} x\right )} e^{9} + {\left (6 \, a^{2} c^{2} d^{4} x^{4} - 24 \, a^{3} c d^{4} x^{2} + a^{4} d^{4}\right )} e^{8} + 8 \, {\left (3 \, a^{2} c^{2} d^{5} x^{3} - 2 \, a^{3} c d^{5} x\right )} e^{7} - 4 \, {\left (a c^{3} d^{6} x^{4} - 9 \, a^{2} c^{2} d^{6} x^{2} + a^{3} c d^{6}\right )} e^{6} - 8 \, {\left (2 \, a c^{3} d^{7} x^{3} - 3 \, a^{2} c^{2} d^{7} x\right )} e^{5} + {\left (c^{4} d^{8} x^{4} - 24 \, a c^{3} d^{8} x^{2} + 6 \, a^{2} c^{2} d^{8}\right )} e^{4} + 4 \, {\left (c^{4} d^{9} x^{3} - 4 \, a c^{3} d^{9} x\right )} e^{3} + 2 \, {\left (3 \, c^{4} d^{10} x^{2} - 2 \, a c^{3} d^{10}\right )} e^{2}\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}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (d + e x\right )^{4}}\, dx \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 = 0.81, size = 252, normalized size = 1.09 \begin {gather*} \frac {32\,c^3\,d^3\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{35\,{\left (a\,e^2-c\,d^2\right )}^4\,\left (d+e\,x\right )}-\frac {2\,e\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{\left (7\,a\,e^3-7\,c\,d^2\,e\right )\,{\left (d+e\,x\right )}^4}-\frac {48\,c^2\,d^2\,e\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{35\,{\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 {12\,c\,d\,e\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{7\,\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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