Optimal. Leaf size=295 \[ \frac {6 a^2 e^3+c d x \left (11 a e^2+5 c d^2\right )}{24 a^2 \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )^2}+\frac {8 a^3 e^5+c d x \left (19 a^2 e^4+16 a c d^2 e^2+5 c^2 d^4\right )}{16 a^3 \left (a+c x^2\right ) \left (a e^2+c d^2\right )^3}+\frac {\sqrt {c} d \left (35 a^3 e^6+35 a^2 c d^2 e^4+21 a c^2 d^4 e^2+5 c^3 d^6\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} \left (a e^2+c d^2\right )^4}+\frac {a e+c d x}{6 a \left (a+c x^2\right )^3 \left (a e^2+c d^2\right )}-\frac {e^7 \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )^4}+\frac {e^7 \log (d+e x)}{\left (a e^2+c d^2\right )^4} \]
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Rubi [A] time = 0.38, antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {741, 823, 801, 635, 205, 260} \[ \frac {c d x \left (19 a^2 e^4+16 a c d^2 e^2+5 c^2 d^4\right )+8 a^3 e^5}{16 a^3 \left (a+c x^2\right ) \left (a e^2+c d^2\right )^3}+\frac {\sqrt {c} d \left (35 a^2 c d^2 e^4+35 a^3 e^6+21 a c^2 d^4 e^2+5 c^3 d^6\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} \left (a e^2+c d^2\right )^4}+\frac {6 a^2 e^3+c d x \left (11 a e^2+5 c d^2\right )}{24 a^2 \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )^2}+\frac {a e+c d x}{6 a \left (a+c x^2\right )^3 \left (a e^2+c d^2\right )}-\frac {e^7 \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )^4}+\frac {e^7 \log (d+e x)}{\left (a e^2+c d^2\right )^4} \]
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 635
Rule 741
Rule 801
Rule 823
Rubi steps
\begin {align*} \int \frac {1}{(d+e x) \left (a+c x^2\right )^4} \, dx &=\frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}-\frac {\int \frac {-5 c d^2-6 a e^2-5 c d e x}{(d+e x) \left (a+c x^2\right )^3} \, dx}{6 a \left (c d^2+a e^2\right )}\\ &=\frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}+\frac {6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {\int \frac {3 c \left (5 c^2 d^4+11 a c d^2 e^2+8 a^2 e^4\right )+3 c^2 d e \left (5 c d^2+11 a e^2\right ) x}{(d+e x) \left (a+c x^2\right )^2} \, dx}{24 a^2 c \left (c d^2+a e^2\right )^2}\\ &=\frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}+\frac {6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {8 a^3 e^5+c d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{16 a^3 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}-\frac {\int \frac {-3 c^2 \left (5 c^3 d^6+16 a c^2 d^4 e^2+19 a^2 c d^2 e^4+16 a^3 e^6\right )-3 c^3 d e \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{(d+e x) \left (a+c x^2\right )} \, dx}{48 a^3 c^2 \left (c d^2+a e^2\right )^3}\\ &=\frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}+\frac {6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {8 a^3 e^5+c d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{16 a^3 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}-\frac {\int \left (-\frac {48 a^3 c^2 e^8}{\left (c d^2+a e^2\right ) (d+e x)}-\frac {3 c^3 \left (5 c^3 d^7+21 a c^2 d^5 e^2+35 a^2 c d^3 e^4+35 a^3 d e^6-16 a^3 e^7 x\right )}{\left (c d^2+a e^2\right ) \left (a+c x^2\right )}\right ) \, dx}{48 a^3 c^2 \left (c d^2+a e^2\right )^3}\\ &=\frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}+\frac {6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {8 a^3 e^5+c d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{16 a^3 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}+\frac {e^7 \log (d+e x)}{\left (c d^2+a e^2\right )^4}+\frac {c \int \frac {5 c^3 d^7+21 a c^2 d^5 e^2+35 a^2 c d^3 e^4+35 a^3 d e^6-16 a^3 e^7 x}{a+c x^2} \, dx}{16 a^3 \left (c d^2+a e^2\right )^4}\\ &=\frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}+\frac {6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {8 a^3 e^5+c d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{16 a^3 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}+\frac {e^7 \log (d+e x)}{\left (c d^2+a e^2\right )^4}-\frac {\left (c e^7\right ) \int \frac {x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^4}+\frac {\left (c d \left (5 c^3 d^6+21 a c^2 d^4 e^2+35 a^2 c d^2 e^4+35 a^3 e^6\right )\right ) \int \frac {1}{a+c x^2} \, dx}{16 a^3 \left (c d^2+a e^2\right )^4}\\ &=\frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}+\frac {6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {8 a^3 e^5+c d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{16 a^3 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}+\frac {\sqrt {c} d \left (5 c^3 d^6+21 a c^2 d^4 e^2+35 a^2 c d^2 e^4+35 a^3 e^6\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} \left (c d^2+a e^2\right )^4}+\frac {e^7 \log (d+e x)}{\left (c d^2+a e^2\right )^4}-\frac {e^7 \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^4}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 265, normalized size = 0.90 \[ \frac {\frac {2 \left (a e^2+c d^2\right )^2 \left (6 a^2 e^3+11 a c d e^2 x+5 c^2 d^3 x\right )}{a^2 \left (a+c x^2\right )^2}+\frac {3 \left (a e^2+c d^2\right ) \left (8 a^3 e^5+19 a^2 c d e^4 x+16 a c^2 d^3 e^2 x+5 c^3 d^5 x\right )}{a^3 \left (a+c x^2\right )}+\frac {3 \sqrt {c} d \left (35 a^3 e^6+35 a^2 c d^2 e^4+21 a c^2 d^4 e^2+5 c^3 d^6\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{7/2}}+\frac {8 \left (a e^2+c d^2\right )^3 (a e+c d x)}{a \left (a+c x^2\right )^3}-24 e^7 \log \left (a+c x^2\right )+48 e^7 \log (d+e x)}{48 \left (a e^2+c d^2\right )^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 43.14, size = 1784, normalized size = 6.05 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 530, normalized size = 1.80 \[ -\frac {e^{7} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )}} + \frac {e^{8} \log \left ({\left | x e + d \right |}\right )}{c^{4} d^{8} e + 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} + 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}} + \frac {{\left (5 \, c^{4} d^{7} + 21 \, a c^{3} d^{5} e^{2} + 35 \, a^{2} c^{2} d^{3} e^{4} + 35 \, a^{3} c d e^{6}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \, {\left (a^{3} c^{4} d^{8} + 4 \, a^{4} c^{3} d^{6} e^{2} + 6 \, a^{5} c^{2} d^{4} e^{4} + 4 \, a^{6} c d^{2} e^{6} + a^{7} e^{8}\right )} \sqrt {a c}} + \frac {8 \, a^{3} c^{3} d^{6} e + 36 \, a^{4} c^{2} d^{4} e^{3} + 72 \, a^{5} c d^{2} e^{5} + 44 \, a^{6} e^{7} + 3 \, {\left (5 \, c^{6} d^{7} + 21 \, a c^{5} d^{5} e^{2} + 35 \, a^{2} c^{4} d^{3} e^{4} + 19 \, a^{3} c^{3} d e^{6}\right )} x^{5} + 24 \, {\left (a^{3} c^{3} d^{2} e^{5} + a^{4} c^{2} e^{7}\right )} x^{4} + 8 \, {\left (5 \, a c^{5} d^{7} + 21 \, a^{2} c^{4} d^{5} e^{2} + 33 \, a^{3} c^{3} d^{3} e^{4} + 17 \, a^{4} c^{2} d e^{6}\right )} x^{3} + 12 \, {\left (a^{3} c^{3} d^{4} e^{3} + 6 \, a^{4} c^{2} d^{2} e^{5} + 5 \, a^{5} c e^{7}\right )} x^{2} + 3 \, {\left (11 \, a^{2} c^{4} d^{7} + 43 \, a^{3} c^{3} d^{5} e^{2} + 61 \, a^{4} c^{2} d^{3} e^{4} + 29 \, a^{5} c d e^{6}\right )} x}{48 \, {\left (c d^{2} + a e^{2}\right )}^{4} {\left (c x^{2} + a\right )}^{3} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 941, normalized size = 3.19 \[ \frac {35 c^{4} d^{3} e^{4} x^{5}}{16 \left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{3} a}+\frac {21 c^{5} d^{5} e^{2} x^{5}}{16 \left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{3} a^{2}}+\frac {5 c^{6} d^{7} x^{5}}{16 \left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{3} a^{3}}+\frac {19 c^{3} d \,e^{6} x^{5}}{16 \left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{3}}+\frac {a \,c^{2} e^{7} x^{4}}{2 \left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{3}}+\frac {c^{3} d^{2} e^{5} x^{4}}{2 \left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{3}}+\frac {17 a \,c^{2} d \,e^{6} x^{3}}{6 \left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{3}}+\frac {7 c^{4} d^{5} e^{2} x^{3}}{2 \left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{3} a}+\frac {5 c^{5} d^{7} x^{3}}{6 \left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{3} a^{2}}+\frac {11 c^{3} d^{3} e^{4} x^{3}}{2 \left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{3}}+\frac {5 a^{2} c \,e^{7} x^{2}}{4 \left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{3}}+\frac {3 a \,c^{2} d^{2} e^{5} x^{2}}{2 \left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{3}}+\frac {c^{3} d^{4} e^{3} x^{2}}{4 \left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{3}}+\frac {29 a^{2} c d \,e^{6} x}{16 \left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{3}}+\frac {61 a \,c^{2} d^{3} e^{4} x}{16 \left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{3}}+\frac {11 c^{4} d^{7} x}{16 \left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{3} a}+\frac {43 c^{3} d^{5} e^{2} x}{16 \left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{3}}+\frac {11 a^{3} e^{7}}{12 \left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{3}}+\frac {3 a^{2} c \,d^{2} e^{5}}{2 \left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{3}}+\frac {3 a \,c^{2} d^{4} e^{3}}{4 \left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{3}}+\frac {c^{3} d^{6} e}{6 \left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{3}}+\frac {35 c^{2} d^{3} e^{4} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \left (a \,e^{2}+c \,d^{2}\right )^{4} \sqrt {a c}\, a}+\frac {21 c^{3} d^{5} e^{2} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \left (a \,e^{2}+c \,d^{2}\right )^{4} \sqrt {a c}\, a^{2}}+\frac {5 c^{4} d^{7} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \left (a \,e^{2}+c \,d^{2}\right )^{4} \sqrt {a c}\, a^{3}}+\frac {35 c d \,e^{6} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \left (a \,e^{2}+c \,d^{2}\right )^{4} \sqrt {a c}}-\frac {e^{7} \ln \left (c \,x^{2}+a \right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{4}}+\frac {e^{7} \ln \left (e x +d \right )}{\left (a \,e^{2}+c \,d^{2}\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 3.36, size = 655, normalized size = 2.22 \[ -\frac {e^{7} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )}} + \frac {e^{7} \log \left (e x + d\right )}{c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} + \frac {{\left (5 \, c^{4} d^{7} + 21 \, a c^{3} d^{5} e^{2} + 35 \, a^{2} c^{2} d^{3} e^{4} + 35 \, a^{3} c d e^{6}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \, {\left (a^{3} c^{4} d^{8} + 4 \, a^{4} c^{3} d^{6} e^{2} + 6 \, a^{5} c^{2} d^{4} e^{4} + 4 \, a^{6} c d^{2} e^{6} + a^{7} e^{8}\right )} \sqrt {a c}} + \frac {24 \, a^{3} c^{2} e^{5} x^{4} + 8 \, a^{3} c^{2} d^{4} e + 28 \, a^{4} c d^{2} e^{3} + 44 \, a^{5} e^{5} + 3 \, {\left (5 \, c^{5} d^{5} + 16 \, a c^{4} d^{3} e^{2} + 19 \, a^{2} c^{3} d e^{4}\right )} x^{5} + 8 \, {\left (5 \, a c^{4} d^{5} + 16 \, a^{2} c^{3} d^{3} e^{2} + 17 \, a^{3} c^{2} d e^{4}\right )} x^{3} + 12 \, {\left (a^{3} c^{2} d^{2} e^{3} + 5 \, a^{4} c e^{5}\right )} x^{2} + 3 \, {\left (11 \, a^{2} c^{3} d^{5} + 32 \, a^{3} c^{2} d^{3} e^{2} + 29 \, a^{4} c d e^{4}\right )} x}{48 \, {\left (a^{6} c^{3} d^{6} + 3 \, a^{7} c^{2} d^{4} e^{2} + 3 \, a^{8} c d^{2} e^{4} + a^{9} e^{6} + {\left (a^{3} c^{6} d^{6} + 3 \, a^{4} c^{5} d^{4} e^{2} + 3 \, a^{5} c^{4} d^{2} e^{4} + a^{6} c^{3} e^{6}\right )} x^{6} + 3 \, {\left (a^{4} c^{5} d^{6} + 3 \, a^{5} c^{4} d^{4} e^{2} + 3 \, a^{6} c^{3} d^{2} e^{4} + a^{7} c^{2} e^{6}\right )} x^{4} + 3 \, {\left (a^{5} c^{4} d^{6} + 3 \, a^{6} c^{3} d^{4} e^{2} + 3 \, a^{7} c^{2} d^{2} e^{4} + a^{8} c e^{6}\right )} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.82, size = 1470, normalized size = 4.98 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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