Optimal. Leaf size=93 \[ \frac {5 d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} \sqrt {c}}+\frac {5 d x}{16 a^3 \left (a+c x^2\right )}+\frac {5 d x}{24 a^2 \left (a+c x^2\right )^2}+\frac {c d x-a e}{6 a c \left (a+c x^2\right )^3} \]
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Rubi [A] time = 0.03, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {639, 199, 205} \[ \frac {5 d x}{16 a^3 \left (a+c x^2\right )}+\frac {5 d x}{24 a^2 \left (a+c x^2\right )^2}+\frac {5 d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} \sqrt {c}}-\frac {a e-c d x}{6 a c \left (a+c x^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 199
Rule 205
Rule 639
Rubi steps
\begin {align*} \int \frac {d+e x}{\left (a+c x^2\right )^4} \, dx &=-\frac {a e-c d x}{6 a c \left (a+c x^2\right )^3}+\frac {(5 d) \int \frac {1}{\left (a+c x^2\right )^3} \, dx}{6 a}\\ &=-\frac {a e-c d x}{6 a c \left (a+c x^2\right )^3}+\frac {5 d x}{24 a^2 \left (a+c x^2\right )^2}+\frac {(5 d) \int \frac {1}{\left (a+c x^2\right )^2} \, dx}{8 a^2}\\ &=-\frac {a e-c d x}{6 a c \left (a+c x^2\right )^3}+\frac {5 d x}{24 a^2 \left (a+c x^2\right )^2}+\frac {5 d x}{16 a^3 \left (a+c x^2\right )}+\frac {(5 d) \int \frac {1}{a+c x^2} \, dx}{16 a^3}\\ &=-\frac {a e-c d x}{6 a c \left (a+c x^2\right )^3}+\frac {5 d x}{24 a^2 \left (a+c x^2\right )^2}+\frac {5 d x}{16 a^3 \left (a+c x^2\right )}+\frac {5 d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 83, normalized size = 0.89 \[ \frac {\frac {\sqrt {a} \left (-8 a^3 e+33 a^2 c d x+40 a c^2 d x^3+15 c^3 d x^5\right )}{\left (a+c x^2\right )^3}+15 \sqrt {c} d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{48 a^{7/2} c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 282, normalized size = 3.03 \[ \left [\frac {30 \, a c^{3} d x^{5} + 80 \, a^{2} c^{2} d x^{3} + 66 \, a^{3} c d x - 16 \, a^{4} e - 15 \, {\left (c^{3} d x^{6} + 3 \, a c^{2} d x^{4} + 3 \, a^{2} c d x^{2} + a^{3} d\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right )}{96 \, {\left (a^{4} c^{4} x^{6} + 3 \, a^{5} c^{3} x^{4} + 3 \, a^{6} c^{2} x^{2} + a^{7} c\right )}}, \frac {15 \, a c^{3} d x^{5} + 40 \, a^{2} c^{2} d x^{3} + 33 \, a^{3} c d x - 8 \, a^{4} e + 15 \, {\left (c^{3} d x^{6} + 3 \, a c^{2} d x^{4} + 3 \, a^{2} c d x^{2} + a^{3} d\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right )}{48 \, {\left (a^{4} c^{4} x^{6} + 3 \, a^{5} c^{3} x^{4} + 3 \, a^{6} c^{2} x^{2} + a^{7} c\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 73, normalized size = 0.78 \[ \frac {5 \, d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} a^{3}} + \frac {15 \, c^{3} d x^{5} + 40 \, a c^{2} d x^{3} + 33 \, a^{2} c d x - 8 \, a^{3} e}{48 \, {\left (c x^{2} + a\right )}^{3} a^{3} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 81, normalized size = 0.87 \[ \frac {5 d x}{24 \left (c \,x^{2}+a \right )^{2} a^{2}}+\frac {5 d x}{16 \left (c \,x^{2}+a \right ) a^{3}}+\frac {5 d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \sqrt {a c}\, a^{3}}+\frac {2 c d x -2 a e}{12 \left (c \,x^{2}+a \right )^{3} a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.96, size = 97, normalized size = 1.04 \[ \frac {15 \, c^{3} d x^{5} + 40 \, a c^{2} d x^{3} + 33 \, a^{2} c d x - 8 \, a^{3} e}{48 \, {\left (a^{3} c^{4} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{5} c^{2} x^{2} + a^{6} c\right )}} + \frac {5 \, d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 87, normalized size = 0.94 \[ \frac {\frac {11\,d\,x}{16\,a}-\frac {e}{6\,c}+\frac {5\,c^2\,d\,x^5}{16\,a^3}+\frac {5\,c\,d\,x^3}{6\,a^2}}{a^3+3\,a^2\,c\,x^2+3\,a\,c^2\,x^4+c^3\,x^6}+\frac {5\,d\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{16\,a^{7/2}\,\sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.62, size = 150, normalized size = 1.61 \[ d \left (- \frac {5 \sqrt {- \frac {1}{a^{7} c}} \log {\left (- a^{4} \sqrt {- \frac {1}{a^{7} c}} + x \right )}}{32} + \frac {5 \sqrt {- \frac {1}{a^{7} c}} \log {\left (a^{4} \sqrt {- \frac {1}{a^{7} c}} + x \right )}}{32}\right ) + \frac {- 8 a^{3} e + 33 a^{2} c d x + 40 a c^{2} d x^{3} + 15 c^{3} d x^{5}}{48 a^{6} c + 144 a^{5} c^{2} x^{2} + 144 a^{4} c^{3} x^{4} + 48 a^{3} c^{4} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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