Optimal. Leaf size=145 \[ \frac {\left (a e^2+5 c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} c^{3/2}}+\frac {x \left (a e^2+5 c d^2\right )}{16 a^3 c \left (a+c x^2\right )}-\frac {4 a d e-x \left (a e^2+5 c d^2\right )}{24 a^2 c \left (a+c x^2\right )^2}-\frac {(d+e x) (a e-c d x)}{6 a c \left (a+c x^2\right )^3} \]
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Rubi [A] time = 0.06, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {739, 639, 199, 205} \[ \frac {\left (a e^2+5 c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} c^{3/2}}+\frac {x \left (a e^2+5 c d^2\right )}{16 a^3 c \left (a+c x^2\right )}-\frac {4 a d e-x \left (a e^2+5 c d^2\right )}{24 a^2 c \left (a+c x^2\right )^2}-\frac {(d+e x) (a e-c d x)}{6 a c \left (a+c x^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 199
Rule 205
Rule 639
Rule 739
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{\left (a+c x^2\right )^4} \, dx &=-\frac {(a e-c d x) (d+e x)}{6 a c \left (a+c x^2\right )^3}+\frac {\int \frac {5 c d^2+a e^2+4 c d e x}{\left (a+c x^2\right )^3} \, dx}{6 a c}\\ &=-\frac {(a e-c d x) (d+e x)}{6 a c \left (a+c x^2\right )^3}-\frac {4 a d e-\left (5 c d^2+a e^2\right ) x}{24 a^2 c \left (a+c x^2\right )^2}+\frac {\left (5 c d^2+a e^2\right ) \int \frac {1}{\left (a+c x^2\right )^2} \, dx}{8 a^2 c}\\ &=-\frac {(a e-c d x) (d+e x)}{6 a c \left (a+c x^2\right )^3}-\frac {4 a d e-\left (5 c d^2+a e^2\right ) x}{24 a^2 c \left (a+c x^2\right )^2}+\frac {\left (5 c d^2+a e^2\right ) x}{16 a^3 c \left (a+c x^2\right )}+\frac {\left (5 c d^2+a e^2\right ) \int \frac {1}{a+c x^2} \, dx}{16 a^3 c}\\ &=-\frac {(a e-c d x) (d+e x)}{6 a c \left (a+c x^2\right )^3}-\frac {4 a d e-\left (5 c d^2+a e^2\right ) x}{24 a^2 c \left (a+c x^2\right )^2}+\frac {\left (5 c d^2+a e^2\right ) x}{16 a^3 c \left (a+c x^2\right )}+\frac {\left (5 c d^2+a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 127, normalized size = 0.88 \[ \frac {\left (a e^2+5 c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} c^{3/2}}+\frac {-a^3 e (16 d+3 e x)+a^2 c x \left (33 d^2+8 e^2 x^2\right )+a c^2 x^3 \left (40 d^2+3 e^2 x^2\right )+15 c^3 d^2 x^5}{48 a^3 c \left (a+c x^2\right )^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 490, normalized size = 3.38 \[ \left [-\frac {32 \, a^{4} c d e - 6 \, {\left (5 \, a c^{4} d^{2} + a^{2} c^{3} e^{2}\right )} x^{5} - 16 \, {\left (5 \, a^{2} c^{3} d^{2} + a^{3} c^{2} e^{2}\right )} x^{3} + 3 \, {\left ({\left (5 \, c^{4} d^{2} + a c^{3} e^{2}\right )} x^{6} + 5 \, a^{3} c d^{2} + a^{4} e^{2} + 3 \, {\left (5 \, a c^{3} d^{2} + a^{2} c^{2} e^{2}\right )} x^{4} + 3 \, {\left (5 \, a^{2} c^{2} d^{2} + a^{3} c e^{2}\right )} x^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) - 6 \, {\left (11 \, a^{3} c^{2} d^{2} - a^{4} c e^{2}\right )} x}{96 \, {\left (a^{4} c^{5} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{6} c^{3} x^{2} + a^{7} c^{2}\right )}}, -\frac {16 \, a^{4} c d e - 3 \, {\left (5 \, a c^{4} d^{2} + a^{2} c^{3} e^{2}\right )} x^{5} - 8 \, {\left (5 \, a^{2} c^{3} d^{2} + a^{3} c^{2} e^{2}\right )} x^{3} - 3 \, {\left ({\left (5 \, c^{4} d^{2} + a c^{3} e^{2}\right )} x^{6} + 5 \, a^{3} c d^{2} + a^{4} e^{2} + 3 \, {\left (5 \, a c^{3} d^{2} + a^{2} c^{2} e^{2}\right )} x^{4} + 3 \, {\left (5 \, a^{2} c^{2} d^{2} + a^{3} c e^{2}\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) - 3 \, {\left (11 \, a^{3} c^{2} d^{2} - a^{4} c e^{2}\right )} x}{48 \, {\left (a^{4} c^{5} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{6} c^{3} x^{2} + a^{7} c^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 123, normalized size = 0.85 \[ \frac {{\left (5 \, c d^{2} + a e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} a^{3} c} + \frac {15 \, c^{3} d^{2} x^{5} + 3 \, a c^{2} x^{5} e^{2} + 40 \, a c^{2} d^{2} x^{3} + 8 \, a^{2} c x^{3} e^{2} + 33 \, a^{2} c d^{2} x - 3 \, a^{3} x e^{2} - 16 \, a^{3} d e}{48 \, {\left (c x^{2} + a\right )}^{3} a^{3} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 129, normalized size = 0.89 \[ \frac {e^{2} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \sqrt {a c}\, a^{2} c}+\frac {5 d^{2} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \sqrt {a c}\, a^{3}}+\frac {\frac {\left (a \,e^{2}+5 c \,d^{2}\right ) c \,x^{5}}{16 a^{3}}+\frac {\left (a \,e^{2}+5 c \,d^{2}\right ) x^{3}}{6 a^{2}}-\frac {d e}{3 c}-\frac {\left (a \,e^{2}-11 c \,d^{2}\right ) x}{16 a c}}{\left (c \,x^{2}+a \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.07, size = 151, normalized size = 1.04 \[ \frac {3 \, {\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )} x^{5} - 16 \, a^{3} d e + 8 \, {\left (5 \, a c^{2} d^{2} + a^{2} c e^{2}\right )} x^{3} + 3 \, {\left (11 \, a^{2} c d^{2} - a^{3} e^{2}\right )} x}{48 \, {\left (a^{3} c^{4} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{5} c^{2} x^{2} + a^{6} c\right )}} + \frac {{\left (5 \, c d^{2} + a e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} a^{3} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.36, size = 132, normalized size = 0.91 \[ \frac {\frac {x^3\,\left (5\,c\,d^2+a\,e^2\right )}{6\,a^2}-\frac {d\,e}{3\,c}-\frac {x\,\left (a\,e^2-11\,c\,d^2\right )}{16\,a\,c}+\frac {c\,x^5\,\left (5\,c\,d^2+a\,e^2\right )}{16\,a^3}}{a^3+3\,a^2\,c\,x^2+3\,a\,c^2\,x^4+c^3\,x^6}+\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (5\,c\,d^2+a\,e^2\right )}{16\,a^{7/2}\,c^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.04, size = 214, normalized size = 1.48 \[ - \frac {\sqrt {- \frac {1}{a^{7} c^{3}}} \left (a e^{2} + 5 c d^{2}\right ) \log {\left (- a^{4} c \sqrt {- \frac {1}{a^{7} c^{3}}} + x \right )}}{32} + \frac {\sqrt {- \frac {1}{a^{7} c^{3}}} \left (a e^{2} + 5 c d^{2}\right ) \log {\left (a^{4} c \sqrt {- \frac {1}{a^{7} c^{3}}} + x \right )}}{32} + \frac {- 16 a^{3} d e + x^{5} \left (3 a c^{2} e^{2} + 15 c^{3} d^{2}\right ) + x^{3} \left (8 a^{2} c e^{2} + 40 a c^{2} d^{2}\right ) + x \left (- 3 a^{3} e^{2} + 33 a^{2} c d^{2}\right )}{48 a^{6} c + 144 a^{5} c^{2} x^{2} + 144 a^{4} c^{3} x^{4} + 48 a^{3} c^{4} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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