Optimal. Leaf size=155 \[ \frac {\left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} c^{5/2}}-\frac {(d+e x) \left (a e^2+5 c d^2\right ) (a e-c d x)}{16 a^3 c^2 \left (a+c x^2\right )}-\frac {(d+e x)^3 (a e-5 c d x)}{24 a^2 c \left (a+c x^2\right )^2}+\frac {x (d+e x)^4}{6 a \left (a+c x^2\right )^3} \]
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Rubi [A] time = 0.08, antiderivative size = 155, 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 = {737, 805, 723, 205} \begin {gather*} -\frac {(d+e x) \left (a e^2+5 c d^2\right ) (a e-c d x)}{16 a^3 c^2 \left (a+c x^2\right )}+\frac {\left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} c^{5/2}}-\frac {(d+e x)^3 (a e-5 c d x)}{24 a^2 c \left (a+c x^2\right )^2}+\frac {x (d+e x)^4}{6 a \left (a+c x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 723
Rule 737
Rule 805
Rubi steps
\begin {align*} \int \frac {(d+e x)^4}{\left (a+c x^2\right )^4} \, dx &=\frac {x (d+e x)^4}{6 a \left (a+c x^2\right )^3}-\frac {\int \frac {(-5 d-e x) (d+e x)^3}{\left (a+c x^2\right )^3} \, dx}{6 a}\\ &=\frac {x (d+e x)^4}{6 a \left (a+c x^2\right )^3}-\frac {(a e-5 c d x) (d+e x)^3}{24 a^2 c \left (a+c x^2\right )^2}+\frac {\left (5 c d^2+a e^2\right ) \int \frac {(d+e x)^2}{\left (a+c x^2\right )^2} \, dx}{8 a^2 c}\\ &=\frac {x (d+e x)^4}{6 a \left (a+c x^2\right )^3}-\frac {(a e-5 c d x) (d+e x)^3}{24 a^2 c \left (a+c x^2\right )^2}-\frac {\left (5 c d^2+a e^2\right ) (a e-c d x) (d+e x)}{16 a^3 c^2 \left (a+c x^2\right )}+\frac {\left (\left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )\right ) \int \frac {1}{a+c x^2} \, dx}{16 a^3 c^2}\\ &=\frac {x (d+e x)^4}{6 a \left (a+c x^2\right )^3}-\frac {(a e-5 c d x) (d+e x)^3}{24 a^2 c \left (a+c x^2\right )^2}-\frac {\left (5 c d^2+a e^2\right ) (a e-c d x) (d+e x)}{16 a^3 c^2 \left (a+c x^2\right )}+\frac {\left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 197, normalized size = 1.27 \begin {gather*} \frac {\left (a^2 e^4+6 a c d^2 e^2+5 c^2 d^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} c^{5/2}}+\frac {-a^4 e^3 (16 d+3 e x)-2 a^3 c e \left (16 d^3+9 d^2 e x+24 d e^2 x^2+4 e^3 x^3\right )+3 a^2 c^2 x \left (11 d^4+16 d^2 e^2 x^2+e^4 x^4\right )+2 a c^3 d^2 x^3 \left (20 d^2+9 e^2 x^2\right )+15 c^4 d^4 x^5}{48 a^3 c^2 \left (a+c x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^4}{\left (a+c x^2\right )^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.42, size = 746, normalized size = 4.81 \begin {gather*} \left [-\frac {96 \, a^{4} c^{2} d e^{3} x^{2} + 64 \, a^{4} c^{2} d^{3} e + 32 \, a^{5} c d e^{3} - 6 \, {\left (5 \, a c^{5} d^{4} + 6 \, a^{2} c^{4} d^{2} e^{2} + a^{3} c^{3} e^{4}\right )} x^{5} - 16 \, {\left (5 \, a^{2} c^{4} d^{4} + 6 \, a^{3} c^{3} d^{2} e^{2} - a^{4} c^{2} e^{4}\right )} x^{3} + 3 \, {\left (5 \, a^{3} c^{2} d^{4} + 6 \, a^{4} c d^{2} e^{2} + a^{5} e^{4} + {\left (5 \, c^{5} d^{4} + 6 \, a c^{4} d^{2} e^{2} + a^{2} c^{3} e^{4}\right )} x^{6} + 3 \, {\left (5 \, a c^{4} d^{4} + 6 \, a^{2} c^{3} d^{2} e^{2} + a^{3} c^{2} e^{4}\right )} x^{4} + 3 \, {\left (5 \, a^{2} c^{3} d^{4} + 6 \, a^{3} c^{2} d^{2} e^{2} + a^{4} c e^{4}\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^{3} d^{4} - 6 \, a^{4} c^{2} d^{2} e^{2} - a^{5} c e^{4}\right )} x}{96 \, {\left (a^{4} c^{6} x^{6} + 3 \, a^{5} c^{5} x^{4} + 3 \, a^{6} c^{4} x^{2} + a^{7} c^{3}\right )}}, -\frac {48 \, a^{4} c^{2} d e^{3} x^{2} + 32 \, a^{4} c^{2} d^{3} e + 16 \, a^{5} c d e^{3} - 3 \, {\left (5 \, a c^{5} d^{4} + 6 \, a^{2} c^{4} d^{2} e^{2} + a^{3} c^{3} e^{4}\right )} x^{5} - 8 \, {\left (5 \, a^{2} c^{4} d^{4} + 6 \, a^{3} c^{3} d^{2} e^{2} - a^{4} c^{2} e^{4}\right )} x^{3} - 3 \, {\left (5 \, a^{3} c^{2} d^{4} + 6 \, a^{4} c d^{2} e^{2} + a^{5} e^{4} + {\left (5 \, c^{5} d^{4} + 6 \, a c^{4} d^{2} e^{2} + a^{2} c^{3} e^{4}\right )} x^{6} + 3 \, {\left (5 \, a c^{4} d^{4} + 6 \, a^{2} c^{3} d^{2} e^{2} + a^{3} c^{2} e^{4}\right )} x^{4} + 3 \, {\left (5 \, a^{2} c^{3} d^{4} + 6 \, a^{3} c^{2} d^{2} e^{2} + a^{4} c e^{4}\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) - 3 \, {\left (11 \, a^{3} c^{3} d^{4} - 6 \, a^{4} c^{2} d^{2} e^{2} - a^{5} c e^{4}\right )} x}{48 \, {\left (a^{4} c^{6} x^{6} + 3 \, a^{5} c^{5} x^{4} + 3 \, a^{6} c^{4} x^{2} + a^{7} c^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 205, normalized size = 1.32 \begin {gather*} \frac {{\left (5 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} a^{3} c^{2}} + \frac {15 \, c^{4} d^{4} x^{5} + 18 \, a c^{3} d^{2} x^{5} e^{2} + 40 \, a c^{3} d^{4} x^{3} + 3 \, a^{2} c^{2} x^{5} e^{4} + 48 \, a^{2} c^{2} d^{2} x^{3} e^{2} + 33 \, a^{2} c^{2} d^{4} x - 8 \, a^{3} c x^{3} e^{4} - 48 \, a^{3} c d x^{2} e^{3} - 18 \, a^{3} c d^{2} x e^{2} - 32 \, a^{3} c d^{3} e - 3 \, a^{4} x e^{4} - 16 \, a^{4} d e^{3}}{48 \, {\left (c x^{2} + a\right )}^{3} a^{3} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 225, normalized size = 1.45 \begin {gather*} \frac {e^{4} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \sqrt {a c}\, a \,c^{2}}+\frac {3 d^{2} e^{2} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \sqrt {a c}\, a^{2} c}+\frac {5 d^{4} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \sqrt {a c}\, a^{3}}+\frac {-\frac {d \,e^{3} x^{2}}{c}+\frac {\left (a^{2} e^{4}+6 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right ) x^{5}}{16 a^{3}}-\frac {\left (a^{2} e^{4}-6 a c \,d^{2} e^{2}-5 c^{2} d^{4}\right ) x^{3}}{6 a^{2} c}-\frac {\left (a \,e^{2}+2 c \,d^{2}\right ) d e}{3 c^{2}}-\frac {\left (a^{2} e^{4}+6 a c \,d^{2} e^{2}-11 c^{2} d^{4}\right ) x}{16 a \,c^{2}}}{\left (c \,x^{2}+a \right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.04, size = 235, normalized size = 1.52 \begin {gather*} -\frac {48 \, a^{3} c d e^{3} x^{2} + 32 \, a^{3} c d^{3} e + 16 \, a^{4} d e^{3} - 3 \, {\left (5 \, c^{4} d^{4} + 6 \, a c^{3} d^{2} e^{2} + a^{2} c^{2} e^{4}\right )} x^{5} - 8 \, {\left (5 \, a c^{3} d^{4} + 6 \, a^{2} c^{2} d^{2} e^{2} - a^{3} c e^{4}\right )} x^{3} - 3 \, {\left (11 \, a^{2} c^{2} d^{4} - 6 \, a^{3} c d^{2} e^{2} - a^{4} e^{4}\right )} x}{48 \, {\left (a^{3} c^{5} x^{6} + 3 \, a^{4} c^{4} x^{4} + 3 \, a^{5} c^{3} x^{2} + a^{6} c^{2}\right )}} + \frac {{\left (5 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} a^{3} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.44, size = 263, normalized size = 1.70 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x\,\left (c\,d^2+a\,e^2\right )\,\left (5\,c\,d^2+a\,e^2\right )}{\sqrt {a}\,\left (a^2\,e^4+6\,a\,c\,d^2\,e^2+5\,c^2\,d^4\right )}\right )\,\left (c\,d^2+a\,e^2\right )\,\left (5\,c\,d^2+a\,e^2\right )}{16\,a^{7/2}\,c^{5/2}}-\frac {\frac {d\,e^3\,x^2}{c}-\frac {x^5\,\left (a^2\,e^4+6\,a\,c\,d^2\,e^2+5\,c^2\,d^4\right )}{16\,a^3}+\frac {x\,\left (a^2\,e^4+6\,a\,c\,d^2\,e^2-11\,c^2\,d^4\right )}{16\,a\,c^2}+\frac {d\,e\,\left (2\,c\,d^2+a\,e^2\right )}{3\,c^2}-\frac {x^3\,\left (-a^2\,e^4+6\,a\,c\,d^2\,e^2+5\,c^2\,d^4\right )}{6\,a^2\,c}}{a^3+3\,a^2\,c\,x^2+3\,a\,c^2\,x^4+c^3\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.15, size = 413, normalized size = 2.66 \begin {gather*} - \frac {\sqrt {- \frac {1}{a^{7} c^{5}}} \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right ) \log {\left (- \frac {a^{4} c^{2} \sqrt {- \frac {1}{a^{7} c^{5}}} \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right )}{a^{2} e^{4} + 6 a c d^{2} e^{2} + 5 c^{2} d^{4}} + x \right )}}{32} + \frac {\sqrt {- \frac {1}{a^{7} c^{5}}} \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right ) \log {\left (\frac {a^{4} c^{2} \sqrt {- \frac {1}{a^{7} c^{5}}} \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right )}{a^{2} e^{4} + 6 a c d^{2} e^{2} + 5 c^{2} d^{4}} + x \right )}}{32} + \frac {- 16 a^{4} d e^{3} - 32 a^{3} c d^{3} e - 48 a^{3} c d e^{3} x^{2} + x^{5} \left (3 a^{2} c^{2} e^{4} + 18 a c^{3} d^{2} e^{2} + 15 c^{4} d^{4}\right ) + x^{3} \left (- 8 a^{3} c e^{4} + 48 a^{2} c^{2} d^{2} e^{2} + 40 a c^{3} d^{4}\right ) + x \left (- 3 a^{4} e^{4} - 18 a^{3} c d^{2} e^{2} + 33 a^{2} c^{2} d^{4}\right )}{48 a^{6} c^{2} + 144 a^{5} c^{3} x^{2} + 144 a^{4} c^{4} x^{4} + 48 a^{3} c^{5} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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