Optimal. Leaf size=203 \[ \frac {x^2 \left (-c (a f+b e)+b^2 f+c^2 d\right )}{2 c^3}+\frac {\log \left (a+b x^2+c x^4\right ) \left (-b c (c d-2 a f)-a c^2 e+b^3 (-f)+b^2 c e\right )}{4 c^4}+\frac {\tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-b^2 c (c d-4 a f)-3 a b c^2 e+2 a c^2 (c d-a f)+b^4 (-f)+b^3 c e\right )}{2 c^4 \sqrt {b^2-4 a c}}+\frac {x^4 (c e-b f)}{4 c^2}+\frac {f x^6}{6 c} \]
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Rubi [A] time = 0.42, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1663, 1628, 634, 618, 206, 628} \[ \frac {x^2 \left (-c (a f+b e)+b^2 f+c^2 d\right )}{2 c^3}+\frac {\log \left (a+b x^2+c x^4\right ) \left (-b c (c d-2 a f)-a c^2 e+b^2 c e+b^3 (-f)\right )}{4 c^4}+\frac {\tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-b^2 c (c d-4 a f)-3 a b c^2 e+2 a c^2 (c d-a f)+b^3 c e+b^4 (-f)\right )}{2 c^4 \sqrt {b^2-4 a c}}+\frac {x^4 (c e-b f)}{4 c^2}+\frac {f x^6}{6 c} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 628
Rule 634
Rule 1628
Rule 1663
Rubi steps
\begin {align*} \int \frac {x^5 \left (d+e x^2+f x^4\right )}{a+b x^2+c x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2 \left (d+e x+f x^2\right )}{a+b x+c x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {c^2 d+b^2 f-c (b e+a f)}{c^3}+\frac {(c e-b f) x}{c^2}+\frac {f x^2}{c}-\frac {a \left (c^2 d+b^2 f-c (b e+a f)\right )-\left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right ) x}{c^3 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {\left (c^2 d+b^2 f-c (b e+a f)\right ) x^2}{2 c^3}+\frac {(c e-b f) x^4}{4 c^2}+\frac {f x^6}{6 c}-\frac {\operatorname {Subst}\left (\int \frac {a \left (c^2 d+b^2 f-c (b e+a f)\right )-\left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 c^3}\\ &=\frac {\left (c^2 d+b^2 f-c (b e+a f)\right ) x^2}{2 c^3}+\frac {(c e-b f) x^4}{4 c^2}+\frac {f x^6}{6 c}+\frac {\left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right ) \operatorname {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^4}-\frac {\left (b^3 c e-3 a b c^2 e-b^4 f-b^2 c (c d-4 a f)+2 a c^2 (c d-a f)\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^4}\\ &=\frac {\left (c^2 d+b^2 f-c (b e+a f)\right ) x^2}{2 c^3}+\frac {(c e-b f) x^4}{4 c^2}+\frac {f x^6}{6 c}+\frac {\left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 c^4}+\frac {\left (b^3 c e-3 a b c^2 e-b^4 f-b^2 c (c d-4 a f)+2 a c^2 (c d-a f)\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 c^4}\\ &=\frac {\left (c^2 d+b^2 f-c (b e+a f)\right ) x^2}{2 c^3}+\frac {(c e-b f) x^4}{4 c^2}+\frac {f x^6}{6 c}+\frac {\left (b^3 c e-3 a b c^2 e-b^4 f-b^2 c (c d-4 a f)+2 a c^2 (c d-a f)\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^4 \sqrt {b^2-4 a c}}+\frac {\left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 c^4}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 193, normalized size = 0.95 \[ \frac {6 c x^2 \left (-c (a f+b e)+b^2 f+c^2 d\right )-3 \log \left (a+b x^2+c x^4\right ) \left (b c (c d-2 a f)+a c^2 e+b^3 f-b^2 c e\right )+\frac {6 \tan ^{-1}\left (\frac {b+2 c x^2}{\sqrt {4 a c-b^2}}\right ) \left (b^2 c (c d-4 a f)+3 a b c^2 e+2 a c^2 (a f-c d)+b^4 f-b^3 c e\right )}{\sqrt {4 a c-b^2}}+3 c^2 x^4 (c e-b f)+2 c^3 f x^6}{12 c^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.96, size = 677, normalized size = 3.33 \[ \left [\frac {2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} f x^{6} + 3 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} e - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} f\right )} x^{4} + 6 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} e + {\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} f\right )} x^{2} + 3 \, \sqrt {b^{2} - 4 \, a c} {\left ({\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d - {\left (b^{3} c - 3 \, a b c^{2}\right )} e + {\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} f\right )} \log \left (\frac {2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c - {\left (2 \, c x^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) - 3 \, {\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d - {\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} e + {\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} f\right )} \log \left (c x^{4} + b x^{2} + a\right )}{12 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )}}, \frac {2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} f x^{6} + 3 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} e - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} f\right )} x^{4} + 6 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} e + {\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} f\right )} x^{2} - 6 \, \sqrt {-b^{2} + 4 \, a c} {\left ({\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d - {\left (b^{3} c - 3 \, a b c^{2}\right )} e + {\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} f\right )} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - 3 \, {\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d - {\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} e + {\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} f\right )} \log \left (c x^{4} + b x^{2} + a\right )}{12 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.00, size = 214, normalized size = 1.05 \[ \frac {2 \, c^{2} f x^{6} - 3 \, b c f x^{4} + 3 \, c^{2} x^{4} e + 6 \, c^{2} d x^{2} + 6 \, b^{2} f x^{2} - 6 \, a c f x^{2} - 6 \, b c x^{2} e}{12 \, c^{3}} - \frac {{\left (b c^{2} d + b^{3} f - 2 \, a b c f - b^{2} c e + a c^{2} e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, c^{4}} + \frac {{\left (b^{2} c^{2} d - 2 \, a c^{3} d + b^{4} f - 4 \, a b^{2} c f + 2 \, a^{2} c^{2} f - b^{3} c e + 3 \, a b c^{2} e\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt {-b^{2} + 4 \, a c} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 474, normalized size = 2.33 \[ \frac {f \,x^{6}}{6 c}-\frac {b f \,x^{4}}{4 c^{2}}+\frac {e \,x^{4}}{4 c}+\frac {a^{2} f \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{2}}-\frac {2 a \,b^{2} f \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{3}}+\frac {3 a b e \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, c^{2}}-\frac {a d \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c}-\frac {a f \,x^{2}}{2 c^{2}}+\frac {b^{4} f \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, c^{4}}-\frac {b^{3} e \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, c^{3}}+\frac {b^{2} d \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, c^{2}}+\frac {b^{2} f \,x^{2}}{2 c^{3}}-\frac {b e \,x^{2}}{2 c^{2}}+\frac {d \,x^{2}}{2 c}+\frac {a b f \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c^{3}}-\frac {a e \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 c^{2}}-\frac {b^{3} f \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 c^{4}}+\frac {b^{2} e \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 c^{3}}-\frac {b d \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 c^{2}} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.63, size = 2295, normalized size = 11.31 \[ \text {result too large to display} \]
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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