Optimal. Leaf size=203 \[ \frac {\left (c^2 d+b^2 f-c (b e+a f)\right ) x^2}{2 c^3}+\frac {(c e-b f) x^4}{4 c^2}+\frac {f x^6}{6 c}+\frac {\left (b^3 c e-3 a b c^2 e-b^4 f-b^2 c (c d-4 a f)+2 a c^2 (c d-a f)\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^4 \sqrt {b^2-4 a c}}+\frac {\left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 c^4} \]
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Rubi [A]
time = 0.29, 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 = {1677, 1642,
648, 632, 212, 642} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 1642
Rule 1677
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} \text {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} \text {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 {\text {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 ) \text {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 ) \text {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 ) \text {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 = 0.10, size = 193, normalized size = 0.95 \begin {gather*} \frac {6 c \left (c^2 d+b^2 f-c (b e+a f)\right ) x^2+3 c^2 (c e-b f) x^4+2 c^3 f x^6+\frac {6 \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 ) \tan ^{-1}\left (\frac {b+2 c x^2}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}-3 \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 )}{12 c^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 224, normalized size = 1.10
method | result | size |
default | \(-\frac {-\frac {1}{3} f \,x^{6} c^{2}+\frac {1}{2} b c f \,x^{4}-\frac {1}{2} c^{2} e \,x^{4}+a c f \,x^{2}-b^{2} f \,x^{2}+b c e \,x^{2}-c^{2} d \,x^{2}}{2 c^{3}}+\frac {\frac {\left (2 a b c f -a \,c^{2} e -b^{3} f +b^{2} c e -b \,c^{2} d \right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c}+\frac {2 \left (a^{2} c f -a \,b^{2} f +a b c e -a \,c^{2} d -\frac {\left (2 a b c f -a \,c^{2} e -b^{3} f +b^{2} c e -b \,c^{2} d \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{2 c^{3}}\) | \(224\) |
risch | \(\text {Expression too large to display}\) | \(4876\) |
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.56, size = 677, normalized size = 3.33 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.28, size = 214, normalized size = 1.05 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} x^4\,\left (\frac {e}{4\,c}-\frac {b\,f}{4\,c^2}\right )-x^2\,\left (\frac {b\,\left (\frac {e}{c}-\frac {b\,f}{c^2}\right )}{2\,c}-\frac {d}{2\,c}+\frac {a\,f}{2\,c^2}\right )+\frac {\ln \left (c\,x^4+b\,x^2+a\right )\,\left (16\,f\,a^2\,b\,c^2-8\,e\,a^2\,c^3-12\,f\,a\,b^3\,c+10\,e\,a\,b^2\,c^2-8\,d\,a\,b\,c^3+2\,f\,b^5-2\,e\,b^4\,c+2\,d\,b^3\,c^2\right )}{2\,\left (16\,a\,c^5-4\,b^2\,c^4\right )}+\frac {f\,x^6}{6\,c}+\frac {\mathrm {atan}\left (\frac {2\,c^6\,\left (4\,a\,c-b^2\right )\,\left (x^2\,\left (\frac {\frac {\left (\frac {4\,f\,a^2\,c^6-16\,f\,a\,b^2\,c^5+10\,e\,a\,b\,c^6-4\,d\,a\,c^7+6\,f\,b^4\,c^4-6\,e\,b^3\,c^5+6\,d\,b^2\,c^6}{c^6}+\frac {4\,b\,c^2\,\left (16\,f\,a^2\,b\,c^2-8\,e\,a^2\,c^3-12\,f\,a\,b^3\,c+10\,e\,a\,b^2\,c^2-8\,d\,a\,b\,c^3+2\,f\,b^5-2\,e\,b^4\,c+2\,d\,b^3\,c^2\right )}{16\,a\,c^5-4\,b^2\,c^4}\right )\,\left (2\,f\,a^2\,c^2-4\,f\,a\,b^2\,c+3\,e\,a\,b\,c^2-2\,d\,a\,c^3+f\,b^4-e\,b^3\,c+d\,b^2\,c^2\right )}{8\,c^4\,\sqrt {4\,a\,c-b^2}}+\frac {b\,\left (2\,f\,a^2\,c^2-4\,f\,a\,b^2\,c+3\,e\,a\,b\,c^2-2\,d\,a\,c^3+f\,b^4-e\,b^3\,c+d\,b^2\,c^2\right )\,\left (16\,f\,a^2\,b\,c^2-8\,e\,a^2\,c^3-12\,f\,a\,b^3\,c+10\,e\,a\,b^2\,c^2-8\,d\,a\,b\,c^3+2\,f\,b^5-2\,e\,b^4\,c+2\,d\,b^3\,c^2\right )}{2\,c^2\,\sqrt {4\,a\,c-b^2}\,\left (16\,a\,c^5-4\,b^2\,c^4\right )}}{a}-\frac {b\,\left (\frac {-2\,a^3\,b\,c^3\,f^2+a^3\,c^4\,e\,f+7\,a^2\,b^3\,c^2\,f^2-8\,a^2\,b^2\,c^3\,e\,f+3\,a^2\,b\,c^4\,d\,f+2\,a^2\,b\,c^4\,e^2-a^2\,c^5\,d\,e-5\,a\,b^5\,c\,f^2+8\,a\,b^4\,c^2\,e\,f-6\,a\,b^3\,c^3\,d\,f-3\,a\,b^3\,c^3\,e^2+4\,a\,b^2\,c^4\,d\,e-a\,b\,c^5\,d^2+b^7\,f^2-2\,b^6\,c\,e\,f+2\,b^5\,c^2\,d\,f+b^5\,c^2\,e^2-2\,b^4\,c^3\,d\,e+b^3\,c^4\,d^2}{c^6}+\frac {\left (\frac {4\,f\,a^2\,c^6-16\,f\,a\,b^2\,c^5+10\,e\,a\,b\,c^6-4\,d\,a\,c^7+6\,f\,b^4\,c^4-6\,e\,b^3\,c^5+6\,d\,b^2\,c^6}{c^6}+\frac {4\,b\,c^2\,\left (16\,f\,a^2\,b\,c^2-8\,e\,a^2\,c^3-12\,f\,a\,b^3\,c+10\,e\,a\,b^2\,c^2-8\,d\,a\,b\,c^3+2\,f\,b^5-2\,e\,b^4\,c+2\,d\,b^3\,c^2\right )}{16\,a\,c^5-4\,b^2\,c^4}\right )\,\left (16\,f\,a^2\,b\,c^2-8\,e\,a^2\,c^3-12\,f\,a\,b^3\,c+10\,e\,a\,b^2\,c^2-8\,d\,a\,b\,c^3+2\,f\,b^5-2\,e\,b^4\,c+2\,d\,b^3\,c^2\right )}{2\,\left (16\,a\,c^5-4\,b^2\,c^4\right )}-\frac {b\,{\left (2\,f\,a^2\,c^2-4\,f\,a\,b^2\,c+3\,e\,a\,b\,c^2-2\,d\,a\,c^3+f\,b^4-e\,b^3\,c+d\,b^2\,c^2\right )}^2}{2\,c^6\,\left (4\,a\,c-b^2\right )}\right )}{2\,a\,\sqrt {4\,a\,c-b^2}}\right )+\frac {\frac {\left (\frac {-16\,f\,a^2\,b\,c^5+8\,e\,a^2\,c^6+8\,f\,a\,b^3\,c^4-8\,e\,a\,b^2\,c^5+8\,d\,a\,b\,c^6}{c^6}+\frac {8\,a\,c^2\,\left (16\,f\,a^2\,b\,c^2-8\,e\,a^2\,c^3-12\,f\,a\,b^3\,c+10\,e\,a\,b^2\,c^2-8\,d\,a\,b\,c^3+2\,f\,b^5-2\,e\,b^4\,c+2\,d\,b^3\,c^2\right )}{16\,a\,c^5-4\,b^2\,c^4}\right )\,\left (2\,f\,a^2\,c^2-4\,f\,a\,b^2\,c+3\,e\,a\,b\,c^2-2\,d\,a\,c^3+f\,b^4-e\,b^3\,c+d\,b^2\,c^2\right )}{8\,c^4\,\sqrt {4\,a\,c-b^2}}+\frac {a\,\left (2\,f\,a^2\,c^2-4\,f\,a\,b^2\,c+3\,e\,a\,b\,c^2-2\,d\,a\,c^3+f\,b^4-e\,b^3\,c+d\,b^2\,c^2\right )\,\left (16\,f\,a^2\,b\,c^2-8\,e\,a^2\,c^3-12\,f\,a\,b^3\,c+10\,e\,a\,b^2\,c^2-8\,d\,a\,b\,c^3+2\,f\,b^5-2\,e\,b^4\,c+2\,d\,b^3\,c^2\right )}{c^2\,\sqrt {4\,a\,c-b^2}\,\left (16\,a\,c^5-4\,b^2\,c^4\right )}}{a}-\frac {b\,\left (\frac {4\,a^3\,b^2\,c^2\,f^2-4\,a^3\,b\,c^3\,e\,f+a^3\,c^4\,e^2-4\,a^2\,b^4\,c\,f^2+6\,a^2\,b^3\,c^2\,e\,f-4\,a^2\,b^2\,c^3\,d\,f-2\,a^2\,b^2\,c^3\,e^2+2\,a^2\,b\,c^4\,d\,e+a\,b^6\,f^2-2\,a\,b^5\,c\,e\,f+2\,a\,b^4\,c^2\,d\,f+a\,b^4\,c^2\,e^2-2\,a\,b^3\,c^3\,d\,e+a\,b^2\,c^4\,d^2}{c^6}+\frac {\left (\frac {-16\,f\,a^2\,b\,c^5+8\,e\,a^2\,c^6+8\,f\,a\,b^3\,c^4-8\,e\,a\,b^2\,c^5+8\,d\,a\,b\,c^6}{c^6}+\frac {8\,a\,c^2\,\left (16\,f\,a^2\,b\,c^2-8\,e\,a^2\,c^3-12\,f\,a\,b^3\,c+10\,e\,a\,b^2\,c^2-8\,d\,a\,b\,c^3+2\,f\,b^5-2\,e\,b^4\,c+2\,d\,b^3\,c^2\right )}{16\,a\,c^5-4\,b^2\,c^4}\right )\,\left (16\,f\,a^2\,b\,c^2-8\,e\,a^2\,c^3-12\,f\,a\,b^3\,c+10\,e\,a\,b^2\,c^2-8\,d\,a\,b\,c^3+2\,f\,b^5-2\,e\,b^4\,c+2\,d\,b^3\,c^2\right )}{2\,\left (16\,a\,c^5-4\,b^2\,c^4\right )}-\frac {a\,{\left (2\,f\,a^2\,c^2-4\,f\,a\,b^2\,c+3\,e\,a\,b\,c^2-2\,d\,a\,c^3+f\,b^4-e\,b^3\,c+d\,b^2\,c^2\right )}^2}{c^6\,\left (4\,a\,c-b^2\right )}\right )}{2\,a\,\sqrt {4\,a\,c-b^2}}\right )}{4\,a^4\,c^4\,f^2-16\,a^3\,b^2\,c^3\,f^2+12\,a^3\,b\,c^4\,e\,f-8\,a^3\,c^5\,d\,f+20\,a^2\,b^4\,c^2\,f^2-28\,a^2\,b^3\,c^3\,e\,f+20\,a^2\,b^2\,c^4\,d\,f+9\,a^2\,b^2\,c^4\,e^2-12\,a^2\,b\,c^5\,d\,e+4\,a^2\,c^6\,d^2-8\,a\,b^6\,c\,f^2+14\,a\,b^5\,c^2\,e\,f-12\,a\,b^4\,c^3\,d\,f-6\,a\,b^4\,c^3\,e^2+10\,a\,b^3\,c^4\,d\,e-4\,a\,b^2\,c^5\,d^2+b^8\,f^2-2\,b^7\,c\,e\,f+2\,b^6\,c^2\,d\,f+b^6\,c^2\,e^2-2\,b^5\,c^3\,d\,e+b^4\,c^4\,d^2}\right )\,\left (2\,f\,a^2\,c^2-4\,f\,a\,b^2\,c+3\,e\,a\,b\,c^2-2\,d\,a\,c^3+f\,b^4-e\,b^3\,c+d\,b^2\,c^2\right )}{2\,c^4\,\sqrt {4\,a\,c-b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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