Optimal. Leaf size=132 \[ \frac {b \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^2 \left (b^2-4 a c\right )^{3/2}}-\frac {b x^2}{2 c \left (b^2-4 a c\right )}+\frac {x^4 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\log \left (a+b x^2+c x^4\right )}{4 c^2} \]
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Rubi [A] time = 0.17, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {1114, 738, 773, 634, 618, 206, 628} \[ \frac {b \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^2 \left (b^2-4 a c\right )^{3/2}}+\frac {x^4 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {b x^2}{2 c \left (b^2-4 a c\right )}+\frac {\log \left (a+b x^2+c x^4\right )}{4 c^2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 628
Rule 634
Rule 738
Rule 773
Rule 1114
Rubi steps
\begin {align*} \int \frac {x^7}{\left (a+b x^2+c x^4\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^3}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac {x^4 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\operatorname {Subst}\left (\int \frac {x (4 a+b x)}{a+b x+c x^2} \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )}\\ &=-\frac {b x^2}{2 c \left (b^2-4 a c\right )}+\frac {x^4 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\operatorname {Subst}\left (\int \frac {-a b+\left (-b^2+4 a c\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 c \left (b^2-4 a c\right )}\\ &=-\frac {b x^2}{2 c \left (b^2-4 a c\right )}+\frac {x^4 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\operatorname {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^2}-\frac {\left (b \left (b^2-6 a c\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^2 \left (b^2-4 a c\right )}\\ &=-\frac {b x^2}{2 c \left (b^2-4 a c\right )}+\frac {x^4 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\log \left (a+b x^2+c x^4\right )}{4 c^2}+\frac {\left (b \left (b^2-6 a c\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 c^2 \left (b^2-4 a c\right )}\\ &=-\frac {b x^2}{2 c \left (b^2-4 a c\right )}+\frac {x^4 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {b \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^2 \left (b^2-4 a c\right )^{3/2}}+\frac {\log \left (a+b x^2+c x^4\right )}{4 c^2}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 121, normalized size = 0.92 \[ \frac {\frac {2 \left (-2 a^2 c+a b \left (b-3 c x^2\right )+b^3 x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {2 b \left (b^2-6 a c\right ) \tan ^{-1}\left (\frac {b+2 c x^2}{\sqrt {4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+\log \left (a+b x^2+c x^4\right )}{4 c^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.20, size = 663, normalized size = 5.02 \[ \left [\frac {2 \, a b^{4} - 12 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + 2 \, {\left (b^{5} - 7 \, a b^{3} c + 12 \, a^{2} b c^{2}\right )} x^{2} + {\left ({\left (b^{3} c - 6 \, a b c^{2}\right )} x^{4} + a b^{3} - 6 \, a^{2} b c + {\left (b^{4} - 6 \, a b^{2} c\right )} x^{2}\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c + {\left (2 \, c x^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) + {\left (a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{4} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{2}\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left (a b^{4} c^{2} - 8 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4} + {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} x^{4} + {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} x^{2}\right )}}, \frac {2 \, a b^{4} - 12 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + 2 \, {\left (b^{5} - 7 \, a b^{3} c + 12 \, a^{2} b c^{2}\right )} x^{2} + 2 \, {\left ({\left (b^{3} c - 6 \, a b c^{2}\right )} x^{4} + a b^{3} - 6 \, a^{2} b c + {\left (b^{4} - 6 \, a b^{2} c\right )} x^{2}\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) + {\left (a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{4} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{2}\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left (a b^{4} c^{2} - 8 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4} + {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} x^{4} + {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.60, size = 152, normalized size = 1.15 \[ -\frac {{\left (b^{3} - 6 \, a b c\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {b^{2} c x^{4} - 4 \, a c^{2} x^{4} - b^{3} x^{2} + 2 \, a b c x^{2} - a b^{2}}{4 \, {\left (c x^{4} + b x^{2} + a\right )} {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}} + \frac {\log \left (c x^{4} + b x^{2} + a\right )}{4 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 222, normalized size = 1.68 \[ -\frac {3 a b \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} c}+\frac {b^{3} \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \left (4 a c -b^{2}\right )^{\frac {3}{2}} c^{2}}+\frac {a \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{\left (4 a c -b^{2}\right ) c}-\frac {b^{2} \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 \left (4 a c -b^{2}\right ) c^{2}}+\frac {\frac {\left (3 a c -b^{2}\right ) b \,x^{2}}{\left (4 a c -b^{2}\right ) c^{2}}+\frac {\left (2 a c -b^{2}\right ) a}{\left (4 a c -b^{2}\right ) c^{2}}}{2 c \,x^{4}+2 b \,x^{2}+2 a} \]
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 = 5.10, size = 1336, normalized size = 10.12 \[ \frac {\frac {a\,\left (2\,a\,c-b^2\right )}{2\,c^2\,\left (4\,a\,c-b^2\right )}+\frac {b\,x^2\,\left (3\,a\,c-b^2\right )}{2\,c^2\,\left (4\,a\,c-b^2\right )}}{c\,x^4+b\,x^2+a}-\frac {\ln \left (c\,x^4+b\,x^2+a\right )\,\left (-128\,a^3\,c^3+96\,a^2\,b^2\,c^2-24\,a\,b^4\,c+2\,b^6\right )}{2\,\left (256\,a^3\,c^5-192\,a^2\,b^2\,c^4+48\,a\,b^4\,c^3-4\,b^6\,c^2\right )}+\frac {b\,\mathrm {atan}\left (\frac {\left (8\,a\,c^3\,{\left (4\,a\,c-b^2\right )}^3-2\,b^2\,c^2\,{\left (4\,a\,c-b^2\right )}^3\right )\,\left (x^2\,\left (\frac {\frac {b\,\left (\frac {6\,b^3\,c^2-28\,a\,b\,c^3}{4\,a\,c^3-b^2\,c^2}+\frac {\left (8\,b^3\,c^4-32\,a\,b\,c^5\right )\,\left (-128\,a^3\,c^3+96\,a^2\,b^2\,c^2-24\,a\,b^4\,c+2\,b^6\right )}{2\,\left (4\,a\,c^3-b^2\,c^2\right )\,\left (256\,a^3\,c^5-192\,a^2\,b^2\,c^4+48\,a\,b^4\,c^3-4\,b^6\,c^2\right )}\right )\,\left (6\,a\,c-b^2\right )}{8\,c^2\,{\left (4\,a\,c-b^2\right )}^{3/2}}+\frac {b\,\left (8\,b^3\,c^4-32\,a\,b\,c^5\right )\,\left (6\,a\,c-b^2\right )\,\left (-128\,a^3\,c^3+96\,a^2\,b^2\,c^2-24\,a\,b^4\,c+2\,b^6\right )}{16\,c^2\,{\left (4\,a\,c-b^2\right )}^{3/2}\,\left (4\,a\,c^3-b^2\,c^2\right )\,\left (256\,a^3\,c^5-192\,a^2\,b^2\,c^4+48\,a\,b^4\,c^3-4\,b^6\,c^2\right )}}{a\,\left (4\,a\,c-b^2\right )}-\frac {b\,\left (\frac {b^3-5\,a\,b\,c}{4\,a\,c^3-b^2\,c^2}+\frac {\left (\frac {6\,b^3\,c^2-28\,a\,b\,c^3}{4\,a\,c^3-b^2\,c^2}+\frac {\left (8\,b^3\,c^4-32\,a\,b\,c^5\right )\,\left (-128\,a^3\,c^3+96\,a^2\,b^2\,c^2-24\,a\,b^4\,c+2\,b^6\right )}{2\,\left (4\,a\,c^3-b^2\,c^2\right )\,\left (256\,a^3\,c^5-192\,a^2\,b^2\,c^4+48\,a\,b^4\,c^3-4\,b^6\,c^2\right )}\right )\,\left (-128\,a^3\,c^3+96\,a^2\,b^2\,c^2-24\,a\,b^4\,c+2\,b^6\right )}{2\,\left (256\,a^3\,c^5-192\,a^2\,b^2\,c^4+48\,a\,b^4\,c^3-4\,b^6\,c^2\right )}-\frac {b^2\,\left (\frac {b^3\,c^4}{2}-2\,a\,b\,c^5\right )\,{\left (6\,a\,c-b^2\right )}^2}{c^4\,{\left (4\,a\,c-b^2\right )}^3\,\left (4\,a\,c^3-b^2\,c^2\right )}\right )}{2\,a\,{\left (4\,a\,c-b^2\right )}^{3/2}}\right )-\frac {\frac {b\,\left (6\,a\,c-b^2\right )\,\left (8\,a+\frac {8\,a\,c^2\,\left (-128\,a^3\,c^3+96\,a^2\,b^2\,c^2-24\,a\,b^4\,c+2\,b^6\right )}{256\,a^3\,c^5-192\,a^2\,b^2\,c^4+48\,a\,b^4\,c^3-4\,b^6\,c^2}\right )}{8\,c^2\,{\left (4\,a\,c-b^2\right )}^{3/2}}+\frac {a\,b\,\left (6\,a\,c-b^2\right )\,\left (-128\,a^3\,c^3+96\,a^2\,b^2\,c^2-24\,a\,b^4\,c+2\,b^6\right )}{{\left (4\,a\,c-b^2\right )}^{3/2}\,\left (256\,a^3\,c^5-192\,a^2\,b^2\,c^4+48\,a\,b^4\,c^3-4\,b^6\,c^2\right )}}{a\,\left (4\,a\,c-b^2\right )}+\frac {b\,\left (\frac {a}{c^2}+\frac {\left (8\,a+\frac {8\,a\,c^2\,\left (-128\,a^3\,c^3+96\,a^2\,b^2\,c^2-24\,a\,b^4\,c+2\,b^6\right )}{256\,a^3\,c^5-192\,a^2\,b^2\,c^4+48\,a\,b^4\,c^3-4\,b^6\,c^2}\right )\,\left (-128\,a^3\,c^3+96\,a^2\,b^2\,c^2-24\,a\,b^4\,c+2\,b^6\right )}{2\,\left (256\,a^3\,c^5-192\,a^2\,b^2\,c^4+48\,a\,b^4\,c^3-4\,b^6\,c^2\right )}-\frac {a\,b^2\,{\left (6\,a\,c-b^2\right )}^2}{c^2\,{\left (4\,a\,c-b^2\right )}^3}\right )}{2\,a\,{\left (4\,a\,c-b^2\right )}^{3/2}}\right )}{36\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}\right )\,\left (6\,a\,c-b^2\right )}{2\,c^2\,{\left (4\,a\,c-b^2\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 40.65, size = 745, normalized size = 5.64 \[ \left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{4 c^{2}}\right ) \log {\left (x^{2} + \frac {- 32 a^{2} c^{3} \left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{4 c^{2}}\right ) + 8 a^{2} c + 16 a b^{2} c^{2} \left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{4 c^{2}}\right ) - a b^{2} - 2 b^{4} c \left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{4 c^{2}}\right )}{6 a b c - b^{3}} \right )} + \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{4 c^{2}}\right ) \log {\left (x^{2} + \frac {- 32 a^{2} c^{3} \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{4 c^{2}}\right ) + 8 a^{2} c + 16 a b^{2} c^{2} \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{4 c^{2}}\right ) - a b^{2} - 2 b^{4} c \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{4 c^{2}}\right )}{6 a b c - b^{3}} \right )} + \frac {2 a^{2} c - a b^{2} + x^{2} \left (3 a b c - b^{3}\right )}{8 a^{2} c^{3} - 2 a b^{2} c^{2} + x^{4} \left (8 a c^{4} - 2 b^{2} c^{3}\right ) + x^{2} \left (8 a b c^{3} - 2 b^{3} c^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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