Optimal. Leaf size=185 \[ \frac {\left (3 a b B-A \left (2 a c+b^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac {x^4 \left (-2 a B-\left (x^2 (b B-2 A c)\right )+A b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {a \left (8 a B c-6 A b c+b^2 B\right )+x^2 \left (4 a A c^2+2 a b B c-4 A b^2 c+b^3 B\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )} \]
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Rubi [A] time = 0.26, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1251, 820, 777, 618, 206} \[ -\frac {x^2 \left (4 a A c^2+2 a b B c-4 A b^2 c+b^3 B\right )+a \left (8 a B c-6 A b c+b^2 B\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {x^4 \left (-2 a B+x^2 (-(b B-2 A c))+A b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {\left (3 a b B-A \left (2 a c+b^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 777
Rule 820
Rule 1251
Rubi steps
\begin {align*} \int \frac {x^5 \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2 (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx,x,x^2\right )\\ &=-\frac {x^4 \left (A b-2 a B-(b B-2 A c) x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\operatorname {Subst}\left (\int \frac {x (-2 (A b-2 a B)-(b B-2 A c) x)}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )}{4 \left (b^2-4 a c\right )}\\ &=-\frac {x^4 \left (A b-2 a B-(b B-2 A c) x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {a \left (b^2 B-6 A b c+8 a B c\right )+\left (b^3 B-4 A b^2 c+2 a b B c+4 a A c^2\right ) x^2}{4 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {\left (3 a b B-A \left (b^2+2 a c\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )^2}\\ &=-\frac {x^4 \left (A b-2 a B-(b B-2 A c) x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {a \left (b^2 B-6 A b c+8 a B c\right )+\left (b^3 B-4 A b^2 c+2 a b B c+4 a A c^2\right ) x^2}{4 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\left (3 a b B-A \left (b^2+2 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 )}{\left (b^2-4 a c\right )^2}\\ &=-\frac {x^4 \left (A b-2 a B-(b B-2 A c) x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {a \left (b^2 B-6 A b c+8 a B c\right )+\left (b^3 B-4 A b^2 c+2 a b B c+4 a A c^2\right ) x^2}{4 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\left (3 a b B-A \left (b^2+2 a c\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 233, normalized size = 1.26 \[ \frac {1}{4} \left (\frac {2 a^2 B c+a \left (b c \left (A+3 B x^2\right )-2 A c^2 x^2+b^2 (-B)\right )+b^2 x^2 (A c-b B)}{c^2 \left (4 a c-b^2\right ) \left (a+b x^2+c x^4\right )^2}+\frac {4 \left (A \left (2 a c+b^2\right )-3 a b B\right ) \tan ^{-1}\left (\frac {b+2 c x^2}{\sqrt {4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}+\frac {b^2 c \left (5 a B+2 A c x^2\right )+2 a b c^2 \left (A-3 B x^2\right )+4 a c^2 \left (A c x^2-4 a B\right )+A b^3 c+b^4 (-B)}{c^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.62, size = 1369, normalized size = 7.40 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 6.59, size = 268, normalized size = 1.45 \[ -\frac {{\left (3 \, B a b - A b^{2} - 2 \, A a c\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {6 \, B a b c^{2} x^{6} - 2 \, A b^{2} c^{2} x^{6} - 4 \, A a c^{3} x^{6} + B b^{4} x^{4} + B a b^{2} c x^{4} - 3 \, A b^{3} c x^{4} + 16 \, B a^{2} c^{2} x^{4} - 6 \, A a b c^{2} x^{4} + 2 \, B a b^{3} x^{2} + 10 \, B a^{2} b c x^{2} - 10 \, A a b^{2} c x^{2} + 4 \, A a^{2} c^{2} x^{2} + B a^{2} b^{2} + 8 \, B a^{3} c - 6 \, A a^{2} b c}{4 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} {\left (c x^{4} + b x^{2} + a\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 411, normalized size = 2.22 \[ \frac {2 A a c \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}}+\frac {A \,b^{2} \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}}-\frac {3 B a b \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}}+\frac {\frac {\left (2 a A c +A \,b^{2}-3 a b B \right ) c \,x^{6}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {\left (6 a A b \,c^{2}+3 A \,b^{3} c -16 a^{2} B \,c^{2}-a \,b^{2} B c -b^{4} B \right ) x^{4}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c}-\frac {\left (2 a A \,c^{2}-5 A \,b^{2} c +5 a b B c +b^{3} B \right ) a \,x^{2}}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c}+\frac {\left (6 A b c -8 a B c -b^{2} B \right ) a^{2}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c}}{2 \left (c \,x^{4}+b \,x^{2}+a \right )^{2}} \]
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 = 0.68, size = 625, normalized size = 3.38 \[ \frac {\mathrm {atan}\left (\frac {\left (x^2\,\left (\frac {\left (A\,b^2\,c^2-3\,B\,a\,b\,c^2+2\,A\,a\,c^3\right )\,\left (A\,b^2-3\,B\,a\,b+2\,A\,a\,c\right )}{a\,{\left (4\,a\,c-b^2\right )}^{9/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {b\,{\left (A\,b^2-3\,B\,a\,b+2\,A\,a\,c\right )}^2\,\left (32\,a^2\,b\,c^4-16\,a\,b^3\,c^3+2\,b^5\,c^2\right )}{2\,a\,{\left (4\,a\,c-b^2\right )}^{15/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )+\frac {2\,b\,c^2\,{\left (A\,b^2-3\,B\,a\,b+2\,A\,a\,c\right )}^2}{{\left (4\,a\,c-b^2\right )}^{15/2}}\right )\,\left (b^4\,{\left (4\,a\,c-b^2\right )}^5+16\,a^2\,c^2\,{\left (4\,a\,c-b^2\right )}^5-8\,a\,b^2\,c\,{\left (4\,a\,c-b^2\right )}^5\right )}{8\,A^2\,a^2\,c^4+8\,A^2\,a\,b^2\,c^3+2\,A^2\,b^4\,c^2-24\,A\,B\,a^2\,b\,c^3-12\,A\,B\,a\,b^3\,c^2+18\,B^2\,a^2\,b^2\,c^2}\right )\,\left (A\,b^2-3\,B\,a\,b+2\,A\,a\,c\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}}-\frac {\frac {x^4\,\left (16\,B\,a^2\,c^2+B\,a\,b^2\,c-6\,A\,a\,b\,c^2+B\,b^4-3\,A\,b^3\,c\right )}{4\,c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}-\frac {c\,x^6\,\left (A\,b^2-3\,B\,a\,b+2\,A\,a\,c\right )}{2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {a\,\left (8\,B\,c\,a^2+B\,a\,b^2-6\,A\,c\,a\,b\right )}{4\,c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {x^2\,\left (5\,B\,a^2\,b\,c+2\,A\,a^2\,c^2+B\,a\,b^3-5\,A\,a\,b^2\,c\right )}{2\,c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{x^4\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^8+2\,a\,b\,x^2+2\,b\,c\,x^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 44.84, size = 833, normalized size = 4.50 \[ \frac {\sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (- 2 A a c - A b^{2} + 3 B a b\right ) \log {\left (x^{2} + \frac {- 2 A a b c - A b^{3} + 3 B a b^{2} - 64 a^{3} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (- 2 A a c - A b^{2} + 3 B a b\right ) + 48 a^{2} b^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (- 2 A a c - A b^{2} + 3 B a b\right ) - 12 a b^{4} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (- 2 A a c - A b^{2} + 3 B a b\right ) + b^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (- 2 A a c - A b^{2} + 3 B a b\right )}{- 4 A a c^{2} - 2 A b^{2} c + 6 B a b c} \right )}}{2} - \frac {\sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (- 2 A a c - A b^{2} + 3 B a b\right ) \log {\left (x^{2} + \frac {- 2 A a b c - A b^{3} + 3 B a b^{2} + 64 a^{3} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (- 2 A a c - A b^{2} + 3 B a b\right ) - 48 a^{2} b^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (- 2 A a c - A b^{2} + 3 B a b\right ) + 12 a b^{4} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (- 2 A a c - A b^{2} + 3 B a b\right ) - b^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (- 2 A a c - A b^{2} + 3 B a b\right )}{- 4 A a c^{2} - 2 A b^{2} c + 6 B a b c} \right )}}{2} + \frac {6 A a^{2} b c - 8 B a^{3} c - B a^{2} b^{2} + x^{6} \left (4 A a c^{3} + 2 A b^{2} c^{2} - 6 B a b c^{2}\right ) + x^{4} \left (6 A a b c^{2} + 3 A b^{3} c - 16 B a^{2} c^{2} - B a b^{2} c - B b^{4}\right ) + x^{2} \left (- 4 A a^{2} c^{2} + 10 A a b^{2} c - 10 B a^{2} b c - 2 B a b^{3}\right )}{64 a^{4} c^{3} - 32 a^{3} b^{2} c^{2} + 4 a^{2} b^{4} c + x^{8} \left (64 a^{2} c^{5} - 32 a b^{2} c^{4} + 4 b^{4} c^{3}\right ) + x^{6} \left (128 a^{2} b c^{4} - 64 a b^{3} c^{3} + 8 b^{5} c^{2}\right ) + x^{4} \left (128 a^{3} c^{4} - 24 a b^{4} c^{2} + 4 b^{6} c\right ) + x^{2} \left (128 a^{3} b c^{3} - 64 a^{2} b^{3} c^{2} + 8 a b^{5} c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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