Optimal. Leaf size=146 \[ \frac {3 a (A b-2 a B) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac {3 x^2 \left (2 a+b x^2\right ) (A b-2 a B)}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {x^6 \left (-2 a B-\left (x^2 (b B-2 A c)\right )+A b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]
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Rubi [A] time = 0.14, antiderivative size = 146, 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, 804, 722, 618, 206} \[ -\frac {x^6 \left (-2 a B+x^2 (-(b B-2 A c))+A b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {3 x^2 \left (2 a+b x^2\right ) (A b-2 a B)}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {3 a (A b-2 a B) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 722
Rule 804
Rule 1251
Rubi steps
\begin {align*} \int \frac {x^7 \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^3 (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx,x,x^2\right )\\ &=-\frac {x^6 \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 {(3 (A b-2 a B)) \operatorname {Subst}\left (\int \frac {x^2}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )}{4 \left (b^2-4 a c\right )}\\ &=-\frac {x^6 \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 {3 (A b-2 a B) x^2 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {(3 a (A b-2 a B)) \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^6 \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 {3 (A b-2 a B) x^2 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {(3 a (A b-2 a B)) \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^6 \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 {3 (A b-2 a B) x^2 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {3 a (A b-2 a B) \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 = 0.27, size = 261, normalized size = 1.79 \[ \frac {1}{4} \left (\frac {a^2 c \left (2 c \left (A+B x^2\right )-3 b B\right )+a b \left (-b c \left (A+4 B x^2\right )+3 A c^2 x^2+b^2 B\right )+b^3 x^2 (b B-A c)}{c^3 \left (4 a c-b^2\right ) \left (a+b x^2+c x^4\right )^2}+\frac {-4 a^2 c^3 \left (4 A+5 B x^2\right )+a b^2 c^2 \left (5 A+16 B x^2\right )+2 a b c^2 \left (11 a B-3 A c x^2\right )-8 a b^3 B c-b^4 c \left (A+2 B x^2\right )+b^5 B}{c^3 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {12 a (A b-2 a B) \tan ^{-1}\left (\frac {b+2 c x^2}{\sqrt {4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 1378, normalized size = 9.44 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 6.46, size = 318, normalized size = 2.18 \[ \frac {3 \, {\left (2 \, B a^{2} - A a b\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {2 \, B b^{4} c x^{6} - 16 \, B a b^{2} c^{2} x^{6} + 20 \, B a^{2} c^{3} x^{6} + 6 \, A a b c^{3} x^{6} + B b^{5} x^{4} - 8 \, B a b^{3} c x^{4} + A b^{4} c x^{4} - 2 \, B a^{2} b c^{2} x^{4} + A a b^{2} c^{2} x^{4} + 16 \, A a^{2} c^{3} x^{4} + 2 \, B a b^{4} x^{2} - 20 \, B a^{2} b^{2} c x^{2} + 2 \, A a b^{3} c x^{2} + 12 \, B a^{3} c^{2} x^{2} + 10 \, A a^{2} b c^{2} x^{2} + B a^{2} b^{3} - 10 \, B a^{3} b c + A a^{2} b^{2} c + 8 \, A a^{3} c^{2}}{4 \, {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} {\left (c x^{4} + b x^{2} + a\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 398, normalized size = 2.73 \[ -\frac {3 A a b \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}}+\frac {6 B \,a^{2} \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}}+\frac {-\frac {\left (3 a A b \,c^{2}+10 a^{2} B \,c^{2}-8 a \,b^{2} B c +b^{4} B \right ) x^{6}}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c}-\frac {\left (16 A \,a^{2} c^{3}+A a \,b^{2} c^{2}+A \,b^{4} c -2 B \,a^{2} b \,c^{2}-8 B a \,b^{3} c +B \,b^{5}\right ) x^{4}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c^{2}}-\frac {\left (5 a A b \,c^{2}+A \,b^{3} c +6 a^{2} B \,c^{2}-10 a \,b^{2} B c +b^{4} B \right ) a \,x^{2}}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c^{2}}-\frac {\left (8 a A \,c^{2}+A \,b^{2} c -10 a b B c +b^{3} B \right ) a^{2}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c^{2}}}{2 \left (c \,x^{4}+b \,x^{2}+a \right )^{2}} \]
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 = 0.69, size = 593, normalized size = 4.06 \[ \frac {3\,a\,\mathrm {atan}\left (\frac {\left (x^2\,\left (\frac {3\,\left (A\,b-2\,B\,a\right )\,\left (6\,B\,a^2\,c^2-3\,A\,a\,b\,c^2\right )}{{\left (4\,a\,c-b^2\right )}^{9/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}-\frac {9\,a\,b\,{\left (A\,b-2\,B\,a\right )}^2\,\left (32\,a^2\,b\,c^4-16\,a\,b^3\,c^3+2\,b^5\,c^2\right )}{2\,{\left (4\,a\,c-b^2\right )}^{15/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )-\frac {18\,a^2\,b\,c^2\,{\left (A\,b-2\,B\,a\right )}^2}{{\left (4\,a\,c-b^2\right )}^{15/2}}\right )\,\left (b^4\,{\left (4\,a\,c-b^2\right )}^5+16\,a^2\,c^2\,{\left (4\,a\,c-b^2\right )}^5-8\,a\,b^2\,c\,{\left (4\,a\,c-b^2\right )}^5\right )}{18\,A^2\,a^2\,b^2\,c^2-72\,A\,B\,a^3\,b\,c^2+72\,B^2\,a^4\,c^2}\right )\,\left (A\,b-2\,B\,a\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}}-\frac {\frac {x^4\,\left (-2\,B\,a^2\,b\,c^2+16\,A\,a^2\,c^3-8\,B\,a\,b^3\,c+A\,a\,b^2\,c^2+B\,b^5+A\,b^4\,c\right )}{4\,c^2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {a^2\,\left (B\,b^3+A\,b^2\,c-10\,B\,a\,b\,c+8\,A\,a\,c^2\right )}{4\,c^2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {x^6\,\left (10\,B\,a^2\,c^2-8\,B\,a\,b^2\,c+3\,A\,a\,b\,c^2+B\,b^4\right )}{2\,c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {a\,x^2\,\left (6\,B\,a^2\,c^2-10\,B\,a\,b^2\,c+5\,A\,a\,b\,c^2+B\,b^4+A\,b^3\,c\right )}{2\,c^2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{x^4\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^8+2\,a\,b\,x^2+2\,b\,c\,x^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 102.04, size = 775, normalized size = 5.31 \[ - \frac {3 a \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (- A b + 2 B a\right ) \log {\left (x^{2} + \frac {- 3 A a b^{2} + 6 B a^{2} b - 192 a^{4} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (- A b + 2 B a\right ) + 144 a^{3} b^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (- A b + 2 B a\right ) - 36 a^{2} b^{4} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (- A b + 2 B a\right ) + 3 a b^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (- A b + 2 B a\right )}{- 6 A a b c + 12 B a^{2} c} \right )}}{2} + \frac {3 a \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (- A b + 2 B a\right ) \log {\left (x^{2} + \frac {- 3 A a b^{2} + 6 B a^{2} b + 192 a^{4} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (- A b + 2 B a\right ) - 144 a^{3} b^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (- A b + 2 B a\right ) + 36 a^{2} b^{4} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (- A b + 2 B a\right ) - 3 a b^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (- A b + 2 B a\right )}{- 6 A a b c + 12 B a^{2} c} \right )}}{2} + \frac {- 8 A a^{3} c^{2} - A a^{2} b^{2} c + 10 B a^{3} b c - B a^{2} b^{3} + x^{6} \left (- 6 A a b c^{3} - 20 B a^{2} c^{3} + 16 B a b^{2} c^{2} - 2 B b^{4} c\right ) + x^{4} \left (- 16 A a^{2} c^{3} - A a b^{2} c^{2} - A b^{4} c + 2 B a^{2} b c^{2} + 8 B a b^{3} c - B b^{5}\right ) + x^{2} \left (- 10 A a^{2} b c^{2} - 2 A a b^{3} c - 12 B a^{3} c^{2} + 20 B a^{2} b^{2} c - 2 B a b^{4}\right )}{64 a^{4} c^{4} - 32 a^{3} b^{2} c^{3} + 4 a^{2} b^{4} c^{2} + x^{8} \left (64 a^{2} c^{6} - 32 a b^{2} c^{5} + 4 b^{4} c^{4}\right ) + x^{6} \left (128 a^{2} b c^{5} - 64 a b^{3} c^{4} + 8 b^{5} c^{3}\right ) + x^{4} \left (128 a^{3} c^{5} - 24 a b^{4} c^{3} + 4 b^{6} c^{2}\right ) + x^{2} \left (128 a^{3} b c^{4} - 64 a^{2} b^{3} c^{3} + 8 a b^{5} c^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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