Optimal. Leaf size=254 \[ -\frac{x^2 \left (x^2 \left (16 a^2 B c^2+6 a A b c^2-15 a b^2 B c+2 b^4 B\right )+2 a \left (6 a A c^2-7 a b B c+b^3 B\right )\right )}{4 c^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\left (-12 a^2 A c^3+30 a^2 b B c^2-10 a b^3 B c+b^5 B\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^3 \left (b^2-4 a c\right )^{5/2}}-\frac{x^6 \left (x^2 \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{4 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{B \log \left (a+b x^2+c x^4\right )}{4 c^3} \]
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Rubi [A] time = 0.404194, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {1251, 818, 634, 618, 206, 628} \[ -\frac{x^2 \left (x^2 \left (16 a^2 B c^2+6 a A b c^2-15 a b^2 B c+2 b^4 B\right )+2 a \left (6 a A c^2-7 a b B c+b^3 B\right )\right )}{4 c^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\left (-12 a^2 A c^3+30 a^2 b B c^2-10 a b^3 B c+b^5 B\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^3 \left (b^2-4 a c\right )^{5/2}}-\frac{x^6 \left (x^2 \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{4 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{B \log \left (a+b x^2+c x^4\right )}{4 c^3} \]
Antiderivative was successfully verified.
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Rule 1251
Rule 818
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{x^9 \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^4 (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx,x,x^2\right )\\ &=-\frac{x^6 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x^2\right )}{4 c \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 \left (3 a (b B-2 A c)+2 B \left (b^2-4 a c\right ) x\right )}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )}{4 c \left (b^2-4 a c\right )}\\ &=-\frac{x^6 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x^2\right )}{4 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{x^2 \left (2 a \left (b^3 B-7 a b B c+6 a A c^2\right )+\left (2 b^4 B-15 a b^2 B c+6 a A b c^2+16 a^2 B c^2\right ) x^2\right )}{4 c^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\operatorname{Subst}\left (\int \frac{2 a \left (b^3 B-7 a b B c+6 a A c^2\right )+2 B \left (b^2-4 a c\right )^2 x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^2 \left (b^2-4 a c\right )^2}\\ &=-\frac{x^6 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x^2\right )}{4 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{x^2 \left (2 a \left (b^3 B-7 a b B c+6 a A c^2\right )+\left (2 b^4 B-15 a b^2 B c+6 a A b c^2+16 a^2 B c^2\right ) x^2\right )}{4 c^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{B \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^3}-\frac{\left (b^5 B-10 a b^3 B c+30 a^2 b B c^2-12 a^2 A c^3\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^3 \left (b^2-4 a c\right )^2}\\ &=-\frac{x^6 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x^2\right )}{4 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{x^2 \left (2 a \left (b^3 B-7 a b B c+6 a A c^2\right )+\left (2 b^4 B-15 a b^2 B c+6 a A b c^2+16 a^2 B c^2\right ) x^2\right )}{4 c^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{B \log \left (a+b x^2+c x^4\right )}{4 c^3}+\frac{\left (b^5 B-10 a b^3 B c+30 a^2 b B c^2-12 a^2 A c^3\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 c^3 \left (b^2-4 a c\right )^2}\\ &=-\frac{x^6 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x^2\right )}{4 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{x^2 \left (2 a \left (b^3 B-7 a b B c+6 a A c^2\right )+\left (2 b^4 B-15 a b^2 B c+6 a A b c^2+16 a^2 B c^2\right ) x^2\right )}{4 c^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\left (b^5 B-10 a b^3 B c+30 a^2 b B c^2-12 a^2 A c^3\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^3 \left (b^2-4 a c\right )^{5/2}}+\frac{B \log \left (a+b x^2+c x^4\right )}{4 c^3}\\ \end{align*}
Mathematica [A] time = 0.525902, size = 354, normalized size = 1.39 \[ \frac{\frac{2 a^2 b c^3 \left (11 A+25 B x^2\right )+4 a^2 c^3 \left (8 a B-5 A c x^2\right )-2 a b^3 c^2 \left (4 A+15 B x^2\right )+a b^2 c^2 \left (16 A c x^2-39 a B\right )+b^4 c \left (11 a B-2 A c x^2\right )+b^5 c \left (A+4 B x^2\right )+b^6 (-B)}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{a^2 c \left (b c \left (3 A+5 B x^2\right )-2 A c^2 x^2-4 b^2 B\right )+2 a^3 B c^2+a b^2 \left (-b c \left (A+5 B x^2\right )+4 A c^2 x^2+b^2 B\right )+b^4 x^2 (b B-A c)}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{2 c \left (-12 a^2 A c^3+30 a^2 b B c^2-10 a b^3 B c+b^5 B\right ) \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}+B c \log \left (a+b x^2+c x^4\right )}{4 c^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 723, normalized size = 2.9 \begin{align*}{\frac{1}{2\, \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{2}} \left ( -{\frac{ \left ( 10\,A{a}^{2}{c}^{3}-8\,Aa{b}^{2}{c}^{2}+A{b}^{4}c-25\,B{a}^{2}b{c}^{2}+15\,Ba{b}^{3}c-2\,B{b}^{5} \right ){x}^{6}}{{c}^{2} \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) }}+{\frac{ \left ( 2\,A{a}^{2}b{c}^{3}+8\,Aa{b}^{3}{c}^{2}-A{b}^{5}c+32\,B{a}^{3}{c}^{3}+11\,B{a}^{2}{b}^{2}{c}^{2}-19\,Ba{b}^{4}c+3\,B{b}^{6} \right ){x}^{4}}{2\,{c}^{3} \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) }}-{\frac{a \left ( 6\,A{a}^{2}{c}^{3}-10\,Aa{b}^{2}{c}^{2}+A{b}^{4}c-31\,B{a}^{2}b{c}^{2}+22\,Ba{b}^{3}c-3\,B{b}^{5} \right ){x}^{2}}{{c}^{3} \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) }}+{\frac{{a}^{2} \left ( 10\,aAb{c}^{2}-A{b}^{3}c+24\,{a}^{2}B{c}^{2}-21\,a{b}^{2}Bc+3\,{b}^{4}B \right ) }{2\,{c}^{3} \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) }} \right ) }+4\,{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){a}^{2}B}{c \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) }}-2\,{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) a{b}^{2}B}{{c}^{2} \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) }}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{4}B}{4\,{c}^{3} \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) }}+6\,{\frac{A{a}^{2}}{ \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-15\,{\frac{B{a}^{2}b}{c \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+5\,{\frac{Ba{b}^{3}}{{c}^{2} \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{B{b}^{5}}{2\,{c}^{3} \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) }\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.04094, size = 4617, normalized size = 18.18 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 39.023, size = 629, normalized size = 2.48 \begin{align*} -\frac{{\left (B b^{5} - 10 \, B a b^{3} c + 30 \, B a^{2} b c^{2} - 12 \, A a^{2} c^{3}\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \,{\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{B \log \left (c x^{4} + b x^{2} + a\right )}{4 \, c^{3}} - \frac{3 \, B b^{4} c^{2} x^{8} - 24 \, B a b^{2} c^{3} x^{8} + 48 \, B a^{2} c^{4} x^{8} - 2 \, B b^{5} c x^{6} + 12 \, B a b^{3} c^{2} x^{6} + 4 \, A b^{4} c^{2} x^{6} - 4 \, B a^{2} b c^{3} x^{6} - 32 \, A a b^{2} c^{3} x^{6} + 40 \, A a^{2} c^{4} x^{6} - 3 \, B b^{6} x^{4} + 20 \, B a b^{4} c x^{4} + 2 \, A b^{5} c x^{4} - 22 \, B a^{2} b^{2} c^{2} x^{4} - 16 \, A a b^{3} c^{2} x^{4} + 32 \, B a^{3} c^{3} x^{4} - 4 \, A a^{2} b c^{3} x^{4} - 6 \, B a b^{5} x^{2} + 40 \, B a^{2} b^{3} c x^{2} + 4 \, A a b^{4} c x^{2} - 28 \, B a^{3} b c^{2} x^{2} - 40 \, A a^{2} b^{2} c^{2} x^{2} + 24 \, A a^{3} c^{3} x^{2} - 3 \, B a^{2} b^{4} + 18 \, B a^{3} b^{2} c + 2 \, A a^{2} b^{3} c - 20 \, A a^{3} b c^{2}}{8 \,{\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )}{\left (c x^{4} + b x^{2} + a\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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