Optimal. Leaf size=412 \[ \frac{\left (-\frac{A c \left (4 a c+b^2\right )+b C \left (b^2-8 a c\right )}{\sqrt{b^2-4 a c}}+C \left (b^2-6 a c\right )+A b c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} c^{3/2} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (\frac{A c \left (4 a c+b^2\right )+b C \left (b^2-8 a c\right )}{\sqrt{b^2-4 a c}}+C \left (b^2-6 a c\right )+A b c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} c^{3/2} \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{x^3 \left (-2 a C+x^2 (2 A c-b C)+A b\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{x (2 A c-b C)}{2 c \left (b^2-4 a c\right )}+\frac{B x^2 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{2 a B \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \]
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Rubi [A] time = 1.33379, antiderivative size = 412, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {1662, 1275, 1279, 1166, 205, 12, 1114, 722, 618, 206} \[ \frac{\left (-\frac{A c \left (4 a c+b^2\right )+b C \left (b^2-8 a c\right )}{\sqrt{b^2-4 a c}}+C \left (b^2-6 a c\right )+A b c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} c^{3/2} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (\frac{A c \left (4 a c+b^2\right )+b C \left (b^2-8 a c\right )}{\sqrt{b^2-4 a c}}+C \left (b^2-6 a c\right )+A b c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} c^{3/2} \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{x^3 \left (-2 a C+x^2 (2 A c-b C)+A b\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{x (2 A c-b C)}{2 c \left (b^2-4 a c\right )}+\frac{B x^2 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{2 a B \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 1662
Rule 1275
Rule 1279
Rule 1166
Rule 205
Rule 12
Rule 1114
Rule 722
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{x^4 \left (A+B x+C x^2\right )}{\left (a+b x^2+c x^4\right )^2} \, dx &=\int \frac{B x^5}{\left (a+b x^2+c x^4\right )^2} \, dx+\int \frac{x^4 \left (A+C x^2\right )}{\left (a+b x^2+c x^4\right )^2} \, dx\\ &=-\frac{x^3 \left (A b-2 a C+(2 A c-b C) x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+B \int \frac{x^5}{\left (a+b x^2+c x^4\right )^2} \, dx+\frac{\int \frac{x^2 \left (3 (A b-2 a C)+(2 A c-b C) x^2\right )}{a+b x^2+c x^4} \, dx}{2 \left (b^2-4 a c\right )}\\ &=\frac{(2 A c-b C) x}{2 c \left (b^2-4 a c\right )}-\frac{x^3 \left (A b-2 a C+(2 A c-b C) x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{1}{2} B \operatorname{Subst}\left (\int \frac{x^2}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )-\frac{\int \frac{a (2 A c-b C)+\left (-A b c-\left (b^2-6 a c\right ) C\right ) x^2}{a+b x^2+c x^4} \, dx}{2 c \left (b^2-4 a c\right )}\\ &=\frac{(2 A c-b C) x}{2 c \left (b^2-4 a c\right )}+\frac{B x^2 \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{x^3 \left (A b-2 a C+(2 A c-b C) x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{(a B) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{b^2-4 a c}+\frac{\left (A b c+\left (b^2-6 a c\right ) C-\frac{A c \left (b^2+4 a c\right )+b \left (b^2-8 a c\right ) C}{\sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{4 c \left (b^2-4 a c\right )}+\frac{\left (A b c+\left (b^2-6 a c\right ) C+\frac{A c \left (b^2+4 a c\right )+b \left (b^2-8 a c\right ) C}{\sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{4 c \left (b^2-4 a c\right )}\\ &=\frac{(2 A c-b C) x}{2 c \left (b^2-4 a c\right )}+\frac{B x^2 \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{x^3 \left (A b-2 a C+(2 A c-b C) x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\left (A b c+\left (b^2-6 a c\right ) C-\frac{A c \left (b^2+4 a c\right )+b \left (b^2-8 a c\right ) C}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} c^{3/2} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (A b c+\left (b^2-6 a c\right ) C+\frac{A c \left (b^2+4 a c\right )+b \left (b^2-8 a c\right ) C}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} c^{3/2} \left (b^2-4 a c\right ) \sqrt{b+\sqrt{b^2-4 a c}}}+\frac{(2 a B) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{b^2-4 a c}\\ &=\frac{(2 A c-b C) x}{2 c \left (b^2-4 a c\right )}+\frac{B x^2 \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{x^3 \left (A b-2 a C+(2 A c-b C) x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\left (A b c+\left (b^2-6 a c\right ) C-\frac{A c \left (b^2+4 a c\right )+b \left (b^2-8 a c\right ) C}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} c^{3/2} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (A b c+\left (b^2-6 a c\right ) C+\frac{A c \left (b^2+4 a c\right )+b \left (b^2-8 a c\right ) C}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} c^{3/2} \left (b^2-4 a c\right ) \sqrt{b+\sqrt{b^2-4 a c}}}+\frac{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 )^{3/2}}\\ \end{align*}
Mathematica [A] time = 1.56148, size = 444, normalized size = 1.08 \[ \frac{1}{4} \left (\frac{2 \left (a (b (B+C x)-2 c x (A+x (B+C x)))+b x^2 (b (B+C x)-A c x)\right )}{c \left (4 a c-b^2\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{2} \left (C \left (b^2 \sqrt{b^2-4 a c}-6 a c \sqrt{b^2-4 a c}+8 a b c-b^3\right )-A c \left (-b \sqrt{b^2-4 a c}+4 a c+b^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \left (A c \left (b \sqrt{b^2-4 a c}+4 a c+b^2\right )+C \left (b^2 \sqrt{b^2-4 a c}-6 a c \sqrt{b^2-4 a c}-8 a b c+b^3\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{4 a B \log \left (\sqrt{b^2-4 a c}-b-2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{4 a B \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.042, size = 1429, normalized size = 3.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (C b^{2} -{\left (2 \, C a + A b\right )} c\right )} x^{3} + B a b +{\left (B b^{2} - 2 \, B a c\right )} x^{2} +{\left (C a b - 2 \, A a c\right )} x}{2 \,{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + a b^{2} c - 4 \, a^{2} c^{2} +{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{2}\right )}} + \frac{-\int \frac{4 \, B a c x - C a b + 2 \, A a c -{\left (C b^{2} -{\left (6 \, C a - A b\right )} c\right )} x^{2}}{c x^{4} + b x^{2} + a}\,{d x}}{2 \,{\left (b^{2} c - 4 \, a c^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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