Optimal. Leaf size=520 \[ -\frac{\left (\frac{b \left (b^2-12 a c\right )}{\sqrt{b^2-4 a c}}-10 a c+b^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{4 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right ) \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{\left (-\frac{b \left (b^2-12 a c\right )}{\sqrt{b^2-4 a c}}-10 a c+b^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{4 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right ) \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{\left (\frac{b \left (b^2-12 a c\right )}{\sqrt{b^2-4 a c}}-10 a c+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{4 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right ) \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{\left (-\frac{b \left (b^2-12 a c\right )}{\sqrt{b^2-4 a c}}-10 a c+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{4 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right ) \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{x^{5/2} \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{b \sqrt{x}}{2 c \left (b^2-4 a c\right )} \]
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Rubi [A] time = 1.37058, antiderivative size = 520, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {1115, 1365, 1502, 1422, 212, 208, 205} \[ -\frac{\left (\frac{b \left (b^2-12 a c\right )}{\sqrt{b^2-4 a c}}-10 a c+b^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{4 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right ) \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{\left (-\frac{b \left (b^2-12 a c\right )}{\sqrt{b^2-4 a c}}-10 a c+b^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{4 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right ) \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{\left (\frac{b \left (b^2-12 a c\right )}{\sqrt{b^2-4 a c}}-10 a c+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{4 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right ) \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{\left (-\frac{b \left (b^2-12 a c\right )}{\sqrt{b^2-4 a c}}-10 a c+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{4 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right ) \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{x^{5/2} \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{b \sqrt{x}}{2 c \left (b^2-4 a c\right )} \]
Antiderivative was successfully verified.
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Rule 1115
Rule 1365
Rule 1502
Rule 1422
Rule 212
Rule 208
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{11/2}}{\left (a+b x^2+c x^4\right )^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^{12}}{\left (a+b x^4+c x^8\right )^2} \, dx,x,\sqrt{x}\right )\\ &=\frac{x^{5/2} \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{\operatorname{Subst}\left (\int \frac{x^4 \left (10 a+b x^4\right )}{a+b x^4+c x^8} \, dx,x,\sqrt{x}\right )}{2 \left (b^2-4 a c\right )}\\ &=-\frac{b \sqrt{x}}{2 c \left (b^2-4 a c\right )}+\frac{x^{5/2} \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{\operatorname{Subst}\left (\int \frac{a b+\left (b^2-10 a c\right ) x^4}{a+b x^4+c x^8} \, dx,x,\sqrt{x}\right )}{2 c \left (b^2-4 a c\right )}\\ &=-\frac{b \sqrt{x}}{2 c \left (b^2-4 a c\right )}+\frac{x^{5/2} \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{\left (b^2-10 a c-\frac{b \left (b^2-12 a c\right )}{\sqrt{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^4} \, dx,x,\sqrt{x}\right )}{4 c \left (b^2-4 a c\right )}+\frac{\left (b^2-10 a c+\frac{b \left (b^2-12 a c\right )}{\sqrt{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^4} \, dx,x,\sqrt{x}\right )}{4 c \left (b^2-4 a c\right )}\\ &=-\frac{b \sqrt{x}}{2 c \left (b^2-4 a c\right )}+\frac{x^{5/2} \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{\left (b^2-10 a c+\frac{b \left (b^2-12 a c\right )}{\sqrt{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-b-\sqrt{b^2-4 a c}}-\sqrt{2} \sqrt{c} x^2} \, dx,x,\sqrt{x}\right )}{4 c \left (b^2-4 a c\right ) \sqrt{-b-\sqrt{b^2-4 a c}}}-\frac{\left (b^2-10 a c+\frac{b \left (b^2-12 a c\right )}{\sqrt{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-b-\sqrt{b^2-4 a c}}+\sqrt{2} \sqrt{c} x^2} \, dx,x,\sqrt{x}\right )}{4 c \left (b^2-4 a c\right ) \sqrt{-b-\sqrt{b^2-4 a c}}}-\frac{\left (b^2-10 a c-\frac{b \left (b^2-12 a c\right )}{\sqrt{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-b+\sqrt{b^2-4 a c}}-\sqrt{2} \sqrt{c} x^2} \, dx,x,\sqrt{x}\right )}{4 c \left (b^2-4 a c\right ) \sqrt{-b+\sqrt{b^2-4 a c}}}-\frac{\left (b^2-10 a c-\frac{b \left (b^2-12 a c\right )}{\sqrt{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-b+\sqrt{b^2-4 a c}}+\sqrt{2} \sqrt{c} x^2} \, dx,x,\sqrt{x}\right )}{4 c \left (b^2-4 a c\right ) \sqrt{-b+\sqrt{b^2-4 a c}}}\\ &=-\frac{b \sqrt{x}}{2 c \left (b^2-4 a c\right )}+\frac{x^{5/2} \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{\left (b^2-10 a c+\frac{b \left (b^2-12 a c\right )}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-b-\sqrt{b^2-4 a c}}}\right )}{4 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right ) \left (-b-\sqrt{b^2-4 a c}\right )^{3/4}}-\frac{\left (b^2-10 a c-\frac{b \left (b^2-12 a c\right )}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-b+\sqrt{b^2-4 a c}}}\right )}{4 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right ) \left (-b+\sqrt{b^2-4 a c}\right )^{3/4}}-\frac{\left (b^2-10 a c+\frac{b \left (b^2-12 a c\right )}{\sqrt{b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-b-\sqrt{b^2-4 a c}}}\right )}{4 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right ) \left (-b-\sqrt{b^2-4 a c}\right )^{3/4}}-\frac{\left (b^2-10 a c-\frac{b \left (b^2-12 a c\right )}{\sqrt{b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-b+\sqrt{b^2-4 a c}}}\right )}{4 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right ) \left (-b+\sqrt{b^2-4 a c}\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.248163, size = 144, normalized size = 0.28 \[ \frac{\text{RootSum}\left [\text{$\#$1}^4 b+\text{$\#$1}^8 c+a\& ,\frac{-10 \text{$\#$1}^4 a c \log \left (\sqrt{x}-\text{$\#$1}\right )+\text{$\#$1}^4 b^2 \log \left (\sqrt{x}-\text{$\#$1}\right )+a b \log \left (\sqrt{x}-\text{$\#$1}\right )}{\text{$\#$1}^3 b+2 \text{$\#$1}^7 c}\& \right ]-\frac{4 \sqrt{x} \left (a \left (b-2 c x^2\right )+b^2 x^2\right )}{a+b x^2+c x^4}}{8 c \left (b^2-4 a c\right )} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.267, size = 146, normalized size = 0.3 \begin{align*} 2\,{\frac{1}{c{x}^{4}+b{x}^{2}+a} \left ( -1/4\,{\frac{ \left ( 2\,ac-{b}^{2} \right ){x}^{5/2}}{c \left ( 4\,ac-{b}^{2} \right ) }}+1/4\,{\frac{ab\sqrt{x}}{c \left ( 4\,ac-{b}^{2} \right ) }} \right ) }+{\frac{1}{8\,c \left ( 4\,ac-{b}^{2} \right ) }\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+{{\it \_Z}}^{4}b+a \right ) }{\frac{ \left ( 10\,ac-{b}^{2} \right ){{\it \_R}}^{4}-ab}{2\,{{\it \_R}}^{7}c+{{\it \_R}}^{3}b}\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \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{b x^{\frac{9}{2}} + 2 \, a x^{\frac{5}{2}}}{2 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} x^{2}\right )}} + \int -\frac{b x^{\frac{7}{2}} + 10 \, a x^{\frac{3}{2}}}{4 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} x^{2}\right )}}\,{d x} \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(-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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