Optimal. Leaf size=533 \[ \frac{c^{3/4} \left (36 b \sqrt{b^2-4 a c}+28 a c+41 b^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{16 \sqrt [4]{2} \left (b^2-4 a c\right )^{5/2} \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{c^{3/4} \left (-36 b \sqrt{b^2-4 a c}+28 a c+41 b^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{16 \sqrt [4]{2} \left (b^2-4 a c\right )^{5/2} \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{c^{3/4} \left (36 b \sqrt{b^2-4 a c}+28 a c+41 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{16 \sqrt [4]{2} \left (b^2-4 a c\right )^{5/2} \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{c^{3/4} \left (-36 b \sqrt{b^2-4 a c}+28 a c+41 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{16 \sqrt [4]{2} \left (b^2-4 a c\right )^{5/2} \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{\sqrt{x} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{\sqrt{x} \left (-4 a c+13 b^2+24 b c x^2\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )} \]
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Rubi [A] time = 1.36301, antiderivative size = 533, 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, 1430, 1422, 212, 208, 205} \[ \frac{c^{3/4} \left (36 b \sqrt{b^2-4 a c}+28 a c+41 b^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{16 \sqrt [4]{2} \left (b^2-4 a c\right )^{5/2} \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{c^{3/4} \left (-36 b \sqrt{b^2-4 a c}+28 a c+41 b^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{16 \sqrt [4]{2} \left (b^2-4 a c\right )^{5/2} \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{c^{3/4} \left (36 b \sqrt{b^2-4 a c}+28 a c+41 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{16 \sqrt [4]{2} \left (b^2-4 a c\right )^{5/2} \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{c^{3/4} \left (-36 b \sqrt{b^2-4 a c}+28 a c+41 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{16 \sqrt [4]{2} \left (b^2-4 a c\right )^{5/2} \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{\sqrt{x} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{\sqrt{x} \left (-4 a c+13 b^2+24 b c x^2\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )} \]
Antiderivative was successfully verified.
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Rule 1115
Rule 1365
Rule 1430
Rule 1422
Rule 212
Rule 208
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{7/2}}{\left (a+b x^2+c x^4\right )^3} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^8}{\left (a+b x^4+c x^8\right )^3} \, dx,x,\sqrt{x}\right )\\ &=\frac{\sqrt{x} \left (2 a+b 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{2 a-11 b x^4}{\left (a+b x^4+c x^8\right )^2} \, dx,x,\sqrt{x}\right )}{4 \left (b^2-4 a c\right )}\\ &=\frac{\sqrt{x} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{\sqrt{x} \left (13 b^2-4 a c+24 b c x^2\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\operatorname{Subst}\left (\int \frac{a \left (5 b^2+28 a c\right )-72 a b c x^4}{a+b x^4+c x^8} \, dx,x,\sqrt{x}\right )}{16 a \left (b^2-4 a c\right )^2}\\ &=\frac{\sqrt{x} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{\sqrt{x} \left (13 b^2-4 a c+24 b c x^2\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\left (c \left (41 b^2+28 a c-36 b \sqrt{b^2-4 a c}\right )\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 )}{16 \left (b^2-4 a c\right )^{5/2}}-\frac{\left (c \left (41 b^2+28 a c+36 b \sqrt{b^2-4 a c}\right )\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 )}{16 \left (b^2-4 a c\right )^{5/2}}\\ &=\frac{\sqrt{x} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{\sqrt{x} \left (13 b^2-4 a c+24 b c x^2\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{\left (c \left (41 b^2+28 a c-36 b \sqrt{b^2-4 a c}\right )\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 )}{16 \left (b^2-4 a c\right )^{5/2} \sqrt{-b+\sqrt{b^2-4 a c}}}-\frac{\left (c \left (41 b^2+28 a c-36 b \sqrt{b^2-4 a c}\right )\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 )}{16 \left (b^2-4 a c\right )^{5/2} \sqrt{-b+\sqrt{b^2-4 a c}}}+\frac{\left (c \left (41 b^2+28 a c+36 b \sqrt{b^2-4 a c}\right )\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 )}{16 \left (b^2-4 a c\right )^{5/2} \sqrt{-b-\sqrt{b^2-4 a c}}}+\frac{\left (c \left (41 b^2+28 a c+36 b \sqrt{b^2-4 a c}\right )\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 )}{16 \left (b^2-4 a c\right )^{5/2} \sqrt{-b-\sqrt{b^2-4 a c}}}\\ &=\frac{\sqrt{x} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{\sqrt{x} \left (13 b^2-4 a c+24 b c x^2\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{c^{3/4} \left (41 b^2+28 a c+36 b \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 )}{16 \sqrt [4]{2} \left (b^2-4 a c\right )^{5/2} \left (-b-\sqrt{b^2-4 a c}\right )^{3/4}}-\frac{c^{3/4} \left (41 b^2+28 a c-36 b \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 )}{16 \sqrt [4]{2} \left (b^2-4 a c\right )^{5/2} \left (-b+\sqrt{b^2-4 a c}\right )^{3/4}}+\frac{c^{3/4} \left (41 b^2+28 a c+36 b \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 )}{16 \sqrt [4]{2} \left (b^2-4 a c\right )^{5/2} \left (-b-\sqrt{b^2-4 a c}\right )^{3/4}}-\frac{c^{3/4} \left (41 b^2+28 a c-36 b \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 )}{16 \sqrt [4]{2} \left (b^2-4 a c\right )^{5/2} \left (-b+\sqrt{b^2-4 a c}\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.443112, size = 177, normalized size = 0.33 \[ -\frac{\text{RootSum}\left [\text{$\#$1}^4 b+\text{$\#$1}^8 c+a\& ,\frac{72 \text{$\#$1}^4 b c \log \left (\sqrt{x}-\text{$\#$1}\right )-28 a c \log \left (\sqrt{x}-\text{$\#$1}\right )-5 b^2 \log \left (\sqrt{x}-\text{$\#$1}\right )}{\text{$\#$1}^3 b+2 \text{$\#$1}^7 c}\& \right ]+\frac{4 \sqrt{x} \left (28 a^2 c+a \left (5 b^2+36 b c x^2-4 c^2 x^4\right )+b x^2 \left (9 b^2+37 b c x^2+24 c^2 x^4\right )\right )}{\left (a+b x^2+c x^4\right )^2}}{64 \left (b^2-4 a c\right )^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.27, size = 237, normalized size = 0.4 \begin{align*} 2\,{\frac{1}{ \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{2}} \left ( -1/32\,{\frac{a \left ( 28\,ac+5\,{b}^{2} \right ) \sqrt{x}}{16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4}}}-{\frac{9\,b \left ( 4\,ac+{b}^{2} \right ){x}^{5/2}}{512\,{a}^{2}{c}^{2}-256\,ac{b}^{2}+32\,{b}^{4}}}+1/32\,{\frac{c \left ( 4\,ac-37\,{b}^{2} \right ){x}^{9/2}}{16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4}}}-3/4\,{\frac{b{c}^{2}{x}^{13/2}}{16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4}}} \right ) }+{\frac{1}{1024\,{a}^{2}{c}^{2}-512\,ac{b}^{2}+64\,{b}^{4}}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+{{\it \_Z}}^{4}b+a \right ) }{\frac{-72\,{{\it \_R}}^{4}bc+28\,ac+5\,{b}^{2}}{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{{\left (5 \, b^{2} c^{2} + 28 \, a c^{3}\right )} x^{\frac{17}{2}} + 2 \,{\left (5 \, b^{3} c + 16 \, a b c^{2}\right )} x^{\frac{13}{2}} +{\left (5 \, b^{4} + a b^{2} c + 60 \, a^{2} c^{2}\right )} x^{\frac{9}{2}} +{\left (a b^{3} + 20 \, a^{2} b c\right )} x^{\frac{5}{2}}}{16 \,{\left ({\left (a b^{4} c^{2} - 8 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4}\right )} x^{8} + a^{3} b^{4} - 8 \, a^{4} b^{2} c + 16 \, a^{5} c^{2} + 2 \,{\left (a b^{5} c - 8 \, a^{2} b^{3} c^{2} + 16 \, a^{3} b c^{3}\right )} x^{6} +{\left (a b^{6} - 6 \, a^{2} b^{4} c + 32 \, a^{4} c^{3}\right )} x^{4} + 2 \,{\left (a^{2} b^{5} - 8 \, a^{3} b^{3} c + 16 \, a^{4} b c^{2}\right )} x^{2}\right )}} - \int \frac{{\left (5 \, b^{2} c + 28 \, a c^{2}\right )} x^{\frac{7}{2}} + 5 \,{\left (b^{3} + 20 \, a b c\right )} x^{\frac{3}{2}}}{32 \,{\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} +{\left (a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3}\right )} x^{4} +{\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\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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