Optimal. Leaf size=594 \[ -\frac{3 c^{3/4} \left (-\frac{b^3}{\sqrt{b^2-4 a c}}+\frac{68 a b c}{\sqrt{b^2-4 a c}}+44 a c+b^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^2 \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{3 c^{3/4} \left (\sqrt{b^2-4 a c} \left (44 a c+b^2\right )-68 a b c+b^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^{5/2} \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{3 c^{3/4} \left (-\frac{b^3}{\sqrt{b^2-4 a c}}+\frac{68 a b c}{\sqrt{b^2-4 a c}}+44 a c+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^2 \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{3 c^{3/4} \left (\sqrt{b^2-4 a c} \left (44 a c+b^2\right )-68 a b c+b^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^{5/2} \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{\sqrt{x} \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{\sqrt{x} \left (c x^2 \left (44 a c+b^2\right )+b \left (20 a c+b^2\right )\right )}{16 a \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )} \]
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Rubi [A] time = 2.36606, antiderivative size = 594, 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, 1364, 1430, 1422, 212, 208, 205} \[ -\frac{3 c^{3/4} \left (-\frac{b^3}{\sqrt{b^2-4 a c}}+\frac{68 a b c}{\sqrt{b^2-4 a c}}+44 a c+b^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^2 \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{3 c^{3/4} \left (\sqrt{b^2-4 a c} \left (44 a c+b^2\right )-68 a b c+b^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^{5/2} \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{3 c^{3/4} \left (-\frac{b^3}{\sqrt{b^2-4 a c}}+\frac{68 a b c}{\sqrt{b^2-4 a c}}+44 a c+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^2 \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{3 c^{3/4} \left (\sqrt{b^2-4 a c} \left (44 a c+b^2\right )-68 a b c+b^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^{5/2} \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{\sqrt{x} \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{\sqrt{x} \left (c x^2 \left (44 a c+b^2\right )+b \left (20 a c+b^2\right )\right )}{16 a \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )} \]
Antiderivative was successfully verified.
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Rule 1115
Rule 1364
Rule 1430
Rule 1422
Rule 212
Rule 208
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{3/2}}{\left (a+b x^2+c x^4\right )^3} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^4}{\left (a+b x^4+c x^8\right )^3} \, dx,x,\sqrt{x}\right )\\ &=-\frac{\sqrt{x} \left (b+2 c 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{b-22 c 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 (b+2 c 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 (b \left (b^2+20 a c\right )+c \left (b^2+44 a c\right ) x^2\right )}{16 a \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{\operatorname{Subst}\left (\int \frac{-3 b \left (b^2-12 a c\right )-3 c \left (b^2+44 a c\right ) 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 (b+2 c 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 (b \left (b^2+20 a c\right )+c \left (b^2+44 a c\right ) x^2\right )}{16 a \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\left (3 c \left (b^2+44 a c+\frac{b^3}{\sqrt{b^2-4 a c}}-\frac{68 a b c}{\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 )}{32 a \left (b^2-4 a c\right )^2}+\frac{\left (3 c \left (b^2+44 a c-\frac{b^3}{\sqrt{b^2-4 a c}}+\frac{68 a b c}{\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 )}{32 a \left (b^2-4 a c\right )^2}\\ &=-\frac{\sqrt{x} \left (b+2 c 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 (b \left (b^2+20 a c\right )+c \left (b^2+44 a c\right ) x^2\right )}{16 a \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{\left (3 c \left (b^2+44 a c-\frac{b^3}{\sqrt{b^2-4 a c}}+\frac{68 a b c}{\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 )}{32 a \left (b^2-4 a c\right )^2 \sqrt{-b-\sqrt{b^2-4 a c}}}-\frac{\left (3 c \left (b^2+44 a c-\frac{b^3}{\sqrt{b^2-4 a c}}+\frac{68 a b c}{\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 )}{32 a \left (b^2-4 a c\right )^2 \sqrt{-b-\sqrt{b^2-4 a c}}}-\frac{\left (3 c \left (b^2+44 a c+\frac{b^3}{\sqrt{b^2-4 a c}}-\frac{68 a b c}{\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 )}{32 a \left (b^2-4 a c\right )^2 \sqrt{-b+\sqrt{b^2-4 a c}}}-\frac{\left (3 c \left (b^2+44 a c+\frac{b^3}{\sqrt{b^2-4 a c}}-\frac{68 a b c}{\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 )}{32 a \left (b^2-4 a c\right )^2 \sqrt{-b+\sqrt{b^2-4 a c}}}\\ &=-\frac{\sqrt{x} \left (b+2 c 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 (b \left (b^2+20 a c\right )+c \left (b^2+44 a c\right ) x^2\right )}{16 a \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{3 c^{3/4} \left (b^2+44 a c-\frac{b^3}{\sqrt{b^2-4 a c}}+\frac{68 a b c}{\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 )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^2 \left (-b-\sqrt{b^2-4 a c}\right )^{3/4}}-\frac{3 c^{3/4} \left (b^2+44 a c+\frac{b^3}{\sqrt{b^2-4 a c}}-\frac{68 a b c}{\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 )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^2 \left (-b+\sqrt{b^2-4 a c}\right )^{3/4}}-\frac{3 c^{3/4} \left (b^2+44 a c-\frac{b^3}{\sqrt{b^2-4 a c}}+\frac{68 a b c}{\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 )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^2 \left (-b-\sqrt{b^2-4 a c}\right )^{3/4}}-\frac{3 c^{3/4} \left (b^2+44 a c+\frac{b^3}{\sqrt{b^2-4 a c}}-\frac{68 a b c}{\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 )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^2 \left (-b+\sqrt{b^2-4 a c}\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.409663, size = 224, normalized size = 0.38 \[ \frac{3 \left (a+b x^2+c x^4\right )^2 \text{RootSum}\left [\text{$\#$1}^4 b+\text{$\#$1}^8 c+a\& ,\frac{44 \text{$\#$1}^4 a c^2 \log \left (\sqrt{x}-\text{$\#$1}\right )+\text{$\#$1}^4 b^2 c \log \left (\sqrt{x}-\text{$\#$1}\right )-12 a b c \log \left (\sqrt{x}-\text{$\#$1}\right )+b^3 \log \left (\sqrt{x}-\text{$\#$1}\right )}{\text{$\#$1}^3 b+2 \text{$\#$1}^7 c}\& \right ]+4 \sqrt{x} \left (20 a b c+44 a c^2 x^2+b^2 c x^2+b^3\right ) \left (a+b x^2+c x^4\right )-16 a \sqrt{x} \left (b^2-4 a c\right ) \left (b+2 c x^2\right )}{64 a \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.268, size = 270, normalized size = 0.5 \begin{align*} 2\,{\frac{1}{ \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{2}} \left ({\frac{3\,b \left ( 12\,ac-{b}^{2} \right ) \sqrt{x}}{512\,{a}^{2}{c}^{2}-256\,ac{b}^{2}+32\,{b}^{4}}}+1/32\,{\frac{ \left ( 76\,{a}^{2}{c}^{2}+13\,ac{b}^{2}+{b}^{4} \right ){x}^{5/2}}{a \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) }}+1/16\,{\frac{bc \left ( 32\,ac+{b}^{2} \right ){x}^{9/2}}{a \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) }}+1/32\,{\frac{{c}^{2} \left ( 44\,ac+{b}^{2} \right ){x}^{13/2}}{a \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) }} \right ) }+{\frac{3}{64\,a \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) }\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+{{\it \_Z}}^{4}b+a \right ) }{\frac{c \left ( 44\,ac+{b}^{2} \right ){{\it \_R}}^{4}-12\,abc+{b}^{3}}{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{3 \,{\left (b^{3} c^{2} - 12 \, a b c^{3}\right )} x^{\frac{17}{2}} +{\left (6 \, b^{4} c - 71 \, a b^{2} c^{2} + 44 \, a^{2} c^{3}\right )} x^{\frac{13}{2}} +{\left (3 \, b^{5} - 28 \, a b^{3} c - 8 \, a^{2} b c^{2}\right )} x^{\frac{9}{2}} +{\left (7 \, a b^{4} - 59 \, a^{2} b^{2} c + 76 \, a^{3} c^{2}\right )} x^{\frac{5}{2}}}{16 \,{\left ({\left (a^{2} b^{4} c^{2} - 8 \, a^{3} b^{2} c^{3} + 16 \, a^{4} c^{4}\right )} x^{8} + a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2} + 2 \,{\left (a^{2} b^{5} c - 8 \, a^{3} b^{3} c^{2} + 16 \, a^{4} b c^{3}\right )} x^{6} +{\left (a^{2} b^{6} - 6 \, a^{3} b^{4} c + 32 \, a^{5} c^{3}\right )} x^{4} + 2 \,{\left (a^{3} b^{5} - 8 \, a^{4} b^{3} c + 16 \, a^{5} b c^{2}\right )} x^{2}\right )}} + \int -\frac{3 \,{\left ({\left (b^{3} c - 12 \, a b c^{2}\right )} x^{\frac{7}{2}} +{\left (b^{4} - 13 \, a b^{2} c - 44 \, a^{2} c^{2}\right )} x^{\frac{3}{2}}\right )}}{32 \,{\left (a^{3} b^{4} - 8 \, a^{4} b^{2} c + 16 \, a^{5} c^{2} +{\left (a^{2} b^{4} c - 8 \, a^{3} b^{2} c^{2} + 16 \, a^{4} c^{3}\right )} x^{4} +{\left (a^{2} b^{5} - 8 \, a^{3} b^{3} c + 16 \, a^{4} 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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