Optimal. Leaf size=503 \[ \frac{c^{3/4} \left (-3 b \sqrt{b^2-4 a c}-28 a c+3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{4 \sqrt [4]{2} a \left (b^2-4 a c\right )^{3/2} \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{c^{3/4} \left (3 b \sqrt{b^2-4 a c}-28 a c+3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{4 \sqrt [4]{2} a \left (b^2-4 a c\right )^{3/2} \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{c^{3/4} \left (-3 b \sqrt{b^2-4 a c}-28 a c+3 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{4 \sqrt [4]{2} a \left (b^2-4 a c\right )^{3/2} \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{c^{3/4} \left (3 b \sqrt{b^2-4 a c}-28 a c+3 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{4 \sqrt [4]{2} a \left (b^2-4 a c\right )^{3/2} \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{\sqrt{x} \left (-2 a c+b^2+b c x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.2756, antiderivative size = 503, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1115, 1345, 1422, 212, 208, 205} \[ \frac{c^{3/4} \left (-3 b \sqrt{b^2-4 a c}-28 a c+3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{4 \sqrt [4]{2} a \left (b^2-4 a c\right )^{3/2} \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{c^{3/4} \left (3 b \sqrt{b^2-4 a c}-28 a c+3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{4 \sqrt [4]{2} a \left (b^2-4 a c\right )^{3/2} \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{c^{3/4} \left (-3 b \sqrt{b^2-4 a c}-28 a c+3 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{4 \sqrt [4]{2} a \left (b^2-4 a c\right )^{3/2} \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{c^{3/4} \left (3 b \sqrt{b^2-4 a c}-28 a c+3 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{4 \sqrt [4]{2} a \left (b^2-4 a c\right )^{3/2} \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{\sqrt{x} \left (-2 a c+b^2+b c x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1115
Rule 1345
Rule 1422
Rule 212
Rule 208
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{x} \left (a+b x^2+c x^4\right )^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^4+c x^8\right )^2} \, dx,x,\sqrt{x}\right )\\ &=\frac{\sqrt{x} \left (b^2-2 a c+b c x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\operatorname{Subst}\left (\int \frac{b^2-2 a c-4 \left (b^2-4 a c\right )-3 b c x^4}{a+b x^4+c x^8} \, dx,x,\sqrt{x}\right )}{2 a \left (b^2-4 a c\right )}\\ &=\frac{\sqrt{x} \left (b^2-2 a c+b c x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\left (c \left (3 b^2-28 a c-3 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 )}{4 a \left (b^2-4 a c\right )^{3/2}}+\frac{\left (c \left (3 b^2-28 a c+3 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 )}{4 a \left (b^2-4 a c\right )^{3/2}}\\ &=\frac{\sqrt{x} \left (b^2-2 a c+b c x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\left (c \left (3 b^2-28 a c-3 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 )}{4 a \left (b^2-4 a c\right )^{3/2} \sqrt{-b-\sqrt{b^2-4 a c}}}+\frac{\left (c \left (3 b^2-28 a c-3 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 )}{4 a \left (b^2-4 a c\right )^{3/2} \sqrt{-b-\sqrt{b^2-4 a c}}}-\frac{\left (c \left (3 b^2-28 a c+3 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 )}{4 a \left (b^2-4 a c\right )^{3/2} \sqrt{-b+\sqrt{b^2-4 a c}}}-\frac{\left (c \left (3 b^2-28 a c+3 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 )}{4 a \left (b^2-4 a c\right )^{3/2} \sqrt{-b+\sqrt{b^2-4 a c}}}\\ &=\frac{\sqrt{x} \left (b^2-2 a c+b c x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{c^{3/4} \left (3 b^2-28 a c-3 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 )}{4 \sqrt [4]{2} a \left (b^2-4 a c\right )^{3/2} \left (-b-\sqrt{b^2-4 a c}\right )^{3/4}}-\frac{c^{3/4} \left (3 b^2-28 a c+3 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 )}{4 \sqrt [4]{2} a \left (b^2-4 a c\right )^{3/2} \left (-b+\sqrt{b^2-4 a c}\right )^{3/4}}+\frac{c^{3/4} \left (3 b^2-28 a c-3 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 )}{4 \sqrt [4]{2} a \left (b^2-4 a c\right )^{3/2} \left (-b-\sqrt{b^2-4 a c}\right )^{3/4}}-\frac{c^{3/4} \left (3 b^2-28 a c+3 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 )}{4 \sqrt [4]{2} a \left (b^2-4 a c\right )^{3/2} \left (-b+\sqrt{b^2-4 a c}\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.234815, size = 153, normalized size = 0.3 \[ -\frac{\left (a+b x^2+c x^4\right ) \text{RootSum}\left [\text{$\#$1}^4 b+\text{$\#$1}^8 c+a\& ,\frac{3 \text{$\#$1}^4 b c \log \left (\sqrt{x}-\text{$\#$1}\right )-14 a c \log \left (\sqrt{x}-\text{$\#$1}\right )+3 b^2 \log \left (\sqrt{x}-\text{$\#$1}\right )}{\text{$\#$1}^3 b+2 \text{$\#$1}^7 c}\& \right ]+4 \sqrt{x} \left (-2 a c+b^2+b c x^2\right )}{8 a \left (4 a c-b^2\right ) \left (a+b x^2+c x^4\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.264, size = 144, normalized size = 0.3 \begin{align*} 2\,{\frac{1}{c{x}^{4}+b{x}^{2}+a} \left ( -1/4\,{\frac{bc{x}^{5/2}}{a \left ( 4\,ac-{b}^{2} \right ) }}+1/4\,{\frac{ \left ( 2\,ac-{b}^{2} \right ) \sqrt{x}}{a \left ( 4\,ac-{b}^{2} \right ) }} \right ) }+{\frac{1}{8\,a \left ( 4\,ac-{b}^{2} \right ) }\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+{{\it \_Z}}^{4}b+a \right ) }{\frac{-3\,{{\it \_R}}^{4}bc+14\,ac-3\,{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.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (3 \, b^{2} c - 14 \, a c^{2}\right )} x^{\frac{9}{2}} +{\left (3 \, b^{3} - 13 \, a b c\right )} x^{\frac{5}{2}} + 4 \,{\left (a b^{2} - 4 \, a^{2} c\right )} \sqrt{x}}{2 \,{\left (a^{3} b^{2} - 4 \, a^{4} c +{\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{4} +{\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{2}\right )}} - \int \frac{{\left (3 \, b^{2} c - 14 \, a c^{2}\right )} x^{\frac{7}{2}} +{\left (3 \, b^{3} - 17 \, a b c\right )} x^{\frac{3}{2}}}{4 \,{\left (a^{3} b^{2} - 4 \, a^{4} c +{\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{4} +{\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{2}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]