Optimal. Leaf size=573 \[ -\frac{5 b^2-18 a c}{2 a^2 \sqrt{x} \left (b^2-4 a c\right )}+\frac{\sqrt [4]{c} \left (-\left (5 b^2-18 a c\right ) \sqrt{b^2-4 a c}-28 a b c+5 b^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-\sqrt{b^2-4 a c}-b}}-\frac{\sqrt [4]{c} \left (\left (5 b^2-18 a c\right ) \sqrt{b^2-4 a c}-28 a b c+5 b^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt [4]{\sqrt{b^2-4 a c}-b}}-\frac{\sqrt [4]{c} \left (-\left (5 b^2-18 a c\right ) \sqrt{b^2-4 a c}-28 a b c+5 b^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-\sqrt{b^2-4 a c}-b}}+\frac{\sqrt [4]{c} \left (\left (5 b^2-18 a c\right ) \sqrt{b^2-4 a c}-28 a b c+5 b^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt [4]{\sqrt{b^2-4 a c}-b}}+\frac{-2 a c+b^2+b c x^2}{2 a \sqrt{x} \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 2.44638, antiderivative size = 573, 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, 1366, 1504, 1510, 298, 205, 208} \[ -\frac{5 b^2-18 a c}{2 a^2 \sqrt{x} \left (b^2-4 a c\right )}+\frac{\sqrt [4]{c} \left (-\left (5 b^2-18 a c\right ) \sqrt{b^2-4 a c}-28 a b c+5 b^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-\sqrt{b^2-4 a c}-b}}-\frac{\sqrt [4]{c} \left (\left (5 b^2-18 a c\right ) \sqrt{b^2-4 a c}-28 a b c+5 b^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt [4]{\sqrt{b^2-4 a c}-b}}-\frac{\sqrt [4]{c} \left (-\left (5 b^2-18 a c\right ) \sqrt{b^2-4 a c}-28 a b c+5 b^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-\sqrt{b^2-4 a c}-b}}+\frac{\sqrt [4]{c} \left (\left (5 b^2-18 a c\right ) \sqrt{b^2-4 a c}-28 a b c+5 b^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt [4]{\sqrt{b^2-4 a c}-b}}+\frac{-2 a c+b^2+b c x^2}{2 a \sqrt{x} \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1115
Rule 1366
Rule 1504
Rule 1510
Rule 298
Rule 205
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x^{3/2} \left (a+b x^2+c x^4\right )^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x^4+c x^8\right )^2} \, dx,x,\sqrt{x}\right )\\ &=\frac{b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) \sqrt{x} \left (a+b x^2+c x^4\right )}-\frac{\operatorname{Subst}\left (\int \frac{-5 b^2+18 a c-5 b c x^4}{x^2 \left (a+b x^4+c x^8\right )} \, dx,x,\sqrt{x}\right )}{2 a \left (b^2-4 a c\right )}\\ &=-\frac{5 b^2-18 a c}{2 a^2 \left (b^2-4 a c\right ) \sqrt{x}}+\frac{b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) \sqrt{x} \left (a+b x^2+c x^4\right )}+\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (-b \left (5 b^2-23 a c\right )-c \left (5 b^2-18 a c\right ) x^4\right )}{a+b x^4+c x^8} \, dx,x,\sqrt{x}\right )}{2 a^2 \left (b^2-4 a c\right )}\\ &=-\frac{5 b^2-18 a c}{2 a^2 \left (b^2-4 a c\right ) \sqrt{x}}+\frac{b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) \sqrt{x} \left (a+b x^2+c x^4\right )}-\frac{\left (c \left (5 b^2-18 a c+\frac{5 b^3}{\sqrt{b^2-4 a c}}-\frac{28 a b c}{\sqrt{b^2-4 a c}}\right )\right ) \operatorname{Subst}\left (\int \frac{x^2}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^4} \, dx,x,\sqrt{x}\right )}{4 a^2 \left (b^2-4 a c\right )}-\frac{\left (c \left (5 b^2-18 a c-\frac{5 b^3}{\sqrt{b^2-4 a c}}+\frac{28 a b c}{\sqrt{b^2-4 a c}}\right )\right ) \operatorname{Subst}\left (\int \frac{x^2}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^4} \, dx,x,\sqrt{x}\right )}{4 a^2 \left (b^2-4 a c\right )}\\ &=-\frac{5 b^2-18 a c}{2 a^2 \left (b^2-4 a c\right ) \sqrt{x}}+\frac{b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) \sqrt{x} \left (a+b x^2+c x^4\right )}+\frac{\left (\sqrt{c} \left (5 b^2-18 a c+\frac{5 b^3}{\sqrt{b^2-4 a c}}-\frac{28 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 )}{4 \sqrt{2} a^2 \left (b^2-4 a c\right )}-\frac{\left (\sqrt{c} \left (5 b^2-18 a c+\frac{5 b^3}{\sqrt{b^2-4 a c}}-\frac{28 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 )}{4 \sqrt{2} a^2 \left (b^2-4 a c\right )}+\frac{\left (\sqrt{c} \left (5 b^2-18 a c-\frac{5 b^3}{\sqrt{b^2-4 a c}}+\frac{28 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 )}{4 \sqrt{2} a^2 \left (b^2-4 a c\right )}-\frac{\left (\sqrt{c} \left (5 b^2-18 a c-\frac{5 b^3}{\sqrt{b^2-4 a c}}+\frac{28 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 )}{4 \sqrt{2} a^2 \left (b^2-4 a c\right )}\\ &=-\frac{5 b^2-18 a c}{2 a^2 \left (b^2-4 a c\right ) \sqrt{x}}+\frac{b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) \sqrt{x} \left (a+b x^2+c x^4\right )}-\frac{\sqrt [4]{c} \left (5 b^2-18 a c-\frac{5 b^3}{\sqrt{b^2-4 a c}}+\frac{28 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 )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right ) \sqrt [4]{-b-\sqrt{b^2-4 a c}}}-\frac{\sqrt [4]{c} \left (5 b^2-18 a c+\frac{5 b^3}{\sqrt{b^2-4 a c}}-\frac{28 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 )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right ) \sqrt [4]{-b+\sqrt{b^2-4 a c}}}+\frac{\sqrt [4]{c} \left (5 b^2-18 a c-\frac{5 b^3}{\sqrt{b^2-4 a c}}+\frac{28 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 )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right ) \sqrt [4]{-b-\sqrt{b^2-4 a c}}}+\frac{\sqrt [4]{c} \left (5 b^2-18 a c+\frac{5 b^3}{\sqrt{b^2-4 a c}}-\frac{28 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 )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right ) \sqrt [4]{-b+\sqrt{b^2-4 a c}}}\\ \end{align*}
Mathematica [C] time = 0.311675, size = 190, normalized size = 0.33 \[ -\frac{\frac{\text{RootSum}\left [\text{$\#$1}^4 b+\text{$\#$1}^8 c+a\& ,\frac{-18 \text{$\#$1}^4 a c^2 \log \left (\sqrt{x}-\text{$\#$1}\right )+5 \text{$\#$1}^4 b^2 c \log \left (\sqrt{x}-\text{$\#$1}\right )-23 a b c \log \left (\sqrt{x}-\text{$\#$1}\right )+5 b^3 \log \left (\sqrt{x}-\text{$\#$1}\right )}{2 \text{$\#$1}^5 c+\text{$\#$1} b}\& \right ]}{b^2-4 a c}+\frac{4 x^{3/2} \left (-3 a b c-2 a c^2 x^2+b^2 c x^2+b^3\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{16}{\sqrt{x}}}{8 a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.305, size = 245, normalized size = 0.4 \begin{align*} -{\frac{{c}^{2}}{a \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }{x}^{{\frac{7}{2}}}}+{\frac{{b}^{2}c}{2\,{a}^{2} \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }{x}^{{\frac{7}{2}}}}-{\frac{3\,bc}{2\,a \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }{x}^{{\frac{3}{2}}}}+{\frac{{b}^{3}}{2\,{a}^{2} \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }{x}^{{\frac{3}{2}}}}-{\frac{1}{8\,{a}^{2} \left ( 4\,ac-{b}^{2} \right ) }\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+{{\it \_Z}}^{4}b+a \right ) }{\frac{c \left ( 18\,ac-5\,{b}^{2} \right ){{\it \_R}}^{6}+b \left ( 23\,ac-5\,{b}^{2} \right ){{\it \_R}}^{2}}{2\,{{\it \_R}}^{7}c+{{\it \_R}}^{3}b}\ln \left ( \sqrt{x}-{\it \_R} \right ) }}-2\,{\frac{1}{{a}^{2}\sqrt{x}}} \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 (5 \, b^{2} c - 18 \, a c^{2}\right )} x^{\frac{7}{2}} +{\left (5 \, b^{3} - 19 \, a b c\right )} x^{\frac{3}{2}} + \frac{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 (5 \, b^{2} c - 18 \, a c^{2}\right )} x^{\frac{5}{2}} +{\left (5 \, b^{3} - 23 \, a b c\right )} \sqrt{x}}{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]