Optimal. Leaf size=163 \[ \frac{b \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{32 a^{3/2}}-\frac{\left (x^2 \left (8 a c+b^2\right )+2 a b\right ) \sqrt{a+b x^2+c x^4}}{16 a x^4}+\frac{1}{2} c^{3/2} \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{6 x^6} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.183787, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {1114, 732, 810, 843, 621, 206, 724} \[ \frac{b \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{32 a^{3/2}}-\frac{\left (x^2 \left (8 a c+b^2\right )+2 a b\right ) \sqrt{a+b x^2+c x^4}}{16 a x^4}+\frac{1}{2} c^{3/2} \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{6 x^6} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1114
Rule 732
Rule 810
Rule 843
Rule 621
Rule 206
Rule 724
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2+c x^4\right )^{3/2}}{x^7} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\left (a+b x+c x^2\right )^{3/2}}{x^4} \, dx,x,x^2\right )\\ &=-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{6 x^6}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{(b+2 c x) \sqrt{a+b x+c x^2}}{x^3} \, dx,x,x^2\right )\\ &=-\frac{\left (2 a b+\left (b^2+8 a c\right ) x^2\right ) \sqrt{a+b x^2+c x^4}}{16 a x^4}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{6 x^6}-\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{2} b \left (b^2-12 a c\right )-8 a c^2 x}{x \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{16 a}\\ &=-\frac{\left (2 a b+\left (b^2+8 a c\right ) x^2\right ) \sqrt{a+b x^2+c x^4}}{16 a x^4}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{6 x^6}+\frac{1}{2} c^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,x^2\right )-\frac{\left (b \left (b^2-12 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{32 a}\\ &=-\frac{\left (2 a b+\left (b^2+8 a c\right ) x^2\right ) \sqrt{a+b x^2+c x^4}}{16 a x^4}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{6 x^6}+c^2 \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x^2}{\sqrt{a+b x^2+c x^4}}\right )+\frac{\left (b \left (b^2-12 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b x^2}{\sqrt{a+b x^2+c x^4}}\right )}{16 a}\\ &=-\frac{\left (2 a b+\left (b^2+8 a c\right ) x^2\right ) \sqrt{a+b x^2+c x^4}}{16 a x^4}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{6 x^6}+\frac{b \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{32 a^{3/2}}+\frac{1}{2} c^{3/2} \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )\\ \end{align*}
Mathematica [A] time = 0.203306, size = 149, normalized size = 0.91 \[ \frac{1}{96} \left (-\frac{2 \sqrt{a+b x^2+c x^4} \left (8 a^2+14 a b x^2+32 a c x^4+3 b^2 x^4\right )}{a x^6}+\frac{3 b \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{a^{3/2}}+48 c^{3/2} \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.177, size = 202, normalized size = 1.2 \begin{align*}{\frac{1}{2}{c}^{{\frac{3}{2}}}\ln \left ({ \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) }-{\frac{3\,bc}{8}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}-{\frac{a}{6\,{x}^{6}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{7\,b}{24\,{x}^{4}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{{b}^{2}}{16\,a{x}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{{b}^{3}}{32}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{2\,c}{3\,{x}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.80551, size = 1804, normalized size = 11.07 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{x^{7}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}}{x^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]