Optimal. Leaf size=102 \[ -\frac{16 c^2 \sqrt{b x^2+c x^4}}{5 b^4 x^2}+\frac{8 c \sqrt{b x^2+c x^4}}{5 b^3 x^4}-\frac{6 \sqrt{b x^2+c x^4}}{5 b^2 x^6}+\frac{1}{b x^4 \sqrt{b x^2+c x^4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.184875, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2015, 2016, 2014} \[ -\frac{16 c^2 \sqrt{b x^2+c x^4}}{5 b^4 x^2}+\frac{8 c \sqrt{b x^2+c x^4}}{5 b^3 x^4}-\frac{6 \sqrt{b x^2+c x^4}}{5 b^2 x^6}+\frac{1}{b x^4 \sqrt{b x^2+c x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2015
Rule 2016
Rule 2014
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (b x^2+c x^4\right )^{3/2}} \, dx &=\frac{1}{b x^4 \sqrt{b x^2+c x^4}}+\frac{6 \int \frac{1}{x^5 \sqrt{b x^2+c x^4}} \, dx}{b}\\ &=\frac{1}{b x^4 \sqrt{b x^2+c x^4}}-\frac{6 \sqrt{b x^2+c x^4}}{5 b^2 x^6}-\frac{(24 c) \int \frac{1}{x^3 \sqrt{b x^2+c x^4}} \, dx}{5 b^2}\\ &=\frac{1}{b x^4 \sqrt{b x^2+c x^4}}-\frac{6 \sqrt{b x^2+c x^4}}{5 b^2 x^6}+\frac{8 c \sqrt{b x^2+c x^4}}{5 b^3 x^4}+\frac{\left (16 c^2\right ) \int \frac{1}{x \sqrt{b x^2+c x^4}} \, dx}{5 b^3}\\ &=\frac{1}{b x^4 \sqrt{b x^2+c x^4}}-\frac{6 \sqrt{b x^2+c x^4}}{5 b^2 x^6}+\frac{8 c \sqrt{b x^2+c x^4}}{5 b^3 x^4}-\frac{16 c^2 \sqrt{b x^2+c x^4}}{5 b^4 x^2}\\ \end{align*}
Mathematica [A] time = 0.0122275, size = 57, normalized size = 0.56 \[ \frac{2 b^2 c x^2-b^3-8 b c^2 x^4-16 c^3 x^6}{5 b^4 x^4 \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.046, size = 59, normalized size = 0.6 \begin{align*} -{\frac{ \left ( c{x}^{2}+b \right ) \left ( 16\,{c}^{3}{x}^{6}+8\,b{c}^{2}{x}^{4}-2\,{b}^{2}c{x}^{2}+{b}^{3} \right ) }{5\,{b}^{4}{x}^{2}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \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 = 1.34104, size = 128, normalized size = 1.25 \begin{align*} -\frac{{\left (16 \, c^{3} x^{6} + 8 \, b c^{2} x^{4} - 2 \, b^{2} c x^{2} + b^{3}\right )} \sqrt{c x^{4} + b x^{2}}}{5 \,{\left (b^{4} c x^{8} + b^{5} x^{6}\right )}} \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{1}{x^{3} \left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}\, 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{1}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]