Optimal. Leaf size=139 \[ \frac{3 a^2 b c x^2 \sqrt{c \left (a+b x^2\right )^2}}{2 \left (a+b x^2\right )}+\frac{a^3 c \log (x) \sqrt{c \left (a+b x^2\right )^2}}{a+b x^2}+\frac{b^3 c x^6 \sqrt{c \left (a+b x^2\right )^2}}{6 \left (a+b x^2\right )}+\frac{3 a b^2 c x^4 \sqrt{c \left (a+b x^2\right )^2}}{4 \left (a+b x^2\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.103352, antiderivative size = 183, normalized size of antiderivative = 1.32, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {1989, 1112, 266, 43} \[ \frac{b^3 c x^6 \sqrt{a^2 c+2 a b c x^2+b^2 c x^4}}{6 \left (a+b x^2\right )}+\frac{3 a b^2 c x^4 \sqrt{a^2 c+2 a b c x^2+b^2 c x^4}}{4 \left (a+b x^2\right )}+\frac{3 a^2 b c x^2 \sqrt{a^2 c+2 a b c x^2+b^2 c x^4}}{2 \left (a+b x^2\right )}+\frac{a^3 c \log (x) \sqrt{a^2 c+2 a b c x^2+b^2 c x^4}}{a+b x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1989
Rule 1112
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (c \left (a+b x^2\right )^2\right )^{3/2}}{x} \, dx &=\int \frac{\left (a^2 c+2 a b c x^2+b^2 c x^4\right )^{3/2}}{x} \, dx\\ &=\frac{\sqrt{a^2 c+2 a b c x^2+b^2 c x^4} \int \frac{\left (a b c+b^2 c x^2\right )^3}{x} \, dx}{b^2 c \left (a b c+b^2 c x^2\right )}\\ &=\frac{\sqrt{a^2 c+2 a b c x^2+b^2 c x^4} \operatorname{Subst}\left (\int \frac{\left (a b c+b^2 c x\right )^3}{x} \, dx,x,x^2\right )}{2 b^2 c \left (a b c+b^2 c x^2\right )}\\ &=\frac{\sqrt{a^2 c+2 a b c x^2+b^2 c x^4} \operatorname{Subst}\left (\int \left (3 a^2 b^4 c^3+\frac{a^3 b^3 c^3}{x}+3 a b^5 c^3 x+b^6 c^3 x^2\right ) \, dx,x,x^2\right )}{2 b^2 c \left (a b c+b^2 c x^2\right )}\\ &=\frac{3 a^2 b c x^2 \sqrt{a^2 c+2 a b c x^2+b^2 c x^4}}{2 \left (a+b x^2\right )}+\frac{3 a b^2 c x^4 \sqrt{a^2 c+2 a b c x^2+b^2 c x^4}}{4 \left (a+b x^2\right )}+\frac{b^3 c x^6 \sqrt{a^2 c+2 a b c x^2+b^2 c x^4}}{6 \left (a+b x^2\right )}+\frac{a^3 c \sqrt{a^2 c+2 a b c x^2+b^2 c x^4} \log (x)}{a+b x^2}\\ \end{align*}
Mathematica [A] time = 0.0226401, size = 62, normalized size = 0.45 \[ \frac{\left (c \left (a+b x^2\right )^2\right )^{3/2} \left (b x^2 \left (18 a^2+9 a b x^2+2 b^2 x^4\right )+12 a^3 \log (x)\right )}{12 \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.009, size = 59, normalized size = 0.4 \begin{align*}{\frac{2\,{b}^{3}{x}^{6}+9\,a{b}^{2}{x}^{4}+18\,{a}^{2}b{x}^{2}+12\,{a}^{3}\ln \left ( x \right ) }{12\, \left ( b{x}^{2}+a \right ) ^{3}} \left ( c \left ( b{x}^{2}+a \right ) ^{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 = 1.44902, size = 163, normalized size = 1.17 \begin{align*} \frac{{\left (2 \, b^{3} c x^{6} + 9 \, a b^{2} c x^{4} + 18 \, a^{2} b c x^{2} + 12 \, a^{3} c \log \left (x\right )\right )} \sqrt{b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c}}{12 \,{\left (b x^{2} + a\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{\left (c \left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.22856, size = 99, normalized size = 0.71 \begin{align*} \frac{1}{12} \,{\left (2 \, b^{3} x^{6} \mathrm{sgn}\left (b x^{2} + a\right ) + 9 \, a b^{2} x^{4} \mathrm{sgn}\left (b x^{2} + a\right ) + 18 \, a^{2} b x^{2} \mathrm{sgn}\left (b x^{2} + a\right ) + 6 \, a^{3} \log \left (x^{2}\right ) \mathrm{sgn}\left (b x^{2} + a\right )\right )} c^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]