Optimal. Leaf size=72 \[ \frac {b \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{6 a^{3/2}}-\frac {\sqrt {a+b x^3+c x^6}}{3 a x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1357, 730, 724, 206} \[ \frac {b \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{6 a^{3/2}}-\frac {\sqrt {a+b x^3+c x^6}}{3 a x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 724
Rule 730
Rule 1357
Rubi steps
\begin {align*} \int \frac {1}{x^4 \sqrt {a+b x^3+c x^6}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {a+b x^3+c x^6}}{3 a x^3}-\frac {b \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{6 a}\\ &=-\frac {\sqrt {a+b x^3+c x^6}}{3 a x^3}+\frac {b \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x^3}{\sqrt {a+b x^3+c x^6}}\right )}{3 a}\\ &=-\frac {\sqrt {a+b x^3+c x^6}}{3 a x^3}+\frac {b \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{6 a^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 72, normalized size = 1.00 \[ \frac {b \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{6 a^{3/2}}-\frac {\sqrt {a+b x^3+c x^6}}{3 a x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.17, size = 179, normalized size = 2.49 \[ \left [\frac {\sqrt {a} b x^{3} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} + 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) - 4 \, \sqrt {c x^{6} + b x^{3} + a} a}{12 \, a^{2} x^{3}}, -\frac {\sqrt {-a} b x^{3} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{6} + a b x^{3} + a^{2}\right )}}\right ) + 2 \, \sqrt {c x^{6} + b x^{3} + a} a}{6 \, a^{2} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c x^{6} + b x^{3} + a} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.56, size = 56, normalized size = 0.78 \[ \frac {b\,\mathrm {atanh}\left (\frac {\frac {b\,x^3}{2}+a}{\sqrt {a}\,\sqrt {c\,x^6+b\,x^3+a}}\right )}{6\,a^{3/2}}-\frac {\sqrt {c\,x^6+b\,x^3+a}}{3\,a\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{4} \sqrt {a + b x^{3} + c x^{6}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________