Optimal. Leaf size=74 \[ \frac {a x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac {b x^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )} \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1343} \begin {gather*} \frac {b x^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac {a x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 1343
Rubi steps
\begin {align*} \int \sqrt {a^2+2 a b x^3+b^2 x^6} \, dx &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (2 a b+2 b^2 x^3\right ) \, dx}{2 a b+2 b^2 x^3}\\ &=\frac {a x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac {b x^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 36, normalized size = 0.49 \begin {gather*} \frac {\sqrt {\left (a+b x^3\right )^2} \left (4 a x+b x^4\right )}{4 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 11.16, size = 36, normalized size = 0.49 \begin {gather*} \frac {\sqrt {\left (a+b x^3\right )^2} \left (4 a x+b x^4\right )}{4 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.06, size = 10, normalized size = 0.14 \begin {gather*} \frac {1}{4} \, b x^{4} + a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.38, size = 20, normalized size = 0.27 \begin {gather*} \frac {1}{4} \, {\left (b x^{4} + 4 \, a x\right )} \mathrm {sgn}\left (b x^{3} + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 33, normalized size = 0.45 \begin {gather*} \frac {\left (b \,x^{3}+4 a \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}\, x}{4 b \,x^{3}+4 a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.44, size = 10, normalized size = 0.14 \begin {gather*} \frac {1}{4} \, b x^{4} + a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {{\left (b\,x^3+a\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.10, size = 8, normalized size = 0.11 \begin {gather*} a x + \frac {b x^{4}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________