Optimal. Leaf size=78 \[ \frac {\left (a+b x^3\right )^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{15 b^2}-\frac {a \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{12 b^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1355, 266, 43} \begin {gather*} \frac {\left (a+b x^3\right )^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{15 b^2}-\frac {a \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{12 b^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rule 1355
Rubi steps
\begin {align*} \int x^5 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int x^5 \left (a b+b^2 x^3\right )^3 \, dx}{b^2 \left (a b+b^2 x^3\right )}\\ &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \operatorname {Subst}\left (\int x \left (a b+b^2 x\right )^3 \, dx,x,x^3\right )}{3 b^2 \left (a b+b^2 x^3\right )}\\ &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \operatorname {Subst}\left (\int \left (-\frac {a \left (a b+b^2 x\right )^3}{b}+\frac {\left (a b+b^2 x\right )^4}{b^2}\right ) \, dx,x,x^3\right )}{3 b^2 \left (a b+b^2 x^3\right )}\\ &=-\frac {a \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{12 b^2}+\frac {\left (a+b x^3\right )^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{15 b^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 61, normalized size = 0.78 \begin {gather*} \frac {x^6 \sqrt {\left (a+b x^3\right )^2} \left (10 a^3+20 a^2 b x^3+15 a b^2 x^6+4 b^3 x^9\right )}{60 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 8.62, size = 61, normalized size = 0.78 \begin {gather*} \frac {x^6 \sqrt {\left (a+b x^3\right )^2} \left (10 a^3+20 a^2 b x^3+15 a b^2 x^6+4 b^3 x^9\right )}{60 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.48, size = 35, normalized size = 0.45 \begin {gather*} \frac {1}{15} \, b^{3} x^{15} + \frac {1}{4} \, a b^{2} x^{12} + \frac {1}{3} \, a^{2} b x^{9} + \frac {1}{6} \, a^{3} x^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.44, size = 45, normalized size = 0.58 \begin {gather*} \frac {1}{60} \, {\left (4 \, b^{3} x^{15} + 15 \, a b^{2} x^{12} + 20 \, a^{2} b x^{9} + 10 \, a^{3} x^{6}\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 = 0.01, size = 58, normalized size = 0.74 \begin {gather*} \frac {\left (4 b^{3} x^{9}+15 a \,b^{2} x^{6}+20 a^{2} b \,x^{3}+10 a^{3}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {3}{2}} x^{6}}{60 \left (b \,x^{3}+a \right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.49, size = 83, normalized size = 1.06 \begin {gather*} -\frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}} a x^{3}}{12 \, b} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}} a^{2}}{12 \, b^{2}} + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}}}{15 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.25, size = 46, normalized size = 0.59 \begin {gather*} \frac {{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{3/2}\,\left (-a^2+3\,a\,b\,x^3+4\,b^2\,x^6\right )}{60\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{5} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________