Optimal. Leaf size=255 \[ \frac {b^5 x^{20} \sqrt {a^2+2 a b x^3+b^2 x^6}}{20 \left (a+b x^3\right )}+\frac {5 a b^4 x^{17} \sqrt {a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )}+\frac {5 a^2 b^3 x^{14} \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac {a^5 x^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}+\frac {5 a^4 b x^8 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac {10 a^3 b^2 x^{11} \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )} \]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1355, 270} \begin {gather*} \frac {b^5 x^{20} \sqrt {a^2+2 a b x^3+b^2 x^6}}{20 \left (a+b x^3\right )}+\frac {5 a b^4 x^{17} \sqrt {a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )}+\frac {5 a^2 b^3 x^{14} \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac {10 a^3 b^2 x^{11} \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac {5 a^4 b x^8 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac {a^5 x^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 270
Rule 1355
Rubi steps
\begin {align*} \int x^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int x^4 \left (a b+b^2 x^3\right )^5 \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (a^5 b^5 x^4+5 a^4 b^6 x^7+10 a^3 b^7 x^{10}+10 a^2 b^8 x^{13}+5 a b^9 x^{16}+b^{10} x^{19}\right ) \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac {a^5 x^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}+\frac {5 a^4 b x^8 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac {10 a^3 b^2 x^{11} \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac {5 a^2 b^3 x^{14} \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac {5 a b^4 x^{17} \sqrt {a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )}+\frac {b^5 x^{20} \sqrt {a^2+2 a b x^3+b^2 x^6}}{20 \left (a+b x^3\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 83, normalized size = 0.33 \begin {gather*} \frac {x^5 \sqrt {\left (a+b x^3\right )^2} \left (10472 a^5+32725 a^4 b x^3+47600 a^3 b^2 x^6+37400 a^2 b^3 x^9+15400 a b^4 x^{12}+2618 b^5 x^{15}\right )}{52360 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 10.54, size = 83, normalized size = 0.33 \begin {gather*} \frac {\sqrt {\left (a+b x^3\right )^2} \left (10472 a^5 x^5+32725 a^4 b x^8+47600 a^3 b^2 x^{11}+37400 a^2 b^3 x^{14}+15400 a b^4 x^{17}+2618 b^5 x^{20}\right )}{52360 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.33, size = 57, normalized size = 0.22 \begin {gather*} \frac {1}{20} \, b^{5} x^{20} + \frac {5}{17} \, a b^{4} x^{17} + \frac {5}{7} \, a^{2} b^{3} x^{14} + \frac {10}{11} \, a^{3} b^{2} x^{11} + \frac {5}{8} \, a^{4} b x^{8} + \frac {1}{5} \, a^{5} x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.42, size = 105, normalized size = 0.41 \begin {gather*} \frac {1}{20} \, b^{5} x^{20} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{17} \, a b^{4} x^{17} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{7} \, a^{2} b^{3} x^{14} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {10}{11} \, a^{3} b^{2} x^{11} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{8} \, a^{4} b x^{8} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {1}{5} \, a^{5} x^{5} \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 = 80, normalized size = 0.31 \begin {gather*} \frac {\left (2618 b^{5} x^{15}+15400 a \,b^{4} x^{12}+37400 a^{2} b^{3} x^{9}+47600 a^{3} b^{2} x^{6}+32725 a^{4} b \,x^{3}+10472 a^{5}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {5}{2}} x^{5}}{52360 \left (b \,x^{3}+a \right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.69, size = 57, normalized size = 0.22 \begin {gather*} \frac {1}{20} \, b^{5} x^{20} + \frac {5}{17} \, a b^{4} x^{17} + \frac {5}{7} \, a^{2} b^{3} x^{14} + \frac {10}{11} \, a^{3} b^{2} x^{11} + \frac {5}{8} \, a^{4} b x^{8} + \frac {1}{5} \, a^{5} x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^4\,{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2} \,d x \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^{4} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________