Optimal. Leaf size=56 \[ \frac {2 \left (a+b \left (c x^2\right )^{3/2}\right )^{5/2}}{15 b^2 c^3}-\frac {2 a \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2}}{9 b^2 c^3} \]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {368, 266, 43} \begin {gather*} \frac {2 \left (a+b \left (c x^2\right )^{3/2}\right )^{5/2}}{15 b^2 c^3}-\frac {2 a \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2}}{9 b^2 c^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rule 368
Rubi steps
\begin {align*} \int x^5 \sqrt {a+b \left (c x^2\right )^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int x^5 \sqrt {a+b x^3} \, dx,x,\sqrt {c x^2}\right )}{c^3}\\ &=\frac {\operatorname {Subst}\left (\int x \sqrt {a+b x} \, dx,x,\left (c x^2\right )^{3/2}\right )}{3 c^3}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a \sqrt {a+b x}}{b}+\frac {(a+b x)^{3/2}}{b}\right ) \, dx,x,\left (c x^2\right )^{3/2}\right )}{3 c^3}\\ &=-\frac {2 a \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2}}{9 b^2 c^3}+\frac {2 \left (a+b \left (c x^2\right )^{3/2}\right )^{5/2}}{15 b^2 c^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 43, normalized size = 0.77 \begin {gather*} \frac {2 \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2} \left (3 b \left (c x^2\right )^{3/2}-2 a\right )}{45 b^2 c^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 2.72, size = 57, normalized size = 1.02 \begin {gather*} -\frac {2 \sqrt {a+b \left (c x^2\right )^{3/2}} \left (2 a^2-a b \left (c x^2\right )^{3/2}-3 b^2 c^3 x^6\right )}{45 b^2 c^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.82, size = 56, normalized size = 1.00 \begin {gather*} \frac {2 \, {\left (3 \, b^{2} c^{3} x^{6} + \sqrt {c x^{2}} a b c x^{2} - 2 \, a^{2}\right )} \sqrt {\sqrt {c x^{2}} b c x^{2} + a}}{45 \, b^{2} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 38, normalized size = 0.68 \begin {gather*} \frac {2 \, {\left (3 \, {\left (b c^{\frac {3}{2}} x^{3} + a\right )}^{\frac {5}{2}} - 5 \, {\left (b c^{\frac {3}{2}} x^{3} + a\right )}^{\frac {3}{2}} a\right )}}{45 \, b^{2} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.23, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a +\left (c \,x^{2}\right )^{\frac {3}{2}} b}\, x^{5}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.53, size = 43, normalized size = 0.77 \begin {gather*} \frac {2 \, {\left (\frac {3 \, {\left (\left (c x^{2}\right )^{\frac {3}{2}} b + a\right )}^{\frac {5}{2}}}{b^{2}} - \frac {5 \, {\left (\left (c x^{2}\right )^{\frac {3}{2}} b + a\right )}^{\frac {3}{2}} a}{b^{2}}\right )}}{45 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x^5\,\sqrt {a+b\,{\left (c\,x^2\right )}^{3/2}} \,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^{5} \sqrt {a + b \left (c x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________