Optimal. Leaf size=151 \[ \frac {3 a b^2 x^8 \sqrt {a^2+2 a b x+b^2 x^2}}{8 (a+b x)}+\frac {3 a^2 b x^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {b^3 x^9 \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {a^3 x^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 (a+b x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {646, 43} \[ \frac {b^3 x^9 \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {3 a b^2 x^8 \sqrt {a^2+2 a b x+b^2 x^2}}{8 (a+b x)}+\frac {3 a^2 b x^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {a^3 x^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 646
Rubi steps
\begin {align*} \int x^5 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int x^5 \left (a b+b^2 x\right )^3 \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a^3 b^3 x^5+3 a^2 b^4 x^6+3 a b^5 x^7+b^6 x^8\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {a^3 x^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 (a+b x)}+\frac {3 a^2 b x^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {3 a b^2 x^8 \sqrt {a^2+2 a b x+b^2 x^2}}{8 (a+b x)}+\frac {b^3 x^9 \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 55, normalized size = 0.36 \[ \frac {x^6 \sqrt {(a+b x)^2} \left (84 a^3+216 a^2 b x+189 a b^2 x^2+56 b^3 x^3\right )}{504 (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.95, size = 35, normalized size = 0.23 \[ \frac {1}{9} \, b^{3} x^{9} + \frac {3}{8} \, a b^{2} x^{8} + \frac {3}{7} \, a^{2} b x^{7} + \frac {1}{6} \, a^{3} x^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.21, size = 73, normalized size = 0.48 \[ \frac {1}{9} \, b^{3} x^{9} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{8} \, a b^{2} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{7} \, a^{2} b x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, a^{3} x^{6} \mathrm {sgn}\left (b x + a\right ) - \frac {a^{9} \mathrm {sgn}\left (b x + a\right )}{504 \, b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 52, normalized size = 0.34 \[ \frac {\left (56 b^{3} x^{3}+189 a \,b^{2} x^{2}+216 a^{2} b x +84 a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} x^{6}}{504 \left (b x +a \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.54, size = 189, normalized size = 1.25 \[ \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} x^{4}}{9 \, b^{2}} - \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a x^{3}}{72 \, b^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{5} x}{4 \, b^{5}} + \frac {37 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} x^{2}}{168 \, b^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{6}}{4 \, b^{6}} - \frac {121 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} x}{504 \, b^{5}} + \frac {125 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{4}}{504 \, b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^5\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{5} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________