Integrand size = 24, antiderivative size = 151 \[ \int x^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {a^3 x^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac {3 a^2 b x^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac {a b^2 x^6 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac {b^3 x^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)} \]
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Time = 0.03 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {660, 45} \[ \int x^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {a b^2 x^6 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac {3 a^2 b x^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac {b^3 x^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {a^3 x^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (a+b x)} \]
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Rule 45
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int x^3 \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^3+3 a^2 b^4 x^4+3 a b^5 x^5+b^6 x^6\right ) \, dx}{b^2 \left (a b+b^2 x\right )} \\ & = \frac {a^3 x^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac {3 a^2 b x^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac {a b^2 x^6 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac {b^3 x^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)} \\ \end{align*}
Time = 0.84 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.68 \[ \int x^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {x^4 \left (35 a^3+84 a^2 b x+70 a b^2 x^2+20 b^3 x^3\right ) \left (\sqrt {a^2} b x+a \left (\sqrt {a^2}-\sqrt {(a+b x)^2}\right )\right )}{140 \left (-a^2-a b x+\sqrt {a^2} \sqrt {(a+b x)^2}\right )} \]
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Time = 2.20 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.34
method | result | size |
gosper | \(\frac {x^{4} \left (20 b^{3} x^{3}+70 a \,b^{2} x^{2}+84 a^{2} b x +35 a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{140 \left (b x +a \right )^{3}}\) | \(52\) |
default | \(\frac {x^{4} \left (20 b^{3} x^{3}+70 a \,b^{2} x^{2}+84 a^{2} b x +35 a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{140 \left (b x +a \right )^{3}}\) | \(52\) |
risch | \(\frac {a^{3} x^{4} \sqrt {\left (b x +a \right )^{2}}}{4 b x +4 a}+\frac {3 a^{2} b \,x^{5} \sqrt {\left (b x +a \right )^{2}}}{5 \left (b x +a \right )}+\frac {a \,b^{2} x^{6} \sqrt {\left (b x +a \right )^{2}}}{2 b x +2 a}+\frac {b^{3} x^{7} \sqrt {\left (b x +a \right )^{2}}}{7 b x +7 a}\) | \(100\) |
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Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.23 \[ \int x^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {1}{7} \, b^{3} x^{7} + \frac {1}{2} \, a b^{2} x^{6} + \frac {3}{5} \, a^{2} b x^{5} + \frac {1}{4} \, a^{3} x^{4} \]
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Time = 0.55 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.22 \[ \int x^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\begin {cases} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} \left (- \frac {a^{6}}{140 b^{4}} + \frac {a^{5} x}{140 b^{3}} - \frac {a^{4} x^{2}}{140 b^{2}} + \frac {a^{3} x^{3}}{140 b} + \frac {17 a^{2} x^{4}}{70} + \frac {5 a b x^{5}}{14} + \frac {b^{2} x^{6}}{7}\right ) & \text {for}\: b^{2} \neq 0 \\\frac {- \frac {a^{6} \left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5} + \frac {3 a^{4} \left (a^{2} + 2 a b x\right )^{\frac {7}{2}}}{7} - \frac {a^{2} \left (a^{2} + 2 a b x\right )^{\frac {9}{2}}}{3} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {11}{2}}}{11}}{8 a^{4} b^{4}} & \text {for}\: a b \neq 0 \\\frac {x^{4} \left (a^{2}\right )^{\frac {3}{2}}}{4} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.87 \[ \int x^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=-\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3} x}{4 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} x^{2}}{7 \, b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{4}}{4 \, b^{4}} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a x}{14 \, b^{3}} + \frac {17 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2}}{70 \, b^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.48 \[ \int x^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {1}{7} \, b^{3} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, a b^{2} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{5} \, a^{2} b x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, a^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) - \frac {a^{7} \mathrm {sgn}\left (b x + a\right )}{140 \, b^{4}} \]
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Timed out. \[ \int x^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\int x^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \]
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