Integrand size = 24, antiderivative size = 144 \[ \int \frac {x^3}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {a^2 x (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a x^2 (a+b x)}{2 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {x^3 (a+b x)}{3 b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a^3 (a+b x) \log (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 144, 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} \[ \int \frac {x^3}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {a x^2 (a+b x)}{2 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {x^3 (a+b x)}{3 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a^2 x (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a^3 (a+b x) \log (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}} \]
[In]
[Out]
Rule 45
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a b+b^2 x\right ) \int \frac {x^3}{a b+b^2 x} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {\left (a b+b^2 x\right ) \int \left (\frac {a^2}{b^4}-\frac {a x}{b^3}+\frac {x^2}{b^2}-\frac {a^3}{b^4 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {a^2 x (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a x^2 (a+b x)}{2 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {x^3 (a+b x)}{3 b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a^3 (a+b x) \log (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 1.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.40 \[ \int \frac {x^3}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {(a+b x) \left (b x \left (6 a^2-3 a b x+2 b^2 x^2\right )-6 a^3 \log (a+b x)\right )}{6 b^4 \sqrt {(a+b x)^2}} \]
[In]
[Out]
Time = 2.08 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.39
method | result | size |
default | \(-\frac {\left (b x +a \right ) \left (-2 b^{3} x^{3}+3 a \,b^{2} x^{2}+6 a^{3} \ln \left (b x +a \right )-6 a^{2} b x \right )}{6 \sqrt {\left (b x +a \right )^{2}}\, b^{4}}\) | \(56\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\frac {1}{3} b^{2} x^{3}-\frac {1}{2} a b \,x^{2}+a^{2} x \right )}{\left (b x +a \right ) b^{3}}-\frac {\sqrt {\left (b x +a \right )^{2}}\, a^{3} \ln \left (b x +a \right )}{\left (b x +a \right ) b^{4}}\) | \(73\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.28 \[ \int \frac {x^3}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {2 \, b^{3} x^{3} - 3 \, a b^{2} x^{2} + 6 \, a^{2} b x - 6 \, a^{3} \log \left (b x + a\right )}{6 \, b^{4}} \]
[In]
[Out]
Time = 0.94 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.19 \[ \int \frac {x^3}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\begin {cases} - \frac {a^{3} \left (\frac {a}{b} + x\right ) \log {\left (\frac {a}{b} + x \right )}}{b^{3} \sqrt {b^{2} \left (\frac {a}{b} + x\right )^{2}}} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} \cdot \left (\frac {11 a^{2}}{6 b^{4}} - \frac {5 a x}{6 b^{3}} + \frac {x^{2}}{3 b^{2}}\right ) & \text {for}\: b^{2} \neq 0 \\\frac {- a^{6} \sqrt {a^{2} + 2 a b x} + a^{4} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}} - \frac {3 a^{2} \left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {7}{2}}}{7}}{8 a^{4} b^{4}} & \text {for}\: a b \neq 0 \\\frac {x^{4}}{4 \sqrt {a^{2}}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.60 \[ \int \frac {x^3}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {5 \, a x^{2}}{6 \, b^{2}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} x^{2}}{3 \, b^{2}} + \frac {5 \, a^{2} x}{3 \, b^{3}} - \frac {a^{3} \log \left (x + \frac {a}{b}\right )}{b^{4}} - \frac {2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2}}{3 \, b^{4}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.47 \[ \int \frac {x^3}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {a^{3} \log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\left (b x + a\right )}{b^{4}} + \frac {2 \, b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a b x^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} x \mathrm {sgn}\left (b x + a\right )}{6 \, b^{3}} \]
[In]
[Out]
Timed out. \[ \int \frac {x^3}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {x^3}{\sqrt {{\left (a+b\,x\right )}^2}} \,d x \]
[In]
[Out]