Integrand size = 25, antiderivative size = 175 \[ \int \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^3 \, dx=-\frac {a d^3 f^2}{2 e \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}+\frac {a d f^2 \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{e}+\frac {a f^2 \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^2}{4 e}+\frac {\left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^4}{8 e}+\frac {3 a d^2 f^2 \log \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 e} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2142, 907} \[ \int \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^3 \, dx=-\frac {a d^3 f^2}{2 e \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}+\frac {3 a d^2 f^2 \log \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}{2 e}+\frac {\left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )^4}{8 e}+\frac {a f^2 \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )^2}{4 e}+\frac {a d f^2 \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}{e} \]
[In]
[Out]
Rule 907
Rule 2142
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^3 \left (d^2+a f^2-2 d x+x^2\right )}{(d-x)^2} \, dx,x,d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 e} \\ & = \frac {\text {Subst}\left (\int \left (2 a d f^2+\frac {a d^3 f^2}{(d-x)^2}-\frac {3 a d^2 f^2}{d-x}+a f^2 x+x^3\right ) \, dx,x,d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 e} \\ & = -\frac {a d^3 f^2}{2 e \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}+\frac {a d f^2 \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{e}+\frac {a f^2 \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^2}{4 e}+\frac {\left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^4}{8 e}+\frac {3 a d^2 f^2 \log \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 e} \\ \end{align*}
Time = 0.76 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.10 \[ \int \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^3 \, dx=\frac {e x \left (2 d^3+6 a d f^2+3 d^2 e x+3 a e f^2 x+4 d e^2 x^2+2 e^3 x^3\right )+\sqrt {a+\frac {e^2 x^2}{f^2}} \left (2 a f^3 (2 d+e x)+e f x \left (3 d^2+4 d e x+2 e^2 x^2\right )\right )-3 a d^2 f^2 \log \left (e \left (\sqrt {a} f+e x-f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )\right )+3 a d^2 f^2 \log \left (-\sqrt {a} f+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 e} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(340\) vs. \(2(157)=314\).
Time = 0.93 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.95
method | result | size |
default | \(f^{3} \left (\frac {x \left (a +\frac {e^{2} x^{2}}{f^{2}}\right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {a +\frac {e^{2} x^{2}}{f^{2}}}}{2}+\frac {a \ln \left (\frac {e^{2} x}{f^{2} \sqrt {\frac {e^{2}}{f^{2}}}}+\sqrt {a +\frac {e^{2} x^{2}}{f^{2}}}\right )}{2 \sqrt {\frac {e^{2}}{f^{2}}}}\right )}{4}\right )+3 f^{2} \left (\frac {e^{3} x^{4}}{4 f^{2}}+\frac {d \,e^{2} x^{3}}{3 f^{2}}+\frac {a e \,x^{2}}{2}+a d x \right )+3 f \left (d^{2} \left (\frac {x \sqrt {a +\frac {e^{2} x^{2}}{f^{2}}}}{2}+\frac {a \ln \left (\frac {e^{2} x}{f^{2} \sqrt {\frac {e^{2}}{f^{2}}}}+\sqrt {a +\frac {e^{2} x^{2}}{f^{2}}}\right )}{2 \sqrt {\frac {e^{2}}{f^{2}}}}\right )+e^{2} \left (\frac {x \left (a +\frac {e^{2} x^{2}}{f^{2}}\right )^{\frac {3}{2}} f^{2}}{4 e^{2}}-\frac {a \,f^{2} \left (\frac {x \sqrt {a +\frac {e^{2} x^{2}}{f^{2}}}}{2}+\frac {a \ln \left (\frac {e^{2} x}{f^{2} \sqrt {\frac {e^{2}}{f^{2}}}}+\sqrt {a +\frac {e^{2} x^{2}}{f^{2}}}\right )}{2 \sqrt {\frac {e^{2}}{f^{2}}}}\right )}{4 e^{2}}\right )+\frac {2 d \,f^{2} \left (\frac {e^{2} x^{2}+a \,f^{2}}{f^{2}}\right )^{\frac {3}{2}}}{3 e}\right )+\frac {\left (e x +d \right )^{4}}{4 e}\) | \(341\) |
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.92 \[ \int \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^3 \, dx=\frac {2 \, e^{4} x^{4} + 4 \, d e^{3} x^{3} - 3 \, a d^{2} f^{2} \log \left (-e x + f \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}}\right ) + 3 \, {\left (a e^{2} f^{2} + d^{2} e^{2}\right )} x^{2} + 2 \, {\left (3 \, a d e f^{2} + d^{3} e\right )} x + {\left (2 \, e^{3} f x^{3} + 4 \, d e^{2} f x^{2} + 4 \, a d f^{3} + {\left (2 \, a e f^{3} + 3 \, d^{2} e f\right )} x\right )} \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}}}{2 \, e} \]
[In]
[Out]
Time = 0.60 (sec) , antiderivative size = 459, normalized size of antiderivative = 2.62 \[ \int \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^3 \, dx=3 a d f^{2} x + \frac {3 a e f^{2} x^{2}}{2} + a f^{3} \left (\begin {cases} \frac {a \left (\begin {cases} \frac {\log {\left (\frac {2 e^{2} x}{f^{2}} + 2 \sqrt {\frac {e^{2}}{f^{2}}} \sqrt {a + \frac {e^{2} x^{2}}{f^{2}}} \right )}}{\sqrt {\frac {e^{2}}{f^{2}}}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {\frac {e^{2} x^{2}}{f^{2}}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {a + \frac {e^{2} x^{2}}{f^{2}}}}{2} & \text {for}\: \frac {e^{2}}{f^{2}} \neq 0 \\\sqrt {a} x & \text {otherwise} \end {cases}\right ) + d^{3} x + \frac {3 d^{2} e x^{2}}{2} + 3 d^{2} f \left (\begin {cases} \frac {a \left (\begin {cases} \frac {\log {\left (\frac {2 e^{2} x}{f^{2}} + 2 \sqrt {\frac {e^{2}}{f^{2}}} \sqrt {a + \frac {e^{2} x^{2}}{f^{2}}} \right )}}{\sqrt {\frac {e^{2}}{f^{2}}}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {\frac {e^{2} x^{2}}{f^{2}}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {a + \frac {e^{2} x^{2}}{f^{2}}}}{2} & \text {for}\: \frac {e^{2}}{f^{2}} \neq 0 \\\sqrt {a} x & \text {otherwise} \end {cases}\right ) + 2 d e^{2} x^{3} + 6 d e f \left (\begin {cases} \sqrt {a + \frac {e^{2} x^{2}}{f^{2}}} \left (\frac {a f^{2}}{3 e^{2}} + \frac {x^{2}}{3}\right ) & \text {for}\: \frac {e^{2}}{f^{2}} \neq 0 \\\frac {\sqrt {a} x^{2}}{2} & \text {otherwise} \end {cases}\right ) + e^{3} x^{4} + 4 e^{2} f \left (\begin {cases} - \frac {a^{2} f^{2} \left (\begin {cases} \frac {\log {\left (\frac {2 e^{2} x}{f^{2}} + 2 \sqrt {\frac {e^{2}}{f^{2}}} \sqrt {a + \frac {e^{2} x^{2}}{f^{2}}} \right )}}{\sqrt {\frac {e^{2}}{f^{2}}}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {\frac {e^{2} x^{2}}{f^{2}}}} & \text {otherwise} \end {cases}\right )}{8 e^{2}} + \sqrt {a + \frac {e^{2} x^{2}}{f^{2}}} \left (\frac {a f^{2} x}{8 e^{2}} + \frac {x^{3}}{4}\right ) & \text {for}\: \frac {e^{2}}{f^{2}} \neq 0 \\\frac {\sqrt {a} x^{3}}{3} & \text {otherwise} \end {cases}\right ) \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.67 \[ \int \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^3 \, dx=\frac {1}{4} \, e^{3} x^{4} + \frac {3 \, {\left (\frac {e^{2} x^{2}}{f^{2}} + a\right )}^{2} f^{4}}{4 \, e} - \frac {3}{8} \, {\left (\frac {a^{2} f^{3} \operatorname {arsinh}\left (\frac {e^{2} x}{\sqrt {a e^{2}} f}\right )}{\sqrt {e^{2}} e^{2}} - \frac {2 \, {\left (\frac {e^{2} x^{2}}{f^{2}} + a\right )}^{\frac {3}{2}} f^{2} x}{e^{2}} + \frac {\sqrt {\frac {e^{2} x^{2}}{f^{2}} + a} a f^{2} x}{e^{2}}\right )} e^{2} f + \frac {1}{8} \, {\left (\frac {3 \, a^{2} f \operatorname {arsinh}\left (\frac {e^{2} x}{\sqrt {a e^{2}} f}\right )}{\sqrt {e^{2}}} + 2 \, {\left (\frac {e^{2} x^{2}}{f^{2}} + a\right )}^{\frac {3}{2}} x + 3 \, \sqrt {\frac {e^{2} x^{2}}{f^{2}} + a} a x\right )} f^{3} + d^{3} x + \frac {3}{2} \, {\left (e x^{2} + {\left (\frac {a f \operatorname {arsinh}\left (\frac {e^{2} x}{\sqrt {a e^{2}} f}\right )}{\sqrt {e^{2}}} + \sqrt {\frac {e^{2} x^{2}}{f^{2}} + a} x\right )} f\right )} d^{2} + {\left (e^{2} x^{3} + \frac {2 \, {\left (\frac {e^{2} x^{2}}{f^{2}} + a\right )}^{\frac {3}{2}} f^{3}}{e} + {\left (\frac {e^{2} x^{3}}{f^{2}} + 3 \, a x\right )} f^{2}\right )} d \]
[In]
[Out]
none
Time = 0.37 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.98 \[ \int \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^3 \, dx=e^{3} x^{4} + \frac {3}{2} \, a e f^{2} x^{2} + 2 \, d e^{2} x^{3} + 3 \, a d f^{2} x + \frac {3}{2} \, d^{2} e x^{2} - \frac {3 \, a d^{2} f {\left | f \right |} \log \left ({\left | -x {\left | e \right |} + \sqrt {e^{2} x^{2} + a f^{2}} \right |}\right )}{2 \, {\left | e \right |}} + d^{3} x + \frac {1}{2} \, \sqrt {e^{2} x^{2} + a f^{2}} {\left (\frac {4 \, a d f {\left | f \right |}}{e} + {\left (2 \, {\left (\frac {e^{2} x {\left | f \right |}}{f} + \frac {2 \, d e {\left | f \right |}}{f}\right )} x + \frac {2 \, a e^{4} f^{4} {\left | f \right |} + 3 \, d^{2} e^{4} f^{2} {\left | f \right |}}{e^{4} f^{3}}\right )} x\right )} \]
[In]
[Out]
Timed out. \[ \int \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^3 \, dx=\int {\left (d+e\,x+f\,\sqrt {a+\frac {e^2\,x^2}{f^2}}\right )}^3 \,d x \]
[In]
[Out]