Integrand size = 28, antiderivative size = 303 \[ \int \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^3 \, dx=\frac {f^2 \left (2 d e-b f^2\right ) \left (4 a e^2-b^2 f^2\right ) \left (e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )}{8 e^4}+\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^2}{16 e^3}+\frac {\left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^4}{8 e}-\frac {f^2 \left (2 d e-b f^2\right )^3 \left (4 a e^2-b^2 f^2\right )}{32 e^5 \left (b f^2+2 e \left (e x+f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}\right )\right )}+\frac {3 f^2 \left (2 d e-b f^2\right )^2 \left (4 a e^2-b^2 f^2\right ) \log \left (b f^2+2 e \left (e x+f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}\right )\right )}{32 e^5} \]
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Time = 0.26 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2141, 907} \[ \int \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^3 \, dx=-\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right )^3}{32 e^5 \left (2 e \left (f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}+\frac {3 f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right )^2 \log \left (2 e \left (f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}{32 e^5}+\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right ) \left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+e x\right )}{8 e^4}+\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x\right )^2}{16 e^3}+\frac {\left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x\right )^4}{8 e} \]
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Rule 907
Rule 2141
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^3 \left (d^2 e-(b d-a e) f^2-\left (2 d e-b f^2\right ) x+e x^2\right )}{\left (-2 d e+b f^2+2 e x\right )^2} \, dx,x,d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {f^2 \left (2 d e-b f^2\right ) \left (4 a e^2-b^2 f^2\right )}{16 e^4}+\frac {f^2 \left (4 a e^2-b^2 f^2\right ) x}{16 e^3}+\frac {x^3}{4 e}+\frac {f^2 \left (2 d e-b f^2\right )^3 \left (4 a e^2-b^2 f^2\right )}{32 e^4 \left (2 d e-b f^2-2 e x\right )^2}-\frac {3 \left (4 a e^2-b^2 f^2\right ) \left (2 d e f-b f^3\right )^2}{32 e^4 \left (2 d e-b f^2-2 e x\right )}\right ) \, dx,x,d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right ) \\ & = \frac {f^2 \left (2 d e-b f^2\right ) \left (4 a e^2-b^2 f^2\right ) \left (e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )}{8 e^4}+\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^2}{16 e^3}+\frac {\left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^4}{8 e}-\frac {f^2 \left (2 d e-b f^2\right )^3 \left (4 a e^2-b^2 f^2\right )}{32 e^5 \left (b f^2+2 e \left (e x+f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}\right )\right )}+\frac {3 f^2 \left (2 d e-b f^2\right )^2 \left (4 a e^2-b^2 f^2\right ) \log \left (b f^2+2 e \left (e x+f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}\right )\right )}{32 e^5} \\ \end{align*}
Time = 1.27 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.86 \[ \int \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^3 \, dx=\frac {1}{16} \left (8 x \left (2 d^3+3 d^2 e x+e x \left (3 a f^2+2 x \left (b f^2+e^2 x\right )\right )+d \left (6 a f^2+x \left (3 b f^2+4 e^2 x\right )\right )\right )+\frac {\sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )} \left (3 b^3 f^7-2 b^2 e f^5 (6 d+e x)+4 b e^2 f^3 \left (3 d^2-2 a f^2+2 d e x+2 e^2 x^2\right )+8 e^3 f \left (2 a f^2 (2 d+e x)+e x \left (3 d^2+4 d e x+2 e^2 x^2\right )\right )\right )}{e^4}+\frac {3 \left (4 a e^2-b^2 f^2\right ) \left (-2 d e f+b f^3\right )^2 \text {arctanh}\left (\frac {e x}{f \left (-\sqrt {a}+\sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )}\right )}{e^5}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(808\) vs. \(2(283)=566\).
Time = 1.26 (sec) , antiderivative size = 809, normalized size of antiderivative = 2.67
method | result | size |
default | \(f^{3} \left (\frac {\left (b +\frac {2 e^{2} x}{f^{2}}\right ) f^{2} \left (a +b x +\frac {e^{2} x^{2}}{f^{2}}\right )^{\frac {3}{2}}}{8 e^{2}}+\frac {3 \left (\frac {4 e^{2} a}{f^{2}}-b^{2}\right ) f^{2} \left (\frac {\left (b +\frac {2 e^{2} x}{f^{2}}\right ) f^{2} \sqrt {a +b x +\frac {e^{2} x^{2}}{f^{2}}}}{4 e^{2}}+\frac {\left (\frac {4 e^{2} a}{f^{2}}-b^{2}\right ) f^{2} \ln \left (\frac {\frac {b}{2}+\frac {e^{2} x}{f^{2}}}{\sqrt {\frac {e^{2}}{f^{2}}}}+\sqrt {a +b x +\frac {e^{2} x^{2}}{f^{2}}}\right )}{8 e^{2} \sqrt {\frac {e^{2}}{f^{2}}}}\right )}{16 e^{2}}\right )+3 f^{2} \left (\frac {e^{3} x^{4}}{4 f^{2}}+\frac {\left (\frac {d \,e^{2}}{f^{2}}+b e \right ) x^{3}}{3}+\frac {\left (a e +b d \right ) x^{2}}{2}+a d x \right )+3 f \left (d^{2} \left (\frac {\left (b +\frac {2 e^{2} x}{f^{2}}\right ) f^{2} \sqrt {a +b x +\frac {e^{2} x^{2}}{f^{2}}}}{4 e^{2}}+\frac {\left (\frac {4 e^{2} a}{f^{2}}-b^{2}\right ) f^{2} \ln \left (\frac {\frac {b}{2}+\frac {e^{2} x}{f^{2}}}{\sqrt {\frac {e^{2}}{f^{2}}}}+\sqrt {a +b x +\frac {e^{2} x^{2}}{f^{2}}}\right )}{8 e^{2} \sqrt {\frac {e^{2}}{f^{2}}}}\right )+e^{2} \left (\frac {x \left (a +b x +\frac {e^{2} x^{2}}{f^{2}}\right )^{\frac {3}{2}} f^{2}}{4 e^{2}}-\frac {5 b \,f^{2} \left (\frac {\left (a +b x +\frac {e^{2} x^{2}}{f^{2}}\right )^{\frac {3}{2}} f^{2}}{3 e^{2}}-\frac {b \,f^{2} \left (\frac {\left (b +\frac {2 e^{2} x}{f^{2}}\right ) f^{2} \sqrt {a +b x +\frac {e^{2} x^{2}}{f^{2}}}}{4 e^{2}}+\frac {\left (\frac {4 e^{2} a}{f^{2}}-b^{2}\right ) f^{2} \ln \left (\frac {\frac {b}{2}+\frac {e^{2} x}{f^{2}}}{\sqrt {\frac {e^{2}}{f^{2}}}}+\sqrt {a +b x +\frac {e^{2} x^{2}}{f^{2}}}\right )}{8 e^{2} \sqrt {\frac {e^{2}}{f^{2}}}}\right )}{2 e^{2}}\right )}{8 e^{2}}-\frac {a \,f^{2} \left (\frac {\left (b +\frac {2 e^{2} x}{f^{2}}\right ) f^{2} \sqrt {a +b x +\frac {e^{2} x^{2}}{f^{2}}}}{4 e^{2}}+\frac {\left (\frac {4 e^{2} a}{f^{2}}-b^{2}\right ) f^{2} \ln \left (\frac {\frac {b}{2}+\frac {e^{2} x}{f^{2}}}{\sqrt {\frac {e^{2}}{f^{2}}}}+\sqrt {a +b x +\frac {e^{2} x^{2}}{f^{2}}}\right )}{8 e^{2} \sqrt {\frac {e^{2}}{f^{2}}}}\right )}{4 e^{2}}\right )+2 e d \left (\frac {\left (a +b x +\frac {e^{2} x^{2}}{f^{2}}\right )^{\frac {3}{2}} f^{2}}{3 e^{2}}-\frac {b \,f^{2} \left (\frac {\left (b +\frac {2 e^{2} x}{f^{2}}\right ) f^{2} \sqrt {a +b x +\frac {e^{2} x^{2}}{f^{2}}}}{4 e^{2}}+\frac {\left (\frac {4 e^{2} a}{f^{2}}-b^{2}\right ) f^{2} \ln \left (\frac {\frac {b}{2}+\frac {e^{2} x}{f^{2}}}{\sqrt {\frac {e^{2}}{f^{2}}}}+\sqrt {a +b x +\frac {e^{2} x^{2}}{f^{2}}}\right )}{8 e^{2} \sqrt {\frac {e^{2}}{f^{2}}}}\right )}{2 e^{2}}\right )\right )+\frac {\left (e x +d \right )^{4}}{4 e}\) | \(809\) |
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Time = 0.33 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.14 \[ \int \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^3 \, dx=\frac {32 \, e^{8} x^{4} + 32 \, {\left (b e^{6} f^{2} + 2 \, d e^{7}\right )} x^{3} + 48 \, {\left (d^{2} e^{6} + {\left (b d e^{5} + a e^{6}\right )} f^{2}\right )} x^{2} + 32 \, {\left (3 \, a d e^{5} f^{2} + d^{3} e^{5}\right )} x + 3 \, {\left (b^{4} f^{8} - 16 \, a d^{2} e^{4} f^{2} - 4 \, {\left (b^{3} d e + a b^{2} e^{2}\right )} f^{6} + 4 \, {\left (b^{2} d^{2} e^{2} + 4 \, a b d e^{3}\right )} f^{4}\right )} \log \left (-b f^{2} - 2 \, e^{2} x + 2 \, e f \sqrt {\frac {b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}}\right ) + 2 \, {\left (3 \, b^{3} e f^{7} + 16 \, e^{7} f x^{3} - 4 \, {\left (3 \, b^{2} d e^{2} + 2 \, a b e^{3}\right )} f^{5} + 4 \, {\left (3 \, b d^{2} e^{3} + 8 \, a d e^{4}\right )} f^{3} + 8 \, {\left (b e^{5} f^{3} + 4 \, d e^{6} f\right )} x^{2} - 2 \, {\left (b^{2} e^{3} f^{5} - 12 \, d^{2} e^{5} f - 4 \, {\left (b d e^{4} + 2 \, a e^{5}\right )} f^{3}\right )} x\right )} \sqrt {\frac {b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}}}{32 \, e^{5}} \]
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Time = 1.14 (sec) , antiderivative size = 1363, normalized size of antiderivative = 4.50 \[ \int \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^3 \, dx=\text {Too large to display} \]
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Exception generated. \[ \int \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^3 \, dx=\text {Exception raised: ValueError} \]
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Time = 0.35 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.31 \[ \int \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^3 \, dx=b e f^{2} x^{3} + e^{3} x^{4} + \frac {3}{2} \, b d f^{2} x^{2} + \frac {3}{2} \, a e f^{2} x^{2} + 2 \, d e^{2} x^{3} + 3 \, a d f^{2} x + \frac {3}{2} \, d^{2} e x^{2} + d^{3} x + \frac {1}{16} \, \sqrt {b f^{2} x + e^{2} x^{2} + a f^{2}} {\left (2 \, {\left (4 \, {\left (\frac {2 \, e^{2} x {\left | f \right |}}{f} + \frac {b e^{6} f^{4} {\left | f \right |} + 4 \, d e^{7} f^{2} {\left | f \right |}}{e^{6} f^{3}}\right )} x - \frac {b^{2} e^{4} f^{6} {\left | f \right |} - 4 \, b d e^{5} f^{4} {\left | f \right |} - 8 \, a e^{6} f^{4} {\left | f \right |} - 12 \, d^{2} e^{6} f^{2} {\left | f \right |}}{e^{6} f^{3}}\right )} x + \frac {3 \, b^{3} e^{2} f^{8} {\left | f \right |} - 12 \, b^{2} d e^{3} f^{6} {\left | f \right |} - 8 \, a b e^{4} f^{6} {\left | f \right |} + 12 \, b d^{2} e^{4} f^{4} {\left | f \right |} + 32 \, a d e^{5} f^{4} {\left | f \right |}}{e^{6} f^{3}}\right )} + \frac {3 \, {\left (b^{4} f^{7} {\left | f \right |} - 4 \, b^{3} d e f^{5} {\left | f \right |} - 4 \, a b^{2} e^{2} f^{5} {\left | f \right |} + 4 \, b^{2} d^{2} e^{2} f^{3} {\left | f \right |} + 16 \, a b d e^{3} f^{3} {\left | f \right |} - 16 \, a d^{2} e^{4} f {\left | f \right |}\right )} \log \left ({\left | -b f^{2} - 2 \, {\left (x {\left | e \right |} - \sqrt {b f^{2} x + e^{2} x^{2} + a f^{2}}\right )} {\left | e \right |} \right |}\right )}{32 \, e^{4} {\left | e \right |}} \]
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Timed out. \[ \int \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^3 \, dx=\int {\left (d+e\,x+f\,\sqrt {a+b\,x+\frac {e^2\,x^2}{f^2}}\right )}^3 \,d x \]
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