Integrand size = 34, antiderivative size = 212 \[ \int \frac {\left (-g+f x^2\right ) \sqrt {c+\sqrt {b+a x}}}{e+d x^2} \, dx=\frac {4 c f \sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}}{15 a d}+\frac {4 f \left (3 b-2 c^2+3 a x\right ) \sqrt {c+\sqrt {b+a x}}}{15 a d}-\frac {a (e f+d g) \text {RootSum}\left [b^2 d-2 b c^2 d+c^4 d+a^2 e+4 b c d \text {$\#$1}^2-4 c^3 d \text {$\#$1}^2-2 b d \text {$\#$1}^4+6 c^2 d \text {$\#$1}^4-4 c d \text {$\#$1}^6+d \text {$\#$1}^8\&,\frac {\log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}}{-b+c^2-2 c \text {$\#$1}^2+\text {$\#$1}^4}\&\right ]}{2 d^2} \]
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\[ \int \frac {\left (-g+f x^2\right ) \sqrt {c+\sqrt {b+a x}}}{e+d x^2} \, dx=\int \frac {\left (-g+f x^2\right ) \sqrt {c+\sqrt {b+a x}}}{e+d x^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {x \sqrt {c+x} \left (-a^2 g+f \left (b-x^2\right )^2\right )}{e+\frac {d \left (b-x^2\right )^2}{a^2}} \, dx,x,\sqrt {b+a x}\right )}{a^3} \\ & = \frac {4 \text {Subst}\left (\int \frac {x^2 \left (-c+x^2\right ) \left (-a^2 g+f \left (b-\left (c-x^2\right )^2\right )^2\right )}{e+\frac {d \left (b-\left (c-x^2\right )^2\right )^2}{a^2}} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a^3} \\ & = \frac {4 \text {Subst}\left (\int \left (-\frac {a^2 c f x^2}{d}+\frac {a^2 f x^4}{d}+\frac {x^2 \left (a^2 c (e f+d g)-a^2 (e f+d g) x^2\right )}{d \left (e+\frac {d \left (b-\left (c-x^2\right )^2\right )^2}{a^2}\right )}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a^3} \\ & = -\frac {4 c f \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a d}+\frac {4 f \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a d}+\frac {4 \text {Subst}\left (\int \frac {x^2 \left (a^2 c (e f+d g)-a^2 (e f+d g) x^2\right )}{e+\frac {d \left (b-\left (c-x^2\right )^2\right )^2}{a^2}} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a^3 d} \\ & = -\frac {4 c f \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a d}+\frac {4 f \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a d}+\frac {4 \text {Subst}\left (\int \frac {a^2 (e f+d g) x^2 \left (c-x^2\right )}{e+\frac {d \left (b-\left (c-x^2\right )^2\right )^2}{a^2}} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a^3 d} \\ & = -\frac {4 c f \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a d}+\frac {4 f \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a d}+\frac {(4 (e f+d g)) \text {Subst}\left (\int \frac {x^2 \left (c-x^2\right )}{e+\frac {d \left (b-\left (c-x^2\right )^2\right )^2}{a^2}} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a d} \\ & = -\frac {4 c f \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a d}+\frac {4 f \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a d}+\frac {(4 (e f+d g)) \text {Subst}\left (\int \left (\frac {a^2 x^4}{-b^2 d \left (1+\frac {-2 b c^2 d+c^4 d+a^2 e}{b^2 d}\right )-4 b c \left (1-\frac {c^2}{b}\right ) d x^2+2 b \left (1-\frac {3 c^2}{b}\right ) d x^4+4 c d x^6-d x^8}+\frac {a^2 c x^2}{b^2 d \left (1+\frac {-2 b c^2 d+c^4 d+a^2 e}{b^2 d}\right )+4 b c \left (1-\frac {c^2}{b}\right ) d x^2-2 b \left (1-\frac {3 c^2}{b}\right ) d x^4-4 c d x^6+d x^8}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a d} \\ & = -\frac {4 c f \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a d}+\frac {4 f \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a d}+\frac {(4 a (e f+d g)) \text {Subst}\left (\int \frac {x^4}{-b^2 d \left (1+\frac {-2 b c^2 d+c^4 d+a^2 e}{b^2 d}\right )-4 b c \left (1-\frac {c^2}{b}\right ) d x^2+2 b \left (1-\frac {3 c^2}{b}\right ) d x^4+4 c d x^6-d x^8} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{d}+\frac {(4 a c (e f+d g)) \text {Subst}\left (\int \frac {x^2}{b^2 d \left (1+\frac {-2 b c^2 d+c^4 d+a^2 e}{b^2 d}\right )+4 b c \left (1-\frac {c^2}{b}\right ) d x^2-2 b \left (1-\frac {3 c^2}{b}\right ) d x^4-4 c d x^6+d x^8} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{d} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-g+f x^2\right ) \sqrt {c+\sqrt {b+a x}}}{e+d x^2} \, dx=-\frac {4 f \sqrt {c+\sqrt {b+a x}} \left (2 c^2-c \sqrt {b+a x}-3 (b+a x)\right )}{15 a d}-\frac {a (e f+d g) \text {RootSum}\left [b^2 d-2 b c^2 d+c^4 d+a^2 e+4 b c d \text {$\#$1}^2-4 c^3 d \text {$\#$1}^2-2 b d \text {$\#$1}^4+6 c^2 d \text {$\#$1}^4-4 c d \text {$\#$1}^6+d \text {$\#$1}^8\&,\frac {\log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}}{-b+c^2-2 c \text {$\#$1}^2+\text {$\#$1}^4}\&\right ]}{2 d^2} \]
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Time = 0.32 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(\frac {-\frac {4 f \left (-\frac {\left (c +\sqrt {a x +b}\right )^{\frac {5}{2}}}{5}+\frac {c \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}\right )}{d}+\frac {a^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{8}-4 d c \,\textit {\_Z}^{6}+\left (6 c^{2} d -2 b d \right ) \textit {\_Z}^{4}+\left (-4 c^{3} d +4 b c d \right ) \textit {\_Z}^{2}+c^{4} d -2 b \,c^{2} d +e \,a^{2}+b^{2} d \right )}{\sum }\frac {\left (\left (-d g -e f \right ) \textit {\_R}^{4}+c \left (d g +e f \right ) \textit {\_R}^{2}\right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5} c +3 \textit {\_R}^{3} c^{2}-\textit {\_R}^{3} b -\textit {\_R} \,c^{3}+\textit {\_R} b c}\right )}{2 d^{2}}}{a}\) | \(198\) |
default | \(-\frac {2 \left (\frac {2 f \left (-\frac {\left (c +\sqrt {a x +b}\right )^{\frac {5}{2}}}{5}+\frac {c \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}\right )}{d}-\frac {a^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{8}-4 d c \,\textit {\_Z}^{6}+\left (6 c^{2} d -2 b d \right ) \textit {\_Z}^{4}+\left (-4 c^{3} d +4 b c d \right ) \textit {\_Z}^{2}+c^{4} d -2 b \,c^{2} d +e \,a^{2}+b^{2} d \right )}{\sum }\frac {\left (\left (-d g -e f \right ) \textit {\_R}^{4}+c \left (d g +e f \right ) \textit {\_R}^{2}\right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5} c +3 \textit {\_R}^{3} c^{2}-\textit {\_R}^{3} b -\textit {\_R} \,c^{3}+\textit {\_R} b c}\right )}{4 d^{2}}\right )}{a}\) | \(198\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.75 (sec) , antiderivative size = 5974, normalized size of antiderivative = 28.18 \[ \int \frac {\left (-g+f x^2\right ) \sqrt {c+\sqrt {b+a x}}}{e+d x^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\left (-g+f x^2\right ) \sqrt {c+\sqrt {b+a x}}}{e+d x^2} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.15 \[ \int \frac {\left (-g+f x^2\right ) \sqrt {c+\sqrt {b+a x}}}{e+d x^2} \, dx=\int { \frac {{\left (f x^{2} - g\right )} \sqrt {c + \sqrt {a x + b}}}{d x^{2} + e} \,d x } \]
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.44 (sec) , antiderivative size = 2428, normalized size of antiderivative = 11.45 \[ \int \frac {\left (-g+f x^2\right ) \sqrt {c+\sqrt {b+a x}}}{e+d x^2} \, dx=\text {Too large to display} \]
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Not integrable
Time = 6.89 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.15 \[ \int \frac {\left (-g+f x^2\right ) \sqrt {c+\sqrt {b+a x}}}{e+d x^2} \, dx=\int -\frac {\sqrt {c+\sqrt {b+a\,x}}\,\left (g-f\,x^2\right )}{d\,x^2+e} \,d x \]
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