Integrand size = 43, antiderivative size = 245 \[ \int \frac {\sqrt {b+a x} \left (-g+f x^2\right )}{\left (e+d x^2\right ) \sqrt {c+\sqrt {b+a x}}} \, dx=-\frac {16 c f \sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}}{15 a d}+\frac {4 f \left (3 b+8 c^2+3 a x\right ) \sqrt {c+\sqrt {b+a x}}}{15 a d}-\frac {a (e f+d g) \text {RootSum}\left [b^2 d-2 b c^2 d+c^4 d+a^2 e+4 b c d \text {$\#$1}^2-4 c^3 d \text {$\#$1}^2-2 b d \text {$\#$1}^4+6 c^2 d \text {$\#$1}^4-4 c d \text {$\#$1}^6+d \text {$\#$1}^8\&,\frac {-c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )+\log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-b \text {$\#$1}+c^2 \text {$\#$1}-2 c \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]}{2 d^2} \]
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Leaf count is larger than twice the leaf count of optimal. \(2139\) vs. \(2(245)=490\).
Time = 11.53 (sec) , antiderivative size = 2139, normalized size of antiderivative = 8.73, number of steps used = 31, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {6872, 200, 6820, 12, 2013, 1108, 648, 632, 212, 642} \[ \int \frac {\sqrt {b+a x} \left (-g+f x^2\right )}{\left (e+d x^2\right ) \sqrt {c+\sqrt {b+a x}}} \, dx=\frac {4 f \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a d}-\frac {8 c f \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a d}+\frac {4 c^2 f \sqrt {c+\sqrt {b+a x}}}{a d}+\frac {\left (b \sqrt {-d}-a \sqrt {e}\right ) (e f+d g) \text {arctanh}\left (\frac {\sqrt {\sqrt [4]{-d} c+\sqrt {\sqrt {-d} c^2-b \sqrt {-d}+a \sqrt {e}}}-\sqrt {2} \sqrt [8]{-d} \sqrt {c+\sqrt {b+a x}}}{\sqrt {c \sqrt [4]{-d}-\sqrt {\sqrt {-d} c^2-b \sqrt {-d}+a \sqrt {e}}}}\right )}{\sqrt {2} (-d)^{13/8} \sqrt {c \sqrt [4]{-d}-\sqrt {\sqrt {-d} c^2-b \sqrt {-d}+a \sqrt {e}}} \sqrt {\sqrt {-d} c^2-b \sqrt {-d}+a \sqrt {e}} \sqrt {e}}-\frac {\left (b \sqrt {-d}-a \sqrt {e}\right ) (e f+d g) \text {arctanh}\left (\frac {\sqrt {\sqrt [4]{-d} c+\sqrt {\sqrt {-d} c^2-b \sqrt {-d}+a \sqrt {e}}}+\sqrt {2} \sqrt [8]{-d} \sqrt {c+\sqrt {b+a x}}}{\sqrt {c \sqrt [4]{-d}-\sqrt {\sqrt {-d} c^2-b \sqrt {-d}+a \sqrt {e}}}}\right )}{\sqrt {2} (-d)^{13/8} \sqrt {c \sqrt [4]{-d}-\sqrt {\sqrt {-d} c^2-b \sqrt {-d}+a \sqrt {e}}} \sqrt {\sqrt {-d} c^2-b \sqrt {-d}+a \sqrt {e}} \sqrt {e}}-\frac {\left (\frac {\sqrt {e} a}{\sqrt {-d}}+b\right ) (e f+d g) \text {arctanh}\left (\frac {\sqrt {(-d)^{3/4} c+\sqrt {d \left (-\sqrt {-d} c^2+b \sqrt {-d}+a \sqrt {e}\right )}}-\sqrt {2} (-d)^{3/8} \sqrt {c+\sqrt {b+a x}}}{\sqrt {c (-d)^{3/4}-\sqrt {d \left (-\sqrt {-d} c^2+b \sqrt {-d}+a \sqrt {e}\right )}}}\right )}{\sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}-\sqrt {d \left (-\sqrt {-d} c^2+b \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {d \left (-\sqrt {-d} c^2+b \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}+\frac {\left (\frac {\sqrt {e} a}{\sqrt {-d}}+b\right ) (e f+d g) \text {arctanh}\left (\frac {\sqrt {(-d)^{3/4} c+\sqrt {d \left (-\sqrt {-d} c^2+b \sqrt {-d}+a \sqrt {e}\right )}}+\sqrt {2} (-d)^{3/8} \sqrt {c+\sqrt {b+a x}}}{\sqrt {c (-d)^{3/4}-\sqrt {d \left (-\sqrt {-d} c^2+b \sqrt {-d}+a \sqrt {e}\right )}}}\right )}{\sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}-\sqrt {d \left (-\sqrt {-d} c^2+b \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {d \left (-\sqrt {-d} c^2+b \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}-\frac {\left (b \sqrt {-d}-a \sqrt {e}\right ) (e f+d g) \log \left (\sqrt [4]{-d} \left (c+\sqrt {b+a x}\right )-\sqrt {2} \sqrt [8]{-d} \sqrt {\sqrt [4]{-d} c+\sqrt {\sqrt {-d} c^2-b \sqrt {-d}+a \sqrt {e}}} \sqrt {c+\sqrt {b+a x}}+\sqrt {\sqrt {-d} c^2-b \sqrt {-d}+a \sqrt {e}}\right )}{2 \sqrt {2} (-d)^{13/8} \sqrt {\sqrt [4]{-d} c+\sqrt {\sqrt {-d} c^2-b \sqrt {-d}+a \sqrt {e}}} \sqrt {\sqrt {-d} c^2-b \sqrt {-d}+a \sqrt {e}} \sqrt {e}}+\frac {\left (b \sqrt {-d}-a \sqrt {e}\right ) (e f+d g) \log \left (\sqrt [4]{-d} \left (c+\sqrt {b+a x}\right )+\sqrt {2} \sqrt [8]{-d} \sqrt {\sqrt [4]{-d} c+\sqrt {\sqrt {-d} c^2-b \sqrt {-d}+a \sqrt {e}}} \sqrt {c+\sqrt {b+a x}}+\sqrt {\sqrt {-d} c^2-b \sqrt {-d}+a \sqrt {e}}\right )}{2 \sqrt {2} (-d)^{13/8} \sqrt {\sqrt [4]{-d} c+\sqrt {\sqrt {-d} c^2-b \sqrt {-d}+a \sqrt {e}}} \sqrt {\sqrt {-d} c^2-b \sqrt {-d}+a \sqrt {e}} \sqrt {e}}+\frac {\left (\frac {\sqrt {e} a}{\sqrt {-d}}+b\right ) (e f+d g) \log \left ((-d)^{3/4} \left (c+\sqrt {b+a x}\right )-\sqrt {2} (-d)^{3/8} \sqrt {(-d)^{3/4} c+\sqrt {d \left (-\sqrt {-d} c^2+b \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {c+\sqrt {b+a x}}+\sqrt {d \left (-\sqrt {-d} c^2+b \sqrt {-d}+a \sqrt {e}\right )}\right )}{2 \sqrt {2} (-d)^{3/8} \sqrt {(-d)^{3/4} c+\sqrt {d \left (-\sqrt {-d} c^2+b \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {d \left (-\sqrt {-d} c^2+b \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}-\frac {\left (\frac {\sqrt {e} a}{\sqrt {-d}}+b\right ) (e f+d g) \log \left ((-d)^{3/4} \left (c+\sqrt {b+a x}\right )+\sqrt {2} (-d)^{3/8} \sqrt {(-d)^{3/4} c+\sqrt {d \left (-\sqrt {-d} c^2+b \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {c+\sqrt {b+a x}}+\sqrt {d \left (-\sqrt {-d} c^2+b \sqrt {-d}+a \sqrt {e}\right )}\right )}{2 \sqrt {2} (-d)^{3/8} \sqrt {(-d)^{3/4} c+\sqrt {d \left (-\sqrt {-d} c^2+b \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {d \left (-\sqrt {-d} c^2+b \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}} \]
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Rule 12
Rule 200
Rule 212
Rule 632
Rule 642
Rule 648
Rule 1108
Rule 2013
Rule 6820
Rule 6872
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {x^2 \left (-a^2 g+f \left (b-x^2\right )^2\right )}{\sqrt {c+x} \left (e+\frac {d \left (b-x^2\right )^2}{a^2}\right )} \, dx,x,\sqrt {b+a x}\right )}{a^3} \\ & = \frac {4 \text {Subst}\left (\int \frac {\left (c-x^2\right )^2 \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 b f}{d}-\frac {a^2 f \left (b-\left (c-x^2\right )^2\right )}{d}-\frac {a^2 b (e f+d g)-a^2 (e f+d g) \left (b-\left (c-x^2\right )^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 b f \sqrt {c+\sqrt {b+a x}}}{a d}-\frac {4 \text {Subst}\left (\int \frac {a^2 b (e f+d g)-a^2 (e f+d g) \left (b-\left (c-x^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 d}-\frac {(4 f) \text {Subst}\left (\int \left (b-\left (c-x^2\right )^2\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a d} \\ & = -\frac {4 \text {Subst}\left (\int \frac {a^2 (e f+d g) \left (c-x^2\right )^2}{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 f) \text {Subst}\left (\int \left (c-x^2\right )^2 \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a d} \\ & = \frac {(4 f) \text {Subst}\left (\int \left (c^2-2 c x^2+x^4\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a d}-\frac {(4 (e f+d g)) \text {Subst}\left (\int \frac {\left (c-x^2\right )^2}{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^2 f \sqrt {c+\sqrt {b+a x}}}{a d}-\frac {8 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 b \sqrt {e}-\frac {a^2 e}{\sqrt {-d}}}{2 e \left (a \sqrt {e}-\sqrt {-d} \left (b-\left (c-x^2\right )^2\right )\right )}+\frac {a b \sqrt {e}+\frac {a^2 e}{\sqrt {-d}}}{2 e \left (a \sqrt {e}+\sqrt {-d} \left (b-\left (c-x^2\right )^2\right )\right )}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a d} \\ & = \frac {4 c^2 f \sqrt {c+\sqrt {b+a x}}}{a d}-\frac {8 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 {\left (2 \left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g)\right ) \text {Subst}\left (\int \frac {1}{a \sqrt {e}+\sqrt {-d} \left (b-\left (c-x^2\right )^2\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{d \sqrt {e}}-\frac {\left (2 \left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g)\right ) \text {Subst}\left (\int \frac {1}{a \sqrt {e}-\sqrt {-d} \left (b-\left (c-x^2\right )^2\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{d \sqrt {e}} \\ & = \frac {4 c^2 f \sqrt {c+\sqrt {b+a x}}}{a d}-\frac {8 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 {\left (2 \left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g)\right ) \text {Subst}\left (\int \frac {1}{b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}+2 c \sqrt {-d} x^2-\sqrt {-d} x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{d \sqrt {e}}-\frac {\left (2 \left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g)\right ) \text {Subst}\left (\int \frac {1}{-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}-2 c \sqrt {-d} x^2+\sqrt {-d} x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{d \sqrt {e}} \\ & = \frac {4 c^2 f \sqrt {c+\sqrt {b+a x}}}{a d}-\frac {8 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 {\left (\left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g)\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {c (-d)^{3/4}+\sqrt {-d \left (-b \sqrt {-d}+c^2 \sqrt {-d}-a \sqrt {e}\right )}}}{(-d)^{3/8}}-x}{\frac {\sqrt {-b (-d)^{3/2}+c^2 (-d)^{3/2}+a d \sqrt {e}}}{(-d)^{3/4}}-\frac {\sqrt {2} \sqrt {c (-d)^{3/4}+\sqrt {-d \left (-b \sqrt {-d}+c^2 \sqrt {-d}-a \sqrt {e}\right )}} x}{(-d)^{3/8}}+x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{\sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}+\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}-\frac {\left (\left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g)\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {c (-d)^{3/4}+\sqrt {-d \left (-b \sqrt {-d}+c^2 \sqrt {-d}-a \sqrt {e}\right )}}}{(-d)^{3/8}}+x}{\frac {\sqrt {-b (-d)^{3/2}+c^2 (-d)^{3/2}+a d \sqrt {e}}}{(-d)^{3/4}}+\frac {\sqrt {2} \sqrt {c (-d)^{3/4}+\sqrt {-d \left (-b \sqrt {-d}+c^2 \sqrt {-d}-a \sqrt {e}\right )}} x}{(-d)^{3/8}}+x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{\sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}+\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}+\frac {\left (\left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g)\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}}{\sqrt [8]{-d}}-x}{\frac {\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}{\sqrt [4]{-d}}-\frac {\sqrt {2} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} x}{\sqrt [8]{-d}}+x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{\sqrt {2} (-d)^{9/8} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}} \sqrt {e}}+\frac {\left (\left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g)\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}}{\sqrt [8]{-d}}+x}{\frac {\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}{\sqrt [4]{-d}}+\frac {\sqrt {2} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} x}{\sqrt [8]{-d}}+x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{\sqrt {2} (-d)^{9/8} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}} \sqrt {e}} \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.43 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {b+a x} \left (-g+f x^2\right )}{\left (e+d x^2\right ) \sqrt {c+\sqrt {b+a x}}} \, dx=\frac {4 f \sqrt {c+\sqrt {b+a x}} \left (8 c^2-4 c \sqrt {b+a x}+3 (b+a x)\right )}{15 a d}-\frac {a (e f+d g) \text {RootSum}\left [b^2 d-2 b c^2 d+c^4 d+a^2 e+4 b c d \text {$\#$1}^2-4 c^3 d \text {$\#$1}^2-2 b d \text {$\#$1}^4+6 c^2 d \text {$\#$1}^4-4 c d \text {$\#$1}^6+d \text {$\#$1}^8\&,\frac {-c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )+\log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-b \text {$\#$1}+c^2 \text {$\#$1}-2 c \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]}{2 d^2} \]
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Time = 0.32 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.92
method | result | size |
derivativedivides | \(\frac {\frac {4 f \left (\frac {\left (c +\sqrt {a x +b}\right )^{\frac {5}{2}}}{5}-\frac {2 c \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}+\sqrt {c +\sqrt {a x +b}}\, c^{2}\right )}{d}-\frac {a^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{8}-4 c d \,\textit {\_Z}^{6}+\left (6 c^{2} d -2 b d \right ) \textit {\_Z}^{4}+\left (-4 c^{3} d +4 b c d \right ) \textit {\_Z}^{2}+c^{4} d -2 b \,c^{2} d +e \,a^{2}+b^{2} d \right )}{\sum }\frac {\left (\left (d g +e f \right ) \textit {\_R}^{4}+2 c \left (-d g -e f \right ) \textit {\_R}^{2}+c^{2} d g +c^{2} e f \right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5} c +3 \textit {\_R}^{3} c^{2}-\textit {\_R}^{3} b -\textit {\_R} \,c^{3}+\textit {\_R} b c}\right )}{2 d^{2}}}{a}\) | \(226\) |
default | \(-\frac {2 \left (-\frac {2 f \left (\frac {\left (c +\sqrt {a x +b}\right )^{\frac {5}{2}}}{5}-\frac {2 c \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}+\sqrt {c +\sqrt {a x +b}}\, c^{2}\right )}{d}+\frac {a^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{8}-4 c d \,\textit {\_Z}^{6}+\left (6 c^{2} d -2 b d \right ) \textit {\_Z}^{4}+\left (-4 c^{3} d +4 b c d \right ) \textit {\_Z}^{2}+c^{4} d -2 b \,c^{2} d +e \,a^{2}+b^{2} d \right )}{\sum }\frac {\left (\left (d g +e f \right ) \textit {\_R}^{4}+2 c \left (-d g -e f \right ) \textit {\_R}^{2}+c^{2} d g +c^{2} e f \right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5} c +3 \textit {\_R}^{3} c^{2}-\textit {\_R}^{3} b -\textit {\_R} \,c^{3}+\textit {\_R} b c}\right )}{4 d^{2}}\right )}{a}\) | \(226\) |
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Timed out. \[ \int \frac {\sqrt {b+a x} \left (-g+f x^2\right )}{\left (e+d x^2\right ) \sqrt {c+\sqrt {b+a x}}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\sqrt {b+a x} \left (-g+f x^2\right )}{\left (e+d x^2\right ) \sqrt {c+\sqrt {b+a x}}} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.16 \[ \int \frac {\sqrt {b+a x} \left (-g+f x^2\right )}{\left (e+d x^2\right ) \sqrt {c+\sqrt {b+a x}}} \, dx=\int { \frac {{\left (f x^{2} - g\right )} \sqrt {a x + b}}{{\left (d x^{2} + e\right )} \sqrt {c + \sqrt {a x + b}}} \,d x } \]
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 76.01 (sec) , antiderivative size = 2635, normalized size of antiderivative = 10.76 \[ \int \frac {\sqrt {b+a x} \left (-g+f x^2\right )}{\left (e+d x^2\right ) \sqrt {c+\sqrt {b+a x}}} \, dx=\text {Too large to display} \]
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Not integrable
Time = 7.33 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.16 \[ \int \frac {\sqrt {b+a x} \left (-g+f x^2\right )}{\left (e+d x^2\right ) \sqrt {c+\sqrt {b+a x}}} \, dx=\int -\frac {\left (g-f\,x^2\right )\,\sqrt {b+a\,x}}{\sqrt {c+\sqrt {b+a\,x}}\,\left (d\,x^2+e\right )} \,d x \]
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