Optimal. Leaf size=348 \[ -\frac {a c f \text {RootSum}\left [\text {$\#$1}^8 e+4 \text {$\#$1}^4 a^2 f-2 \text {$\#$1}^4 b^2 e+b^4 e\& ,\frac {\text {$\#$1}^4 \left (-\log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-\text {$\#$1}\right )\right )-b^2 \log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-\text {$\#$1}\right )}{\text {$\#$1}^5 (-e)-2 \text {$\#$1} a^2 f+\text {$\#$1} b^2 e}\& \right ]}{2 e}+\frac {1}{2} a d \text {RootSum}\left [\text {$\#$1}^8 e+4 \text {$\#$1}^4 a^2 f-2 \text {$\#$1}^4 b^2 e+b^4 e\& ,\frac {\text {$\#$1}^4 \left (-\log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-\text {$\#$1}\right )\right )-b^2 \log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-\text {$\#$1}\right )}{\text {$\#$1}^5 (-e)-2 \text {$\#$1} a^2 f+\text {$\#$1} b^2 e}\& \right ]-\frac {b^2 c}{a e \sqrt {\sqrt {a^2 x^2+b^2}+a x}}+\frac {c \left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}}{3 a e} \]
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Rubi [B] time = 3.87, antiderivative size = 1370, normalized size of antiderivative = 3.94, number of steps used = 33, number of rules used = 11, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.275, Rules used = {6725, 2117, 14, 2119, 1628, 826, 1169, 634, 618, 204, 628} \begin {gather*} -\frac {c b^2}{a e \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a e}+\frac {\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} (d e-c f) \tan ^{-1}\left (\frac {\sqrt {-e} \left (\sqrt {\sqrt {f} a+\sqrt {-b^2} \sqrt {-e}}-\sqrt {2} \sqrt [4]{-e} \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}}}\right )}{\sqrt {2} (-e)^{9/4} \sqrt {f}}-\frac {\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} (d e-c f) \tan ^{-1}\left (\frac {\sqrt {-e} \left (\sqrt {\sqrt {f} a+\sqrt {-b^2} \sqrt {-e}}+\sqrt {2} \sqrt [4]{-e} \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}}}\right )}{\sqrt {2} (-e)^{9/4} \sqrt {f}}-\frac {\sqrt {\sqrt {f} a+\sqrt {-b^2} \sqrt {-e}} (d e-c f) \tan ^{-1}\left (\frac {\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}}-\sqrt {2} (-e)^{3/4} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {-e} \sqrt {\sqrt {f} a+\sqrt {-b^2} \sqrt {-e}}}\right )}{\sqrt {2} (-e)^{7/4} \sqrt {f}}+\frac {\sqrt {\sqrt {f} a+\sqrt {-b^2} \sqrt {-e}} (d e-c f) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a x+\sqrt {b^2+a^2 x^2}} (-e)^{3/4}+\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}}}{\sqrt {-e} \sqrt {\sqrt {f} a+\sqrt {-b^2} \sqrt {-e}}}\right )}{\sqrt {2} (-e)^{7/4} \sqrt {f}}+\frac {\sqrt {\sqrt {f} a+\sqrt {-b^2} \sqrt {-e}} (d e-c f) \log \left (\sqrt [4]{-e} \left (a x+\sqrt {b^2+a^2 x^2}\right )-\sqrt {2} \sqrt {\sqrt {f} a+\sqrt {-b^2} \sqrt {-e}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt {-b^2} \sqrt [4]{-e}\right )}{2 \sqrt {2} (-e)^{7/4} \sqrt {f}}-\frac {\sqrt {\sqrt {f} a+\sqrt {-b^2} \sqrt {-e}} (d e-c f) \log \left (\sqrt [4]{-e} \left (a x+\sqrt {b^2+a^2 x^2}\right )+\sqrt {2} \sqrt {\sqrt {f} a+\sqrt {-b^2} \sqrt {-e}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt {-b^2} \sqrt [4]{-e}\right )}{2 \sqrt {2} (-e)^{7/4} \sqrt {f}}-\frac {\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} (d e-c f) \log \left ((-e)^{3/4} \left (a x+\sqrt {b^2+a^2 x^2}\right )-\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt {-b^2} (-e)^{3/4}\right )}{2 \sqrt {2} (-e)^{9/4} \sqrt {f}}+\frac {\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} (d e-c f) \log \left ((-e)^{3/4} \left (a x+\sqrt {b^2+a^2 x^2}\right )+\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt {-b^2} (-e)^{3/4}\right )}{2 \sqrt {2} (-e)^{9/4} \sqrt {f}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 204
Rule 618
Rule 628
Rule 634
Rule 826
Rule 1169
Rule 1628
Rule 2117
Rule 2119
Rule 6725
Rubi steps
\begin {align*} \int \frac {\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{f+e x^2} \, dx &=\int \left (\frac {c \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{e}+\frac {(d e-c f) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{e \left (f+e x^2\right )}\right ) \, dx\\ &=\frac {c \int \sqrt {a x+\sqrt {b^2+a^2 x^2}} \, dx}{e}+\frac {(d e-c f) \int \frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{f+e x^2} \, dx}{e}\\ &=\frac {c \operatorname {Subst}\left (\int \frac {b^2+x^2}{x^{3/2}} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 a e}+\frac {(d e-c f) \int \left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{2 \sqrt {f} \left (\sqrt {f}-\sqrt {-e} x\right )}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{2 \sqrt {f} \left (\sqrt {f}+\sqrt {-e} x\right )}\right ) \, dx}{e}\\ &=\frac {c \operatorname {Subst}\left (\int \left (\frac {b^2}{x^{3/2}}+\sqrt {x}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 a e}+\frac {(d e-c f) \int \frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {f}-\sqrt {-e} x} \, dx}{2 e \sqrt {f}}+\frac {(d e-c f) \int \frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {f}+\sqrt {-e} x} \, dx}{2 e \sqrt {f}}\\ &=-\frac {b^2 c}{a e \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a e}+\frac {(d e-c f) \operatorname {Subst}\left (\int \frac {b^2+x^2}{\sqrt {x} \left (b^2 \sqrt {-e}+2 a \sqrt {f} x-\sqrt {-e} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 e \sqrt {f}}+\frac {(d e-c f) \operatorname {Subst}\left (\int \frac {b^2+x^2}{\sqrt {x} \left (-b^2 \sqrt {-e}+2 a \sqrt {f} x+\sqrt {-e} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 e \sqrt {f}}\\ &=-\frac {b^2 c}{a e \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a e}+\frac {(d e-c f) \operatorname {Subst}\left (\int \left (-\frac {1}{\sqrt {-e} \sqrt {x}}+\frac {2 \left (b^2 e-a \sqrt {-e} \sqrt {f} x\right )}{e \sqrt {x} \left (b^2 \sqrt {-e}+2 a \sqrt {f} x-\sqrt {-e} x^2\right )}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 e \sqrt {f}}+\frac {(d e-c f) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt {-e} \sqrt {x}}+\frac {2 \left (b^2 e+a \sqrt {-e} \sqrt {f} x\right )}{e \sqrt {x} \left (-b^2 \sqrt {-e}+2 a \sqrt {f} x+\sqrt {-e} x^2\right )}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 e \sqrt {f}}\\ &=-\frac {b^2 c}{a e \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a e}+\frac {(d e-c f) \operatorname {Subst}\left (\int \frac {b^2 e-a \sqrt {-e} \sqrt {f} x}{\sqrt {x} \left (b^2 \sqrt {-e}+2 a \sqrt {f} x-\sqrt {-e} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{e^2 \sqrt {f}}+\frac {(d e-c f) \operatorname {Subst}\left (\int \frac {b^2 e+a \sqrt {-e} \sqrt {f} x}{\sqrt {x} \left (-b^2 \sqrt {-e}+2 a \sqrt {f} x+\sqrt {-e} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{e^2 \sqrt {f}}\\ &=-\frac {b^2 c}{a e \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a e}+\frac {(2 (d e-c f)) \operatorname {Subst}\left (\int \frac {b^2 e-a \sqrt {-e} \sqrt {f} x^2}{b^2 \sqrt {-e}+2 a \sqrt {f} x^2-\sqrt {-e} x^4} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{e^2 \sqrt {f}}+\frac {(2 (d e-c f)) \operatorname {Subst}\left (\int \frac {b^2 e+a \sqrt {-e} \sqrt {f} x^2}{-b^2 \sqrt {-e}+2 a \sqrt {f} x^2+\sqrt {-e} x^4} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{e^2 \sqrt {f}}\\ &=-\frac {b^2 c}{a e \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a e}-\frac {(d e-c f) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} b^2 e \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}}}{\sqrt [4]{-e}}-\left (b^2 e+a \sqrt {-b^2} \sqrt {-e} \sqrt {f}\right ) x}{\sqrt {-b^2}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} x}{\sqrt [4]{-e}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {2} \sqrt {-b^2} (-e)^{9/4} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} \sqrt {f}}-\frac {(d e-c f) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} b^2 e \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}}}{\sqrt [4]{-e}}+\left (b^2 e+a \sqrt {-b^2} \sqrt {-e} \sqrt {f}\right ) x}{\sqrt {-b^2}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} x}{\sqrt [4]{-e}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {2} \sqrt {-b^2} (-e)^{9/4} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} \sqrt {f}}+\frac {(d e-c f) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} b^2 e \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}}}{(-e)^{3/4}}-\left (b^2 e-a \sqrt {-b^2} \sqrt {-e} \sqrt {f}\right ) x}{\sqrt {-b^2}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} x}{(-e)^{3/4}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {2} \sqrt {-b^2} (-e)^{7/4} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} \sqrt {f}}+\frac {(d e-c f) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} b^2 e \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}}}{(-e)^{3/4}}+\left (b^2 e-a \sqrt {-b^2} \sqrt {-e} \sqrt {f}\right ) x}{\sqrt {-b^2}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} x}{(-e)^{3/4}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {2} \sqrt {-b^2} (-e)^{7/4} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} \sqrt {f}}\\ &=-\frac {b^2 c}{a e \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a e}-\frac {\left (\left (\sqrt {-b^2} \sqrt {-e}-a \sqrt {f}\right ) (d e-c f)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b^2}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} x}{\sqrt [4]{-e}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 e^2 \sqrt {f}}-\frac {\left (\left (\sqrt {-b^2} \sqrt {-e}-a \sqrt {f}\right ) (d e-c f)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b^2}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} x}{\sqrt [4]{-e}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 e^2 \sqrt {f}}+\frac {\left (\sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} (d e-c f)\right ) \operatorname {Subst}\left (\int \frac {-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}}}{\sqrt [4]{-e}}+2 x}{\sqrt {-b^2}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} x}{\sqrt [4]{-e}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 \sqrt {2} (-e)^{7/4} \sqrt {f}}-\frac {\left (\sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} (d e-c f)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}}}{\sqrt [4]{-e}}+2 x}{\sqrt {-b^2}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} x}{\sqrt [4]{-e}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 \sqrt {2} (-e)^{7/4} \sqrt {f}}+\frac {\left (\left (\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}\right ) (d e-c f)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b^2}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} x}{(-e)^{3/4}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 e^2 \sqrt {f}}+\frac {\left (\left (\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}\right ) (d e-c f)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b^2}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} x}{(-e)^{3/4}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 e^2 \sqrt {f}}-\frac {\left (\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} (d e-c f)\right ) \operatorname {Subst}\left (\int \frac {-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}}}{(-e)^{3/4}}+2 x}{\sqrt {-b^2}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} x}{(-e)^{3/4}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 \sqrt {2} (-e)^{9/4} \sqrt {f}}+\frac {\left (\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} (d e-c f)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}}}{(-e)^{3/4}}+2 x}{\sqrt {-b^2}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} x}{(-e)^{3/4}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 \sqrt {2} (-e)^{9/4} \sqrt {f}}\\ &=-\frac {b^2 c}{a e \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a e}+\frac {\sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} (d e-c f) \log \left (\sqrt {-b^2} \sqrt [4]{-e}-\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt [4]{-e} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} (-e)^{7/4} \sqrt {f}}-\frac {\sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} (d e-c f) \log \left (\sqrt {-b^2} \sqrt [4]{-e}+\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt [4]{-e} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} (-e)^{7/4} \sqrt {f}}-\frac {\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} (d e-c f) \log \left (\sqrt {-b^2} (-e)^{3/4}-\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+(-e)^{3/4} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} (-e)^{9/4} \sqrt {f}}+\frac {\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} (d e-c f) \log \left (\sqrt {-b^2} (-e)^{3/4}+\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+(-e)^{3/4} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} (-e)^{9/4} \sqrt {f}}+\frac {\left (\left (\sqrt {-b^2} \sqrt {-e}-a \sqrt {f}\right ) (d e-c f)\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {2 \left (\sqrt {-b^2} e+a \sqrt {-e} \sqrt {f}\right )}{e}-x^2} \, dx,x,-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}}}{\sqrt [4]{-e}}+2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{e^2 \sqrt {f}}+\frac {\left (\left (\sqrt {-b^2} \sqrt {-e}-a \sqrt {f}\right ) (d e-c f)\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {2 \left (\sqrt {-b^2} e+a \sqrt {-e} \sqrt {f}\right )}{e}-x^2} \, dx,x,\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}}}{\sqrt [4]{-e}}+2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{e^2 \sqrt {f}}-\frac {\left (\left (\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}\right ) (d e-c f)\right ) \operatorname {Subst}\left (\int \frac {1}{-2 \left (\sqrt {-b^2}+\frac {a \sqrt {f}}{\sqrt {-e}}\right )-x^2} \, dx,x,-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}}}{(-e)^{3/4}}+2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{e^2 \sqrt {f}}-\frac {\left (\left (\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}\right ) (d e-c f)\right ) \operatorname {Subst}\left (\int \frac {1}{-2 \left (\sqrt {-b^2}+\frac {a \sqrt {f}}{\sqrt {-e}}\right )-x^2} \, dx,x,\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}}}{(-e)^{3/4}}+2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{e^2 \sqrt {f}}\\ &=-\frac {b^2 c}{a e \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a e}+\frac {\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} (d e-c f) \tan ^{-1}\left (\frac {(-e)^{3/4} \left (\frac {\sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}}}{\sqrt [4]{-e}}-\sqrt {2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}}}\right )}{\sqrt {2} (-e)^{9/4} \sqrt {f}}-\frac {\sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} (d e-c f) \tan ^{-1}\left (\frac {\sqrt [4]{-e} \left (\frac {\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}}}{(-e)^{3/4}}-\sqrt {2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}}}\right )}{\sqrt {2} (-e)^{7/4} \sqrt {f}}-\frac {\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} (d e-c f) \tan ^{-1}\left (\frac {(-e)^{3/4} \left (\frac {\sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}}}{\sqrt [4]{-e}}+\sqrt {2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}}}\right )}{\sqrt {2} (-e)^{9/4} \sqrt {f}}+\frac {\sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} (d e-c f) \tan ^{-1}\left (\frac {\sqrt [4]{-e} \left (\frac {\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}}}{(-e)^{3/4}}+\sqrt {2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}}}\right )}{\sqrt {2} (-e)^{7/4} \sqrt {f}}+\frac {\sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} (d e-c f) \log \left (\sqrt {-b^2} \sqrt [4]{-e}-\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt [4]{-e} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} (-e)^{7/4} \sqrt {f}}-\frac {\sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} (d e-c f) \log \left (\sqrt {-b^2} \sqrt [4]{-e}+\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt [4]{-e} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} (-e)^{7/4} \sqrt {f}}-\frac {\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} (d e-c f) \log \left (\sqrt {-b^2} (-e)^{3/4}-\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+(-e)^{3/4} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} (-e)^{9/4} \sqrt {f}}+\frac {\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} (d e-c f) \log \left (\sqrt {-b^2} (-e)^{3/4}+\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+(-e)^{3/4} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} (-e)^{9/4} \sqrt {f}}\\ \end {align*}
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Mathematica [A] time = 0.72, size = 205, normalized size = 0.59 \begin {gather*} \frac {3 a (d e-c f) \text {RootSum}\left [\text {$\#$1}^8 e+4 \text {$\#$1}^4 a^2 f-2 \text {$\#$1}^4 b^2 e+b^4 e\&,\frac {\text {$\#$1}^4 \log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-\text {$\#$1}\right )+b^2 \log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-\text {$\#$1}\right )}{\text {$\#$1}^5 e+2 \text {$\#$1} a^2 f-\text {$\#$1} b^2 e}\&\right ]+\frac {4 c \left (a x \left (\sqrt {a^2 x^2+b^2}+a x\right )-b^2\right )}{a \sqrt {\sqrt {a^2 x^2+b^2}+a x}}}{6 e} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.45, size = 348, normalized size = 1.00 \begin {gather*} -\frac {b^2 c}{a e \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a e}+\frac {1}{2} a d \text {RootSum}\left [b^4 e-2 b^2 e \text {$\#$1}^4+4 a^2 f \text {$\#$1}^4+e \text {$\#$1}^8\&,\frac {-b^2 \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-\log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^4}{b^2 e \text {$\#$1}-2 a^2 f \text {$\#$1}-e \text {$\#$1}^5}\&\right ]-\frac {a c f \text {RootSum}\left [b^4 e-2 b^2 e \text {$\#$1}^4+4 a^2 f \text {$\#$1}^4+e \text {$\#$1}^8\&,\frac {-b^2 \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-\log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^4}{b^2 e \text {$\#$1}-2 a^2 f \text {$\#$1}-e \text {$\#$1}^5}\&\right ]}{2 e} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (c x^{2} + d\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{e x^{2} + f}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (c \,x^{2}+d \right ) \sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}{e \,x^{2}+f}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (c x^{2} + d\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{e x^{2} + f}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}\,\left (c\,x^2+d\right )}{e\,x^2+f} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} \left (c x^{2} + d\right )}{e x^{2} + f}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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