Optimal. Leaf size=140 \[ \frac {1}{2} a \text {RootSum}\left [\text {$\#$1}^8 c+4 \text {$\#$1}^4 a^2 d-2 \text {$\#$1}^4 b^2 c+b^4 c\& ,\frac {\text {$\#$1}^4 \left (-\log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-\text {$\#$1}\right )\right )-b^2 \log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-\text {$\#$1}\right )}{\text {$\#$1}^7 (-c)-2 \text {$\#$1}^3 a^2 d+\text {$\#$1}^3 b^2 c}\& \right ] \]
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Rubi [B] time = 4.16, antiderivative size = 1309, normalized size of antiderivative = 9.35, number of steps used = 30, number of rules used = 10, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {6725, 2119, 1628, 828, 826, 1169, 634, 618, 204, 628} \begin {gather*} \frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \tan ^{-1}\left (\frac {\sqrt {-c} \left (\sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}}-\sqrt {2} \sqrt [4]{-c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{5/4} \sqrt {d}}-\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \tan ^{-1}\left (\frac {\sqrt {-c} \left (\sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}}+\sqrt {2} \sqrt [4]{-c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{5/4} \sqrt {d}}-\frac {\sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}} \tan ^{-1}\left (\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}-\sqrt {2} (-c)^{3/4} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {-c} \sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}}}\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{3/4} \sqrt {d}}+\frac {\sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a x+\sqrt {b^2+a^2 x^2}} (-c)^{3/4}+\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}{\sqrt {-c} \sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}}}\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{3/4} \sqrt {d}}-\frac {\sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}} \log \left (\sqrt [4]{-c} \left (a x+\sqrt {b^2+a^2 x^2}\right )-\sqrt {2} \sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt {-b^2} \sqrt [4]{-c}\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{3/4} \sqrt {d}}+\frac {\sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}} \log \left (\sqrt [4]{-c} \left (a x+\sqrt {b^2+a^2 x^2}\right )+\sqrt {2} \sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt {-b^2} \sqrt [4]{-c}\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{3/4} \sqrt {d}}+\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \log \left ((-c)^{3/4} \left (a x+\sqrt {b^2+a^2 x^2}\right )-\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt {-b^2} (-c)^{3/4}\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{5/4} \sqrt {d}}-\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \log \left ((-c)^{3/4} \left (a x+\sqrt {b^2+a^2 x^2}\right )+\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt {-b^2} (-c)^{3/4}\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{5/4} \sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 826
Rule 828
Rule 1169
Rule 1628
Rule 2119
Rule 6725
Rubi steps
\begin {align*} \int \frac {1}{\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx &=\int \left (\frac {1}{2 \sqrt {d} \left (\sqrt {d}-\sqrt {-c} x\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {1}{2 \sqrt {d} \left (\sqrt {d}+\sqrt {-c} x\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}\right ) \, dx\\ &=\frac {\int \frac {1}{\left (\sqrt {d}-\sqrt {-c} x\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx}{2 \sqrt {d}}+\frac {\int \frac {1}{\left (\sqrt {d}+\sqrt {-c} x\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx}{2 \sqrt {d}}\\ &=\frac {\operatorname {Subst}\left (\int \frac {b^2+x^2}{x^{3/2} \left (b^2 \sqrt {-c}+2 a \sqrt {d} x-\sqrt {-c} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 \sqrt {d}}+\frac {\operatorname {Subst}\left (\int \frac {b^2+x^2}{x^{3/2} \left (-b^2 \sqrt {-c}+2 a \sqrt {d} x+\sqrt {-c} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 \sqrt {d}}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {1}{\sqrt {-c} x^{3/2}}+\frac {2 \left (b^2 c-a \sqrt {-c} \sqrt {d} x\right )}{c x^{3/2} \left (b^2 \sqrt {-c}+2 a \sqrt {d} x-\sqrt {-c} x^2\right )}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 \sqrt {d}}+\frac {\operatorname {Subst}\left (\int \left (\frac {1}{\sqrt {-c} x^{3/2}}+\frac {2 \left (b^2 c+a \sqrt {-c} \sqrt {d} x\right )}{c x^{3/2} \left (-b^2 \sqrt {-c}+2 a \sqrt {d} x+\sqrt {-c} x^2\right )}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 \sqrt {d}}\\ &=\frac {\operatorname {Subst}\left (\int \frac {b^2 c-a \sqrt {-c} \sqrt {d} x}{x^{3/2} \left (b^2 \sqrt {-c}+2 a \sqrt {d} x-\sqrt {-c} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{c \sqrt {d}}+\frac {\operatorname {Subst}\left (\int \frac {b^2 c+a \sqrt {-c} \sqrt {d} x}{x^{3/2} \left (-b^2 \sqrt {-c}+2 a \sqrt {d} x+\sqrt {-c} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{c \sqrt {d}}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {-a b^2 c \sqrt {d}-b^2 (-c)^{3/2} x}{\sqrt {x} \left (b^2 \sqrt {-c}+2 a \sqrt {d} x-\sqrt {-c} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{b^2 (-c)^{3/2} \sqrt {d}}+\frac {\operatorname {Subst}\left (\int \frac {-a b^2 c \sqrt {d}+b^2 (-c)^{3/2} x}{\sqrt {x} \left (-b^2 \sqrt {-c}+2 a \sqrt {d} x+\sqrt {-c} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{b^2 (-c)^{3/2} \sqrt {d}}\\ &=-\frac {2 \operatorname {Subst}\left (\int \frac {-a b^2 c \sqrt {d}-b^2 (-c)^{3/2} x^2}{b^2 \sqrt {-c}+2 a \sqrt {d} x^2-\sqrt {-c} x^4} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{b^2 (-c)^{3/2} \sqrt {d}}+\frac {2 \operatorname {Subst}\left (\int \frac {-a b^2 c \sqrt {d}+b^2 (-c)^{3/2} x^2}{-b^2 \sqrt {-c}+2 a \sqrt {d} x^2+\sqrt {-c} x^4} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{b^2 (-c)^{3/2} \sqrt {d}}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {-\frac {\sqrt {2} a b^2 c \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {d}}{\sqrt [4]{-c}}-\left (b^2 \sqrt {-b^2} (-c)^{3/2}-a b^2 c \sqrt {d}\right ) x}{\sqrt {-b^2}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} x}{\sqrt [4]{-c}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {2} \left (-b^2\right )^{3/2} (-c)^{7/4} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {d}}-\frac {\operatorname {Subst}\left (\int \frac {-\frac {\sqrt {2} a b^2 c \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {d}}{\sqrt [4]{-c}}+\left (b^2 \sqrt {-b^2} (-c)^{3/2}-a b^2 c \sqrt {d}\right ) x}{\sqrt {-b^2}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} x}{\sqrt [4]{-c}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {2} \left (-b^2\right )^{3/2} (-c)^{7/4} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {d}}-\frac {\operatorname {Subst}\left (\int \frac {-\frac {\sqrt {2} a b^2 c \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {d}}{(-c)^{3/4}}-\left (-b^2 \sqrt {-b^2} (-c)^{3/2}-a b^2 c \sqrt {d}\right ) x}{\sqrt {-b^2}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} x}{(-c)^{3/4}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {2} \left (-b^2\right )^{3/2} (-c)^{5/4} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {d}}-\frac {\operatorname {Subst}\left (\int \frac {-\frac {\sqrt {2} a b^2 c \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {d}}{(-c)^{3/4}}+\left (-b^2 \sqrt {-b^2} (-c)^{3/2}-a b^2 c \sqrt {d}\right ) x}{\sqrt {-b^2}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} x}{(-c)^{3/4}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {2} \left (-b^2\right )^{3/2} (-c)^{5/4} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {d}}\\ &=-\frac {\left (\frac {a}{\sqrt {-b^2}}-\frac {\sqrt {-c}}{\sqrt {d}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b^2}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} x}{\sqrt [4]{-c}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 c}-\frac {\left (\frac {a}{\sqrt {-b^2}}-\frac {\sqrt {-c}}{\sqrt {d}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b^2}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} x}{\sqrt [4]{-c}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 c}-\frac {\left (\frac {a}{\sqrt {-b^2}}+\frac {\sqrt {-c}}{\sqrt {d}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b^2}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} x}{(-c)^{3/4}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 c}-\frac {\left (\frac {a}{\sqrt {-b^2}}+\frac {\sqrt {-c}}{\sqrt {d}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b^2}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} x}{(-c)^{3/4}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 c}-\frac {\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \operatorname {Subst}\left (\int \frac {-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}{\sqrt [4]{-c}}+2 x}{\sqrt {-b^2}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} x}{\sqrt [4]{-c}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{3/4} \sqrt {d}}+\frac {\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}{\sqrt [4]{-c}}+2 x}{\sqrt {-b^2}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} x}{\sqrt [4]{-c}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{3/4} \sqrt {d}}+\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \operatorname {Subst}\left (\int \frac {-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}{(-c)^{3/4}}+2 x}{\sqrt {-b^2}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} x}{(-c)^{3/4}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{5/4} \sqrt {d}}-\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}{(-c)^{3/4}}+2 x}{\sqrt {-b^2}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} x}{(-c)^{3/4}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{5/4} \sqrt {d}}\\ &=-\frac {\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \log \left (\sqrt {-b^2} \sqrt [4]{-c}-\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt [4]{-c} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{3/4} \sqrt {d}}+\frac {\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \log \left (\sqrt {-b^2} \sqrt [4]{-c}+\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt [4]{-c} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{3/4} \sqrt {d}}+\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \log \left (\sqrt {-b^2} (-c)^{3/4}-\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+(-c)^{3/4} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{5/4} \sqrt {d}}-\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \log \left (\sqrt {-b^2} (-c)^{3/4}+\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+(-c)^{3/4} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{5/4} \sqrt {d}}+\frac {\left (\frac {a}{\sqrt {-b^2}}-\frac {\sqrt {-c}}{\sqrt {d}}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {2 \left (\sqrt {-b^2} c+a \sqrt {-c} \sqrt {d}\right )}{c}-x^2} \, dx,x,-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}{\sqrt [4]{-c}}+2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{c}+\frac {\left (\frac {a}{\sqrt {-b^2}}-\frac {\sqrt {-c}}{\sqrt {d}}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {2 \left (\sqrt {-b^2} c+a \sqrt {-c} \sqrt {d}\right )}{c}-x^2} \, dx,x,\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}{\sqrt [4]{-c}}+2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{c}+\frac {\left (\frac {a}{\sqrt {-b^2}}+\frac {\sqrt {-c}}{\sqrt {d}}\right ) \operatorname {Subst}\left (\int \frac {1}{-2 \left (\sqrt {-b^2}+\frac {a \sqrt {d}}{\sqrt {-c}}\right )-x^2} \, dx,x,-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}{(-c)^{3/4}}+2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{c}+\frac {\left (\frac {a}{\sqrt {-b^2}}+\frac {\sqrt {-c}}{\sqrt {d}}\right ) \operatorname {Subst}\left (\int \frac {1}{-2 \left (\sqrt {-b^2}+\frac {a \sqrt {d}}{\sqrt {-c}}\right )-x^2} \, dx,x,\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}{(-c)^{3/4}}+2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{c}\\ &=\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \tan ^{-1}\left (\frac {(-c)^{3/4} \left (\frac {\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}{\sqrt [4]{-c}}-\sqrt {2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{5/4} \sqrt {d}}-\frac {\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \tan ^{-1}\left (\frac {\sqrt [4]{-c} \left (\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}{(-c)^{3/4}}-\sqrt {2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{3/4} \sqrt {d}}-\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \tan ^{-1}\left (\frac {(-c)^{3/4} \left (\frac {\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}{\sqrt [4]{-c}}+\sqrt {2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{5/4} \sqrt {d}}+\frac {\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \tan ^{-1}\left (\frac {\sqrt [4]{-c} \left (\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}{(-c)^{3/4}}+\sqrt {2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{3/4} \sqrt {d}}-\frac {\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \log \left (\sqrt {-b^2} \sqrt [4]{-c}-\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt [4]{-c} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{3/4} \sqrt {d}}+\frac {\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \log \left (\sqrt {-b^2} \sqrt [4]{-c}+\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt [4]{-c} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{3/4} \sqrt {d}}+\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \log \left (\sqrt {-b^2} (-c)^{3/4}-\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+(-c)^{3/4} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{5/4} \sqrt {d}}-\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \log \left (\sqrt {-b^2} (-c)^{3/4}+\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+(-c)^{3/4} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} \sqrt {-b^2} (-c)^{5/4} \sqrt {d}}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 135, normalized size = 0.96 \begin {gather*} -\frac {1}{2} a \text {RootSum}\left [\text {$\#$1}^8 b^4 c+4 \text {$\#$1}^4 a^2 d-2 \text {$\#$1}^4 b^2 c+c\&,\frac {\text {$\#$1}^4 b^2 \log \left (\frac {1}{\sqrt {\sqrt {a^2 x^2+b^2}+a x}}-\text {$\#$1}\right )+\log \left (\frac {1}{\sqrt {\sqrt {a^2 x^2+b^2}+a x}}-\text {$\#$1}\right )}{\text {$\#$1}^5 b^4 c+2 \text {$\#$1} a^2 d-\text {$\#$1} b^2 c}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.21, size = 140, normalized size = 1.00 \begin {gather*} \frac {1}{2} a \text {RootSum}\left [b^4 c-2 b^2 c \text {$\#$1}^4+4 a^2 d \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {-b^2 \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-\log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^4}{b^2 c \text {$\#$1}^3-2 a^2 d \text {$\#$1}^3-c \text {$\#$1}^7}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.72, size = 1800, normalized size = 12.86
result too large to display
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 {1}{{\left (c x^{2} + d\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}\,{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 \[\int \frac {1}{\left (c \,x^{2}+d \right ) \sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}\, dx\]
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 {1}{{\left (c x^{2} + d\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}\,{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 = -0.01 \begin {gather*} \int \frac {1}{\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}\,\left (c\,x^2+d\right )} \,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 {1}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} \left (c x^{2} + d\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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