Optimal. Leaf size=197 \[ -a d \text {RootSum}\left [\text {$\#$1}^8 c+4 \text {$\#$1}^4 a^2 d-2 \text {$\#$1}^4 b^2 c+b^4 c\& ,\frac {\text {$\#$1}^4 \left (-\log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-\text {$\#$1}\right )\right )-b^2 \log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-\text {$\#$1}\right )}{\text {$\#$1}^5 (-c)-2 \text {$\#$1} a^2 d+\text {$\#$1} b^2 c}\& \right ]-\frac {b^2}{a \sqrt {\sqrt {a^2 x^2+b^2}+a x}}+\frac {\left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}}{3 a} \]
________________________________________________________________________________________
Rubi [B] time = 4.34, antiderivative size = 1288, normalized size of antiderivative = 6.54, number of steps used = 33, number of rules used = 11, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.262, Rules used = {6725, 2117, 14, 2119, 1628, 826, 1169, 634, 618, 204, 628} \begin {gather*} -\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \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 )}{(-c)^{5/4}}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \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 )}{(-c)^{5/4}}-\frac {\sqrt {2} \sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}} \sqrt {d} \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 )}{(-c)^{3/4}}+\frac {\sqrt {2} \sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}} \sqrt {d} \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 )}{(-c)^{3/4}}+\frac {\sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}} \sqrt {d} \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 )}{\sqrt {2} (-c)^{3/4}}-\frac {\sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}} \sqrt {d} \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 )}{\sqrt {2} (-c)^{3/4}}-\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \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 )}{\sqrt {2} (-c)^{5/4}}+\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \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 )}{\sqrt {2} (-c)^{5/4}} \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}}}{d+c x^2} \, dx &=\int \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\frac {2 d \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{d+c x^2}\right ) \, dx\\ &=-\left ((2 d) \int \frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{d+c x^2} \, dx\right )+\int \sqrt {a x+\sqrt {b^2+a^2 x^2}} \, dx\\ &=\frac {\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}-(2 d) \int \left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{2 \sqrt {d} \left (\sqrt {d}-\sqrt {-c} x\right )}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{2 \sqrt {d} \left (\sqrt {d}+\sqrt {-c} x\right )}\right ) \, dx\\ &=\frac {\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}-\sqrt {d} \int \frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {d}-\sqrt {-c} x} \, dx-\sqrt {d} \int \frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {d}+\sqrt {-c} x} \, dx\\ &=-\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}-\sqrt {d} \operatorname {Subst}\left (\int \frac {b^2+x^2}{\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 )-\sqrt {d} \operatorname {Subst}\left (\int \frac {b^2+x^2}{\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 )\\ &=-\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}-\sqrt {d} \operatorname {Subst}\left (\int \left (-\frac {1}{\sqrt {-c} \sqrt {x}}+\frac {2 \left (b^2 c-a \sqrt {-c} \sqrt {d} x\right )}{c \sqrt {x} \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 )-\sqrt {d} \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt {-c} \sqrt {x}}+\frac {2 \left (b^2 c+a \sqrt {-c} \sqrt {d} x\right )}{c \sqrt {x} \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 )\\ &=-\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}-\frac {\left (2 \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {b^2 c-a \sqrt {-c} \sqrt {d} 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 )}{c}-\frac {\left (2 \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {b^2 c+a \sqrt {-c} \sqrt {d} 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 )}{c}\\ &=-\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}-\frac {\left (4 \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {b^2 c-a \sqrt {-c} \sqrt {d} 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 )}{c}-\frac {\left (4 \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {b^2 c+a \sqrt {-c} \sqrt {d} 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 )}{c}\\ &=-\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}-\frac {\left (\sqrt {2} \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} b^2 c \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}{\sqrt [4]{-c}}-\left (b^2 c+a \sqrt {-b^2} \sqrt {-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 {-b^2} (-c)^{5/4} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}-\frac {\left (\sqrt {2} \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} b^2 c \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}{\sqrt [4]{-c}}+\left (b^2 c+a \sqrt {-b^2} \sqrt {-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 {-b^2} (-c)^{5/4} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}+\frac {\left (\sqrt {2} \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} b^2 c \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}{(-c)^{3/4}}-\left (b^2 c-a \sqrt {-b^2} \sqrt {-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 {-b^2} (-c)^{3/4} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}+\frac {\left (\sqrt {2} \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} b^2 c \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}{(-c)^{3/4}}+\left (b^2 c-a \sqrt {-b^2} \sqrt {-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 {-b^2} (-c)^{3/4} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}\\ &=-\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}+\frac {\left (\left (\sqrt {-b^2} \sqrt {-c}-a \sqrt {d}\right ) \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 )}{c}+\frac {\left (\left (\sqrt {-b^2} \sqrt {-c}-a \sqrt {d}\right ) \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 )}{c}+\frac {\left (\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {d}\right ) \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 )}{\sqrt {2} (-c)^{3/4}}-\frac {\left (\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {d}\right ) \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 )}{\sqrt {2} (-c)^{3/4}}-\frac {\left (\left (\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}\right ) \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 )}{c}-\frac {\left (\left (\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}\right ) \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 )}{c}-\frac {\left (\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {d}\right ) \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 )}{\sqrt {2} (-c)^{5/4}}+\frac {\left (\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {d}\right ) \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 )}{\sqrt {2} (-c)^{5/4}}\\ &=-\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}+\frac {\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \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 )}{\sqrt {2} (-c)^{3/4}}-\frac {\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \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 )}{\sqrt {2} (-c)^{3/4}}-\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \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 )}{\sqrt {2} (-c)^{5/4}}+\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \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 )}{\sqrt {2} (-c)^{5/4}}-\frac {\left (2 \left (\sqrt {-b^2} \sqrt {-c}-a \sqrt {d}\right ) \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 (2 \left (\sqrt {-b^2} \sqrt {-c}-a \sqrt {d}\right ) \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 (2 \left (\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}\right ) \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 (2 \left (\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}\right ) \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 {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \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 )}{(-c)^{5/4}}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \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 )}{(-c)^{3/4}}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \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 )}{(-c)^{5/4}}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \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 )}{(-c)^{3/4}}+\frac {\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \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 )}{\sqrt {2} (-c)^{3/4}}-\frac {\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \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 )}{\sqrt {2} (-c)^{3/4}}-\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \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 )}{\sqrt {2} (-c)^{5/4}}+\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \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 )}{\sqrt {2} (-c)^{5/4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.54, size = 191, normalized size = 0.97 \begin {gather*} -a d \text {RootSum}\left [\text {$\#$1}^8 c+4 \text {$\#$1}^4 a^2 d-2 \text {$\#$1}^4 b^2 c+b^4 c\&,\frac {\text {$\#$1}^4 \log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-\text {$\#$1}\right )+b^2 \log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-\text {$\#$1}\right )}{\text {$\#$1}^5 c+2 \text {$\#$1} a^2 d-\text {$\#$1} b^2 c}\&\right ]-\frac {2 \left (b^2-a x \left (\sqrt {a^2 x^2+b^2}+a x\right )\right )}{3 a \sqrt {\sqrt {a^2 x^2+b^2}+a x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.36, size = 197, normalized size = 1.00 \begin {gather*} -\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}-a d \text {RootSum}\left [b^4 c-2 b^2 c \text {$\#$1}^4+4 a^2 d \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {-b^2 \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-\log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^4}{b^2 c \text {$\#$1}-2 a^2 d \text {$\#$1}-c \text {$\#$1}^5}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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}}}}{c x^{2} + d}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (c \,x^{2}-d \right ) \sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}{c \,x^{2}+d}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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}}}}{c x^{2} + d}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}\,\left (d-c\,x^2\right )}{c\,x^2+d} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 )}{c x^{2} + d}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________