Optimal. Leaf size=200 \[ -\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}-a d \text {RootSum}\left [b^4 c-2 b^2 c \text {$\#$1}^4+4 a^2 d \text {$\#$1}^4+c \text {$\#$1}^8\& ,\frac {-b^2 \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-\log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^4}{b^2 c \text {$\#$1}^3-2 a^2 d \text {$\#$1}^3-c \text {$\#$1}^7}\& \right ] \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(1359\) vs. \(2(200)=400\).
time = 3.38, antiderivative size = 1359, normalized size of antiderivative = 6.80, number of steps
used = 35, number of rules used = 12, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6857, 2142,
14, 2144, 1642, 842, 840, 1183, 648, 632, 210, 642} \begin {gather*} -\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {d} \text {ArcTan}\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 {-b^2} (-c)^{5/4}}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {d} \text {ArcTan}\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 {-b^2} (-c)^{5/4}}+\frac {\sqrt {2} \sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}} \sqrt {d} \text {ArcTan}\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 {-b^2} (-c)^{3/4}}-\frac {\sqrt {2} \sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}} \sqrt {d} \text {ArcTan}\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 {-b^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} \sqrt {-b^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} \sqrt {-b^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} \sqrt {-b^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} \sqrt {-b^2} (-c)^{5/4}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 210
Rule 632
Rule 642
Rule 648
Rule 840
Rule 842
Rule 1183
Rule 1642
Rule 2142
Rule 2144
Rule 6857
Rubi steps
\begin {align*} \int \frac {-d+c x^2}{\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx &=\int \left (\frac {1}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}}-\frac {2 d}{\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}\right ) \, dx\\ &=-\left ((2 d) \int \frac {1}{\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\right )+\int \frac {1}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\\ &=\frac {\text {Subst}\left (\int \frac {b^2+x^2}{x^{5/2}} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 a}-(2 d) \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 {\text {Subst}\left (\int \left (\frac {b^2}{x^{5/2}}+\frac {1}{\sqrt {x}}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 a}-\sqrt {d} \int \frac {1}{\left (\sqrt {d}-\sqrt {-c} x\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx-\sqrt {d} \int \frac {1}{\left (\sqrt {d}+\sqrt {-c} x\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\\ &=-\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}-\sqrt {d} \text {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 )-\sqrt {d} \text {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 )\\ &=-\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}-\sqrt {d} \text {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 )-\sqrt {d} \text {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 )\\ &=-\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}-\frac {\left (2 \sqrt {d}\right ) \text {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}-\frac {\left (2 \sqrt {d}\right ) \text {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}\\ &=-\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}+\frac {\left (2 \sqrt {d}\right ) \text {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}}-\frac {\left (2 \sqrt {d}\right ) \text {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}}\\ &=-\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}+\frac {\left (4 \sqrt {d}\right ) \text {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}}-\frac {\left (4 \sqrt {d}\right ) \text {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}}\\ &=-\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}+\frac {\left (\sqrt {2} \sqrt {d}\right ) \text {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 )}{\left (-b^2\right )^{3/2} (-c)^{7/4} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}+\frac {\left (\sqrt {2} \sqrt {d}\right ) \text {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 )}{\left (-b^2\right )^{3/2} (-c)^{7/4} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}+\frac {\left (\sqrt {2} \sqrt {d}\right ) \text {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 )}{\left (-b^2\right )^{3/2} (-c)^{5/4} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}+\frac {\left (\sqrt {2} \sqrt {d}\right ) \text {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 )}{\left (-b^2\right )^{3/2} (-c)^{5/4} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}\\ &=-\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}+\frac {\left (\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {d}\right ) \text {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} \sqrt {-b^2} (-c)^{3/4}}-\frac {\left (\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {d}\right ) \text {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} \sqrt {-b^2} (-c)^{3/4}}+\frac {\left (\left (b^2 \sqrt {-c}-a \sqrt {-b^2} \sqrt {d}\right ) \sqrt {d}\right ) \text {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 )}{b^2 c}+\frac {\left (\left (b^2 \sqrt {-c}-a \sqrt {-b^2} \sqrt {d}\right ) \sqrt {d}\right ) \text {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 )}{b^2 c}-\frac {\left (\left (b^2 \sqrt {-c}+a \sqrt {-b^2} \sqrt {d}\right ) \sqrt {d}\right ) \text {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 )}{b^2 c}-\frac {\left (\left (b^2 \sqrt {-c}+a \sqrt {-b^2} \sqrt {d}\right ) \sqrt {d}\right ) \text {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 )}{b^2 c}-\frac {\left (\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {d}\right ) \text {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} \sqrt {-b^2} (-c)^{5/4}}+\frac {\left (\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {d}\right ) \text {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} \sqrt {-b^2} (-c)^{5/4}}\\ &=-\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{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} \sqrt {-b^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} \sqrt {-b^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} \sqrt {-b^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} \sqrt {-b^2} (-c)^{5/4}}-\frac {\left (2 \left (b^2 \sqrt {-c}-a \sqrt {-b^2} \sqrt {d}\right ) \sqrt {d}\right ) \text {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 )}{b^2 c}-\frac {\left (2 \left (b^2 \sqrt {-c}-a \sqrt {-b^2} \sqrt {d}\right ) \sqrt {d}\right ) \text {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 )}{b^2 c}+\frac {\left (2 \left (b^2 \sqrt {-c}+a \sqrt {-b^2} \sqrt {d}\right ) \sqrt {d}\right ) \text {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 )}{b^2 c}+\frac {\left (2 \left (b^2 \sqrt {-c}+a \sqrt {-b^2} \sqrt {d}\right ) \sqrt {d}\right ) \text {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 )}{b^2 c}\\ &=-\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{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 )}{\sqrt {-b^2} (-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 )}{\sqrt {-b^2} (-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 )}{\sqrt {-b^2} (-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 )}{\sqrt {-b^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} \sqrt {-b^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} \sqrt {-b^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} \sqrt {-b^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} \sqrt {-b^2} (-c)^{5/4}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 195, normalized size = 0.98 \begin {gather*} \frac {2 \left (b^2+3 a x \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}-a d \text {RootSum}\left [b^4 c-2 b^2 c \text {$\#$1}^4+4 a^2 d \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {b^2 \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )+\log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^4}{-b^2 c \text {$\#$1}^3+2 a^2 d \text {$\#$1}^3+c \text {$\#$1}^7}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {c \,x^{2}-d}{\left (c \,x^{2}+d \right ) \sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order
1.
time = 0.51, size = 1950, normalized size = 9.75 \begin {gather*} \frac {12 \, a b^{2} \left (-\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}} \arctan \left (\frac {{\left (\sqrt {a^{3} d^{6} x + \sqrt {a^{2} x^{2} + b^{2}} a^{2} d^{6} - {\left (2 \, a^{2} b^{4} c^{3} d^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} + a^{2} b^{2} c d^{5} - 2 \, a^{4} d^{6}\right )} \sqrt {-\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}}} {\left (b^{4} c^{4} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} - a^{2} c d^{3}\right )} \sqrt {-\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}} - {\left (a b^{4} c^{4} d^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} - a^{3} c d^{6}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} \sqrt {-\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}}\right )} \left (-\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}}}{a^{2} d^{7}}\right ) - 12 \, a b^{2} \left (\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {a^{3} d^{6} x + \sqrt {a^{2} x^{2} + b^{2}} a^{2} d^{6} + {\left (2 \, a^{2} b^{4} c^{3} d^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} - a^{2} b^{2} c d^{5} + 2 \, a^{4} d^{6}\right )} \sqrt {\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}}} {\left (b^{4} c^{4} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} + a^{2} c d^{3}\right )} \left (\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {3}{4}} - {\left (a b^{4} c^{4} d^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} + a^{3} c d^{6}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} \left (\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {3}{4}}}{a^{2} d^{7}}\right ) - 3 \, a b^{2} \left (\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}} \log \left (8 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a d^{3} + 8 \, {\left (b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} + a^{2} d^{3}\right )} \left (\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}}\right ) + 3 \, a b^{2} \left (\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}} \log \left (8 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a d^{3} - 8 \, {\left (b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} + a^{2} d^{3}\right )} \left (\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}}\right ) + 3 \, a b^{2} \left (-\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}} \log \left (8 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a d^{3} + 8 \, {\left (b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} - a^{2} d^{3}\right )} \left (-\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}}\right ) - 3 \, a b^{2} \left (-\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}} \log \left (8 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a d^{3} - 8 \, {\left (b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} - a^{2} d^{3}\right )} \left (-\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}}\right ) - 2 \, {\left (a^{2} x^{2} - \sqrt {a^{2} x^{2} + b^{2}} a x - b^{2}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{3 \, a b^{2}} \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 {c x^{2} - d}{\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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {d-c\,x^2}{\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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