Integrand size = 42, antiderivative size = 196 \[ \int \frac {\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-d+c x^2} \, dx=-\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}+a d \text {RootSum}\left [b^4 c-2 b^2 c \text {$\#$1}^4-4 a^2 d \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {-b^2 \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-\log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^4}{b^2 c \text {$\#$1}+2 a^2 d \text {$\#$1}-c \text {$\#$1}^5}\&\right ] \]
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Leaf count is larger than twice the leaf count of optimal. \(454\) vs. \(2(196)=392\).
Time = 0.70 (sec) , antiderivative size = 454, normalized size of antiderivative = 2.32, number of steps used = 21, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6857, 2142, 14, 2144, 1642, 840, 1180, 214, 211} \[ \int \frac {\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-d+c x^2} \, dx=\frac {2 \sqrt {d} \sqrt {a \sqrt {d}-\sqrt {a^2 d+b^2 c}} \arctan \left (\frac {\sqrt [4]{c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {a \sqrt {d}-\sqrt {a^2 d+b^2 c}}}\right )}{c^{3/4}}+\frac {2 \sqrt {d} \sqrt {\sqrt {a^2 d+b^2 c}+a \sqrt {d}} \arctan \left (\frac {\sqrt [4]{c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {\sqrt {a^2 d+b^2 c}+a \sqrt {d}}}\right )}{c^{3/4}}-\frac {2 \sqrt {d} \sqrt {a \sqrt {d}-\sqrt {a^2 d+b^2 c}} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {a \sqrt {d}-\sqrt {a^2 d+b^2 c}}}\right )}{c^{3/4}}-\frac {2 \sqrt {d} \sqrt {\sqrt {a^2 d+b^2 c}+a \sqrt {d}} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {\sqrt {a^2 d+b^2 c}+a \sqrt {d}}}\right )}{c^{3/4}}-\frac {b^2}{a \sqrt {\sqrt {a^2 x^2+b^2}+a x}}+\frac {\left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}}{3 a} \]
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Rule 14
Rule 211
Rule 214
Rule 840
Rule 1180
Rule 1642
Rule 2142
Rule 2144
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \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 \\ & = (2 d) \int \frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-d+c x^2} \, dx+\int \sqrt {a x+\sqrt {b^2+a^2 x^2}} \, dx \\ & = \frac {\text {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 {\text {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} \text {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} \text {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} \text {Subst}\left (\int \left (-\frac {1}{\sqrt {c} \sqrt {x}}+\frac {2 \left (b^2 \sqrt {c}+a \sqrt {d} x\right )}{\sqrt {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} \text {Subst}\left (\int \left (\frac {1}{\sqrt {c} \sqrt {x}}+\frac {2 \left (b^2 \sqrt {c}-a \sqrt {d} x\right )}{\sqrt {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 ) \text {Subst}\left (\int \frac {b^2 \sqrt {c}+a \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 )}{\sqrt {c}}-\frac {\left (2 \sqrt {d}\right ) \text {Subst}\left (\int \frac {b^2 \sqrt {c}-a \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 )}{\sqrt {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 ) \text {Subst}\left (\int \frac {b^2 \sqrt {c}+a \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 )}{\sqrt {c}}-\frac {\left (4 \sqrt {d}\right ) \text {Subst}\left (\int \frac {b^2 \sqrt {c}-a \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 )}{\sqrt {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 (2 \sqrt {d} \left (a \sqrt {d}-\sqrt {b^2 c+a^2 d}\right )\right ) \text {Subst}\left (\int \frac {1}{a \sqrt {d}-\sqrt {b^2 c+a^2 d}-\sqrt {c} x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {c}}+\frac {\left (2 \sqrt {d} \left (a \sqrt {d}-\sqrt {b^2 c+a^2 d}\right )\right ) \text {Subst}\left (\int \frac {1}{a \sqrt {d}-\sqrt {b^2 c+a^2 d}+\sqrt {c} x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {c}}-\frac {\left (2 \sqrt {d} \left (a \sqrt {d}+\sqrt {b^2 c+a^2 d}\right )\right ) \text {Subst}\left (\int \frac {1}{a \sqrt {d}+\sqrt {b^2 c+a^2 d}-\sqrt {c} x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {c}}+\frac {\left (2 \sqrt {d} \left (a \sqrt {d}+\sqrt {b^2 c+a^2 d}\right )\right ) \text {Subst}\left (\int \frac {1}{a \sqrt {d}+\sqrt {b^2 c+a^2 d}+\sqrt {c} x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {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 {2 \sqrt {d} \sqrt {a \sqrt {d}-\sqrt {b^2 c+a^2 d}} \arctan \left (\frac {\sqrt [4]{c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a \sqrt {d}-\sqrt {b^2 c+a^2 d}}}\right )}{c^{3/4}}+\frac {2 \sqrt {d} \sqrt {a \sqrt {d}+\sqrt {b^2 c+a^2 d}} \arctan \left (\frac {\sqrt [4]{c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a \sqrt {d}+\sqrt {b^2 c+a^2 d}}}\right )}{c^{3/4}}-\frac {2 \sqrt {d} \sqrt {a \sqrt {d}-\sqrt {b^2 c+a^2 d}} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a \sqrt {d}-\sqrt {b^2 c+a^2 d}}}\right )}{c^{3/4}}-\frac {2 \sqrt {d} \sqrt {a \sqrt {d}+\sqrt {b^2 c+a^2 d}} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a \sqrt {d}+\sqrt {b^2 c+a^2 d}}}\right )}{c^{3/4}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-d+c x^2} \, dx=-\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}+a d \text {RootSum}\left [b^4 c-2 b^2 c \text {$\#$1}^4-4 a^2 d \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {-b^2 \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-\log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^4}{b^2 c \text {$\#$1}+2 a^2 d \text {$\#$1}-c \text {$\#$1}^5}\&\right ] \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.19
\[\int \frac {\left (c \,x^{2}+d \right ) \sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}{c \,x^{2}-d}d x\]
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.32 (sec) , antiderivative size = 1727, normalized size of antiderivative = 8.81 \[ \int \frac {\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-d+c x^2} \, dx=\text {Too large to display} \]
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Not integrable
Time = 2.52 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.17 \[ \int \frac {\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-d+c x^2} \, dx=\int \frac {\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} \left (c x^{2} + d\right )}{c x^{2} - d}\, dx \]
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Not integrable
Time = 0.29 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.20 \[ \int \frac {\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-d+c x^2} \, dx=\int { \frac {{\left (c x^{2} + d\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{c x^{2} - d} \,d x } \]
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Not integrable
Time = 0.53 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.20 \[ \int \frac {\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-d+c x^2} \, dx=\int { \frac {{\left (c x^{2} + d\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{c x^{2} - d} \,d x } \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.20 \[ \int \frac {\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-d+c x^2} \, dx=\int -\frac {\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}\,\left (c\,x^2+d\right )}{d-c\,x^2} \,d x \]
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