Integrand size = 49, antiderivative size = 390 \[ \int \frac {\sqrt {-b+a^2 x^2} \left (d+c x^4\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x^2} \, dx=\frac {63 a^3 b^3 d-72 a b^4 c x^2+126 a^5 b^2 d x^2+210 a^3 b^3 c x^4-2016 a^7 b d x^4+126 a^5 b^2 c x^6+2016 a^9 d x^6-504 a^7 b c x^8+224 a^9 c x^{10}+\sqrt {-b+a^2 x^2} \left (-16 b^4 c x-126 a^4 b^2 d x+154 a^2 b^3 c x^3-1008 a^6 b d x^3-42 a^4 b^2 c x^5+2016 a^8 d x^5-392 a^6 b c x^7+224 a^8 c x^9\right )}{63 a^3 x \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}+\frac {a \sqrt [4]{b} d \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{-\sqrt {b}+a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2}}-\frac {a \sqrt [4]{b} d \text {arctanh}\left (\frac {\frac {\sqrt [4]{b}}{\sqrt {2}}+\frac {a x}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt {-b+a^2 x^2}}{\sqrt {2} \sqrt [4]{b}}}{\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{\sqrt {2}} \]
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Time = 1.27 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.11, number of steps used = 20, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6874, 2145, 474, 12, 327, 335, 217, 1179, 642, 1176, 631, 210, 265, 276} \[ \int \frac {\sqrt {-b+a^2 x^2} \left (d+c x^4\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x^2} \, dx=\frac {a \sqrt [4]{b} d \arctan \left (1-\frac {\sqrt {2} \sqrt {\sqrt {a^2 x^2-b}+a x}}{\sqrt [4]{b}}\right )}{\sqrt {2}}-\frac {a \sqrt [4]{b} d \arctan \left (\frac {\sqrt {2} \sqrt {\sqrt {a^2 x^2-b}+a x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2}}+2 a d \sqrt {\sqrt {a^2 x^2-b}+a x}+\frac {2 a b d \sqrt {\sqrt {a^2 x^2-b}+a x}}{\left (\sqrt {a^2 x^2-b}+a x\right )^2+b}+\frac {a \sqrt [4]{b} d \log \left (\sqrt {a^2 x^2-b}-\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {a^2 x^2-b}+a x}+a x+\sqrt {b}\right )}{2 \sqrt {2}}-\frac {a \sqrt [4]{b} d \log \left (\sqrt {a^2 x^2-b}+\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {a^2 x^2-b}+a x}+a x+\sqrt {b}\right )}{2 \sqrt {2}}-\frac {b^4 c}{56 a^3 \left (\sqrt {a^2 x^2-b}+a x\right )^{7/2}}-\frac {b^2 c \sqrt {\sqrt {a^2 x^2-b}+a x}}{4 a^3}+\frac {c \left (\sqrt {a^2 x^2-b}+a x\right )^{9/2}}{72 a^3} \]
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Rule 12
Rule 210
Rule 217
Rule 265
Rule 276
Rule 327
Rule 335
Rule 474
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 2145
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d \sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x^2}+c x^2 \sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}}\right ) \, dx \\ & = c \int x^2 \sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}} \, dx+d \int \frac {\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x^2} \, dx \\ & = \frac {c \text {Subst}\left (\int \frac {\left (-b+x^2\right )^2 \left (b+x^2\right )^2}{x^{9/2}} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{16 a^3}+(a d) \text {Subst}\left (\int \frac {\left (-b+x^2\right )^2}{\sqrt {x} \left (b+x^2\right )^2} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right ) \\ & = \frac {2 a b d \sqrt {a x+\sqrt {-b+a^2 x^2}}}{b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2}+\frac {c \text {Subst}\left (\int \frac {\left (-b^2+x^4\right )^2}{x^{9/2}} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{16 a^3}-\frac {(a d) \text {Subst}\left (\int -\frac {2 b x^{3/2}}{b+x^2} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{2 b} \\ & = \frac {2 a b d \sqrt {a x+\sqrt {-b+a^2 x^2}}}{b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2}+\frac {c \text {Subst}\left (\int \left (\frac {b^4}{x^{9/2}}-\frac {2 b^2}{\sqrt {x}}+x^{7/2}\right ) \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{16 a^3}+(a d) \text {Subst}\left (\int \frac {x^{3/2}}{b+x^2} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right ) \\ & = -\frac {b^4 c}{56 a^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/2}}-\frac {b^2 c \sqrt {a x+\sqrt {-b+a^2 x^2}}}{4 a^3}+2 a d \sqrt {a x+\sqrt {-b+a^2 x^2}}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}{72 a^3}+\frac {2 a b d \sqrt {a x+\sqrt {-b+a^2 x^2}}}{b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2}-(a b d) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (b+x^2\right )} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right ) \\ & = -\frac {b^4 c}{56 a^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/2}}-\frac {b^2 c \sqrt {a x+\sqrt {-b+a^2 x^2}}}{4 a^3}+2 a d \sqrt {a x+\sqrt {-b+a^2 x^2}}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}{72 a^3}+\frac {2 a b d \sqrt {a x+\sqrt {-b+a^2 x^2}}}{b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2}-(2 a b d) \text {Subst}\left (\int \frac {1}{b+x^4} \, dx,x,\sqrt {a x+\sqrt {-b+a^2 x^2}}\right ) \\ & = -\frac {b^4 c}{56 a^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/2}}-\frac {b^2 c \sqrt {a x+\sqrt {-b+a^2 x^2}}}{4 a^3}+2 a d \sqrt {a x+\sqrt {-b+a^2 x^2}}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}{72 a^3}+\frac {2 a b d \sqrt {a x+\sqrt {-b+a^2 x^2}}}{b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2}-\left (a \sqrt {b} d\right ) \text {Subst}\left (\int \frac {\sqrt {b}-x^2}{b+x^4} \, dx,x,\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )-\left (a \sqrt {b} d\right ) \text {Subst}\left (\int \frac {\sqrt {b}+x^2}{b+x^4} \, dx,x,\sqrt {a x+\sqrt {-b+a^2 x^2}}\right ) \\ & = -\frac {b^4 c}{56 a^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/2}}-\frac {b^2 c \sqrt {a x+\sqrt {-b+a^2 x^2}}}{4 a^3}+2 a d \sqrt {a x+\sqrt {-b+a^2 x^2}}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}{72 a^3}+\frac {2 a b d \sqrt {a x+\sqrt {-b+a^2 x^2}}}{b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2}+\frac {\left (a \sqrt [4]{b} d\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}+2 x}{-\sqrt {b}-\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{2 \sqrt {2}}+\frac {\left (a \sqrt [4]{b} d\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}-2 x}{-\sqrt {b}+\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{2 \sqrt {2}}-\frac {1}{2} \left (a \sqrt {b} d\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )-\frac {1}{2} \left (a \sqrt {b} d\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt {a x+\sqrt {-b+a^2 x^2}}\right ) \\ & = -\frac {b^4 c}{56 a^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/2}}-\frac {b^2 c \sqrt {a x+\sqrt {-b+a^2 x^2}}}{4 a^3}+2 a d \sqrt {a x+\sqrt {-b+a^2 x^2}}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}{72 a^3}+\frac {2 a b d \sqrt {a x+\sqrt {-b+a^2 x^2}}}{b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2}+\frac {a \sqrt [4]{b} d \log \left (\sqrt {b}+a x+\sqrt {-b+a^2 x^2}-\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{2 \sqrt {2}}-\frac {a \sqrt [4]{b} d \log \left (\sqrt {b}+a x+\sqrt {-b+a^2 x^2}+\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{2 \sqrt {2}}-\frac {\left (a \sqrt [4]{b} d\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{b}}\right )}{\sqrt {2}}+\frac {\left (a \sqrt [4]{b} d\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{b}}\right )}{\sqrt {2}} \\ & = -\frac {b^4 c}{56 a^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/2}}-\frac {b^2 c \sqrt {a x+\sqrt {-b+a^2 x^2}}}{4 a^3}+2 a d \sqrt {a x+\sqrt {-b+a^2 x^2}}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}{72 a^3}+\frac {2 a b d \sqrt {a x+\sqrt {-b+a^2 x^2}}}{b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2}+\frac {a \sqrt [4]{b} d \arctan \left (1-\frac {\sqrt {2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{b}}\right )}{\sqrt {2}}-\frac {a \sqrt [4]{b} d \arctan \left (1+\frac {\sqrt {2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{b}}\right )}{\sqrt {2}}+\frac {a \sqrt [4]{b} d \log \left (\sqrt {b}+a x+\sqrt {-b+a^2 x^2}-\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{2 \sqrt {2}}-\frac {a \sqrt [4]{b} d \log \left (\sqrt {b}+a x+\sqrt {-b+a^2 x^2}+\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{2 \sqrt {2}} \\ \end{align*}
Time = 0.77 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {-b+a^2 x^2} \left (d+c x^4\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x^2} \, dx=\frac {63 a^3 b^3 d-72 a b^4 c x^2+126 a^5 b^2 d x^2+210 a^3 b^3 c x^4-2016 a^7 b d x^4+126 a^5 b^2 c x^6+2016 a^9 d x^6-504 a^7 b c x^8+224 a^9 c x^{10}-2 x \sqrt {-b+a^2 x^2} \left (8 b^4 c-77 a^2 b^3 c x^2+21 a^4 b^2 \left (3 d+c x^4\right )-112 a^8 x^4 \left (9 d+c x^4\right )+28 a^6 b x^2 \left (18 d+7 c x^4\right )\right )}{63 a^3 x \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}+\frac {a \sqrt [4]{b} d \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{-\sqrt {b}+a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2}}-\frac {a \sqrt [4]{b} d \text {arctanh}\left (\frac {\sqrt {b}+a x+\sqrt {-b+a^2 x^2}}{\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{\sqrt {2}} \]
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\[\int \frac {\sqrt {a^{2} x^{2}-b}\, \left (c \,x^{4}+d \right ) \sqrt {a x +\sqrt {a^{2} x^{2}-b}}}{x^{2}}d x\]
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {-b+a^2 x^2} \left (d+c x^4\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x^2} \, dx=-\frac {63 \, \left (-a^{4} b d^{4}\right )^{\frac {1}{4}} a^{3} x \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} - b}} a d + \left (-a^{4} b d^{4}\right )^{\frac {1}{4}}\right ) + 63 i \, \left (-a^{4} b d^{4}\right )^{\frac {1}{4}} a^{3} x \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} - b}} a d + i \, \left (-a^{4} b d^{4}\right )^{\frac {1}{4}}\right ) - 63 i \, \left (-a^{4} b d^{4}\right )^{\frac {1}{4}} a^{3} x \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} - b}} a d - i \, \left (-a^{4} b d^{4}\right )^{\frac {1}{4}}\right ) - 63 \, \left (-a^{4} b d^{4}\right )^{\frac {1}{4}} a^{3} x \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} - b}} a d - \left (-a^{4} b d^{4}\right )^{\frac {1}{4}}\right ) + 2 \, {\left (2 \, a^{4} c x^{5} - 2 \, a^{2} b c x^{3} - {\left (189 \, a^{4} d - 16 \, b^{2} c\right )} x - {\left (16 \, a^{3} c x^{4} - 8 \, a b c x^{2} - 63 \, a^{3} d\right )} \sqrt {a^{2} x^{2} - b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}{126 \, a^{3} x} \]
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\[ \int \frac {\sqrt {-b+a^2 x^2} \left (d+c x^4\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x^2} \, dx=\int \frac {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} \sqrt {a^{2} x^{2} - b} \left (c x^{4} + d\right )}{x^{2}}\, dx \]
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\[ \int \frac {\sqrt {-b+a^2 x^2} \left (d+c x^4\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x^2} \, dx=\int { \frac {{\left (c x^{4} + d\right )} \sqrt {a^{2} x^{2} - b} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}{x^{2}} \,d x } \]
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\[ \int \frac {\sqrt {-b+a^2 x^2} \left (d+c x^4\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x^2} \, dx=\int { \frac {{\left (c x^{4} + d\right )} \sqrt {a^{2} x^{2} - b} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {-b+a^2 x^2} \left (d+c x^4\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x^2} \, dx=\int \frac {\sqrt {a\,x+\sqrt {a^2\,x^2-b}}\,\left (c\,x^4+d\right )\,\sqrt {a^2\,x^2-b}}{x^2} \,d x \]
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