Integrand size = 26, antiderivative size = 333 \[ \int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2 \, dx=-\frac {3 b d^3 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{10 a^2 \left (\frac {d}{x}\right )^{5/2}}+\frac {7 b d^2 \left (28 a c-15 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{480 a^4 \left (\frac {d}{x}\right )^{3/2}}+\frac {\left (16 a^2 c^2-56 a b^2 c d+21 b^4 d^2\right ) \left (2 a+b \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x}{256 a^5}-\frac {\left (20 a c-21 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2} x^2}{80 a^3}+\frac {\left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2} x^3}{3 a}+\frac {\left (4 a c-b^2 d\right ) \left (16 a^2 c^2-56 a b^2 c d+21 b^4 d^2\right ) \text {arctanh}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{512 a^{11/2}} \]
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Time = 0.48 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {1994, 1371, 758, 848, 820, 734, 738, 212} \[ \int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2 \, dx=\frac {7 b d^2 \left (28 a c-15 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{480 a^4 \left (\frac {d}{x}\right )^{3/2}}-\frac {x^2 \left (20 a c-21 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{80 a^3}-\frac {3 b d^3 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{10 a^2 \left (\frac {d}{x}\right )^{5/2}}+\frac {\left (4 a c-b^2 d\right ) \left (16 a^2 c^2-56 a b^2 c d+21 b^4 d^2\right ) \text {arctanh}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{512 a^{11/2}}+\frac {x \left (16 a^2 c^2-56 a b^2 c d+21 b^4 d^2\right ) \left (2 a+b \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{256 a^5}+\frac {x^3 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{3 a} \]
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Rule 212
Rule 734
Rule 738
Rule 758
Rule 820
Rule 848
Rule 1371
Rule 1994
Rubi steps \begin{align*} \text {integral}& = -\left (d^3 \text {Subst}\left (\int \frac {\sqrt {a+b \sqrt {x}+\frac {c x}{d}}}{x^4} \, dx,x,\frac {d}{x}\right )\right ) \\ & = -\left (\left (2 d^3\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x+\frac {c x^2}{d}}}{x^7} \, dx,x,\sqrt {\frac {d}{x}}\right )\right ) \\ & = \frac {\left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2} x^3}{3 a}+\frac {d^3 \text {Subst}\left (\int \frac {\left (\frac {9 b}{2}+\frac {3 c x}{d}\right ) \sqrt {a+b x+\frac {c x^2}{d}}}{x^6} \, dx,x,\sqrt {\frac {d}{x}}\right )}{3 a} \\ & = -\frac {3 b d^3 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{10 a^2 \left (\frac {d}{x}\right )^{5/2}}+\frac {\left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2} x^3}{3 a}-\frac {d^3 \text {Subst}\left (\int \frac {\left (\frac {3}{4} \left (21 b^2-\frac {20 a c}{d}\right )+\frac {9 b c x}{d}\right ) \sqrt {a+b x+\frac {c x^2}{d}}}{x^5} \, dx,x,\sqrt {\frac {d}{x}}\right )}{15 a^2} \\ & = -\frac {3 b d^3 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{10 a^2 \left (\frac {d}{x}\right )^{5/2}}-\frac {\left (20 a c-21 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2} x^2}{80 a^3}+\frac {\left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2} x^3}{3 a}+\frac {d^3 \text {Subst}\left (\int \frac {\left (-\frac {21 b \left (28 a c-15 b^2 d\right )}{8 d}-\frac {3 c \left (20 a c-21 b^2 d\right ) x}{4 d^2}\right ) \sqrt {a+b x+\frac {c x^2}{d}}}{x^4} \, dx,x,\sqrt {\frac {d}{x}}\right )}{60 a^3} \\ & = -\frac {3 b d^3 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{10 a^2 \left (\frac {d}{x}\right )^{5/2}}+\frac {7 b d^2 \left (28 a c-15 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{480 a^4 \left (\frac {d}{x}\right )^{3/2}}-\frac {\left (20 a c-21 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2} x^2}{80 a^3}+\frac {\left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2} x^3}{3 a}-\frac {\left (d \left (16 a^2 c^2-56 a b^2 c d+21 b^4 d^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x+\frac {c x^2}{d}}}{x^3} \, dx,x,\sqrt {\frac {d}{x}}\right )}{64 a^4} \\ & = -\frac {3 b d^3 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{10 a^2 \left (\frac {d}{x}\right )^{5/2}}+\frac {7 b d^2 \left (28 a c-15 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{480 a^4 \left (\frac {d}{x}\right )^{3/2}}+\frac {\left (16 a^2 c^2-56 a b^2 c d+21 b^4 d^2\right ) \left (2 a+b \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x}{256 a^5}-\frac {\left (20 a c-21 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2} x^2}{80 a^3}+\frac {\left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2} x^3}{3 a}-\frac {\left (\left (4 a c-b^2 d\right ) \left (16 a^2 c^2-56 a b^2 c d+21 b^4 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{512 a^5} \\ & = -\frac {3 b d^3 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{10 a^2 \left (\frac {d}{x}\right )^{5/2}}+\frac {7 b d^2 \left (28 a c-15 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{480 a^4 \left (\frac {d}{x}\right )^{3/2}}+\frac {\left (16 a^2 c^2-56 a b^2 c d+21 b^4 d^2\right ) \left (2 a+b \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x}{256 a^5}-\frac {\left (20 a c-21 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2} x^2}{80 a^3}+\frac {\left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2} x^3}{3 a}+\frac {\left (\left (4 a c-b^2 d\right ) \left (16 a^2 c^2-56 a b^2 c d+21 b^4 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b \sqrt {\frac {d}{x}}}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{256 a^5} \\ & = -\frac {3 b d^3 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{10 a^2 \left (\frac {d}{x}\right )^{5/2}}+\frac {7 b d^2 \left (28 a c-15 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{480 a^4 \left (\frac {d}{x}\right )^{3/2}}+\frac {\left (16 a^2 c^2-56 a b^2 c d+21 b^4 d^2\right ) \left (2 a+b \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x}{256 a^5}-\frac {\left (20 a c-21 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2} x^2}{80 a^3}+\frac {\left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2} x^3}{3 a}+\frac {\left (4 a c-b^2 d\right ) \left (16 a^2 c^2-56 a b^2 c d+21 b^4 d^2\right ) \tanh ^{-1}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{512 a^{11/2}} \\ \end{align*}
Time = 2.68 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.94 \[ \int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2 \, dx=\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \left (\sqrt {a} x \left (-210 a b^3 d \left (b d+8 c \sqrt {\frac {d}{x}}\right )+315 b^5 d \left (\frac {d}{x}\right )^{3/2} x+1280 a^5 x^2+64 a^4 x \left (5 c+2 b \sqrt {\frac {d}{x}} x\right )-16 a^3 \left (30 c^2+9 b^2 d x+34 b c \sqrt {\frac {d}{x}} x\right )+8 a^2 b \left (112 b c d+226 c^2 \sqrt {\frac {d}{x}}+21 b^2 d \sqrt {\frac {d}{x}} x\right )\right )+\frac {15 \sqrt {d} \left (-64 a^3 c^3+240 a^2 b^2 c^2 d-140 a b^4 c d^2+21 b^6 d^3\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {d}{x}}-\sqrt {\frac {d \left (c+a x+b \sqrt {\frac {d}{x}} x\right )}{x}}}{\sqrt {a} \sqrt {d}}\right )}{\sqrt {\frac {d \left (c+\left (a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}}}\right )}{3840 a^{11/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(654\) vs. \(2(283)=566\).
Time = 0.30 (sec) , antiderivative size = 655, normalized size of antiderivative = 1.97
method | result | size |
default | \(\frac {\sqrt {\frac {b \sqrt {\frac {d}{x}}\, x +a x +c}{x}}\, \sqrt {x}\, \left (630 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {3}{2}} \left (\frac {d}{x}\right )^{\frac {5}{2}} x^{\frac {5}{2}} b^{5}+2560 x^{\frac {3}{2}} \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} a^{\frac {11}{2}}-2304 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} a^{\frac {9}{2}} \sqrt {\frac {d}{x}}\, x^{\frac {3}{2}} b -1680 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} a^{\frac {5}{2}} \left (\frac {d}{x}\right )^{\frac {3}{2}} x^{\frac {3}{2}} b^{3}+1260 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {5}{2}} d^{2} \sqrt {x}\, b^{4}-315 d^{3} \ln \left (\frac {\sqrt {\frac {d}{x}}\, \sqrt {x}\, b +2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) a \,b^{6}+2016 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} a^{\frac {7}{2}} d \sqrt {x}\, b^{2}-1680 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {5}{2}} \left (\frac {d}{x}\right )^{\frac {3}{2}} x^{\frac {3}{2}} b^{3} c -3360 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {7}{2}} d \sqrt {x}\, b^{2} c -1920 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} a^{\frac {9}{2}} c \sqrt {x}+3136 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} a^{\frac {7}{2}} \sqrt {\frac {d}{x}}\, \sqrt {x}\, b c +2100 d^{2} \ln \left (\frac {\sqrt {\frac {d}{x}}\, \sqrt {x}\, b +2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) a^{2} b^{4} c +960 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {9}{2}} c^{2} \sqrt {x}+480 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {7}{2}} \sqrt {\frac {d}{x}}\, \sqrt {x}\, b \,c^{2}-3600 d \ln \left (\frac {\sqrt {\frac {d}{x}}\, \sqrt {x}\, b +2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) a^{3} b^{2} c^{2}+960 \ln \left (\frac {\sqrt {\frac {d}{x}}\, \sqrt {x}\, b +2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) a^{4} c^{3}\right )}{7680 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {13}{2}}}\) | \(655\) |
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Timed out. \[ \int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2 \, dx=\text {Timed out} \]
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\[ \int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2 \, dx=\int x^{2} \sqrt {a + b \sqrt {\frac {d}{x}} + \frac {c}{x}}\, dx \]
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\[ \int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2 \, dx=\int { \sqrt {b \sqrt {\frac {d}{x}} + a + \frac {c}{x}} x^{2} \,d x } \]
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\[ \int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2 \, dx=\int { \sqrt {b \sqrt {\frac {d}{x}} + a + \frac {c}{x}} x^{2} \,d x } \]
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Timed out. \[ \int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2 \, dx=\int x^2\,\sqrt {a+\frac {c}{x}+b\,\sqrt {\frac {d}{x}}} \,d x \]
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