Optimal. Leaf size=371 \[ \frac {b \sqrt {d} \left (4 a c-b^2 d\right ) \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right ) \tanh ^{-1}\left (\frac {b d+2 c \sqrt {\frac {d}{x}}}{2 \sqrt {c} \sqrt {d} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{1024 c^{13/2}}+\frac {b \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right ) \left (b d+2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{512 c^6}-\frac {\left (1024 a^2 c^2+18 b c \sqrt {\frac {d}{x}} \left (148 a c-77 b^2 d\right )-3276 a b^2 c d+1155 b^4 d^2\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{6720 c^5}+\frac {\left (32 a c-33 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{140 c^3 x}+\frac {11 b \left (\frac {d}{x}\right )^{3/2} \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{42 c^2 d}-\frac {2 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{7 c x^2} \]
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Rubi [A] time = 0.66, antiderivative size = 371, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {1970, 1357, 742, 832, 779, 612, 621, 206} \[ -\frac {\left (1024 a^2 c^2+18 b c \sqrt {\frac {d}{x}} \left (148 a c-77 b^2 d\right )-3276 a b^2 c d+1155 b^4 d^2\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{6720 c^5}+\frac {b \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right ) \left (b d+2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{512 c^6}+\frac {b \sqrt {d} \left (4 a c-b^2 d\right ) \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right ) \tanh ^{-1}\left (\frac {b d+2 c \sqrt {\frac {d}{x}}}{2 \sqrt {c} \sqrt {d} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{1024 c^{13/2}}+\frac {\left (32 a c-33 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{140 c^3 x}+\frac {11 b \left (\frac {d}{x}\right )^{3/2} \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{42 c^2 d}-\frac {2 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{7 c x^2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 742
Rule 779
Rule 832
Rule 1357
Rule 1970
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^4} \, dx &=-\frac {\operatorname {Subst}\left (\int x^2 \sqrt {a+b \sqrt {x}+\frac {c x}{d}} \, dx,x,\frac {d}{x}\right )}{d^3}\\ &=-\frac {2 \operatorname {Subst}\left (\int x^5 \sqrt {a+b x+\frac {c x^2}{d}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{d^3}\\ &=-\frac {2 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{7 c x^2}-\frac {2 \operatorname {Subst}\left (\int x^3 \left (-4 a-\frac {11 b x}{2}\right ) \sqrt {a+b x+\frac {c x^2}{d}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{7 c d^2}\\ &=\frac {11 b \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2} \left (\frac {d}{x}\right )^{3/2}}{42 c^2 d}-\frac {2 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{7 c x^2}-\frac {\operatorname {Subst}\left (\int x^2 \left (\frac {33 a b}{2}-\frac {3 \left (32 a c-33 b^2 d\right ) x}{4 d}\right ) \sqrt {a+b x+\frac {c x^2}{d}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{21 c^2 d}\\ &=\frac {11 b \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2} \left (\frac {d}{x}\right )^{3/2}}{42 c^2 d}-\frac {2 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{7 c x^2}+\frac {\left (32 a c-33 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{140 c^3 x}-\frac {\operatorname {Subst}\left (\int x \left (-\frac {3}{2} a \left (33 b^2-\frac {32 a c}{d}\right )+\frac {9 b \left (148 a c-77 b^2 d\right ) x}{8 d}\right ) \sqrt {a+b x+\frac {c x^2}{d}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{105 c^3}\\ &=-\frac {\left (1024 a^2 c^2-3276 a b^2 c d+1155 b^4 d^2+18 b c \left (148 a c-77 b^2 d\right ) \sqrt {\frac {d}{x}}\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{6720 c^5}+\frac {11 b \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2} \left (\frac {d}{x}\right )^{3/2}}{42 c^2 d}-\frac {2 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{7 c x^2}+\frac {\left (32 a c-33 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{140 c^3 x}+\frac {\left (b \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right )\right ) \operatorname {Subst}\left (\int \sqrt {a+b x+\frac {c x^2}{d}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{128 c^5}\\ &=\frac {b \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right ) \left (b d+2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{512 c^6}-\frac {\left (1024 a^2 c^2-3276 a b^2 c d+1155 b^4 d^2+18 b c \left (148 a c-77 b^2 d\right ) \sqrt {\frac {d}{x}}\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{6720 c^5}+\frac {11 b \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2} \left (\frac {d}{x}\right )^{3/2}}{42 c^2 d}-\frac {2 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{7 c x^2}+\frac {\left (32 a c-33 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{140 c^3 x}+\frac {\left (b \left (4 a c-b^2 d\right ) \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{1024 c^6}\\ &=\frac {b \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right ) \left (b d+2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{512 c^6}-\frac {\left (1024 a^2 c^2-3276 a b^2 c d+1155 b^4 d^2+18 b c \left (148 a c-77 b^2 d\right ) \sqrt {\frac {d}{x}}\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{6720 c^5}+\frac {11 b \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2} \left (\frac {d}{x}\right )^{3/2}}{42 c^2 d}-\frac {2 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{7 c x^2}+\frac {\left (32 a c-33 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{140 c^3 x}+\frac {\left (b \left (4 a c-b^2 d\right ) \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {4 c}{d}-x^2} \, dx,x,\frac {b+\frac {2 c \sqrt {\frac {d}{x}}}{d}}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{512 c^6}\\ &=\frac {b \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right ) \left (b d+2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{512 c^6}-\frac {\left (1024 a^2 c^2-3276 a b^2 c d+1155 b^4 d^2+18 b c \left (148 a c-77 b^2 d\right ) \sqrt {\frac {d}{x}}\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{6720 c^5}+\frac {11 b \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2} \left (\frac {d}{x}\right )^{3/2}}{42 c^2 d}-\frac {2 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{7 c x^2}+\frac {\left (32 a c-33 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{140 c^3 x}+\frac {b \sqrt {d} \left (4 a c-b^2 d\right ) \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \left (b+\frac {2 c \sqrt {\frac {d}{x}}}{d}\right )}{2 \sqrt {c} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{1024 c^{13/2}}\\ \end {align*}
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Mathematica [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^4} \, dx \]
Verification is Not applicable to the result.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.15, size = 979, normalized size = 2.64 \[ \frac {\sqrt {\frac {a x +\sqrt {\frac {d}{x}}\, b x +c}{x}}\, \left (-3465 \left (\frac {d}{x}\right )^{\frac {7}{2}} b^{7} \sqrt {c}\, x^{7} \ln \left (\frac {\sqrt {\frac {d}{x}}\, b x +2 c +2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {c}}{\sqrt {x}}\right )+26460 \left (\frac {d}{x}\right )^{\frac {5}{2}} a \,b^{5} c^{\frac {3}{2}} x^{6} \ln \left (\frac {\sqrt {\frac {d}{x}}\, b x +2 c +2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {c}}{\sqrt {x}}\right )+6930 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, a \,b^{6} d^{3} x^{4}+6930 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \left (\frac {d}{x}\right )^{\frac {7}{2}} b^{7} x^{7}-58800 \left (\frac {d}{x}\right )^{\frac {3}{2}} a^{2} b^{3} c^{\frac {5}{2}} x^{5} \ln \left (\frac {\sqrt {\frac {d}{x}}\, b x +2 c +2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {c}}{\sqrt {x}}\right )-25200 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, a^{2} b^{4} c \,d^{2} x^{4}-39060 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \left (\frac {d}{x}\right )^{\frac {5}{2}} a \,b^{5} c \,x^{6}+33600 \sqrt {\frac {d}{x}}\, a^{3} b \,c^{\frac {7}{2}} x^{4} \ln \left (\frac {\sqrt {\frac {d}{x}}\, b x +2 c +2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {c}}{\sqrt {x}}\right )+16800 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, a^{3} b^{2} c^{2} d \,x^{4}+67200 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \left (\frac {d}{x}\right )^{\frac {3}{2}} a^{2} b^{3} c^{2} x^{5}-6930 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} b^{6} d^{3} x^{3}-33600 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {\frac {d}{x}}\, a^{3} b \,c^{3} x^{4}+25200 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} a \,b^{4} c \,d^{2} x^{3}+13860 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} \left (\frac {d}{x}\right )^{\frac {5}{2}} b^{5} c \,x^{5}-16800 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} a^{2} b^{2} c^{2} d \,x^{3}-50400 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} \left (\frac {d}{x}\right )^{\frac {3}{2}} a \,b^{3} c^{2} x^{4}-18480 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} b^{4} c^{2} d^{2} x^{2}+33600 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} \sqrt {\frac {d}{x}}\, a^{2} b \,c^{3} x^{3}+52416 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} a \,b^{2} c^{3} d \,x^{2}+22176 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} \left (\frac {d}{x}\right )^{\frac {3}{2}} b^{3} c^{3} x^{3}-16384 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} a^{2} c^{4} x^{2}-42624 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} \sqrt {\frac {d}{x}}\, a b \,c^{4} x^{2}-25344 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} b^{2} c^{4} d x +24576 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} a \,c^{5} x +28160 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} \sqrt {\frac {d}{x}}\, b \,c^{5} x -30720 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} c^{6}\right )}{107520 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, c^{7} x^{3}} \]
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 \[ \int \frac {\sqrt {b \sqrt {\frac {d}{x}} + a + \frac {c}{x}}}{x^{4}}\,{d x} \]
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 \[ \int \frac {\sqrt {a+\frac {c}{x}+b\,\sqrt {\frac {d}{x}}}}{x^4} \,d x \]
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 \[ \int \frac {\sqrt {a + b \sqrt {\frac {d}{x}} + \frac {c}{x}}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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