Optimal. Leaf size=289 \[ -\frac {\left (1024 a^2 c^2-2940 a b^2 c d+945 b^4 d^2+14 b c \left (92 a c-45 b^2 d\right ) \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{960 c^5}+\frac {9 b \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \left (\frac {d}{x}\right )^{3/2}}{20 c^2 d}-\frac {2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{5 c x^2}+\frac {\left (64 a c-63 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{120 c^3 x}+\frac {b \sqrt {d} \left (240 a^2 c^2-280 a b^2 c d+63 b^4 d^2\right ) \tanh ^{-1}\left (\frac {b d+2 c \sqrt {\frac {d}{x}}}{2 \sqrt {c} \sqrt {d} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{128 c^{11/2}} \]
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Rubi [A]
time = 0.37, antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1994, 1371,
756, 846, 793, 635, 212} \begin {gather*} \frac {b \sqrt {d} \left (240 a^2 c^2-280 a b^2 c d+63 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 )}{128 c^{11/2}}-\frac {\left (1024 a^2 c^2+14 b c \sqrt {\frac {d}{x}} \left (92 a c-45 b^2 d\right )-2940 a b^2 c d+945 b^4 d^2\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{960 c^5}+\frac {\left (64 a c-63 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{120 c^3 x}+\frac {9 b \left (\frac {d}{x}\right )^{3/2} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{20 c^2 d}-\frac {2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{5 c x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 756
Rule 793
Rule 846
Rule 1371
Rule 1994
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^4} \, dx &=-\frac {\text {Subst}\left (\int \frac {x^2}{\sqrt {a+b \sqrt {x}+\frac {c x}{d}}} \, dx,x,\frac {d}{x}\right )}{d^3}\\ &=-\frac {2 \text {Subst}\left (\int \frac {x^5}{\sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{d^3}\\ &=-\frac {2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{5 c x^2}-\frac {2 \text {Subst}\left (\int \frac {x^3 \left (-4 a-\frac {9 b x}{2}\right )}{\sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{5 c d^2}\\ &=\frac {9 b \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \left (\frac {d}{x}\right )^{3/2}}{20 c^2 d}-\frac {2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{5 c x^2}-\frac {\text {Subst}\left (\int \frac {x^2 \left (\frac {27 a b}{2}-\frac {\left (64 a c-63 b^2 d\right ) x}{4 d}\right )}{\sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{10 c^2 d}\\ &=\frac {9 b \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \left (\frac {d}{x}\right )^{3/2}}{20 c^2 d}-\frac {2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{5 c x^2}+\frac {\left (64 a c-63 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{120 c^3 x}-\frac {\text {Subst}\left (\int \frac {x \left (-\frac {1}{2} a \left (63 b^2-\frac {64 a c}{d}\right )+\frac {7 b \left (92 a c-45 b^2 d\right ) x}{8 d}\right )}{\sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{30 c^3}\\ &=-\frac {\left (1024 a^2 c^2-2940 a b^2 c d+945 b^4 d^2+14 b c \left (92 a c-45 b^2 d\right ) \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{960 c^5}+\frac {9 b \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \left (\frac {d}{x}\right )^{3/2}}{20 c^2 d}-\frac {2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{5 c x^2}+\frac {\left (64 a c-63 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{120 c^3 x}+\frac {\left (b \left (240 a^2 c^2-280 a b^2 c d+63 b^4 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{128 c^5}\\ &=-\frac {\left (1024 a^2 c^2-2940 a b^2 c d+945 b^4 d^2+14 b c \left (92 a c-45 b^2 d\right ) \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{960 c^5}+\frac {9 b \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \left (\frac {d}{x}\right )^{3/2}}{20 c^2 d}-\frac {2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{5 c x^2}+\frac {\left (64 a c-63 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{120 c^3 x}+\frac {\left (b \left (240 a^2 c^2-280 a b^2 c d+63 b^4 d^2\right )\right ) \text {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 )}{64 c^5}\\ &=-\frac {\left (1024 a^2 c^2-2940 a b^2 c d+945 b^4 d^2+14 b c \left (92 a c-45 b^2 d\right ) \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{960 c^5}+\frac {9 b \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \left (\frac {d}{x}\right )^{3/2}}{20 c^2 d}-\frac {2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{5 c x^2}+\frac {\left (64 a c-63 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{120 c^3 x}+\frac {b \sqrt {d} \left (240 a^2 c^2-280 a b^2 c d+63 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 )}{128 c^{11/2}}\\ \end {align*}
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Mathematica [A]
time = 1.01, size = 324, normalized size = 1.12 \begin {gather*} \frac {-2 \sqrt {c} \left (384 c^5-16 c^4 \left (8 a+3 b \sqrt {\frac {d}{x}}\right ) x+945 b^4 d^2 \left (a+b \sqrt {\frac {d}{x}}\right ) x^3-105 b^2 c d x^2 \left (-3 b^2 d+28 a^2 x+34 a b \sqrt {\frac {d}{x}} x\right )+8 c^3 x \left (9 b^2 d+64 a^2 x+43 a b \sqrt {\frac {d}{x}} x\right )+2 c^2 x^2 \left (-574 a b^2 d-63 b^3 d \sqrt {\frac {d}{x}}+512 a^3 x+1156 a^2 b \sqrt {\frac {d}{x}} x\right )\right )-15 b \left (240 a^2 c^2-280 a b^2 c d+63 b^4 d^2\right ) x^3 \sqrt {\frac {d \left (c+\left (a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}} \log \left (b d+2 c \sqrt {\frac {d}{x}}-2 \sqrt {c} \sqrt {\frac {d \left (c+a x+b \sqrt {\frac {d}{x}} x\right )}{x}}\right )}{1920 c^{11/2} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 487, normalized size = 1.69
method | result | size |
default | \(\frac {\sqrt {\frac {b \sqrt {\frac {d}{x}}\, x +a x +c}{x}}\, \left (945 \ln \left (\frac {2 c +b \sqrt {\frac {d}{x}}\, x +2 \sqrt {c}\, \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}}{\sqrt {x}}\right ) \left (\frac {d}{x}\right )^{\frac {5}{2}} x^{5} b^{5} c -1890 d^{2} c^{\frac {3}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, x^{2} b^{4}-4200 \ln \left (\frac {2 c +b \sqrt {\frac {d}{x}}\, x +2 \sqrt {c}\, \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}}{\sqrt {x}}\right ) \left (\frac {d}{x}\right )^{\frac {3}{2}} x^{4} a \,b^{3} c^{2}+1260 c^{\frac {5}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \left (\frac {d}{x}\right )^{\frac {3}{2}} x^{3} b^{3}+5880 d \,c^{\frac {5}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, x^{2} a \,b^{2}-1008 d \,c^{\frac {7}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, x \,b^{2}+3600 \ln \left (\frac {2 c +b \sqrt {\frac {d}{x}}\, x +2 \sqrt {c}\, \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}}{\sqrt {x}}\right ) \sqrt {\frac {d}{x}}\, x^{3} a^{2} b \,c^{3}-2576 c^{\frac {7}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {\frac {d}{x}}\, x^{2} a b -2048 c^{\frac {7}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{2} x^{2}+864 c^{\frac {9}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {\frac {d}{x}}\, x b +1024 c^{\frac {9}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a x -768 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, c^{\frac {11}{2}}\right )}{1920 x^{2} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, c^{\frac {13}{2}}}\) | \(487\) |
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 [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {1}{x^{4} \sqrt {a + b \sqrt {\frac {d}{x}} + \frac {c}{x}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {1}{x^4\,\sqrt {a+\frac {c}{x}+b\,\sqrt {\frac {d}{x}}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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