Optimal. Leaf size=248 \[ -\frac {7 b d^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{12 a^2 \left (\frac {d}{x}\right )^{3/2}}+\frac {5 b d \left (44 a c-21 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{96 a^4 \sqrt {\frac {d}{x}}}-\frac {\left (36 a c-35 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x}{48 a^3}+\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2}{2 a}+\frac {\left (48 a^2 c^2-120 a b^2 c d+35 b^4 d^2\right ) \tanh ^{-1}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{64 a^{9/2}} \]
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Rubi [A]
time = 0.31, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1994, 1371,
758, 848, 820, 738, 212} \begin {gather*} \frac {5 b d \left (44 a c-21 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{96 a^4 \sqrt {\frac {d}{x}}}-\frac {x \left (36 a c-35 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{48 a^3}-\frac {7 b d^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{12 a^2 \left (\frac {d}{x}\right )^{3/2}}+\frac {\left (48 a^2 c^2-120 a b^2 c d+35 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 )}{64 a^{9/2}}+\frac {x^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 738
Rule 758
Rule 820
Rule 848
Rule 1371
Rule 1994
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx &=-\left (d^2 \text {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b \sqrt {x}+\frac {c x}{d}}} \, dx,x,\frac {d}{x}\right )\right )\\ &=-\left (\left (2 d^2\right ) \text {Subst}\left (\int \frac {1}{x^5 \sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )\right )\\ &=\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2}{2 a}+\frac {d^2 \text {Subst}\left (\int \frac {\frac {7 b}{2}+\frac {3 c x}{d}}{x^4 \sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{2 a}\\ &=-\frac {7 b d^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{12 a^2 \left (\frac {d}{x}\right )^{3/2}}+\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2}{2 a}-\frac {d^2 \text {Subst}\left (\int \frac {\frac {1}{4} \left (35 b^2-\frac {36 a c}{d}\right )+\frac {7 b c x}{d}}{x^3 \sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{6 a^2}\\ &=-\frac {7 b d^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{12 a^2 \left (\frac {d}{x}\right )^{3/2}}-\frac {\left (36 a c-35 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x}{48 a^3}+\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2}{2 a}+\frac {d^2 \text {Subst}\left (\int \frac {-\frac {5 b \left (44 a c-21 b^2 d\right )}{8 d}-\frac {c \left (36 a c-35 b^2 d\right ) x}{4 d^2}}{x^2 \sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{12 a^3}\\ &=-\frac {7 b d^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{12 a^2 \left (\frac {d}{x}\right )^{3/2}}+\frac {5 b d \left (44 a c-21 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{96 a^4 \sqrt {\frac {d}{x}}}-\frac {\left (36 a c-35 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x}{48 a^3}+\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2}{2 a}-\frac {\left (48 a^2 c^2-120 a b^2 c d+35 b^4 d^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{64 a^4}\\ &=-\frac {7 b d^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{12 a^2 \left (\frac {d}{x}\right )^{3/2}}+\frac {5 b d \left (44 a c-21 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{96 a^4 \sqrt {\frac {d}{x}}}-\frac {\left (36 a c-35 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x}{48 a^3}+\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2}{2 a}+\frac {\left (48 a^2 c^2-120 a b^2 c d+35 b^4 d^2\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 )}{32 a^4}\\ &=-\frac {7 b d^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{12 a^2 \left (\frac {d}{x}\right )^{3/2}}+\frac {5 b d \left (44 a c-21 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{96 a^4 \sqrt {\frac {d}{x}}}-\frac {\left (36 a c-35 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x}{48 a^3}+\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2}{2 a}+\frac {\left (48 a^2 c^2-120 a b^2 c d+35 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 )}{64 a^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 1.11, size = 279, normalized size = 1.12 \begin {gather*} \frac {\sqrt {a} d \left (-105 b^3 d \left (b d+c \sqrt {\frac {d}{x}}\right )+48 a^4 x^2-8 a^3 x \left (3 c+b \sqrt {\frac {d}{x}} x\right )+a^2 \left (-72 c^2+14 b^2 d x+92 b c \sqrt {\frac {d}{x}} x\right )-5 a b \left (-58 b c d-44 c^2 \sqrt {\frac {d}{x}}+7 b^2 d \sqrt {\frac {d}{x}} x\right )\right )-3 \sqrt {d} \left (48 a^2 c^2-120 a b^2 c d+35 b^4 d^2\right ) \sqrt {\frac {d \left (c+\left (a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}} \tanh ^{-1}\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 )}{96 a^{9/2} d \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 398, normalized size = 1.60
method | result | size |
default | \(-\frac {\sqrt {\frac {b \sqrt {\frac {d}{x}}\, x +a x +c}{x}}\, \sqrt {x}\, \left (-105 d^{2} \ln \left (\frac {b \sqrt {\frac {d}{x}}\, \sqrt {x}+2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) a \,b^{4}+210 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {3}{2}} \left (\frac {d}{x}\right )^{\frac {3}{2}} x^{\frac {3}{2}} b^{3}-140 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, d \,a^{\frac {5}{2}} \sqrt {x}\, b^{2}+112 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {7}{2}} \sqrt {\frac {d}{x}}\, x^{\frac {3}{2}} b -96 x^{\frac {3}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {9}{2}}+360 d \ln \left (\frac {b \sqrt {\frac {d}{x}}\, \sqrt {x}+2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) a^{2} b^{2} c -440 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {5}{2}} \sqrt {\frac {d}{x}}\, \sqrt {x}\, b c +144 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {7}{2}} c \sqrt {x}-144 \ln \left (\frac {b \sqrt {\frac {d}{x}}\, \sqrt {x}+2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) a^{3} c^{2}\right )}{192 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {11}{2}}}\) | \(398\) |
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 {x}{\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 [A]
time = 1.93, size = 363, normalized size = 1.46 \begin {gather*} -\frac {2 \, \sqrt {a d^{2} x + \sqrt {d x} b d^{2} + c d^{2}} {\left (2 \, \sqrt {d x} {\left (4 \, \sqrt {d x} {\left (\frac {7 \, b}{a^{2}} - \frac {6 \, \sqrt {d x}}{a d}\right )} - \frac {35 \, a b^{2} d^{2} - 36 \, a^{2} c d}{a^{4} d}\right )} + \frac {5 \, {\left (21 \, b^{3} d^{3} - 44 \, a b c d^{2}\right )}}{a^{4} d}\right )} + \frac {3 \, {\left (35 \, b^{4} d^{4} - 120 \, a b^{2} c d^{3} + 48 \, a^{2} c^{2} d^{2}\right )} \log \left ({\left | -b d^{2} - 2 \, \sqrt {a d} {\left (\sqrt {a d} \sqrt {d x} - \sqrt {a d^{2} x + \sqrt {d x} b d^{2} + c d^{2}}\right )} \right |}\right )}{\sqrt {a d} a^{4}} - \frac {105 \, b^{4} d^{4} \log \left ({\left | -b d^{2} + 2 \, \sqrt {c d^{2}} \sqrt {a d} \right |}\right ) - 360 \, a b^{2} c d^{3} \log \left ({\left | -b d^{2} + 2 \, \sqrt {c d^{2}} \sqrt {a d} \right |}\right ) + 144 \, a^{2} c^{2} d^{2} \log \left ({\left | -b d^{2} + 2 \, \sqrt {c d^{2}} \sqrt {a d} \right |}\right ) + 210 \, \sqrt {c d^{2}} \sqrt {a d} b^{3} d^{2} - 440 \, \sqrt {c d^{2}} \sqrt {a d} a b c d}{\sqrt {a d} a^{4}}}{192 \, d^{\frac {3}{2}} \mathrm {sgn}\left (x\right )} \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 {x}{\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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