Optimal. Leaf size=169 \[ \frac {b c^2 \sqrt {a+b \sqrt {\frac {c}{x}}}}{12 a \left (\frac {c}{x}\right )^{3/2}}+\frac {5 b^3 c^2 \sqrt {a+b \sqrt {\frac {c}{x}}}}{32 a^3 \sqrt {\frac {c}{x}}}-\frac {5 b^2 c \sqrt {a+b \sqrt {\frac {c}{x}}} x}{48 a^2}+\frac {1}{2} \sqrt {a+b \sqrt {\frac {c}{x}}} x^2-\frac {5 b^4 c^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{\sqrt {a}}\right )}{32 a^{7/2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 172, normalized size of antiderivative = 1.02, number of steps
used = 8, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {376, 272, 43,
44, 65, 214} \begin {gather*} -\frac {5 b^4 c^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{\sqrt {a}}\right )}{32 a^{7/2}}+\frac {5 b^3 c^2 \sqrt {a+b \sqrt {\frac {c}{x}}}}{32 a^3 \sqrt {\frac {c}{x}}}-\frac {5 b^2 c x \sqrt {a+b \sqrt {\frac {c}{x}}}}{48 a^2}+\frac {b x^3 \left (\frac {c}{x}\right )^{3/2} \sqrt {a+b \sqrt {\frac {c}{x}}}}{12 a c}+\frac {1}{2} x^2 \sqrt {a+b \sqrt {\frac {c}{x}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 44
Rule 65
Rule 214
Rule 272
Rule 376
Rubi steps
\begin {align*} \int \sqrt {a+b \sqrt {\frac {c}{x}}} x \, dx &=\text {Subst}\left (\int \sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}} x \, dx,\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=-\text {Subst}\left (2 \text {Subst}\left (\int \frac {\sqrt {a+b \sqrt {c} x}}{x^5} \, dx,x,\frac {1}{\sqrt {x}}\right ),\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=\frac {1}{2} \sqrt {a+b \sqrt {\frac {c}{x}}} x^2-\text {Subst}\left (\frac {1}{4} \left (b \sqrt {c}\right ) \text {Subst}\left (\int \frac {1}{x^4 \sqrt {a+b \sqrt {c} x}} \, dx,x,\frac {1}{\sqrt {x}}\right ),\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=\frac {1}{2} \sqrt {a+b \sqrt {\frac {c}{x}}} x^2+\frac {b \sqrt {a+b \sqrt {\frac {c}{x}}} \left (\frac {c}{x}\right )^{3/2} x^3}{12 a c}+\text {Subst}\left (\frac {\left (5 b^2 c\right ) \text {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b \sqrt {c} x}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{24 a},\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=-\frac {5 b^2 c \sqrt {a+b \sqrt {\frac {c}{x}}} x}{48 a^2}+\frac {1}{2} \sqrt {a+b \sqrt {\frac {c}{x}}} x^2+\frac {b \sqrt {a+b \sqrt {\frac {c}{x}}} \left (\frac {c}{x}\right )^{3/2} x^3}{12 a c}-\text {Subst}\left (\frac {\left (5 b^3 c^{3/2}\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b \sqrt {c} x}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{32 a^2},\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=\frac {5 b^3 c^2 \sqrt {a+b \sqrt {\frac {c}{x}}}}{32 a^3 \sqrt {\frac {c}{x}}}-\frac {5 b^2 c \sqrt {a+b \sqrt {\frac {c}{x}}} x}{48 a^2}+\frac {1}{2} \sqrt {a+b \sqrt {\frac {c}{x}}} x^2+\frac {b \sqrt {a+b \sqrt {\frac {c}{x}}} \left (\frac {c}{x}\right )^{3/2} x^3}{12 a c}+\text {Subst}\left (\frac {\left (5 b^4 c^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b \sqrt {c} x}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{64 a^3},\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=\frac {5 b^3 c^2 \sqrt {a+b \sqrt {\frac {c}{x}}}}{32 a^3 \sqrt {\frac {c}{x}}}-\frac {5 b^2 c \sqrt {a+b \sqrt {\frac {c}{x}}} x}{48 a^2}+\frac {1}{2} \sqrt {a+b \sqrt {\frac {c}{x}}} x^2+\frac {b \sqrt {a+b \sqrt {\frac {c}{x}}} \left (\frac {c}{x}\right )^{3/2} x^3}{12 a c}+\text {Subst}\left (\frac {\left (5 b^3 c^{3/2}\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b \sqrt {c}}+\frac {x^2}{b \sqrt {c}}} \, dx,x,\sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}\right )}{32 a^3},\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=\frac {5 b^3 c^2 \sqrt {a+b \sqrt {\frac {c}{x}}}}{32 a^3 \sqrt {\frac {c}{x}}}-\frac {5 b^2 c \sqrt {a+b \sqrt {\frac {c}{x}}} x}{48 a^2}+\frac {1}{2} \sqrt {a+b \sqrt {\frac {c}{x}}} x^2+\frac {b \sqrt {a+b \sqrt {\frac {c}{x}}} \left (\frac {c}{x}\right )^{3/2} x^3}{12 a c}-\frac {5 b^4 c^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{\sqrt {a}}\right )}{32 a^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 112, normalized size = 0.66 \begin {gather*} \frac {\sqrt {a+b \sqrt {\frac {c}{x}}} \left (48 a^3+8 a^2 b \sqrt {\frac {c}{x}}+15 b^3 \left (\frac {c}{x}\right )^{3/2}-\frac {10 a b^2 c}{x}\right ) x^2}{96 a^3}-\frac {5 b^4 c^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{\sqrt {a}}\right )}{32 a^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 211, normalized size = 1.25
method | result | size |
default | \(-\frac {\sqrt {a +b \sqrt {\frac {c}{x}}}\, \sqrt {x}\, \left (15 \ln \left (\frac {b \sqrt {\frac {c}{x}}\, \sqrt {x}+2 \sqrt {a x +b \sqrt {\frac {c}{x}}\, x}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) c^{2} a \,b^{4}-30 a^{\frac {3}{2}} \sqrt {a x +b \sqrt {\frac {c}{x}}\, x}\, \left (\frac {c}{x}\right )^{\frac {3}{2}} x^{\frac {3}{2}} b^{3}-60 c \,a^{\frac {5}{2}} \sqrt {a x +b \sqrt {\frac {c}{x}}\, x}\, \sqrt {x}\, b^{2}+80 a^{\frac {5}{2}} \left (a x +b \sqrt {\frac {c}{x}}\, x \right )^{\frac {3}{2}} \sqrt {\frac {c}{x}}\, \sqrt {x}\, b -96 \sqrt {x}\, \left (a x +b \sqrt {\frac {c}{x}}\, x \right )^{\frac {3}{2}} a^{\frac {7}{2}}\right )}{192 \sqrt {x \left (a +b \sqrt {\frac {c}{x}}\right )}\, a^{\frac {9}{2}}}\) | \(211\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 211, normalized size = 1.25 \begin {gather*} \frac {1}{192} \, {\left (\frac {15 \, b^{4} \log \left (\frac {\sqrt {b \sqrt {\frac {c}{x}} + a} - \sqrt {a}}{\sqrt {b \sqrt {\frac {c}{x}} + a} + \sqrt {a}}\right )}{a^{\frac {7}{2}}} + \frac {2 \, {\left (15 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {7}{2}} b^{4} - 55 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {5}{2}} a b^{4} + 73 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {3}{2}} a^{2} b^{4} + 15 \, \sqrt {b \sqrt {\frac {c}{x}} + a} a^{3} b^{4}\right )}}{{\left (b \sqrt {\frac {c}{x}} + a\right )}^{4} a^{3} - 4 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{3} a^{4} + 6 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{2} a^{5} - 4 \, {\left (b \sqrt {\frac {c}{x}} + a\right )} a^{6} + a^{7}}\right )} c^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 224, normalized size = 1.33 \begin {gather*} \left [\frac {15 \, \sqrt {a} b^{4} c^{2} \log \left (-2 \, \sqrt {b \sqrt {\frac {c}{x}} + a} \sqrt {a} x \sqrt {\frac {c}{x}} + 2 \, a x \sqrt {\frac {c}{x}} + b c\right ) - 2 \, {\left (10 \, a^{2} b^{2} c x - 48 \, a^{4} x^{2} - {\left (15 \, a b^{3} c x + 8 \, a^{3} b x^{2}\right )} \sqrt {\frac {c}{x}}\right )} \sqrt {b \sqrt {\frac {c}{x}} + a}}{192 \, a^{4}}, \frac {15 \, \sqrt {-a} b^{4} c^{2} \arctan \left (\frac {\sqrt {b \sqrt {\frac {c}{x}} + a} \sqrt {-a}}{a}\right ) - {\left (10 \, a^{2} b^{2} c x - 48 \, a^{4} x^{2} - {\left (15 \, a b^{3} c x + 8 \, a^{3} b x^{2}\right )} \sqrt {\frac {c}{x}}\right )} \sqrt {b \sqrt {\frac {c}{x}} + a}}{96 \, a^{4}}\right ] \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 x \sqrt {a + b \sqrt {\frac {c}{x}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 = -0.01 \begin {gather*} \int x\,\sqrt {a+b\,\sqrt {\frac {c}{x}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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