Optimal. Leaf size=230 \[ \frac {x^{m+1} \sqrt {\frac {2 c \sqrt {\frac {d}{x}}}{\sqrt {d} \left (b \sqrt {d}-\sqrt {b^2 d-4 a c}\right )}+1} \sqrt {\frac {2 c \sqrt {\frac {d}{x}}}{\sqrt {d} \left (\sqrt {b^2 d-4 a c}+b \sqrt {d}\right )}+1} F_1\left (-2 (m+1);\frac {1}{2},\frac {1}{2};-2 m-1;-\frac {2 c \sqrt {\frac {d}{x}}}{\sqrt {d} \left (b \sqrt {d}-\sqrt {b^2 d-4 a c}\right )},-\frac {2 c \sqrt {\frac {d}{x}}}{\sqrt {d} \left (\sqrt {d} b+\sqrt {b^2 d-4 a c}\right )}\right )}{(m+1) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \]
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Rubi [A] time = 0.47, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1971, 1379, 759, 133} \[ \frac {x^{m+1} \sqrt {\frac {2 c \sqrt {\frac {d}{x}}}{\sqrt {d} \left (b \sqrt {d}-\sqrt {b^2 d-4 a c}\right )}+1} \sqrt {\frac {2 c \sqrt {\frac {d}{x}}}{\sqrt {d} \left (\sqrt {b^2 d-4 a c}+b \sqrt {d}\right )}+1} F_1\left (-2 (m+1);\frac {1}{2},\frac {1}{2};-2 m-1;-\frac {2 c \sqrt {\frac {d}{x}}}{\sqrt {d} \left (b \sqrt {d}-\sqrt {b^2 d-4 a c}\right )},-\frac {2 c \sqrt {\frac {d}{x}}}{\sqrt {d} \left (\sqrt {d} b+\sqrt {b^2 d-4 a c}\right )}\right )}{(m+1) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \]
Antiderivative was successfully verified.
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Rule 133
Rule 759
Rule 1379
Rule 1971
Rubi steps
\begin {align*} \int \frac {x^m}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx &=-\left (\left (d \left (\frac {d}{x}\right )^m x^m\right ) \operatorname {Subst}\left (\int \frac {x^{-2-m}}{\sqrt {a+b \sqrt {x}+\frac {c x}{d}}} \, dx,x,\frac {d}{x}\right )\right )\\ &=-\left (\left (2 d \left (\frac {d}{x}\right )^m x^m\right ) \operatorname {Subst}\left (\int \frac {x^{-1+2 (-1-m)}}{\sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )\right )\\ &=-\frac {\left (2 d \sqrt {1+\frac {2 c \sqrt {\frac {d}{x}}}{d \left (b-\frac {\sqrt {-4 a c+b^2 d}}{\sqrt {d}}\right )}} \sqrt {1+\frac {2 c \sqrt {\frac {d}{x}}}{d \left (b+\frac {\sqrt {-4 a c+b^2 d}}{\sqrt {d}}\right )}} \left (\frac {d}{x}\right )^m x^m\right ) \operatorname {Subst}\left (\int \frac {x^{-1+2 (-1-m)}}{\sqrt {1+\frac {2 c x}{\sqrt {d} \left (b \sqrt {d}-\sqrt {-4 a c+b^2 d}\right )}} \sqrt {1+\frac {2 c x}{\sqrt {d} \left (b \sqrt {d}+\sqrt {-4 a c+b^2 d}\right )}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\\ &=\frac {\sqrt {1+\frac {2 c \sqrt {\frac {d}{x}}}{\sqrt {d} \left (b \sqrt {d}-\sqrt {-4 a c+b^2 d}\right )}} \sqrt {1+\frac {2 c \sqrt {\frac {d}{x}}}{\sqrt {d} \left (b \sqrt {d}+\sqrt {-4 a c+b^2 d}\right )}} x^{1+m} F_1\left (-2 (1+m);\frac {1}{2},\frac {1}{2};-1-2 m;-\frac {2 c \sqrt {\frac {d}{x}}}{\sqrt {d} \left (b \sqrt {d}-\sqrt {-4 a c+b^2 d}\right )},-\frac {2 c \sqrt {\frac {d}{x}}}{\sqrt {d} \left (b \sqrt {d}+\sqrt {-4 a c+b^2 d}\right )}\right )}{(1+m) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\\ \end {align*}
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Mathematica [F] time = 0.26, size = 0, normalized size = 0.00 \[ \int \frac {x^m}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx \]
Verification is Not applicable to the result.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\sqrt {a +\sqrt {\frac {d}{x}}\, b +\frac {c}{x}}}\, dx \]
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 \[ \int \frac {x^{m}}{\sqrt {b \sqrt {\frac {d}{x}} + a + \frac {c}{x}}}\,{d x} \]
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 \[ \int \frac {x^m}{\sqrt {a+\frac {c}{x}+b\,\sqrt {\frac {d}{x}}}} \,d x \]
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 \[ \int \frac {x^{m}}{\sqrt {a + b \sqrt {\frac {d}{x}} + \frac {c}{x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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