Optimal. Leaf size=87 \[ -\frac {2 b (d x)^{m+1} \left (a+\frac {b}{\sqrt {c x^2}}\right )^{3/2} \left (-\frac {b}{a \sqrt {c x^2}}\right )^m \, _2F_1\left (\frac {3}{2},m+2;\frac {5}{2};\frac {b}{a \sqrt {c x^2}}+1\right )}{3 a^2 d \sqrt {c x^2}} \]
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Rubi [A] time = 0.07, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {368, 339, 67, 65} \[ -\frac {2 b (d x)^{m+1} \left (a+\frac {b}{\sqrt {c x^2}}\right )^{3/2} \left (-\frac {b}{a \sqrt {c x^2}}\right )^m \, _2F_1\left (\frac {3}{2},m+2;\frac {5}{2};\frac {b}{a \sqrt {c x^2}}+1\right )}{3 a^2 d \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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Rule 65
Rule 67
Rule 339
Rule 368
Rubi steps
\begin {align*} \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^2}}} \, dx &=\frac {\left ((d x)^{1+m} \left (c x^2\right )^{\frac {1}{2} (-1-m)}\right ) \operatorname {Subst}\left (\int \sqrt {a+\frac {b}{x}} x^m \, dx,x,\sqrt {c x^2}\right )}{d}\\ &=-\frac {\left ((d x)^{1+m} \left (c x^2\right )^{\frac {1}{2} (-1-m)}\right ) \operatorname {Subst}\left (\int x^{-2-m} \sqrt {a+b x} \, dx,x,\frac {1}{\sqrt {c x^2}}\right )}{d}\\ &=-\frac {\left (b^2 (d x)^{1+m} \left (c x^2\right )^{\frac {1}{2} (-1-m)+\frac {m}{2}} \left (-\frac {b}{a \sqrt {c x^2}}\right )^m\right ) \operatorname {Subst}\left (\int \left (-\frac {b x}{a}\right )^{-2-m} \sqrt {a+b x} \, dx,x,\frac {1}{\sqrt {c x^2}}\right )}{a^2 d}\\ &=-\frac {2 b (d x)^{1+m} \left (-\frac {b}{a \sqrt {c x^2}}\right )^m \left (a+\frac {b}{\sqrt {c x^2}}\right )^{3/2} \, _2F_1\left (\frac {3}{2},2+m;\frac {5}{2};1+\frac {b}{a \sqrt {c x^2}}\right )}{3 a^2 d \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 81, normalized size = 0.93 \[ \frac {2 x (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^2}}} \, _2F_1\left (-\frac {1}{2},m+\frac {1}{2};m+\frac {3}{2};-\frac {a \sqrt {c x^2}}{b}\right )}{(2 m+1) \sqrt {\frac {a \sqrt {c x^2}}{b}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.21, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (d x\right )^{m} \sqrt {\frac {a c x^{2} + \sqrt {c x^{2}} b}{c x^{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d x\right )^{m} \sqrt {a + \frac {b}{\sqrt {c x^{2}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.23, size = 0, normalized size = 0.00 \[ \int \sqrt {a +\frac {b}{\sqrt {c \,x^{2}}}}\, \left (d x \right )^{m}\, dx \]
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 \[ \int \left (d x\right )^{m} \sqrt {a + \frac {b}{\sqrt {c x^{2}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (d\,x\right )}^m\,\sqrt {a+\frac {b}{\sqrt {c\,x^2}}} \,d x \]
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 \[ \int \left (d x\right )^{m} \sqrt {a + \frac {b}{\sqrt {c x^{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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