Optimal. Leaf size=137 \[ -\frac {\sqrt {a+b \sqrt {c+d x}}}{x}+\frac {b d \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )}{2 \sqrt {c} \sqrt {a-b \sqrt {c}}}-\frac {b d \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )}{2 \sqrt {c} \sqrt {a+b \sqrt {c}}} \]
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Rubi [A] time = 0.17, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {371, 1398, 821, 12, 708, 1093, 207} \[ -\frac {\sqrt {a+b \sqrt {c+d x}}}{x}+\frac {b d \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )}{2 \sqrt {c} \sqrt {a-b \sqrt {c}}}-\frac {b d \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )}{2 \sqrt {c} \sqrt {a+b \sqrt {c}}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 207
Rule 371
Rule 708
Rule 821
Rule 1093
Rule 1398
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \sqrt {c+d x}}}{x^2} \, dx &=d \operatorname {Subst}\left (\int \frac {\sqrt {a+b \sqrt {x}}}{(-c+x)^2} \, dx,x,c+d x\right )\\ &=(2 d) \operatorname {Subst}\left (\int \frac {x \sqrt {a+b x}}{\left (-c+x^2\right )^2} \, dx,x,\sqrt {c+d x}\right )\\ &=-\frac {\sqrt {a+b \sqrt {c+d x}}}{x}-\frac {d \operatorname {Subst}\left (\int -\frac {b c}{2 \sqrt {a+b x} \left (-c+x^2\right )} \, dx,x,\sqrt {c+d x}\right )}{c}\\ &=-\frac {\sqrt {a+b \sqrt {c+d x}}}{x}+\frac {1}{2} (b d) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x} \left (-c+x^2\right )} \, dx,x,\sqrt {c+d x}\right )\\ &=-\frac {\sqrt {a+b \sqrt {c+d x}}}{x}+\left (b^2 d\right ) \operatorname {Subst}\left (\int \frac {1}{a^2-b^2 c-2 a x^2+x^4} \, dx,x,\sqrt {a+b \sqrt {c+d x}}\right )\\ &=-\frac {\sqrt {a+b \sqrt {c+d x}}}{x}+\frac {(b d) \operatorname {Subst}\left (\int \frac {1}{-a-b \sqrt {c}+x^2} \, dx,x,\sqrt {a+b \sqrt {c+d x}}\right )}{2 \sqrt {c}}-\frac {(b d) \operatorname {Subst}\left (\int \frac {1}{-a+b \sqrt {c}+x^2} \, dx,x,\sqrt {a+b \sqrt {c+d x}}\right )}{2 \sqrt {c}}\\ &=-\frac {\sqrt {a+b \sqrt {c+d x}}}{x}+\frac {b d \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )}{2 \sqrt {a-b \sqrt {c}} \sqrt {c}}-\frac {b d \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )}{2 \sqrt {a+b \sqrt {c}} \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 181, normalized size = 1.32 \[ \frac {\left (a-b \sqrt {c}\right ) \left (2 \sqrt {c} \left (a+b \sqrt {c}\right ) \sqrt {a+b \sqrt {c+d x}}+b d x \sqrt {a+b \sqrt {c}} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )\right )-b d x \sqrt {a-b \sqrt {c}} \left (a+b \sqrt {c}\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )}{2 \sqrt {c} x \left (b^2 c-a^2\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.60, size = 1003, normalized size = 7.32 \[ -\frac {x \sqrt {-\frac {a b^{2} d^{2} + \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (b^{2} c^{2} - a^{2} c\right )}}{b^{2} c^{2} - a^{2} c}} \log \left (\sqrt {\sqrt {d x + c} b + a} b^{4} d^{3} + {\left (b^{4} c d^{2} - \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (a b^{2} c^{2} - a^{3} c\right )}\right )} \sqrt {-\frac {a b^{2} d^{2} + \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (b^{2} c^{2} - a^{2} c\right )}}{b^{2} c^{2} - a^{2} c}}\right ) - x \sqrt {-\frac {a b^{2} d^{2} + \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (b^{2} c^{2} - a^{2} c\right )}}{b^{2} c^{2} - a^{2} c}} \log \left (\sqrt {\sqrt {d x + c} b + a} b^{4} d^{3} - {\left (b^{4} c d^{2} - \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (a b^{2} c^{2} - a^{3} c\right )}\right )} \sqrt {-\frac {a b^{2} d^{2} + \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (b^{2} c^{2} - a^{2} c\right )}}{b^{2} c^{2} - a^{2} c}}\right ) + x \sqrt {-\frac {a b^{2} d^{2} - \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (b^{2} c^{2} - a^{2} c\right )}}{b^{2} c^{2} - a^{2} c}} \log \left (\sqrt {\sqrt {d x + c} b + a} b^{4} d^{3} + {\left (b^{4} c d^{2} + \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (a b^{2} c^{2} - a^{3} c\right )}\right )} \sqrt {-\frac {a b^{2} d^{2} - \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (b^{2} c^{2} - a^{2} c\right )}}{b^{2} c^{2} - a^{2} c}}\right ) - x \sqrt {-\frac {a b^{2} d^{2} - \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (b^{2} c^{2} - a^{2} c\right )}}{b^{2} c^{2} - a^{2} c}} \log \left (\sqrt {\sqrt {d x + c} b + a} b^{4} d^{3} - {\left (b^{4} c d^{2} + \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (a b^{2} c^{2} - a^{3} c\right )}\right )} \sqrt {-\frac {a b^{2} d^{2} - \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (b^{2} c^{2} - a^{2} c\right )}}{b^{2} c^{2} - a^{2} c}}\right ) + 4 \, \sqrt {\sqrt {d x + c} b + a}}{4 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.66, size = 232, normalized size = 1.69 \[ \frac {\frac {2 \, \sqrt {\sqrt {d x + c} b + a} b^{3} d^{2}}{b^{2} c - {\left (\sqrt {d x + c} b + a\right )}^{2} + 2 \, {\left (\sqrt {d x + c} b + a\right )} a - a^{2}} - \frac {{\left (b^{3} c d^{2} {\left | b \right |} + a b^{3} \sqrt {c} d^{2}\right )} \arctan \left (\frac {\sqrt {\sqrt {d x + c} b + a}}{\sqrt {-a + \sqrt {b^{2} c}}}\right )}{{\left (b c^{\frac {3}{2}} + a c\right )} \sqrt {b \sqrt {c} - a} {\left | b \right |}} + \frac {{\left (b^{3} c d^{2} {\left | b \right |} - a b^{3} \sqrt {c} d^{2}\right )} \arctan \left (\frac {\sqrt {\sqrt {d x + c} b + a}}{\sqrt {-a - \sqrt {b^{2} c}}}\right )}{{\left (b c^{\frac {3}{2}} - a c\right )} \sqrt {-b \sqrt {c} - a} {\left | b \right |}}}{2 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 151, normalized size = 1.10 \[ \frac {b^{2} d \arctan \left (\frac {\sqrt {a +\sqrt {d x +c}\, b}}{\sqrt {-a -\sqrt {b^{2} c}}}\right )}{2 \sqrt {b^{2} c}\, \sqrt {-a -\sqrt {b^{2} c}}}-\frac {b^{2} d \arctan \left (\frac {\sqrt {a +\sqrt {d x +c}\, b}}{\sqrt {-a +\sqrt {b^{2} c}}}\right )}{2 \sqrt {b^{2} c}\, \sqrt {-a +\sqrt {b^{2} c}}}-\frac {\sqrt {a +\sqrt {d x +c}\, b}\, b^{2} d}{-b^{2} c +\left (d x +c \right ) b^{2}} \]
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 \[ \int \frac {\sqrt {\sqrt {d x + c} b + a}}{x^{2}}\,{d x} \]
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 \[ \int \frac {\sqrt {a+b\,\sqrt {c+d\,x}}}{x^2} \,d x \]
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 \[ \int \frac {\sqrt {a + b \sqrt {c + d x}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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