Optimal. Leaf size=133 \[ \frac {2 b x}{(a-c)^2}-\frac {2 \sqrt {a+b x} \sqrt {b x+c}}{(a-c)^2}-\frac {2 (a+c) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {b x+c}}\right )}{(a-c)^2}+\frac {4 \sqrt {a} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {b x+c}}\right )}{(a-c)^2}+\frac {(a+c) \log (x)}{(a-c)^2} \]
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Rubi [A] time = 0.23, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {6689, 101, 157, 63, 217, 206, 93, 208} \[ \frac {2 b x}{(a-c)^2}-\frac {2 \sqrt {a+b x} \sqrt {b x+c}}{(a-c)^2}-\frac {2 (a+c) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {b x+c}}\right )}{(a-c)^2}+\frac {4 \sqrt {a} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {b x+c}}\right )}{(a-c)^2}+\frac {(a+c) \log (x)}{(a-c)^2} \]
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 101
Rule 157
Rule 206
Rule 208
Rule 217
Rule 6689
Rubi steps
\begin {align*} \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^2} \, dx &=\frac {\int \left (2 b+\frac {a \left (1+\frac {c}{a}\right )}{x}-\frac {2 \sqrt {a+b x} \sqrt {c+b x}}{x}\right ) \, dx}{(a-c)^2}\\ &=\frac {2 b x}{(a-c)^2}+\frac {(a+c) \log (x)}{(a-c)^2}-\frac {2 \int \frac {\sqrt {a+b x} \sqrt {c+b x}}{x} \, dx}{(a-c)^2}\\ &=\frac {2 b x}{(a-c)^2}-\frac {2 \sqrt {a+b x} \sqrt {c+b x}}{(a-c)^2}+\frac {(a+c) \log (x)}{(a-c)^2}+\frac {2 \int \frac {-a c-\frac {1}{2} b (a+c) x}{x \sqrt {a+b x} \sqrt {c+b x}} \, dx}{(a-c)^2}\\ &=\frac {2 b x}{(a-c)^2}-\frac {2 \sqrt {a+b x} \sqrt {c+b x}}{(a-c)^2}+\frac {(a+c) \log (x)}{(a-c)^2}-\frac {(2 a c) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+b x}} \, dx}{(a-c)^2}-\frac {(b (a+c)) \int \frac {1}{\sqrt {a+b x} \sqrt {c+b x}} \, dx}{(a-c)^2}\\ &=\frac {2 b x}{(a-c)^2}-\frac {2 \sqrt {a+b x} \sqrt {c+b x}}{(a-c)^2}+\frac {(a+c) \log (x)}{(a-c)^2}-\frac {(4 a c) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+b x}}\right )}{(a-c)^2}-\frac {(2 (a+c)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a+c+x^2}} \, dx,x,\sqrt {a+b x}\right )}{(a-c)^2}\\ &=\frac {2 b x}{(a-c)^2}-\frac {2 \sqrt {a+b x} \sqrt {c+b x}}{(a-c)^2}+\frac {4 \sqrt {a} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+b x}}\right )}{(a-c)^2}+\frac {(a+c) \log (x)}{(a-c)^2}-\frac {(2 (a+c)) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+b x}}\right )}{(a-c)^2}\\ &=\frac {2 b x}{(a-c)^2}-\frac {2 \sqrt {a+b x} \sqrt {c+b x}}{(a-c)^2}-\frac {2 (a+c) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {c+b x}}\right )}{(a-c)^2}+\frac {4 \sqrt {a} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+b x}}\right )}{(a-c)^2}+\frac {(a+c) \log (x)}{(a-c)^2}\\ \end {align*}
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Mathematica [A] time = 1.07, size = 195, normalized size = 1.47 \[ \frac {\sqrt {b} \left (-2 \left (c \sqrt {a+b x}+b x \left (\sqrt {a+b x}-\sqrt {b x+c}\right )\right )+(a+c) \log (x) \sqrt {b x+c}+4 \sqrt {a} \sqrt {c} \sqrt {b x+c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {b x+c}}\right )\right )-2 (a+c) \sqrt {b (c-a)} \sqrt {-\frac {b x+c}{a-c}} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {a+b x}}{\sqrt {b (c-a)}}\right )}{\sqrt {b} (a-c)^2 \sqrt {b x+c}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 290, normalized size = 2.18 \[ \left [\frac {2 \, b x + {\left (a + c\right )} \log \left (-2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b x + c} - a - c\right ) + {\left (a + c\right )} \log \relax (x) + 2 \, \sqrt {a c} \log \left (\frac {2 \, a^{2} c + 2 \, a c^{2} + 2 \, {\left (2 \, a c + \sqrt {a c} {\left (a + c\right )}\right )} \sqrt {b x + a} \sqrt {b x + c} + {\left (a^{2} b + 2 \, a b c + b c^{2}\right )} x + 2 \, {\left (2 \, a c + {\left (a b + b c\right )} x\right )} \sqrt {a c}}{x}\right ) - 2 \, \sqrt {b x + a} \sqrt {b x + c}}{a^{2} - 2 \, a c + c^{2}}, \frac {2 \, b x + {\left (a + c\right )} \log \left (-2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b x + c} - a - c\right ) + {\left (a + c\right )} \log \relax (x) - 4 \, \sqrt {-a c} \arctan \left (-\frac {\sqrt {-a c} b x - \sqrt {-a c} \sqrt {b x + a} \sqrt {b x + c}}{a c}\right ) - 2 \, \sqrt {b x + a} \sqrt {b x + c}}{a^{2} - 2 \, a c + c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.67, size = 194, normalized size = 1.46 \[ \frac {4 \, a c \arctan \left (\frac {{\left (\sqrt {b x + a} - \sqrt {b x + c}\right )}^{2} - a - c}{2 \, \sqrt {-a c}}\right )}{{\left (a^{2} - 2 \, a c + c^{2}\right )} \sqrt {-a c}} - \frac {2 \, {\left (a^{2} - 2 \, a c + c^{2}\right )} \sqrt {b x + a} \sqrt {b x + c}}{a^{4} - 4 \, a^{3} c + 6 \, a^{2} c^{2} - 4 \, a c^{3} + c^{4}} + \frac {{\left (a + c\right )} \log \left ({\left (\sqrt {b x + a} - \sqrt {b x + c}\right )}^{2}\right )}{a^{2} - 2 \, a c + c^{2}} + \frac {{\left (a + c\right )} \log \left ({\left | b x \right |}\right )}{a^{2} - 2 \, a c + c^{2}} + \frac {2 \, {\left (b x + a\right )}}{a^{2} - 2 \, a c + c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 258, normalized size = 1.94 \[ \frac {a \ln \relax (x )}{\left (a -c \right )^{2}}+\frac {2 b x}{\left (a -c \right )^{2}}+\frac {c \ln \relax (x )}{\left (a -c \right )^{2}}+\frac {\sqrt {b x +a}\, \sqrt {b x +c}\, \left (2 a c \,\mathrm {csgn}\relax (b ) \ln \left (\frac {a b x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b^{2} x^{2}+a b x +b c x +a c}}{x}\right )-\sqrt {a c}\, a \ln \left (\frac {\left (2 b x +a +c +2 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, \mathrm {csgn}\relax (b )\right ) \mathrm {csgn}\relax (b )}{2}\right )-\sqrt {a c}\, c \ln \left (\frac {\left (2 b x +a +c +2 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, \mathrm {csgn}\relax (b )\right ) \mathrm {csgn}\relax (b )}{2}\right )-2 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, \sqrt {a c}\, \mathrm {csgn}\relax (b )\right ) \mathrm {csgn}\relax (b )}{\left (a -c \right )^{2} \sqrt {a c}\, \sqrt {b^{2} x^{2}+a b x +b c x +a c}} \]
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 \[ \int \frac {1}{x {\left (\sqrt {b x + a} + \sqrt {b x + c}\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.14, size = 524, normalized size = 3.94 \[ \frac {2\,b\,x}{{\left (a-c\right )}^2}-\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {c+b\,x}-\sqrt {c}}+1\right )\,\left (\frac {4\,c}{{\left (a-c\right )}^2}+\frac {2}{a-c}\right )-\frac {\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3\,\left (4\,a+4\,c\right )}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^3\,\left (a^2-2\,a\,c+c^2\right )}+\frac {\left (\sqrt {a+b\,x}-\sqrt {a}\right )\,\left (4\,a+4\,c\right )}{\left (\sqrt {c+b\,x}-\sqrt {c}\right )\,\left (a^2-2\,a\,c+c^2\right )}-\frac {16\,\sqrt {a}\,\sqrt {c}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^2\,\left (a^2-2\,a\,c+c^2\right )}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^4}-\frac {2\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^2}+1}+\frac {2\,\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {c+b\,x}-\sqrt {c}}-1\right )\,\left (a+c\right )}{{\left (a-c\right )}^2}+\frac {\ln \relax (x)\,\left (a+c\right )}{a^2-2\,a\,c+c^2}+\frac {2\,\sqrt {a}\,\sqrt {c}\,\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {c+b\,x}-\sqrt {c}}\right )}{{\left (a-c\right )}^2}-\frac {2\,\sqrt {a}\,\sqrt {c}\,\ln \left (\frac {a\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {c+b\,x}-\sqrt {c}}-\sqrt {a}\,\sqrt {c}+\frac {c\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {c+b\,x}-\sqrt {c}}-\frac {\sqrt {a}\,\sqrt {c}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^2}\right )}{a^2-2\,a\,c+c^2} \]
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 \[ \int \frac {1}{x \left (\sqrt {a + b x} + \sqrt {b x + c}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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