Optimal. Leaf size=57 \[ \log (x) \left (a^2+b^2 c\right )+4 a b \sqrt {c+d x}-4 a b \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )+b^2 d x \]
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Rubi [A] time = 0.07, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {371, 1398, 801, 635, 207, 260} \[ \log (x) \left (a^2+b^2 c\right )+4 a b \sqrt {c+d x}-4 a b \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )+b^2 d x \]
Antiderivative was successfully verified.
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Rule 207
Rule 260
Rule 371
Rule 635
Rule 801
Rule 1398
Rubi steps
\begin {align*} \int \frac {\left (a+b \sqrt {c+d x}\right )^2}{x} \, dx &=\operatorname {Subst}\left (\int \frac {\left (a+b \sqrt {x}\right )^2}{-c+x} \, dx,x,c+d x\right )\\ &=2 \operatorname {Subst}\left (\int \frac {x (a+b x)^2}{-c+x^2} \, dx,x,\sqrt {c+d x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (2 a b+b^2 x+\frac {2 a b c+\left (a^2+b^2 c\right ) x}{-c+x^2}\right ) \, dx,x,\sqrt {c+d x}\right )\\ &=b^2 d x+4 a b \sqrt {c+d x}+2 \operatorname {Subst}\left (\int \frac {2 a b c+\left (a^2+b^2 c\right ) x}{-c+x^2} \, dx,x,\sqrt {c+d x}\right )\\ &=b^2 d x+4 a b \sqrt {c+d x}+(4 a b c) \operatorname {Subst}\left (\int \frac {1}{-c+x^2} \, dx,x,\sqrt {c+d x}\right )+\left (2 \left (a^2+b^2 c\right )\right ) \operatorname {Subst}\left (\int \frac {x}{-c+x^2} \, dx,x,\sqrt {c+d x}\right )\\ &=b^2 d x+4 a b \sqrt {c+d x}-4 a b \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )+\left (a^2+b^2 c\right ) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.16, size = 79, normalized size = 1.39 \[ b \left (4 a \sqrt {c+d x}+b d x\right )+\left (a-b \sqrt {c}\right )^2 \log \left (\sqrt {c+d x}+\sqrt {c}\right )+\left (a+b \sqrt {c}\right )^2 \log \left (\sqrt {c}-\sqrt {c+d x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 118, normalized size = 2.07 \[ \left [b^{2} d x + 2 \, a b \sqrt {c} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 4 \, \sqrt {d x + c} a b + {\left (b^{2} c + a^{2}\right )} \log \relax (x), b^{2} d x + 4 \, a b \sqrt {-c} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + 4 \, \sqrt {d x + c} a b + {\left (b^{2} c + a^{2}\right )} \log \relax (x)\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 59, normalized size = 1.04 \[ \frac {4 \, a b c \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} + {\left (d x + c\right )} b^{2} + 4 \, \sqrt {d x + c} a b + {\left (b^{2} c + a^{2}\right )} \log \left (d x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 51, normalized size = 0.89 \[ b^{2} c \ln \relax (x )+b^{2} d x -4 a b \sqrt {c}\, \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )+a^{2} \ln \relax (x )+4 \sqrt {d x +c}\, a b \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.93, size = 70, normalized size = 1.23 \[ 2 \, a b \sqrt {c} \log \left (\frac {\sqrt {d x + c} - \sqrt {c}}{\sqrt {d x + c} + \sqrt {c}}\right ) + {\left (d x + c\right )} b^{2} + 4 \, \sqrt {d x + c} a b + {\left (b^{2} c + a^{2}\right )} \log \left (d x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 130, normalized size = 2.28 \[ \ln \left (\left (2\,a^2+2\,c\,b^2\right )\,\sqrt {c+d\,x}-2\,{\left (a+b\,\sqrt {c}\right )}^2\,\sqrt {c+d\,x}+4\,a\,b\,c\right )\,{\left (a+b\,\sqrt {c}\right )}^2+\ln \left (\left (2\,a^2+2\,c\,b^2\right )\,\sqrt {c+d\,x}-2\,{\left (a-b\,\sqrt {c}\right )}^2\,\sqrt {c+d\,x}+4\,a\,b\,c\right )\,{\left (a-b\,\sqrt {c}\right )}^2+4\,a\,b\,\sqrt {c+d\,x}+b^2\,d\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 27.29, size = 65, normalized size = 1.14 \[ a^{2} \log {\relax (x )} - 2 a b \left (- \frac {2 c \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c}} \right )}}{\sqrt {- c}} - 2 \sqrt {c + d x}\right ) + b^{2} c \log {\relax (x )} + b^{2} d x \]
Verification of antiderivative is not currently implemented for this CAS.
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