Optimal. Leaf size=82 \[ \frac {2 b \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^2-b^2 c}+\frac {a \log (x)}{a^2-b^2 c}-\frac {2 a \log \left (a+b \sqrt {c+d x}\right )}{a^2-b^2 c} \]
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Rubi [A]
time = 0.06, antiderivative size = 82, 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 = {378, 1412, 815,
649, 212, 266} \begin {gather*} -\frac {2 a \log \left (a+b \sqrt {c+d x}\right )}{a^2-b^2 c}+\frac {2 b \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^2-b^2 c}+\frac {a \log (x)}{a^2-b^2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 266
Rule 378
Rule 649
Rule 815
Rule 1412
Rubi steps
\begin {align*} \int \frac {1}{x \left (a+b \sqrt {c+d x}\right )} \, dx &=\text {Subst}\left (\int \frac {1}{\left (a+b \sqrt {x}\right ) (-c+x)} \, dx,x,c+d x\right )\\ &=2 \text {Subst}\left (\int \frac {x}{(a+b x) \left (-c+x^2\right )} \, dx,x,\sqrt {c+d x}\right )\\ &=2 \text {Subst}\left (\int \left (-\frac {a b}{\left (a^2-b^2 c\right ) (a+b x)}+\frac {b c-a x}{\left (a^2-b^2 c\right ) \left (c-x^2\right )}\right ) \, dx,x,\sqrt {c+d x}\right )\\ &=-\frac {2 a \log \left (a+b \sqrt {c+d x}\right )}{a^2-b^2 c}+\frac {2 \text {Subst}\left (\int \frac {b c-a x}{c-x^2} \, dx,x,\sqrt {c+d x}\right )}{a^2-b^2 c}\\ &=-\frac {2 a \log \left (a+b \sqrt {c+d x}\right )}{a^2-b^2 c}-\frac {(2 a) \text {Subst}\left (\int \frac {x}{c-x^2} \, dx,x,\sqrt {c+d x}\right )}{a^2-b^2 c}+\frac {(2 b c) \text {Subst}\left (\int \frac {1}{c-x^2} \, dx,x,\sqrt {c+d x}\right )}{a^2-b^2 c}\\ &=\frac {2 b \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^2-b^2 c}+\frac {a \log (x)}{a^2-b^2 c}-\frac {2 a \log \left (a+b \sqrt {c+d x}\right )}{a^2-b^2 c}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 62, normalized size = 0.76 \begin {gather*} \frac {2 b \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )+a \log (-d x)-2 a \log \left (a+b \sqrt {c+d x}\right )}{a^2-b^2 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 69, normalized size = 0.84
method | result | size |
derivativedivides | \(-\frac {2 a \ln \left (a +b \sqrt {d x +c}\right )}{-b^{2} c +a^{2}}+\frac {a \ln \left (-d x \right )+2 b \sqrt {c}\, \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{-b^{2} c +a^{2}}\) | \(69\) |
default | \(-\frac {2 a \ln \left (a +b \sqrt {d x +c}\right )}{-b^{2} c +a^{2}}+\frac {a \ln \left (-d x \right )+2 b \sqrt {c}\, \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{-b^{2} c +a^{2}}\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 95, normalized size = 1.16 \begin {gather*} \frac {b \sqrt {c} \log \left (\frac {\sqrt {d x + c} - \sqrt {c}}{\sqrt {d x + c} + \sqrt {c}}\right )}{b^{2} c - a^{2}} - \frac {a \log \left (d x\right )}{b^{2} c - a^{2}} + \frac {2 \, a \log \left (\sqrt {d x + c} b + a\right )}{b^{2} c - a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 125, normalized size = 1.52 \begin {gather*} \left [\frac {b \sqrt {c} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, a \log \left (\sqrt {d x + c} b + a\right ) - a \log \left (x\right )}{b^{2} c - a^{2}}, \frac {2 \, b \sqrt {-c} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + 2 \, a \log \left (\sqrt {d x + c} b + a\right ) - a \log \left (x\right )}{b^{2} c - a^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 5.89, size = 85, normalized size = 1.04 \begin {gather*} - \frac {2 a b \left (\begin {cases} \frac {\sqrt {c + d x}}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + b \sqrt {c + d x} \right )}}{b} & \text {otherwise} \end {cases}\right )}{a^{2} - b^{2} c} - \frac {2 \left (- \frac {a \log {\left (- d x \right )}}{2} + \frac {b c \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c}} \right )}}{\sqrt {- c}}\right )}{a^{2} - b^{2} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.83, size = 88, normalized size = 1.07 \begin {gather*} \frac {2 \, a b \log \left ({\left | \sqrt {d x + c} b + a \right |}\right )}{b^{3} c - a^{2} b} + \frac {2 \, b c \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{{\left (b^{2} c - a^{2}\right )} \sqrt {-c}} - \frac {a \log \left (d x\right )}{b^{2} c - a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.28, size = 181, normalized size = 2.21 \begin {gather*} \frac {\ln \left (2\,b^3\,c^{3/2}-2\,b^3\,c\,\sqrt {c+d\,x}-6\,a\,b^2\,c+6\,a\,b^2\,\sqrt {c}\,\sqrt {c+d\,x}\right )}{a+b\,\sqrt {c}}+\frac {\ln \left (-2\,b^3\,c^{3/2}-2\,b^3\,c\,\sqrt {c+d\,x}-6\,a\,b^2\,c-6\,a\,b^2\,\sqrt {c}\,\sqrt {c+d\,x}\right )}{a-b\,\sqrt {c}}+\frac {2\,a\,\ln \left (4\,b^5\,c^2\,\sqrt {c+d\,x}-36\,a^3\,b^2\,c+4\,a\,b^4\,c^2-36\,a^2\,b^3\,c\,\sqrt {c+d\,x}\right )}{b^2\,c-a^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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