Optimal. Leaf size=110 \[ \frac {2 \left (a^3 b^3+a b^3-2\right ) \tanh ^{-1}\left (\frac {a b^2+2 \sqrt {a+b x}}{\sqrt {a} \sqrt {a b^4+4}}\right )}{\sqrt {a} \sqrt {a b^4+4}}+\left (a^2 b+b\right ) \log \left (a b^2 \sqrt {a+b x}+b x\right )-\frac {2 a \sqrt {a+b x}}{b} \]
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Rubi [A] time = 0.56, antiderivative size = 104, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {1984, 1657, 634, 618, 206, 628} \begin {gather*} -\frac {2 \left (2-a \left (a^2+1\right ) b^3\right ) \tanh ^{-1}\left (\frac {a b^2+2 \sqrt {a+b x}}{\sqrt {a} \sqrt {a b^4+4}}\right )}{\sqrt {a} \sqrt {a b^4+4}}+\left (a^2+1\right ) b \log \left (a b \sqrt {a+b x}+x\right )-\frac {2 a \sqrt {a+b x}}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 628
Rule 634
Rule 1657
Rule 1984
Rubi steps
\begin {align*} \int \frac {1-a x+b \sqrt {a+b x}}{\sqrt {a+b x} \left (x+a b \sqrt {a+b x}\right )} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {a^2+b+b^2 x-a x^2}{x^2+a \left (-1+b^2 x\right )} \, dx,x,\sqrt {a+b x}\right )}{b}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {a^2+b+b^2 x-a x^2}{-a+a b^2 x+x^2} \, dx,x,\sqrt {a+b x}\right )}{b}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (-a+\frac {b+\left (1+a^2\right ) b^2 x}{-a+a b^2 x+x^2}\right ) \, dx,x,\sqrt {a+b x}\right )}{b}\\ &=-\frac {2 a \sqrt {a+b x}}{b}+\frac {2 \operatorname {Subst}\left (\int \frac {b+\left (1+a^2\right ) b^2 x}{-a+a b^2 x+x^2} \, dx,x,\sqrt {a+b x}\right )}{b}\\ &=-\frac {2 a \sqrt {a+b x}}{b}+\left (\left (1+a^2\right ) b\right ) \operatorname {Subst}\left (\int \frac {a b^2+2 x}{-a+a b^2 x+x^2} \, dx,x,\sqrt {a+b x}\right )+\left (2-a \left (1+a^2\right ) b^3\right ) \operatorname {Subst}\left (\int \frac {1}{-a+a b^2 x+x^2} \, dx,x,\sqrt {a+b x}\right )\\ &=-\frac {2 a \sqrt {a+b x}}{b}+\left (1+a^2\right ) b \log \left (x+a b \sqrt {a+b x}\right )-\left (2 \left (2-a \left (1+a^2\right ) b^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a \left (4+a b^4\right )-x^2} \, dx,x,a b^2+2 \sqrt {a+b x}\right )\\ &=-\frac {2 a \sqrt {a+b x}}{b}-\frac {2 \left (2-a \left (1+a^2\right ) b^3\right ) \tanh ^{-1}\left (\frac {a b^2+2 \sqrt {a+b x}}{\sqrt {a} \sqrt {4+a b^4}}\right )}{\sqrt {a} \sqrt {4+a b^4}}+\left (1+a^2\right ) b \log \left (x+a b \sqrt {a+b x}\right )\\ \end {align*}
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Mathematica [A] time = 0.21, size = 106, normalized size = 0.96 \begin {gather*} -\frac {2 \left (a^3 b^3+a b^3-2\right ) \tanh ^{-1}\left (\frac {-a b^2-2 \sqrt {a+b x}}{\sqrt {a} \sqrt {a b^4+4}}\right )}{\sqrt {a} \sqrt {a b^4+4}}+\left (a^2+1\right ) b \log \left (a b \sqrt {a+b x}+x\right )-\frac {2 a \sqrt {a+b x}}{b} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.16, size = 124, normalized size = 1.13 \begin {gather*} -\frac {2 a \sqrt {a+b x}}{b}+\frac {2 \left (-2+a b^3+a^3 b^3\right ) \tanh ^{-1}\left (\frac {\sqrt {a} b^2}{\sqrt {4+a b^4}}+\frac {2 \sqrt {a+b x}}{\sqrt {a} \sqrt {4+a b^4}}\right )}{\sqrt {a} \sqrt {4+a b^4}}+\left (b+a^2 b\right ) \log \left (b x+a b^2 \sqrt {a+b x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 392, normalized size = 3.56 \begin {gather*} \left [\frac {\sqrt {a^{2} b^{4} + 4 \, a} {\left ({\left (a^{3} + a\right )} b^{4} - 2 \, b\right )} \log \left (\frac {2 \, a^{3} b^{3} - 2 \, b x^{2} + {\left (a^{2} b^{4} - 4 \, a\right )} x + \sqrt {a^{2} b^{4} + 4 \, a} {\left (a b^{2} x + 2 \, a^{2} b\right )} + {\left (a^{3} b^{5} + 4 \, a^{2} b + \sqrt {a^{2} b^{4} + 4 \, a} {\left (a^{2} b^{3} - 2 \, x\right )}\right )} \sqrt {b x + a}}{a^{2} b^{3} x + a^{3} b^{2} - x^{2}}\right ) + {\left ({\left (a^{4} + a^{2}\right )} b^{6} + 4 \, {\left (a^{3} + a\right )} b^{2}\right )} \log \left (\sqrt {b x + a} a b + x\right ) - 2 \, {\left (a^{3} b^{4} + 4 \, a^{2}\right )} \sqrt {b x + a}}{a^{2} b^{5} + 4 \, a b}, -\frac {2 \, \sqrt {-a^{2} b^{4} - 4 \, a} {\left ({\left (a^{3} + a\right )} b^{4} - 2 \, b\right )} \arctan \left (\frac {\sqrt {-a^{2} b^{4} - 4 \, a} a b^{2} + 2 \, \sqrt {-a^{2} b^{4} - 4 \, a} \sqrt {b x + a}}{a^{2} b^{4} + 4 \, a}\right ) - {\left ({\left (a^{4} + a^{2}\right )} b^{6} + 4 \, {\left (a^{3} + a\right )} b^{2}\right )} \log \left (\sqrt {b x + a} a b + x\right ) + 2 \, {\left (a^{3} b^{4} + 4 \, a^{2}\right )} \sqrt {b x + a}}{a^{2} b^{5} + 4 \, a b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 100, normalized size = 0.91 \begin {gather*} {\left (a^{2} b + b\right )} \log \left (\sqrt {b x + a} a b^{2} + b x\right ) - \frac {2 \, {\left (a^{3} b^{3} + a b^{3} - 2\right )} \arctan \left (\frac {a b^{2} + 2 \, \sqrt {b x + a}}{\sqrt {-a^{2} b^{4} - 4 \, a}}\right )}{\sqrt {-a^{2} b^{4} - 4 \, a}} - \frac {2 \, \sqrt {b x + a} a}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 106, normalized size = 0.96
method | result | size |
derivativedivides | \(-\frac {2 \left (a \sqrt {b x +a}-b \left (\frac {\left (a^{2} b +b \right ) \ln \left (b x +a \,b^{2} \sqrt {b x +a}\right )}{2}-\frac {2 \left (1-\frac {\left (a^{2} b +b \right ) a \,b^{2}}{2}\right ) \arctanh \left (\frac {a \,b^{2}+2 \sqrt {b x +a}}{\sqrt {a^{2} b^{4}+4 a}}\right )}{\sqrt {a^{2} b^{4}+4 a}}\right )\right )}{b}\) | \(106\) |
default | \(-\frac {2 \left (a \sqrt {b x +a}-b \left (\frac {\left (a^{2} b +b \right ) \ln \left (b x +a \,b^{2} \sqrt {b x +a}\right )}{2}-\frac {2 \left (1-\frac {\left (a^{2} b +b \right ) a \,b^{2}}{2}\right ) \arctanh \left (\frac {a \,b^{2}+2 \sqrt {b x +a}}{\sqrt {a^{2} b^{4}+4 a}}\right )}{\sqrt {a^{2} b^{4}+4 a}}\right )\right )}{b}\) | \(106\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 125, normalized size = 1.14 \begin {gather*} \frac {{\left (a^{2} + 1\right )} b^{2} \log \left (\sqrt {b x + a} a b^{2} + b x\right ) - 2 \, \sqrt {b x + a} a - \frac {{\left ({\left (a^{3} + a\right )} b^{4} - 2 \, b\right )} \log \left (\frac {a b^{2} - \sqrt {{\left (a b^{4} + 4\right )} a} + 2 \, \sqrt {b x + a}}{a b^{2} + \sqrt {{\left (a b^{4} + 4\right )} a} + 2 \, \sqrt {b x + a}}\right )}{\sqrt {{\left (a b^{4} + 4\right )} a}}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 279, normalized size = 2.54 \begin {gather*} \frac {\ln \left (4\,a+2\,\sqrt {a\,\left (a\,b^4+4\right )}\,\sqrt {a+b\,x}+a^2\,b^4+a\,b^2\,\sqrt {a\,\left (a\,b^4+4\right )}\right )\,\left (a\,b\,\left (a\,b^4+4\right )-2\,\sqrt {a\,\left (a\,b^4+4\right )}+a^3\,b\,\left (a\,b^4+4\right )+a\,b^3\,\sqrt {a\,\left (a\,b^4+4\right )}+a^3\,b^3\,\sqrt {a\,\left (a\,b^4+4\right )}\right )}{a\,\left (a\,b^4+4\right )}-\frac {2\,a\,\sqrt {a+b\,x}}{b}+\frac {\ln \left (4\,a-2\,\sqrt {a\,\left (a\,b^4+4\right )}\,\sqrt {a+b\,x}+a^2\,b^4-a\,b^2\,\sqrt {a\,\left (a\,b^4+4\right )}\right )\,\left (2\,\sqrt {a\,\left (a\,b^4+4\right )}+a\,b\,\left (a\,b^4+4\right )+a^3\,b\,\left (a\,b^4+4\right )-a\,b^3\,\sqrt {a\,\left (a\,b^4+4\right )}-a^3\,b^3\,\sqrt {a\,\left (a\,b^4+4\right )}\right )}{a\,\left (a\,b^4+4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 92.60, size = 144, normalized size = 1.31 \begin {gather*} - \frac {2 a \sqrt {a + b x}}{b} - 2 b \left (a^{2} + 1\right ) \log {\left (\frac {1}{\sqrt {a + b x}} \right )} - \frac {\left (- a^{3} b - a b\right ) \log {\left (\frac {a b^{2}}{\sqrt {a + b x}} - \frac {a}{a + b x} + 1 \right )}}{a} - \frac {4 \left (a^{3} b^{3} + a b^{3} + \frac {b^{2} \left (- a^{3} b - a b\right )}{2} - 1\right ) \operatorname {atan}{\left (\frac {2 \left (- \frac {b^{2}}{2} + \frac {1}{\sqrt {a + b x}}\right )}{\sqrt {- \frac {a b^{4} + 4}{a}}} \right )}}{a \sqrt {- \frac {a b^{4} + 4}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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