Optimal. Leaf size=247 \[ \frac {\sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (a+b x)}{a \sqrt {-c}-b \sqrt {d}}\right )}{2 (-c)^{3/2}}-\frac {\sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (a+b x)}{\sqrt {-c} a+b \sqrt {d}}\right )}{2 (-c)^{3/2}}-\frac {\sqrt {d} \log (a+b x) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{a \sqrt {-c}+b \sqrt {d}}\right )}{2 (-c)^{3/2}}+\frac {\sqrt {d} \log (a+b x) \log \left (-\frac {b \left (\sqrt {-c} x+\sqrt {d}\right )}{a \sqrt {-c}-b \sqrt {d}}\right )}{2 (-c)^{3/2}}+\frac {(a+b x) \log (a+b x)}{b c}-\frac {x}{c} \]
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Rubi [A] time = 0.34, antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2409, 2389, 2295, 2394, 2393, 2391} \[ \frac {\sqrt {d} \text {PolyLog}\left (2,\frac {\sqrt {-c} (a+b x)}{a \sqrt {-c}-b \sqrt {d}}\right )}{2 (-c)^{3/2}}-\frac {\sqrt {d} \text {PolyLog}\left (2,\frac {\sqrt {-c} (a+b x)}{a \sqrt {-c}+b \sqrt {d}}\right )}{2 (-c)^{3/2}}-\frac {\sqrt {d} \log (a+b x) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{a \sqrt {-c}+b \sqrt {d}}\right )}{2 (-c)^{3/2}}+\frac {\sqrt {d} \log (a+b x) \log \left (-\frac {b \left (\sqrt {-c} x+\sqrt {d}\right )}{a \sqrt {-c}-b \sqrt {d}}\right )}{2 (-c)^{3/2}}+\frac {(a+b x) \log (a+b x)}{b c}-\frac {x}{c} \]
Antiderivative was successfully verified.
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Rule 2295
Rule 2389
Rule 2391
Rule 2393
Rule 2394
Rule 2409
Rubi steps
\begin {align*} \int \frac {\log (a+b x)}{c+\frac {d}{x^2}} \, dx &=\int \left (\frac {\log (a+b x)}{c}-\frac {d \log (a+b x)}{c \left (d+c x^2\right )}\right ) \, dx\\ &=\frac {\int \log (a+b x) \, dx}{c}-\frac {d \int \frac {\log (a+b x)}{d+c x^2} \, dx}{c}\\ &=\frac {\operatorname {Subst}(\int \log (x) \, dx,x,a+b x)}{b c}-\frac {d \int \left (\frac {\log (a+b x)}{2 \sqrt {d} \left (\sqrt {d}-\sqrt {-c} x\right )}+\frac {\log (a+b x)}{2 \sqrt {d} \left (\sqrt {d}+\sqrt {-c} x\right )}\right ) \, dx}{c}\\ &=-\frac {x}{c}+\frac {(a+b x) \log (a+b x)}{b c}-\frac {\sqrt {d} \int \frac {\log (a+b x)}{\sqrt {d}-\sqrt {-c} x} \, dx}{2 c}-\frac {\sqrt {d} \int \frac {\log (a+b x)}{\sqrt {d}+\sqrt {-c} x} \, dx}{2 c}\\ &=-\frac {x}{c}+\frac {(a+b x) \log (a+b x)}{b c}-\frac {\sqrt {d} \log (a+b x) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{a \sqrt {-c}+b \sqrt {d}}\right )}{2 (-c)^{3/2}}+\frac {\sqrt {d} \log (a+b x) \log \left (-\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{a \sqrt {-c}-b \sqrt {d}}\right )}{2 (-c)^{3/2}}+\frac {\left (b \sqrt {d}\right ) \int \frac {\log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{a \sqrt {-c}+b \sqrt {d}}\right )}{a+b x} \, dx}{2 (-c)^{3/2}}-\frac {\left (b \sqrt {d}\right ) \int \frac {\log \left (\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{-a \sqrt {-c}+b \sqrt {d}}\right )}{a+b x} \, dx}{2 (-c)^{3/2}}\\ &=-\frac {x}{c}+\frac {(a+b x) \log (a+b x)}{b c}-\frac {\sqrt {d} \log (a+b x) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{a \sqrt {-c}+b \sqrt {d}}\right )}{2 (-c)^{3/2}}+\frac {\sqrt {d} \log (a+b x) \log \left (-\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{a \sqrt {-c}-b \sqrt {d}}\right )}{2 (-c)^{3/2}}-\frac {\sqrt {d} \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-c} x}{-a \sqrt {-c}+b \sqrt {d}}\right )}{x} \, dx,x,a+b x\right )}{2 (-c)^{3/2}}+\frac {\sqrt {d} \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-c} x}{a \sqrt {-c}+b \sqrt {d}}\right )}{x} \, dx,x,a+b x\right )}{2 (-c)^{3/2}}\\ &=-\frac {x}{c}+\frac {(a+b x) \log (a+b x)}{b c}-\frac {\sqrt {d} \log (a+b x) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{a \sqrt {-c}+b \sqrt {d}}\right )}{2 (-c)^{3/2}}+\frac {\sqrt {d} \log (a+b x) \log \left (-\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{a \sqrt {-c}-b \sqrt {d}}\right )}{2 (-c)^{3/2}}+\frac {\sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (a+b x)}{a \sqrt {-c}-b \sqrt {d}}\right )}{2 (-c)^{3/2}}-\frac {\sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (a+b x)}{a \sqrt {-c}+b \sqrt {d}}\right )}{2 (-c)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 247, normalized size = 1.00 \[ \frac {\sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (a+b x)}{a \sqrt {-c}-b \sqrt {d}}\right )}{2 (-c)^{3/2}}-\frac {\sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (a+b x)}{\sqrt {-c} a+b \sqrt {d}}\right )}{2 (-c)^{3/2}}-\frac {\sqrt {d} \log (a+b x) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{a \sqrt {-c}+b \sqrt {d}}\right )}{2 (-c)^{3/2}}+\frac {\sqrt {d} \log (a+b x) \log \left (-\frac {b \left (\sqrt {-c} x+\sqrt {d}\right )}{a \sqrt {-c}-b \sqrt {d}}\right )}{2 (-c)^{3/2}}+\frac {(a+b x) \log (a+b x)}{b c}-\frac {x}{c} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2} \log \left (b x + a\right )}{c x^{2} + d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (b x + a\right )}{c + \frac {d}{x^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 248, normalized size = 1.00 \[ -\frac {d \ln \left (\frac {a c +\sqrt {-c d}\, b -\left (b x +a \right ) c}{a c +\sqrt {-c d}\, b}\right ) \ln \left (b x +a \right )}{2 \sqrt {-c d}\, c}+\frac {d \ln \left (\frac {-a c +\sqrt {-c d}\, b +\left (b x +a \right ) c}{-a c +\sqrt {-c d}\, b}\right ) \ln \left (b x +a \right )}{2 \sqrt {-c d}\, c}-\frac {d \dilog \left (\frac {a c +\sqrt {-c d}\, b -\left (b x +a \right ) c}{a c +\sqrt {-c d}\, b}\right )}{2 \sqrt {-c d}\, c}+\frac {d \dilog \left (\frac {-a c +\sqrt {-c d}\, b +\left (b x +a \right ) c}{-a c +\sqrt {-c d}\, b}\right )}{2 \sqrt {-c d}\, c}+\frac {x \ln \left (b x +a \right )}{c}+\frac {a \ln \left (b x +a \right )}{b c}-\frac {x}{c}-\frac {a}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.24, size = 298, normalized size = 1.21 \[ -{\left (\frac {d \arctan \left (\frac {c x}{\sqrt {c d}}\right )}{\sqrt {c d} c} - \frac {x}{c}\right )} \log \left (b x + a\right ) - \frac {2 \, b c x - 2 \, a c \log \left (b x + a\right ) + {\left (b \arctan \left (\frac {{\left (b^{2} x + a b\right )} \sqrt {c} \sqrt {d}}{a^{2} c + b^{2} d}, \frac {a b c x + a^{2} c}{a^{2} c + b^{2} d}\right ) \log \left (c x^{2} + d\right ) - b \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \log \left (\frac {b^{2} c x^{2} + 2 \, a b c x + a^{2} c}{a^{2} c + b^{2} d}\right ) + i \, b {\rm Li}_2\left (-\frac {a b c x + b^{2} d + {\left (i \, b^{2} x - i \, a b\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 i \, a b \sqrt {c} \sqrt {d} - b^{2} d}\right ) - i \, b {\rm Li}_2\left (-\frac {a b c x + b^{2} d - {\left (i \, b^{2} x - i \, a b\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 i \, a b \sqrt {c} \sqrt {d} - b^{2} d}\right )\right )} \sqrt {c} \sqrt {d}}{2 \, b c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (a+b\,x\right )}{c+\frac {d}{x^2}} \,d x \]
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 \[ \int \frac {x^{2} \log {\left (a + b x \right )}}{c x^{2} + d}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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