Optimal. Leaf size=537 \[ -\frac{\sqrt{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 (-a)^{3/2}}-\frac{\sqrt{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt{-\sqrt{-a}} d+\sqrt [4]{b} c}\right )}{4 (-a)^{3/2}}+\frac{\sqrt{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}+\frac{\sqrt{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 (-a)^{3/2}}-\frac{\sqrt{b} \log (c+d x) \log \left (\frac{d \left (\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x\right )}{\sqrt{-\sqrt{-a}} d+\sqrt [4]{b} c}\right )}{4 (-a)^{3/2}}+\frac{\sqrt{b} \log (c+d x) \log \left (\frac{d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 (-a)^{3/2}}-\frac{\sqrt{b} \log (c+d x) \log \left (-\frac{d \left (\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 (-a)^{3/2}}+\frac{\sqrt{b} \log (c+d x) \log \left (-\frac{d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}-\frac{d^2 \log (x)}{2 a c^2}+\frac{d^2 \log (c+d x)}{2 a c^2}-\frac{\log (c+d x)}{2 a x^2}-\frac{d}{2 a c x} \]
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Rubi [A] time = 0.650977, antiderivative size = 537, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526, Rules used = {275, 325, 205, 2416, 2395, 44, 260, 2394, 2393, 2391} \[ -\frac{\sqrt{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 (-a)^{3/2}}-\frac{\sqrt{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt{-\sqrt{-a}} d+\sqrt [4]{b} c}\right )}{4 (-a)^{3/2}}+\frac{\sqrt{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}+\frac{\sqrt{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 (-a)^{3/2}}-\frac{\sqrt{b} \log (c+d x) \log \left (\frac{d \left (\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x\right )}{\sqrt{-\sqrt{-a}} d+\sqrt [4]{b} c}\right )}{4 (-a)^{3/2}}+\frac{\sqrt{b} \log (c+d x) \log \left (\frac{d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 (-a)^{3/2}}-\frac{\sqrt{b} \log (c+d x) \log \left (-\frac{d \left (\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 (-a)^{3/2}}+\frac{\sqrt{b} \log (c+d x) \log \left (-\frac{d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}-\frac{d^2 \log (x)}{2 a c^2}+\frac{d^2 \log (c+d x)}{2 a c^2}-\frac{\log (c+d x)}{2 a x^2}-\frac{d}{2 a c x} \]
Antiderivative was successfully verified.
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Rule 275
Rule 325
Rule 205
Rule 2416
Rule 2395
Rule 44
Rule 260
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{\log (c+d x)}{x^3 \left (a+b x^4\right )} \, dx &=\int \left (\frac{\log (c+d x)}{a x^3}-\frac{b x \log (c+d x)}{a \left (a+b x^4\right )}\right ) \, dx\\ &=\frac{\int \frac{\log (c+d x)}{x^3} \, dx}{a}-\frac{b \int \frac{x \log (c+d x)}{a+b x^4} \, dx}{a}\\ &=-\frac{\log (c+d x)}{2 a x^2}-\frac{b \int \left (-\frac{\sqrt{b} x \log (c+d x)}{2 \sqrt{-a} \left (\sqrt{-a} \sqrt{b}-b x^2\right )}-\frac{\sqrt{b} x \log (c+d x)}{2 \sqrt{-a} \left (\sqrt{-a} \sqrt{b}+b x^2\right )}\right ) \, dx}{a}+\frac{d \int \frac{1}{x^2 (c+d x)} \, dx}{2 a}\\ &=-\frac{\log (c+d x)}{2 a x^2}-\frac{b^{3/2} \int \frac{x \log (c+d x)}{\sqrt{-a} \sqrt{b}-b x^2} \, dx}{2 (-a)^{3/2}}-\frac{b^{3/2} \int \frac{x \log (c+d x)}{\sqrt{-a} \sqrt{b}+b x^2} \, dx}{2 (-a)^{3/2}}+\frac{d \int \left (\frac{1}{c x^2}-\frac{d}{c^2 x}+\frac{d^2}{c^2 (c+d x)}\right ) \, dx}{2 a}\\ &=-\frac{d}{2 a c x}-\frac{d^2 \log (x)}{2 a c^2}+\frac{d^2 \log (c+d x)}{2 a c^2}-\frac{\log (c+d x)}{2 a x^2}-\frac{b^{3/2} \int \left (-\frac{\log (c+d x)}{2 b^{3/4} \left (\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x\right )}+\frac{\log (c+d x)}{2 b^{3/4} \left (\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x\right )}\right ) \, dx}{2 (-a)^{3/2}}-\frac{b^{3/2} \int \left (\frac{\log (c+d x)}{2 b^{3/4} \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}-\frac{\log (c+d x)}{2 b^{3/4} \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}\right ) \, dx}{2 (-a)^{3/2}}\\ &=-\frac{d}{2 a c x}-\frac{d^2 \log (x)}{2 a c^2}+\frac{d^2 \log (c+d x)}{2 a c^2}-\frac{\log (c+d x)}{2 a x^2}+\frac{b^{3/4} \int \frac{\log (c+d x)}{\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x} \, dx}{4 (-a)^{3/2}}-\frac{b^{3/4} \int \frac{\log (c+d x)}{\sqrt [4]{-a}-\sqrt [4]{b} x} \, dx}{4 (-a)^{3/2}}-\frac{b^{3/4} \int \frac{\log (c+d x)}{\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x} \, dx}{4 (-a)^{3/2}}+\frac{b^{3/4} \int \frac{\log (c+d x)}{\sqrt [4]{-a}+\sqrt [4]{b} x} \, dx}{4 (-a)^{3/2}}\\ &=-\frac{d}{2 a c x}-\frac{d^2 \log (x)}{2 a c^2}+\frac{d^2 \log (c+d x)}{2 a c^2}-\frac{\log (c+d x)}{2 a x^2}-\frac{\sqrt{b} \log \left (\frac{d \left (\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d}\right ) \log (c+d x)}{4 (-a)^{3/2}}+\frac{\sqrt{b} \log \left (\frac{d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{3/2}}-\frac{\sqrt{b} \log \left (-\frac{d \left (\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right ) \log (c+d x)}{4 (-a)^{3/2}}+\frac{\sqrt{b} \log \left (-\frac{d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{3/2}}+\frac{\left (\sqrt{b} d\right ) \int \frac{\log \left (\frac{d \left (\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d}\right )}{c+d x} \, dx}{4 (-a)^{3/2}}-\frac{\left (\sqrt{b} d\right ) \int \frac{\log \left (\frac{d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{c+d x} \, dx}{4 (-a)^{3/2}}+\frac{\left (\sqrt{b} d\right ) \int \frac{\log \left (\frac{d \left (\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x\right )}{-\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d}\right )}{c+d x} \, dx}{4 (-a)^{3/2}}-\frac{\left (\sqrt{b} d\right ) \int \frac{\log \left (\frac{d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{-\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{c+d x} \, dx}{4 (-a)^{3/2}}\\ &=-\frac{d}{2 a c x}-\frac{d^2 \log (x)}{2 a c^2}+\frac{d^2 \log (c+d x)}{2 a c^2}-\frac{\log (c+d x)}{2 a x^2}-\frac{\sqrt{b} \log \left (\frac{d \left (\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d}\right ) \log (c+d x)}{4 (-a)^{3/2}}+\frac{\sqrt{b} \log \left (\frac{d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{3/2}}-\frac{\sqrt{b} \log \left (-\frac{d \left (\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right ) \log (c+d x)}{4 (-a)^{3/2}}+\frac{\sqrt{b} \log \left (-\frac{d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{3/2}}+\frac{\sqrt{b} \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [4]{b} x}{-\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d}\right )}{x} \, dx,x,c+d x\right )}{4 (-a)^{3/2}}+\frac{\sqrt{b} \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt [4]{b} x}{\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d}\right )}{x} \, dx,x,c+d x\right )}{4 (-a)^{3/2}}-\frac{\sqrt{b} \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [4]{b} x}{-\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{x} \, dx,x,c+d x\right )}{4 (-a)^{3/2}}-\frac{\sqrt{b} \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt [4]{b} x}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{x} \, dx,x,c+d x\right )}{4 (-a)^{3/2}}\\ &=-\frac{d}{2 a c x}-\frac{d^2 \log (x)}{2 a c^2}+\frac{d^2 \log (c+d x)}{2 a c^2}-\frac{\log (c+d x)}{2 a x^2}-\frac{\sqrt{b} \log \left (\frac{d \left (\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d}\right ) \log (c+d x)}{4 (-a)^{3/2}}+\frac{\sqrt{b} \log \left (\frac{d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{3/2}}-\frac{\sqrt{b} \log \left (-\frac{d \left (\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right ) \log (c+d x)}{4 (-a)^{3/2}}+\frac{\sqrt{b} \log \left (-\frac{d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{3/2}}-\frac{\sqrt{b} \text{Li}_2\left (\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 (-a)^{3/2}}-\frac{\sqrt{b} \text{Li}_2\left (\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d}\right )}{4 (-a)^{3/2}}+\frac{\sqrt{b} \text{Li}_2\left (\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}+\frac{\sqrt{b} \text{Li}_2\left (\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.196353, size = 506, normalized size = 0.94 \[ \frac{\sqrt{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}-\frac{\sqrt{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}-\frac{\sqrt{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}+\frac{\sqrt{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 (-a)^{3/2}}-\frac{\sqrt{b} \log (c+d x) \log \left (\frac{d \left (-\sqrt [4]{b} x+i \sqrt [4]{-a}\right )}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}+\frac{\sqrt{b} \log (c+d x) \log \left (\frac{d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 (-a)^{3/2}}-\frac{\sqrt{b} \log (c+d x) \log \left (-\frac{d \left (\sqrt [4]{b} x+i \sqrt [4]{-a}\right )}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}+\frac{\sqrt{b} \log (c+d x) \log \left (-\frac{d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}-\frac{d \left (\frac{d \log (x)}{c^2}-\frac{d \log (c+d x)}{c^2}+\frac{1}{c x}\right )}{2 a}-\frac{\log (c+d x)}{2 a x^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.422, size = 161, normalized size = 0.3 \begin{align*} -{\frac{{d}^{2}}{4\,a}\sum _{{\it \_R1}={\it RootOf} \left ( b{{\it \_Z}}^{4}-4\,{{\it \_Z}}^{3}bc+6\,{{\it \_Z}}^{2}b{c}^{2}-4\,{\it \_Z}\,b{c}^{3}+a{d}^{4}+b{c}^{4} \right ) }{\frac{1}{{{\it \_R1}}^{2}-2\,{\it \_R1}\,c+{c}^{2}} \left ( \ln \left ( dx+c \right ) \ln \left ({\frac{-dx+{\it \_R1}-c}{{\it \_R1}}} \right ) +{\it dilog} \left ({\frac{-dx+{\it \_R1}-c}{{\it \_R1}}} \right ) \right ) }}-{\frac{{d}^{2}\ln \left ( dx \right ) }{2\,{c}^{2}a}}-{\frac{d}{2\,acx}}+{\frac{{d}^{2}\ln \left ( dx+c \right ) }{2\,{c}^{2}a}}-{\frac{\ln \left ( dx+c \right ) }{2\,a{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (d x + c\right )}{b x^{7} + a x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (d x + c\right )}{{\left (b x^{4} + a\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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