Optimal. Leaf size=530 \[ -\frac{\sqrt{-a} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 b^{3/2}}-\frac{\sqrt{-a} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt{-\sqrt{-a}} d+\sqrt [4]{b} c}\right )}{4 b^{3/2}}+\frac{\sqrt{-a} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{3/2}}+\frac{\sqrt{-a} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 b^{3/2}}-\frac{\sqrt{-a} \log (c+d x) \log \left (\frac{d \left (\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x\right )}{\sqrt{-\sqrt{-a}} d+\sqrt [4]{b} c}\right )}{4 b^{3/2}}+\frac{\sqrt{-a} \log (c+d x) \log \left (\frac{d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 b^{3/2}}-\frac{\sqrt{-a} \log (c+d x) \log \left (-\frac{d \left (\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 b^{3/2}}+\frac{\sqrt{-a} \log (c+d x) \log \left (-\frac{d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{3/2}}-\frac{c^2 \log (c+d x)}{2 b d^2}+\frac{x^2 \log (c+d x)}{2 b}+\frac{c x}{2 b d}-\frac{x^2}{4 b} \]
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Rubi [A] time = 0.647887, antiderivative size = 530, 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, 321, 205, 2416, 2395, 43, 260, 2394, 2393, 2391} \[ -\frac{\sqrt{-a} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 b^{3/2}}-\frac{\sqrt{-a} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt{-\sqrt{-a}} d+\sqrt [4]{b} c}\right )}{4 b^{3/2}}+\frac{\sqrt{-a} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{3/2}}+\frac{\sqrt{-a} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 b^{3/2}}-\frac{\sqrt{-a} \log (c+d x) \log \left (\frac{d \left (\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x\right )}{\sqrt{-\sqrt{-a}} d+\sqrt [4]{b} c}\right )}{4 b^{3/2}}+\frac{\sqrt{-a} \log (c+d x) \log \left (\frac{d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 b^{3/2}}-\frac{\sqrt{-a} \log (c+d x) \log \left (-\frac{d \left (\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 b^{3/2}}+\frac{\sqrt{-a} \log (c+d x) \log \left (-\frac{d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{3/2}}-\frac{c^2 \log (c+d x)}{2 b d^2}+\frac{x^2 \log (c+d x)}{2 b}+\frac{c x}{2 b d}-\frac{x^2}{4 b} \]
Antiderivative was successfully verified.
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Rule 275
Rule 321
Rule 205
Rule 2416
Rule 2395
Rule 43
Rule 260
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{x^5 \log (c+d x)}{a+b x^4} \, dx &=\int \left (\frac{x \log (c+d x)}{b}-\frac{a x \log (c+d x)}{b \left (a+b x^4\right )}\right ) \, dx\\ &=\frac{\int x \log (c+d x) \, dx}{b}-\frac{a \int \frac{x \log (c+d x)}{a+b x^4} \, dx}{b}\\ &=\frac{x^2 \log (c+d x)}{2 b}-\frac{a \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}{b}-\frac{d \int \frac{x^2}{c+d x} \, dx}{2 b}\\ &=\frac{x^2 \log (c+d x)}{2 b}-\frac{\sqrt{-a} \int \frac{x \log (c+d x)}{\sqrt{-a} \sqrt{b}-b x^2} \, dx}{2 \sqrt{b}}-\frac{\sqrt{-a} \int \frac{x \log (c+d x)}{\sqrt{-a} \sqrt{b}+b x^2} \, dx}{2 \sqrt{b}}-\frac{d \int \left (-\frac{c}{d^2}+\frac{x}{d}+\frac{c^2}{d^2 (c+d x)}\right ) \, dx}{2 b}\\ &=\frac{c x}{2 b d}-\frac{x^2}{4 b}-\frac{c^2 \log (c+d x)}{2 b d^2}+\frac{x^2 \log (c+d x)}{2 b}-\frac{\sqrt{-a} \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 \sqrt{b}}-\frac{\sqrt{-a} \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 \sqrt{b}}\\ &=\frac{c x}{2 b d}-\frac{x^2}{4 b}-\frac{c^2 \log (c+d x)}{2 b d^2}+\frac{x^2 \log (c+d x)}{2 b}+\frac{\sqrt{-a} \int \frac{\log (c+d x)}{\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x} \, dx}{4 b^{5/4}}-\frac{\sqrt{-a} \int \frac{\log (c+d x)}{\sqrt [4]{-a}-\sqrt [4]{b} x} \, dx}{4 b^{5/4}}-\frac{\sqrt{-a} \int \frac{\log (c+d x)}{\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x} \, dx}{4 b^{5/4}}+\frac{\sqrt{-a} \int \frac{\log (c+d x)}{\sqrt [4]{-a}+\sqrt [4]{b} x} \, dx}{4 b^{5/4}}\\ &=\frac{c x}{2 b d}-\frac{x^2}{4 b}-\frac{c^2 \log (c+d x)}{2 b d^2}+\frac{x^2 \log (c+d x)}{2 b}-\frac{\sqrt{-a} \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 b^{3/2}}+\frac{\sqrt{-a} \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 b^{3/2}}-\frac{\sqrt{-a} \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 b^{3/2}}+\frac{\sqrt{-a} \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 b^{3/2}}+\frac{\left (\sqrt{-a} 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 b^{3/2}}-\frac{\left (\sqrt{-a} 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 b^{3/2}}+\frac{\left (\sqrt{-a} 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 b^{3/2}}-\frac{\left (\sqrt{-a} 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 b^{3/2}}\\ &=\frac{c x}{2 b d}-\frac{x^2}{4 b}-\frac{c^2 \log (c+d x)}{2 b d^2}+\frac{x^2 \log (c+d x)}{2 b}-\frac{\sqrt{-a} \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 b^{3/2}}+\frac{\sqrt{-a} \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 b^{3/2}}-\frac{\sqrt{-a} \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 b^{3/2}}+\frac{\sqrt{-a} \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 b^{3/2}}+\frac{\sqrt{-a} \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 b^{3/2}}+\frac{\sqrt{-a} \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 b^{3/2}}-\frac{\sqrt{-a} \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 b^{3/2}}-\frac{\sqrt{-a} \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 b^{3/2}}\\ &=\frac{c x}{2 b d}-\frac{x^2}{4 b}-\frac{c^2 \log (c+d x)}{2 b d^2}+\frac{x^2 \log (c+d x)}{2 b}-\frac{\sqrt{-a} \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 b^{3/2}}+\frac{\sqrt{-a} \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 b^{3/2}}-\frac{\sqrt{-a} \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 b^{3/2}}+\frac{\sqrt{-a} \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 b^{3/2}}-\frac{\sqrt{-a} \text{Li}_2\left (\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 b^{3/2}}-\frac{\sqrt{-a} \text{Li}_2\left (\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d}\right )}{4 b^{3/2}}+\frac{\sqrt{-a} \text{Li}_2\left (\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{3/2}}+\frac{\sqrt{-a} \text{Li}_2\left (\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.221593, size = 484, normalized size = 0.91 \[ \frac{\sqrt{-a} d^2 \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )-\sqrt{-a} d^2 \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right )-\sqrt{-a} d^2 \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right )+\sqrt{-a} d^2 \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )+\sqrt{-a} d^2 \log (c+d x) \log \left (\frac{d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )-\sqrt{-a} d^2 \log (c+d x) \log \left (\frac{d \left (\sqrt [4]{-a}-i \sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+i \sqrt [4]{b} c}\right )-\sqrt{-a} d^2 \log (c+d x) \log \left (\frac{d \left (\sqrt [4]{-a}+i \sqrt [4]{b} x\right )}{\sqrt [4]{-a} d-i \sqrt [4]{b} c}\right )+\sqrt{-a} d^2 \log (c+d x) \log \left (\frac{d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d-\sqrt [4]{b} c}\right )-2 \sqrt{b} c^2 \log (c+d x)+2 \sqrt{b} d^2 x^2 \log (c+d x)+2 \sqrt{b} c d x-\sqrt{b} d^2 x^2}{4 b^{3/2} d^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.376, size = 164, normalized size = 0.3 \begin{align*}{\frac{{x}^{2}\ln \left ( dx+c \right ) }{2\,b}}-{\frac{{c}^{2}\ln \left ( dx+c \right ) }{2\,b{d}^{2}}}-{\frac{{x}^{2}}{4\,b}}+{\frac{cx}{2\,bd}}+{\frac{3\,{c}^{2}}{4\,b{d}^{2}}}-{\frac{a{d}^{2}}{4\,{b}^{2}}\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 ) }} \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{x^{5} \log \left (d x + c\right )}{b x^{4} + a}, 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{x^{5} \log \left (d x + c\right )}{b x^{4} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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