Optimal. Leaf size=521 \[ -\frac {\sqrt {-\sqrt {-a}} \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^{5/4}}+\frac {\sqrt {-\sqrt {-a}} \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 b^{5/4}}-\frac {\sqrt [4]{-a} \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{5/4}}+\frac {\sqrt [4]{-a} \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^{5/4}}+\frac {\sqrt {-\sqrt {-a}} \log (c+d x) \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt {-\sqrt {-a}} d+\sqrt [4]{b} c}\right )}{4 b^{5/4}}+\frac {\sqrt [4]{-a} \log (c+d x) \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 b^{5/4}}-\frac {\sqrt {-\sqrt {-a}} \log (c+d x) \log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^{5/4}}-\frac {\sqrt [4]{-a} \log (c+d x) \log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{5/4}}+\frac {(c+d x) \log (c+d x)}{b d}-\frac {x}{b} \]
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Rubi [A] time = 0.76, antiderivative size = 521, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 14, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.737, Rules used = {321, 211, 1165, 628, 1162, 617, 204, 2416, 2389, 2295, 2409, 2394, 2393, 2391} \[ -\frac {\sqrt {-\sqrt {-a}} \text {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^{5/4}}+\frac {\sqrt {-\sqrt {-a}} \text {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt {-\sqrt {-a}} d+\sqrt [4]{b} c}\right )}{4 b^{5/4}}-\frac {\sqrt [4]{-a} \text {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{5/4}}+\frac {\sqrt [4]{-a} \text {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 b^{5/4}}+\frac {\sqrt {-\sqrt {-a}} \log (c+d x) \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt {-\sqrt {-a}} d+\sqrt [4]{b} c}\right )}{4 b^{5/4}}+\frac {\sqrt [4]{-a} \log (c+d x) \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 b^{5/4}}-\frac {\sqrt {-\sqrt {-a}} \log (c+d x) \log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^{5/4}}-\frac {\sqrt [4]{-a} \log (c+d x) \log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{5/4}}+\frac {(c+d x) \log (c+d x)}{b d}-\frac {x}{b} \]
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 321
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 2295
Rule 2389
Rule 2391
Rule 2393
Rule 2394
Rule 2409
Rule 2416
Rubi steps
\begin {align*} \int \frac {x^4 \log (c+d x)}{a+b x^4} \, dx &=\int \left (\frac {\log (c+d x)}{b}-\frac {a \log (c+d x)}{b \left (a+b x^4\right )}\right ) \, dx\\ &=\frac {\int \log (c+d x) \, dx}{b}-\frac {a \int \frac {\log (c+d x)}{a+b x^4} \, dx}{b}\\ &=-\frac {a \int \left (\frac {\sqrt {-a} \log (c+d x)}{2 a \left (\sqrt {-a}-\sqrt {b} x^2\right )}+\frac {\sqrt {-a} \log (c+d x)}{2 a \left (\sqrt {-a}+\sqrt {b} x^2\right )}\right ) \, dx}{b}+\frac {\operatorname {Subst}(\int \log (x) \, dx,x,c+d x)}{b d}\\ &=-\frac {x}{b}+\frac {(c+d x) \log (c+d x)}{b d}-\frac {\sqrt {-a} \int \frac {\log (c+d x)}{\sqrt {-a}-\sqrt {b} x^2} \, dx}{2 b}-\frac {\sqrt {-a} \int \frac {\log (c+d x)}{\sqrt {-a}+\sqrt {b} x^2} \, dx}{2 b}\\ &=-\frac {x}{b}+\frac {(c+d x) \log (c+d x)}{b d}-\frac {\sqrt {-a} \int \left (\frac {\sqrt {-\sqrt {-a}} \log (c+d x)}{2 \sqrt {-a} \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}+\frac {\sqrt {-\sqrt {-a}} \log (c+d x)}{2 \sqrt {-a} \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}\right ) \, dx}{2 b}-\frac {\sqrt {-a} \int \left (\frac {\log (c+d x)}{2 \sqrt [4]{-a} \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}+\frac {\log (c+d x)}{2 \sqrt [4]{-a} \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}\right ) \, dx}{2 b}\\ &=-\frac {x}{b}+\frac {(c+d x) \log (c+d x)}{b d}-\frac {\sqrt {-\sqrt {-a}} \int \frac {\log (c+d x)}{\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x} \, dx}{4 b}-\frac {\sqrt {-\sqrt {-a}} \int \frac {\log (c+d x)}{\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x} \, dx}{4 b}-\frac {\sqrt [4]{-a} \int \frac {\log (c+d x)}{\sqrt [4]{-a}-\sqrt [4]{b} x} \, dx}{4 b}-\frac {\sqrt [4]{-a} \int \frac {\log (c+d x)}{\sqrt [4]{-a}+\sqrt [4]{b} x} \, dx}{4 b}\\ &=-\frac {x}{b}+\frac {(c+d x) \log (c+d x)}{b d}+\frac {\sqrt {-\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^{5/4}}+\frac {\sqrt [4]{-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^{5/4}}-\frac {\sqrt {-\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^{5/4}}-\frac {\sqrt [4]{-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^{5/4}}-\frac {\left (\sqrt {-\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^{5/4}}+\frac {\left (\sqrt {-\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^{5/4}}-\frac {\left (\sqrt [4]{-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^{5/4}}+\frac {\left (\sqrt [4]{-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^{5/4}}\\ &=-\frac {x}{b}+\frac {(c+d x) \log (c+d x)}{b d}+\frac {\sqrt {-\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^{5/4}}+\frac {\sqrt [4]{-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^{5/4}}-\frac {\sqrt {-\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^{5/4}}-\frac {\sqrt [4]{-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^{5/4}}+\frac {\sqrt {-\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^{5/4}}-\frac {\sqrt {-\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^{5/4}}+\frac {\sqrt [4]{-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^{5/4}}-\frac {\sqrt [4]{-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^{5/4}}\\ &=-\frac {x}{b}+\frac {(c+d x) \log (c+d x)}{b d}+\frac {\sqrt {-\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^{5/4}}+\frac {\sqrt [4]{-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^{5/4}}-\frac {\sqrt {-\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^{5/4}}-\frac {\sqrt [4]{-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^{5/4}}-\frac {\sqrt {-\sqrt {-a}} \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^{5/4}}+\frac {\sqrt {-\sqrt {-a}} \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 b^{5/4}}-\frac {\sqrt [4]{-a} \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{5/4}}+\frac {\sqrt [4]{-a} \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^{5/4}}\\ \end {align*}
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Mathematica [C] time = 0.24, size = 458, normalized size = 0.88 \[ \frac {-\sqrt [4]{-a} d \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )-i \sqrt [4]{-a} d \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right )+i \sqrt [4]{-a} d \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right )+\sqrt [4]{-a} d \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )+\sqrt [4]{-a} d \log (c+d x) \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )-i \sqrt [4]{-a} d \log (c+d x) \log \left (\frac {d \left (\sqrt [4]{-a}-i \sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+i \sqrt [4]{b} c}\right )+i \sqrt [4]{-a} d \log (c+d x) \log \left (\frac {d \left (\sqrt [4]{-a}+i \sqrt [4]{b} x\right )}{\sqrt [4]{-a} d-i \sqrt [4]{b} c}\right )-\sqrt [4]{-a} d \log (c+d x) \log \left (\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d-\sqrt [4]{b} c}\right )+4 \sqrt [4]{b} d x \log (c+d x)+4 \sqrt [4]{b} c \log (c+d x)-4 \sqrt [4]{b} d x}{4 b^{5/4} d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{4} \log \left (d x + c\right )}{b x^{4} + a}, x\right ) \]
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 \[ \int \frac {x^{4} \log \left (d x + c\right )}{b x^{4} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.26, size = 154, normalized size = 0.30 \[ -\frac {a \,d^{3} \left (\ln \left (\frac {-d x +\RootOf \left (\textit {\_Z}^{4} b -4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )-c}{\RootOf \left (\textit {\_Z}^{4} b -4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}\right ) \ln \left (d x +c \right )+\dilog \left (\frac {-d x +\RootOf \left (\textit {\_Z}^{4} b -4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )-c}{\RootOf \left (\textit {\_Z}^{4} b -4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}\right )\right )}{4 b^{2} \left (\RootOf \left (\textit {\_Z}^{4} b -4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )^{3}-3 \RootOf \left (\textit {\_Z}^{4} b -4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )^{2} c +3 \RootOf \left (\textit {\_Z}^{4} b -4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right ) c^{2}-c^{3}\right )}+\frac {x \ln \left (d x +c \right )}{b}+\frac {c \ln \left (d x +c \right )}{b d}-\frac {x}{b}-\frac {c}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \log \left (d x + c\right )}{b x^{4} + a}\,{d x} \]
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 \[ \int \frac {x^4\,\ln \left (c+d\,x\right )}{b\,x^4+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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