Optimal. Leaf size=530 \[ \frac {c x}{2 b d}-\frac {x^2}{4 b}-\frac {c^2 \log (c+d x)}{2 b d^2}+\frac {x^2 \log (c+d x)}{2 b}-\frac {\sqrt {-a} \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 b^{3/2}}+\frac {\sqrt {-a} \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b^{3/2}}-\frac {\sqrt {-a} \log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 b^{3/2}}+\frac {\sqrt {-a} \log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b^{3/2}}-\frac {\sqrt {-a} \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^{3/2}}-\frac {\sqrt {-a} \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 b^{3/2}}+\frac {\sqrt {-a} \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{3/2}}+\frac {\sqrt {-a} \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.55, antiderivative size = 530, normalized size of antiderivative = 1.00, 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 = {281, 327,
211, 2463, 2442, 45, 266, 2441, 2440, 2438} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 211
Rule 266
Rule 281
Rule 327
Rule 2438
Rule 2440
Rule 2441
Rule 2442
Rule 2463
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} \text {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} \text {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} \text {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} \text {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] Result contains complex when optimal does not.
time = 0.17, size = 484, normalized size = 0.91 \begin {gather*} \frac {2 \sqrt {b} c d x-\sqrt {b} d^2 x^2-2 \sqrt {b} c^2 \log (c+d x)+2 \sqrt {b} d^2 x^2 \log (c+d x)+\sqrt {-a} d^2 \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)-\sqrt {-a} d^2 \log \left (\frac {d \left (\sqrt [4]{-a}-i \sqrt [4]{b} x\right )}{i \sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)-\sqrt {-a} d^2 \log \left (\frac {d \left (\sqrt [4]{-a}+i \sqrt [4]{b} x\right )}{-i \sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)+\sqrt {-a} d^2 \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)+\sqrt {-a} d^2 \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )-\sqrt {-a} d^2 \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right )-\sqrt {-a} d^2 \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right )+\sqrt {-a} d^2 \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^{3/2} d^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.47, size = 163, normalized size = 0.31
method | result | size |
derivativedivides | \(\frac {-\frac {\left (-\frac {\left (d x +c \right )^{2} \ln \left (d x +c \right )}{2}+\frac {\left (d x +c \right )^{2}}{4}+c \left (\left (d x +c \right ) \ln \left (d x +c \right )-d x -c \right )\right ) d^{4}}{b}-\frac {\left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\dilog \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right ) a \,d^{8}}{4 b^{2}}}{d^{6}}\) | \(163\) |
default | \(\frac {-\frac {\left (-\frac {\left (d x +c \right )^{2} \ln \left (d x +c \right )}{2}+\frac {\left (d x +c \right )^{2}}{4}+c \left (\left (d x +c \right ) \ln \left (d x +c \right )-d x -c \right )\right ) d^{4}}{b}-\frac {\left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\dilog \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right ) a \,d^{8}}{4 b^{2}}}{d^{6}}\) | \(163\) |
risch | \(\frac {x^{2} \ln \left (d x +c \right )}{2 b}-\frac {c^{2} \ln \left (d x +c \right )}{2 b \,d^{2}}-\frac {x^{2}}{4 b}+\frac {c x}{2 b d}+\frac {3 c^{2}}{4 d^{2} b}-\frac {d^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\dilog \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right ) a}{4 b^{2}}\) | \(164\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^5\,\ln \left (c+d\,x\right )}{b\,x^4+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________