Optimal. Leaf size=119 \[ \frac {a i x}{g}-\frac {b i n x}{g}+\frac {b i (d+e x) \log \left (c (d+e x)^n\right )}{e g}+\frac {(g h-f i) \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^2}+\frac {b (g h-f i) n \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.10, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2465, 2436,
2332, 2441, 2440, 2438} \begin {gather*} \frac {b n (g h-f i) \text {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g^2}+\frac {(g h-f i) \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}+\frac {a i x}{g}+\frac {b i (d+e x) \log \left (c (d+e x)^n\right )}{e g}-\frac {b i n x}{g} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2332
Rule 2436
Rule 2438
Rule 2440
Rule 2441
Rule 2465
Rubi steps
\begin {align*} \int \frac {(h+219 x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x} \, dx &=\int \left (\frac {219 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}+\frac {(-219 f+g h) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g (f+g x)}\right ) \, dx\\ &=\frac {219 \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{g}+\frac {(-219 f+g h) \int \frac {a+b \log \left (c (d+e x)^n\right )}{f+g x} \, dx}{g}\\ &=\frac {219 a x}{g}-\frac {(219 f-g h) \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^2}+\frac {(219 b) \int \log \left (c (d+e x)^n\right ) \, dx}{g}+\frac {(b e (219 f-g h) n) \int \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{g^2}\\ &=\frac {219 a x}{g}-\frac {(219 f-g h) \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^2}+\frac {(219 b) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e g}+\frac {(b (219 f-g h) n) \text {Subst}\left (\int \frac {\log \left (1+\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g^2}\\ &=\frac {219 a x}{g}-\frac {219 b n x}{g}+\frac {219 b (d+e x) \log \left (c (d+e x)^n\right )}{e g}-\frac {(219 f-g h) \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^2}-\frac {b (219 f-g h) n \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.08, size = 110, normalized size = 0.92 \begin {gather*} \frac {a g i x-b g i n x+\frac {b g i (d+e x) \log \left (c (d+e x)^n\right )}{e}+(g h-f i) \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )+b (g h-f i) n \text {Li}_2\left (\frac {g (d+e x)}{-e f+d g}\right )}{g^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 = 0.43, size = 750, normalized size = 6.30
method | result | size |
risch | \(\frac {b \ln \left (\left (e x +d \right )^{n}\right ) i x}{g}-\frac {b n i f}{g^{2}}+\frac {a \ln \left (g x +f \right ) h}{g}-\frac {a \ln \left (g x +f \right ) f i}{g^{2}}+\frac {i b \pi \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} \ln \left (g x +f \right ) f i}{2 g^{2}}+\frac {i b \pi \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \ln \left (g x +f \right ) h}{2 g}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \ln \left (g x +f \right ) h}{2 g}+\frac {i b \pi \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} i x}{2 g}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} i x}{2 g}-\frac {b n \dilog \left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right ) h}{g}-\frac {b \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (g x +f \right ) f i}{g^{2}}+\frac {b n \dilog \left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right ) f i}{g^{2}}-\frac {b n \ln \left (g x +f \right ) \ln \left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right ) h}{g}+\frac {b \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (g x +f \right ) h}{g}-\frac {i b \pi \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} \ln \left (g x +f \right ) h}{2 g}-\frac {i b \pi \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} i x}{2 g}+\frac {b \ln \left (c \right ) i x}{g}+\frac {b \ln \left (c \right ) \ln \left (g x +f \right ) h}{g}+\frac {b n i d \ln \left (\left (g x +f \right ) e +d g -e f \right )}{e g}+\frac {b n \ln \left (g x +f \right ) \ln \left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right ) f i}{g^{2}}-\frac {b \ln \left (c \right ) \ln \left (g x +f \right ) f i}{g^{2}}+\frac {a i x}{g}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right ) \ln \left (g x +f \right ) f i}{2 g^{2}}-\frac {b i n x}{g}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right ) \ln \left (g x +f \right ) h}{2 g}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \ln \left (g x +f \right ) f i}{2 g^{2}}-\frac {i b \pi \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \ln \left (g x +f \right ) f i}{2 g^{2}}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right ) i x}{2 g}\) | \(750\) |
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right ) \left (h + i x\right )}{f + g x}\, dx \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 = -0.01 \begin {gather*} \int \frac {\left (h+i\,x\right )\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{f+g\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________