Optimal. Leaf size=175 \[ \frac {a^3 p \log (a x+b)}{3 e (a d-b e)^3}-\frac {b p \left (3 a^2 d^2-3 a b d e+b^2 e^2\right ) \log (d+e x)}{3 d^3 (a d-b e)^3}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 e (d+e x)^3}+\frac {b p (2 a d-b e)}{3 d^2 (d+e x) (a d-b e)^2}+\frac {b p}{6 d (d+e x)^2 (a d-b e)}-\frac {p \log (x)}{3 d^3 e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.17, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2463, 514, 72} \[ -\frac {b p \left (3 a^2 d^2-3 a b d e+b^2 e^2\right ) \log (d+e x)}{3 d^3 (a d-b e)^3}+\frac {a^3 p \log (a x+b)}{3 e (a d-b e)^3}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 e (d+e x)^3}+\frac {b p (2 a d-b e)}{3 d^2 (d+e x) (a d-b e)^2}+\frac {b p}{6 d (d+e x)^2 (a d-b e)}-\frac {p \log (x)}{3 d^3 e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 72
Rule 514
Rule 2463
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{(d+e x)^4} \, dx &=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 e (d+e x)^3}-\frac {(b p) \int \frac {1}{\left (a+\frac {b}{x}\right ) x^2 (d+e x)^3} \, dx}{3 e}\\ &=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 e (d+e x)^3}-\frac {(b p) \int \frac {1}{x (b+a x) (d+e x)^3} \, dx}{3 e}\\ &=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 e (d+e x)^3}-\frac {(b p) \int \left (\frac {1}{b d^3 x}+\frac {a^4}{b (-a d+b e)^3 (b+a x)}+\frac {e^2}{d (a d-b e) (d+e x)^3}+\frac {e^2 (2 a d-b e)}{d^2 (a d-b e)^2 (d+e x)^2}+\frac {e^2 \left (3 a^2 d^2-3 a b d e+b^2 e^2\right )}{d^3 (a d-b e)^3 (d+e x)}\right ) \, dx}{3 e}\\ &=\frac {b p}{6 d (a d-b e) (d+e x)^2}+\frac {b (2 a d-b e) p}{3 d^2 (a d-b e)^2 (d+e x)}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 e (d+e x)^3}-\frac {p \log (x)}{3 d^3 e}+\frac {a^3 p \log (b+a x)}{3 e (a d-b e)^3}-\frac {b \left (3 a^2 d^2-3 a b d e+b^2 e^2\right ) p \log (d+e x)}{3 d^3 (a d-b e)^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.28, size = 164, normalized size = 0.94 \[ \frac {\frac {a^3 p \log (a x+b)}{(a d-b e)^3}-\frac {b e p \left (3 a^2 d^2-3 a b d e+b^2 e^2\right ) \log (d+e x)}{d^3 (a d-b e)^3}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{(d+e x)^3}+\frac {b e p (2 a d-b e)}{d^2 (d+e x) (a d-b e)^2}+\frac {b e p}{2 d (d+e x)^2 (a d-b e)}-\frac {p \log (x)}{d^3}}{3 e} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 6.46, size = 818, normalized size = 4.67 \[ \frac {2 \, {\left (2 \, a^{2} b d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + b^{3} d e^{5}\right )} p x^{2} + {\left (9 \, a^{2} b d^{4} e^{2} - 14 \, a b^{2} d^{3} e^{3} + 5 \, b^{3} d^{2} e^{4}\right )} p x - 2 \, {\left (a^{3} d^{6} - 3 \, a^{2} b d^{5} e + 3 \, a b^{2} d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} p \log \left (\frac {a x + b}{x}\right ) + {\left (5 \, a^{2} b d^{5} e - 8 \, a b^{2} d^{4} e^{2} + 3 \, b^{3} d^{3} e^{3}\right )} p + 2 \, {\left (a^{3} d^{3} e^{3} p x^{3} + 3 \, a^{3} d^{4} e^{2} p x^{2} + 3 \, a^{3} d^{5} e p x + a^{3} d^{6} p\right )} \log \left (a x + b\right ) - 2 \, {\left ({\left (3 \, a^{2} b d^{2} e^{4} - 3 \, a b^{2} d e^{5} + b^{3} e^{6}\right )} p x^{3} + 3 \, {\left (3 \, a^{2} b d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + b^{3} d e^{5}\right )} p x^{2} + 3 \, {\left (3 \, a^{2} b d^{4} e^{2} - 3 \, a b^{2} d^{3} e^{3} + b^{3} d^{2} e^{4}\right )} p x + {\left (3 \, a^{2} b d^{5} e - 3 \, a b^{2} d^{4} e^{2} + b^{3} d^{3} e^{3}\right )} p\right )} \log \left (e x + d\right ) - 2 \, {\left (a^{3} d^{6} - 3 \, a^{2} b d^{5} e + 3 \, a b^{2} d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} \log \relax (c) - 2 \, {\left ({\left (a^{3} d^{3} e^{3} - 3 \, a^{2} b d^{2} e^{4} + 3 \, a b^{2} d e^{5} - b^{3} e^{6}\right )} p x^{3} + 3 \, {\left (a^{3} d^{4} e^{2} - 3 \, a^{2} b d^{3} e^{3} + 3 \, a b^{2} d^{2} e^{4} - b^{3} d e^{5}\right )} p x^{2} + 3 \, {\left (a^{3} d^{5} e - 3 \, a^{2} b d^{4} e^{2} + 3 \, a b^{2} d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} p x + {\left (a^{3} d^{6} - 3 \, a^{2} b d^{5} e + 3 \, a b^{2} d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} p\right )} \log \relax (x)}{6 \, {\left (a^{3} d^{9} e - 3 \, a^{2} b d^{8} e^{2} + 3 \, a b^{2} d^{7} e^{3} - b^{3} d^{6} e^{4} + {\left (a^{3} d^{6} e^{4} - 3 \, a^{2} b d^{5} e^{5} + 3 \, a b^{2} d^{4} e^{6} - b^{3} d^{3} e^{7}\right )} x^{3} + 3 \, {\left (a^{3} d^{7} e^{3} - 3 \, a^{2} b d^{6} e^{4} + 3 \, a b^{2} d^{5} e^{5} - b^{3} d^{4} e^{6}\right )} x^{2} + 3 \, {\left (a^{3} d^{8} e^{2} - 3 \, a^{2} b d^{7} e^{3} + 3 \, a b^{2} d^{6} e^{4} - b^{3} d^{5} e^{5}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.32, size = 1841, normalized size = 10.52 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.41, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (c \left (a +\frac {b}{x}\right )^{p}\right )}{\left (e x +d \right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.52, size = 299, normalized size = 1.71 \[ \frac {{\left (\frac {2 \, a^{3} \log \left (a x + b\right )}{a^{3} b d^{3} - 3 \, a^{2} b^{2} d^{2} e + 3 \, a b^{3} d e^{2} - b^{4} e^{3}} - \frac {2 \, {\left (3 \, a^{2} d^{2} e - 3 \, a b d e^{2} + b^{2} e^{3}\right )} \log \left (e x + d\right )}{a^{3} d^{6} - 3 \, a^{2} b d^{5} e + 3 \, a b^{2} d^{4} e^{2} - b^{3} d^{3} e^{3}} + \frac {5 \, a d^{2} e - 3 \, b d e^{2} + 2 \, {\left (2 \, a d e^{2} - b e^{3}\right )} x}{a^{2} d^{6} - 2 \, a b d^{5} e + b^{2} d^{4} e^{2} + {\left (a^{2} d^{4} e^{2} - 2 \, a b d^{3} e^{3} + b^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (a^{2} d^{5} e - 2 \, a b d^{4} e^{2} + b^{2} d^{3} e^{3}\right )} x} - \frac {2 \, \log \relax (x)}{b d^{3}}\right )} b p}{6 \, e} - \frac {\log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right )}{3 \, {\left (e x + d\right )}^{3} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.85, size = 662, normalized size = 3.78 \[ \frac {p\,\ln \left (d+e\,x\right )}{3\,d^3\,e}-\frac {3\,b^2\,e^2\,p}{2\,\left (3\,a^2\,d^5\,e+6\,a^2\,d^4\,e^2\,x+3\,a^2\,d^3\,e^3\,x^2-6\,a\,b\,d^4\,e^2-12\,a\,b\,d^3\,e^3\,x-6\,a\,b\,d^2\,e^4\,x^2+3\,b^2\,d^3\,e^3+6\,b^2\,d^2\,e^4\,x+3\,b^2\,d\,e^5\,x^2\right )}-\frac {p\,\ln \relax (x)}{3\,d^3\,e}-\frac {a^3\,p\,\ln \left (b+a\,x\right )}{-3\,a^3\,d^3\,e+9\,a^2\,b\,d^2\,e^2-9\,a\,b^2\,d\,e^3+3\,b^3\,e^4}-\frac {\ln \left (c\,{\left (\frac {b+a\,x}{x}\right )}^p\right )}{3\,\left (d^3\,e+3\,d^2\,e^2\,x+3\,d\,e^3\,x^2+e^4\,x^3\right )}-\frac {b^2\,e^3\,p\,x}{3\,a^2\,d^6\,e+6\,a^2\,d^5\,e^2\,x+3\,a^2\,d^4\,e^3\,x^2-6\,a\,b\,d^5\,e^2-12\,a\,b\,d^4\,e^3\,x-6\,a\,b\,d^3\,e^4\,x^2+3\,b^2\,d^4\,e^3+6\,b^2\,d^3\,e^4\,x+3\,b^2\,d^2\,e^5\,x^2}-\frac {a^3\,d^3\,p\,\ln \left (d+e\,x\right )}{3\,a^3\,d^6\,e-9\,a^2\,b\,d^5\,e^2+9\,a\,b^2\,d^4\,e^3-3\,b^3\,d^3\,e^4}+\frac {5\,a\,b\,d\,e\,p}{2\,\left (3\,a^2\,d^5\,e+6\,a^2\,d^4\,e^2\,x+3\,a^2\,d^3\,e^3\,x^2-6\,a\,b\,d^4\,e^2-12\,a\,b\,d^3\,e^3\,x-6\,a\,b\,d^2\,e^4\,x^2+3\,b^2\,d^3\,e^3+6\,b^2\,d^2\,e^4\,x+3\,b^2\,d\,e^5\,x^2\right )}+\frac {2\,a\,b\,d\,e^2\,p\,x}{3\,a^2\,d^6\,e+6\,a^2\,d^5\,e^2\,x+3\,a^2\,d^4\,e^3\,x^2-6\,a\,b\,d^5\,e^2-12\,a\,b\,d^4\,e^3\,x-6\,a\,b\,d^3\,e^4\,x^2+3\,b^2\,d^4\,e^3+6\,b^2\,d^3\,e^4\,x+3\,b^2\,d^2\,e^5\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________