Optimal. Leaf size=260 \[ \frac {b p r}{6 h (b g-a h) (g+h x)^2}+\frac {d q r}{6 h (d g-c h) (g+h x)^2}+\frac {b^2 p r}{3 h (b g-a h)^2 (g+h x)}+\frac {d^2 q r}{3 h (d g-c h)^2 (g+h x)}+\frac {b^3 p r \log (a+b x)}{3 h (b g-a h)^3}+\frac {d^3 q r \log (c+d x)}{3 h (d g-c h)^3}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h (g+h x)^3}-\frac {b^3 p r \log (g+h x)}{3 h (b g-a h)^3}-\frac {d^3 q r \log (g+h x)}{3 h (d g-c h)^3} \]
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Rubi [A]
time = 0.11, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2581, 46}
\begin {gather*} \frac {b^3 p r \log (a+b x)}{3 h (b g-a h)^3}-\frac {b^3 p r \log (g+h x)}{3 h (b g-a h)^3}+\frac {b^2 p r}{3 h (g+h x) (b g-a h)^2}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h (g+h x)^3}+\frac {b p r}{6 h (g+h x)^2 (b g-a h)}+\frac {d^3 q r \log (c+d x)}{3 h (d g-c h)^3}-\frac {d^3 q r \log (g+h x)}{3 h (d g-c h)^3}+\frac {d^2 q r}{3 h (g+h x) (d g-c h)^2}+\frac {d q r}{6 h (g+h x)^2 (d g-c h)} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 2581
Rubi steps
\begin {align*} \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^4} \, dx &=-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h (g+h x)^3}+\frac {(b p r) \int \frac {1}{(a+b x) (g+h x)^3} \, dx}{3 h}+\frac {(d q r) \int \frac {1}{(c+d x) (g+h x)^3} \, dx}{3 h}\\ &=-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h (g+h x)^3}+\frac {(b p r) \int \left (\frac {b^3}{(b g-a h)^3 (a+b x)}-\frac {h}{(b g-a h) (g+h x)^3}-\frac {b h}{(b g-a h)^2 (g+h x)^2}-\frac {b^2 h}{(b g-a h)^3 (g+h x)}\right ) \, dx}{3 h}+\frac {(d q r) \int \left (\frac {d^3}{(d g-c h)^3 (c+d x)}-\frac {h}{(d g-c h) (g+h x)^3}-\frac {d h}{(d g-c h)^2 (g+h x)^2}-\frac {d^2 h}{(d g-c h)^3 (g+h x)}\right ) \, dx}{3 h}\\ &=\frac {b p r}{6 h (b g-a h) (g+h x)^2}+\frac {d q r}{6 h (d g-c h) (g+h x)^2}+\frac {b^2 p r}{3 h (b g-a h)^2 (g+h x)}+\frac {d^2 q r}{3 h (d g-c h)^2 (g+h x)}+\frac {b^3 p r \log (a+b x)}{3 h (b g-a h)^3}+\frac {d^3 q r \log (c+d x)}{3 h (d g-c h)^3}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h (g+h x)^3}-\frac {b^3 p r \log (g+h x)}{3 h (b g-a h)^3}-\frac {d^3 q r \log (g+h x)}{3 h (d g-c h)^3}\\ \end {align*}
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Mathematica [A]
time = 0.49, size = 254, normalized size = 0.98 \begin {gather*} \frac {-2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\frac {r (g+h x) \left ((b g-a h)^2 (d g-c h)^2 (b d g (p+q)-h (b c p+a d q))-(g+h x) \left ((b g-a h) (d g-c h) \left (4 a b d^2 g h q-2 a^2 d^2 h^2 q-2 b^2 \left (-2 c d g h p+c^2 h^2 p+d^2 g^2 (p+q)\right )\right )-2 (g+h x) \left (b^3 (d g-c h)^3 p (\log (a+b x)-\log (g+h x))+d^3 (b g-a h)^3 q (\log (c+d x)-\log (g+h x))\right )\right )\right )}{(b g-a h)^3 (d g-c h)^3}}{6 h (g+h x)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.16, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}{\left (h x +g \right )^{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 457, normalized size = 1.76 \begin {gather*} \frac {{\left ({\left (\frac {2 \, b^{2} \log \left (b x + a\right )}{b^{3} g^{3} - 3 \, a b^{2} g^{2} h + 3 \, a^{2} b g h^{2} - a^{3} h^{3}} - \frac {2 \, b^{2} \log \left (h x + g\right )}{b^{3} g^{3} - 3 \, a b^{2} g^{2} h + 3 \, a^{2} b g h^{2} - a^{3} h^{3}} + \frac {2 \, b h x + 3 \, b g - a h}{b^{2} g^{4} - 2 \, a b g^{3} h + a^{2} g^{2} h^{2} + {\left (b^{2} g^{2} h^{2} - 2 \, a b g h^{3} + a^{2} h^{4}\right )} x^{2} + 2 \, {\left (b^{2} g^{3} h - 2 \, a b g^{2} h^{2} + a^{2} g h^{3}\right )} x}\right )} b f p + {\left (\frac {2 \, d^{2} \log \left (d x + c\right )}{d^{3} g^{3} - 3 \, c d^{2} g^{2} h + 3 \, c^{2} d g h^{2} - c^{3} h^{3}} - \frac {2 \, d^{2} \log \left (h x + g\right )}{d^{3} g^{3} - 3 \, c d^{2} g^{2} h + 3 \, c^{2} d g h^{2} - c^{3} h^{3}} + \frac {2 \, d h x + 3 \, d g - c h}{d^{2} g^{4} - 2 \, c d g^{3} h + c^{2} g^{2} h^{2} + {\left (d^{2} g^{2} h^{2} - 2 \, c d g h^{3} + c^{2} h^{4}\right )} x^{2} + 2 \, {\left (d^{2} g^{3} h - 2 \, c d g^{2} h^{2} + c^{2} g h^{3}\right )} x}\right )} d f q\right )} r}{6 \, f h} - \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{3 \, {\left (h x + g\right )}^{3} h} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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.
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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.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1765 vs.
\(2 (243) = 486\).
time = 5.31, size = 1765, normalized size = 6.79 \begin {gather*} \frac {b^{4} p r \log \left ({\left | b x + a \right |}\right )}{3 \, {\left (b^{4} g^{3} h - 3 \, a b^{3} g^{2} h^{2} + 3 \, a^{2} b^{2} g h^{3} - a^{3} b h^{4}\right )}} + \frac {d^{4} q r \log \left ({\left | d x + c \right |}\right )}{3 \, {\left (d^{4} g^{3} h - 3 \, c d^{3} g^{2} h^{2} + 3 \, c^{2} d^{2} g h^{3} - c^{3} d h^{4}\right )}} - \frac {p r \log \left (b x + a\right )}{3 \, {\left (h^{4} x^{3} + 3 \, g h^{3} x^{2} + 3 \, g^{2} h^{2} x + g^{3} h\right )}} - \frac {q r \log \left (d x + c\right )}{3 \, {\left (h^{4} x^{3} + 3 \, g h^{3} x^{2} + 3 \, g^{2} h^{2} x + g^{3} h\right )}} - \frac {{\left (b^{3} d^{3} g^{3} p r - 3 \, b^{3} c d^{2} g^{2} h p r + 3 \, b^{3} c^{2} d g h^{2} p r - b^{3} c^{3} h^{3} p r + b^{3} d^{3} g^{3} q r - 3 \, a b^{2} d^{3} g^{2} h q r + 3 \, a^{2} b d^{3} g h^{2} q r - a^{3} d^{3} h^{3} q r\right )} \log \left (h x + g\right )}{3 \, {\left (b^{3} d^{3} g^{6} h - 3 \, b^{3} c d^{2} g^{5} h^{2} - 3 \, a b^{2} d^{3} g^{5} h^{2} + 3 \, b^{3} c^{2} d g^{4} h^{3} + 9 \, a b^{2} c d^{2} g^{4} h^{3} + 3 \, a^{2} b d^{3} g^{4} h^{3} - b^{3} c^{3} g^{3} h^{4} - 9 \, a b^{2} c^{2} d g^{3} h^{4} - 9 \, a^{2} b c d^{2} g^{3} h^{4} - a^{3} d^{3} g^{3} h^{4} + 3 \, a b^{2} c^{3} g^{2} h^{5} + 9 \, a^{2} b c^{2} d g^{2} h^{5} + 3 \, a^{3} c d^{2} g^{2} h^{5} - 3 \, a^{2} b c^{3} g h^{6} - 3 \, a^{3} c^{2} d g h^{6} + a^{3} c^{3} h^{7}\right )}} + \frac {2 \, b^{2} d^{2} g^{2} h^{2} p r x^{2} - 4 \, b^{2} c d g h^{3} p r x^{2} + 2 \, b^{2} c^{2} h^{4} p r x^{2} + 2 \, b^{2} d^{2} g^{2} h^{2} q r x^{2} - 4 \, a b d^{2} g h^{3} q r x^{2} + 2 \, a^{2} d^{2} h^{4} q r x^{2} + 5 \, b^{2} d^{2} g^{3} h p r x - 10 \, b^{2} c d g^{2} h^{2} p r x - a b d^{2} g^{2} h^{2} p r x + 5 \, b^{2} c^{2} g h^{3} p r x + 2 \, a b c d g h^{3} p r x - a b c^{2} h^{4} p r x + 5 \, b^{2} d^{2} g^{3} h q r x - b^{2} c d g^{2} h^{2} q r x - 10 \, a b d^{2} g^{2} h^{2} q r x + 2 \, a b c d g h^{3} q r x + 5 \, a^{2} d^{2} g h^{3} q r x - a^{2} c d h^{4} q r x + 3 \, b^{2} d^{2} g^{4} p r - 6 \, b^{2} c d g^{3} h p r - a b d^{2} g^{3} h p r + 3 \, b^{2} c^{2} g^{2} h^{2} p r + 2 \, a b c d g^{2} h^{2} p r - a b c^{2} g h^{3} p r + 3 \, b^{2} d^{2} g^{4} q r - b^{2} c d g^{3} h q r - 6 \, a b d^{2} g^{3} h q r + 2 \, a b c d g^{2} h^{2} q r + 3 \, a^{2} d^{2} g^{2} h^{2} q r - a^{2} c d g h^{3} q r - 2 \, b^{2} d^{2} g^{4} r \log \left (f\right ) + 4 \, b^{2} c d g^{3} h r \log \left (f\right ) + 4 \, a b d^{2} g^{3} h r \log \left (f\right ) - 2 \, b^{2} c^{2} g^{2} h^{2} r \log \left (f\right ) - 8 \, a b c d g^{2} h^{2} r \log \left (f\right ) - 2 \, a^{2} d^{2} g^{2} h^{2} r \log \left (f\right ) + 4 \, a b c^{2} g h^{3} r \log \left (f\right ) + 4 \, a^{2} c d g h^{3} r \log \left (f\right ) - 2 \, a^{2} c^{2} h^{4} r \log \left (f\right ) - 2 \, b^{2} d^{2} g^{4} + 4 \, b^{2} c d g^{3} h + 4 \, a b d^{2} g^{3} h - 2 \, b^{2} c^{2} g^{2} h^{2} - 8 \, a b c d g^{2} h^{2} - 2 \, a^{2} d^{2} g^{2} h^{2} + 4 \, a b c^{2} g h^{3} + 4 \, a^{2} c d g h^{3} - 2 \, a^{2} c^{2} h^{4}}{6 \, {\left (b^{2} d^{2} g^{4} h^{4} x^{3} - 2 \, b^{2} c d g^{3} h^{5} x^{3} - 2 \, a b d^{2} g^{3} h^{5} x^{3} + b^{2} c^{2} g^{2} h^{6} x^{3} + 4 \, a b c d g^{2} h^{6} x^{3} + a^{2} d^{2} g^{2} h^{6} x^{3} - 2 \, a b c^{2} g h^{7} x^{3} - 2 \, a^{2} c d g h^{7} x^{3} + a^{2} c^{2} h^{8} x^{3} + 3 \, b^{2} d^{2} g^{5} h^{3} x^{2} - 6 \, b^{2} c d g^{4} h^{4} x^{2} - 6 \, a b d^{2} g^{4} h^{4} x^{2} + 3 \, b^{2} c^{2} g^{3} h^{5} x^{2} + 12 \, a b c d g^{3} h^{5} x^{2} + 3 \, a^{2} d^{2} g^{3} h^{5} x^{2} - 6 \, a b c^{2} g^{2} h^{6} x^{2} - 6 \, a^{2} c d g^{2} h^{6} x^{2} + 3 \, a^{2} c^{2} g h^{7} x^{2} + 3 \, b^{2} d^{2} g^{6} h^{2} x - 6 \, b^{2} c d g^{5} h^{3} x - 6 \, a b d^{2} g^{5} h^{3} x + 3 \, b^{2} c^{2} g^{4} h^{4} x + 12 \, a b c d g^{4} h^{4} x + 3 \, a^{2} d^{2} g^{4} h^{4} x - 6 \, a b c^{2} g^{3} h^{5} x - 6 \, a^{2} c d g^{3} h^{5} x + 3 \, a^{2} c^{2} g^{2} h^{6} x + b^{2} d^{2} g^{7} h - 2 \, b^{2} c d g^{6} h^{2} - 2 \, a b d^{2} g^{6} h^{2} + b^{2} c^{2} g^{5} h^{3} + 4 \, a b c d g^{5} h^{3} + a^{2} d^{2} g^{5} h^{3} - 2 \, a b c^{2} g^{4} h^{4} - 2 \, a^{2} c d g^{4} h^{4} + a^{2} c^{2} g^{3} h^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.48, size = 977, normalized size = 3.76 \begin {gather*} \frac {\frac {3\,b^2\,d^2\,g^3\,p\,r+3\,b^2\,d^2\,g^3\,q\,r-a\,b\,c^2\,h^3\,p\,r-a^2\,c\,d\,h^3\,q\,r+3\,b^2\,c^2\,g\,h^2\,p\,r+3\,a^2\,d^2\,g\,h^2\,q\,r-a\,b\,d^2\,g^2\,h\,p\,r-6\,a\,b\,d^2\,g^2\,h\,q\,r-6\,b^2\,c\,d\,g^2\,h\,p\,r-b^2\,c\,d\,g^2\,h\,q\,r+2\,a\,b\,c\,d\,g\,h^2\,p\,r+2\,a\,b\,c\,d\,g\,h^2\,q\,r}{2\,\left (a^2\,c^2\,h^4-2\,a^2\,c\,d\,g\,h^3+a^2\,d^2\,g^2\,h^2-2\,a\,b\,c^2\,g\,h^3+4\,a\,b\,c\,d\,g^2\,h^2-2\,a\,b\,d^2\,g^3\,h+b^2\,c^2\,g^2\,h^2-2\,b^2\,c\,d\,g^3\,h+b^2\,d^2\,g^4\right )}+\frac {x\,\left (b^2\,c^2\,h^3\,p\,r+a^2\,d^2\,h^3\,q\,r+b^2\,d^2\,g^2\,h\,p\,r+b^2\,d^2\,g^2\,h\,q\,r-2\,a\,b\,d^2\,g\,h^2\,q\,r-2\,b^2\,c\,d\,g\,h^2\,p\,r\right )}{a^2\,c^2\,h^4-2\,a^2\,c\,d\,g\,h^3+a^2\,d^2\,g^2\,h^2-2\,a\,b\,c^2\,g\,h^3+4\,a\,b\,c\,d\,g^2\,h^2-2\,a\,b\,d^2\,g^3\,h+b^2\,c^2\,g^2\,h^2-2\,b^2\,c\,d\,g^3\,h+b^2\,d^2\,g^4}}{3\,g^2\,h+6\,g\,h^2\,x+3\,h^3\,x^2}+\frac {\ln \left (g+h\,x\right )\,\left (g^2\,\left (3\,c\,h\,p\,r\,b^3\,d^2+3\,a\,h\,q\,r\,b^2\,d^3\right )-g^3\,\left (b^3\,d^3\,p\,r+b^3\,d^3\,q\,r\right )-g\,\left (3\,q\,r\,a^2\,b\,d^3\,h^2+3\,p\,r\,b^3\,c^2\,d\,h^2\right )+b^3\,c^3\,h^3\,p\,r+a^3\,d^3\,h^3\,q\,r\right )}{3\,a^3\,c^3\,h^7-9\,a^3\,c^2\,d\,g\,h^6+9\,a^3\,c\,d^2\,g^2\,h^5-3\,a^3\,d^3\,g^3\,h^4-9\,a^2\,b\,c^3\,g\,h^6+27\,a^2\,b\,c^2\,d\,g^2\,h^5-27\,a^2\,b\,c\,d^2\,g^3\,h^4+9\,a^2\,b\,d^3\,g^4\,h^3+9\,a\,b^2\,c^3\,g^2\,h^5-27\,a\,b^2\,c^2\,d\,g^3\,h^4+27\,a\,b^2\,c\,d^2\,g^4\,h^3-9\,a\,b^2\,d^3\,g^5\,h^2-3\,b^3\,c^3\,g^3\,h^4+9\,b^3\,c^2\,d\,g^4\,h^3-9\,b^3\,c\,d^2\,g^5\,h^2+3\,b^3\,d^3\,g^6\,h}-\frac {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )\,\left (\frac {x}{3}+\frac {g}{3\,h}\right )}{{\left (g+h\,x\right )}^4}-\frac {b^3\,p\,r\,\ln \left (a+b\,x\right )}{3\,a^3\,h^4-9\,a^2\,b\,g\,h^3+9\,a\,b^2\,g^2\,h^2-3\,b^3\,g^3\,h}-\frac {d^3\,q\,r\,\ln \left (c+d\,x\right )}{3\,c^3\,h^4-9\,c^2\,d\,g\,h^3+9\,c\,d^2\,g^2\,h^2-3\,d^3\,g^3\,h} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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