Optimal. Leaf size=334 \[ -\frac {(b g-a h)^4 p r x}{5 b^4}-\frac {(d g-c h)^4 q r x}{5 d^4}-\frac {(b g-a h)^3 p r (g+h x)^2}{10 b^3 h}-\frac {(d g-c h)^3 q r (g+h x)^2}{10 d^3 h}-\frac {(b g-a h)^2 p r (g+h x)^3}{15 b^2 h}-\frac {(d g-c h)^2 q r (g+h x)^3}{15 d^2 h}-\frac {(b g-a h) p r (g+h x)^4}{20 b h}-\frac {(d g-c h) q r (g+h x)^4}{20 d h}-\frac {p r (g+h x)^5}{25 h}-\frac {q r (g+h x)^5}{25 h}-\frac {(b g-a h)^5 p r \log (a+b x)}{5 b^5 h}-\frac {(d g-c h)^5 q r \log (c+d x)}{5 d^5 h}+\frac {(g+h x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 h} \]
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Rubi [A]
time = 0.13, antiderivative size = 334, 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, 45}
\begin {gather*} -\frac {p r (b g-a h)^5 \log (a+b x)}{5 b^5 h}-\frac {p r x (b g-a h)^4}{5 b^4}-\frac {p r (g+h x)^2 (b g-a h)^3}{10 b^3 h}-\frac {p r (g+h x)^3 (b g-a h)^2}{15 b^2 h}+\frac {(g+h x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 h}-\frac {p r (g+h x)^4 (b g-a h)}{20 b h}-\frac {q r (d g-c h)^5 \log (c+d x)}{5 d^5 h}-\frac {q r x (d g-c h)^4}{5 d^4}-\frac {q r (g+h x)^2 (d g-c h)^3}{10 d^3 h}-\frac {q r (g+h x)^3 (d g-c h)^2}{15 d^2 h}-\frac {q r (g+h x)^4 (d g-c h)}{20 d h}-\frac {p r (g+h x)^5}{25 h}-\frac {q r (g+h x)^5}{25 h} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2581
Rubi steps
\begin {align*} \int (g+h x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx &=\frac {(g+h x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 h}-\frac {(b p r) \int \frac {(g+h x)^5}{a+b x} \, dx}{5 h}-\frac {(d q r) \int \frac {(g+h x)^5}{c+d x} \, dx}{5 h}\\ &=\frac {(g+h x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 h}-\frac {(b p r) \int \left (\frac {h (b g-a h)^4}{b^5}+\frac {(b g-a h)^5}{b^5 (a+b x)}+\frac {h (b g-a h)^3 (g+h x)}{b^4}+\frac {h (b g-a h)^2 (g+h x)^2}{b^3}+\frac {h (b g-a h) (g+h x)^3}{b^2}+\frac {h (g+h x)^4}{b}\right ) \, dx}{5 h}-\frac {(d q r) \int \left (\frac {h (d g-c h)^4}{d^5}+\frac {(d g-c h)^5}{d^5 (c+d x)}+\frac {h (d g-c h)^3 (g+h x)}{d^4}+\frac {h (d g-c h)^2 (g+h x)^2}{d^3}+\frac {h (d g-c h) (g+h x)^3}{d^2}+\frac {h (g+h x)^4}{d}\right ) \, dx}{5 h}\\ &=-\frac {(b g-a h)^4 p r x}{5 b^4}-\frac {(d g-c h)^4 q r x}{5 d^4}-\frac {(b g-a h)^3 p r (g+h x)^2}{10 b^3 h}-\frac {(d g-c h)^3 q r (g+h x)^2}{10 d^3 h}-\frac {(b g-a h)^2 p r (g+h x)^3}{15 b^2 h}-\frac {(d g-c h)^2 q r (g+h x)^3}{15 d^2 h}-\frac {(b g-a h) p r (g+h x)^4}{20 b h}-\frac {(d g-c h) q r (g+h x)^4}{20 d h}-\frac {p r (g+h x)^5}{25 h}-\frac {q r (g+h x)^5}{25 h}-\frac {(b g-a h)^5 p r \log (a+b x)}{5 b^5 h}-\frac {(d g-c h)^5 q r \log (c+d x)}{5 d^5 h}+\frac {(g+h x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 h}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 275, normalized size = 0.82 \begin {gather*} \frac {-\frac {p r \left (60 b h (b g-a h)^4 x+30 b^2 (b g-a h)^3 (g+h x)^2+20 b^3 (b g-a h)^2 (g+h x)^3+15 b^4 (b g-a h) (g+h x)^4+12 b^5 (g+h x)^5+60 (b g-a h)^5 \log (a+b x)\right )}{60 b^5}-\frac {q r \left (60 d h (d g-c h)^4 x+30 d^2 (d g-c h)^3 (g+h x)^2+20 d^3 (d g-c h)^2 (g+h x)^3+15 d^4 (d g-c h) (g+h x)^4+12 d^5 (g+h x)^5+60 (d g-c h)^5 \log (c+d x)\right )}{60 d^5}+(g+h x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 h} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.16, size = 0, normalized size = 0.00 \[\int \left (h x +g \right )^{4} \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 625 vs.
\(2 (309) = 618\).
time = 0.29, size = 625, normalized size = 1.87 \begin {gather*} \frac {1}{5} \, {\left (h^{4} x^{5} + 5 \, g h^{3} x^{4} + 10 \, g^{2} h^{2} x^{3} + 10 \, g^{3} h x^{2} + 5 \, g^{4} x\right )} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) + \frac {r {\left (\frac {60 \, {\left (5 \, a b^{4} f g^{4} p - 10 \, a^{2} b^{3} f g^{3} h p + 10 \, a^{3} b^{2} f g^{2} h^{2} p - 5 \, a^{4} b f g h^{3} p + a^{5} f h^{4} p\right )} \log \left (b x + a\right )}{b^{5}} + \frac {60 \, {\left (5 \, c d^{4} f g^{4} q - 10 \, c^{2} d^{3} f g^{3} h q + 10 \, c^{3} d^{2} f g^{2} h^{2} q - 5 \, c^{4} d f g h^{3} q + c^{5} f h^{4} q\right )} \log \left (d x + c\right )}{d^{5}} - \frac {12 \, b^{4} d^{4} f h^{4} {\left (p + q\right )} x^{5} - 15 \, {\left (a b^{3} d^{4} f h^{4} p - {\left (5 \, d^{4} f g h^{3} {\left (p + q\right )} - c d^{3} f h^{4} q\right )} b^{4}\right )} x^{4} - 20 \, {\left (5 \, a b^{3} d^{4} f g h^{3} p - a^{2} b^{2} d^{4} f h^{4} p - {\left (10 \, d^{4} f g^{2} h^{2} {\left (p + q\right )} - 5 \, c d^{3} f g h^{3} q + c^{2} d^{2} f h^{4} q\right )} b^{4}\right )} x^{3} - 30 \, {\left (10 \, a b^{3} d^{4} f g^{2} h^{2} p - 5 \, a^{2} b^{2} d^{4} f g h^{3} p + a^{3} b d^{4} f h^{4} p - {\left (10 \, d^{4} f g^{3} h {\left (p + q\right )} - 10 \, c d^{3} f g^{2} h^{2} q + 5 \, c^{2} d^{2} f g h^{3} q - c^{3} d f h^{4} q\right )} b^{4}\right )} x^{2} - 60 \, {\left (10 \, a b^{3} d^{4} f g^{3} h p - 10 \, a^{2} b^{2} d^{4} f g^{2} h^{2} p + 5 \, a^{3} b d^{4} f g h^{3} p - a^{4} d^{4} f h^{4} p - {\left (5 \, d^{4} f g^{4} {\left (p + q\right )} - 10 \, c d^{3} f g^{3} h q + 10 \, c^{2} d^{2} f g^{2} h^{2} q - 5 \, c^{3} d f g h^{3} q + c^{4} f h^{4} q\right )} b^{4}\right )} x}{b^{4} d^{4}}\right )}}{300 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 943 vs.
\(2 (309) = 618\).
time = 0.41, size = 943, normalized size = 2.82 \begin {gather*} \frac {12 \, {\left (5 \, b^{5} d^{5} h^{4} - {\left (b^{5} d^{5} h^{4} p + b^{5} d^{5} h^{4} q\right )} r\right )} x^{5} + 15 \, {\left (20 \, b^{5} d^{5} g h^{3} - {\left ({\left (5 \, b^{5} d^{5} g h^{3} - a b^{4} d^{5} h^{4}\right )} p + {\left (5 \, b^{5} d^{5} g h^{3} - b^{5} c d^{4} h^{4}\right )} q\right )} r\right )} x^{4} + 20 \, {\left (30 \, b^{5} d^{5} g^{2} h^{2} - {\left ({\left (10 \, b^{5} d^{5} g^{2} h^{2} - 5 \, a b^{4} d^{5} g h^{3} + a^{2} b^{3} d^{5} h^{4}\right )} p + {\left (10 \, b^{5} d^{5} g^{2} h^{2} - 5 \, b^{5} c d^{4} g h^{3} + b^{5} c^{2} d^{3} h^{4}\right )} q\right )} r\right )} x^{3} + 30 \, {\left (20 \, b^{5} d^{5} g^{3} h - {\left ({\left (10 \, b^{5} d^{5} g^{3} h - 10 \, a b^{4} d^{5} g^{2} h^{2} + 5 \, a^{2} b^{3} d^{5} g h^{3} - a^{3} b^{2} d^{5} h^{4}\right )} p + {\left (10 \, b^{5} d^{5} g^{3} h - 10 \, b^{5} c d^{4} g^{2} h^{2} + 5 \, b^{5} c^{2} d^{3} g h^{3} - b^{5} c^{3} d^{2} h^{4}\right )} q\right )} r\right )} x^{2} + 60 \, {\left (5 \, b^{5} d^{5} g^{4} - {\left ({\left (5 \, b^{5} d^{5} g^{4} - 10 \, a b^{4} d^{5} g^{3} h + 10 \, a^{2} b^{3} d^{5} g^{2} h^{2} - 5 \, a^{3} b^{2} d^{5} g h^{3} + a^{4} b d^{5} h^{4}\right )} p + {\left (5 \, b^{5} d^{5} g^{4} - 10 \, b^{5} c d^{4} g^{3} h + 10 \, b^{5} c^{2} d^{3} g^{2} h^{2} - 5 \, b^{5} c^{3} d^{2} g h^{3} + b^{5} c^{4} d h^{4}\right )} q\right )} r\right )} x + 60 \, {\left (b^{5} d^{5} h^{4} p r x^{5} + 5 \, b^{5} d^{5} g h^{3} p r x^{4} + 10 \, b^{5} d^{5} g^{2} h^{2} p r x^{3} + 10 \, b^{5} d^{5} g^{3} h p r x^{2} + 5 \, b^{5} d^{5} g^{4} p r x + {\left (5 \, a b^{4} d^{5} g^{4} - 10 \, a^{2} b^{3} d^{5} g^{3} h + 10 \, a^{3} b^{2} d^{5} g^{2} h^{2} - 5 \, a^{4} b d^{5} g h^{3} + a^{5} d^{5} h^{4}\right )} p r\right )} \log \left (b x + a\right ) + 60 \, {\left (b^{5} d^{5} h^{4} q r x^{5} + 5 \, b^{5} d^{5} g h^{3} q r x^{4} + 10 \, b^{5} d^{5} g^{2} h^{2} q r x^{3} + 10 \, b^{5} d^{5} g^{3} h q r x^{2} + 5 \, b^{5} d^{5} g^{4} q r x + {\left (5 \, b^{5} c d^{4} g^{4} - 10 \, b^{5} c^{2} d^{3} g^{3} h + 10 \, b^{5} c^{3} d^{2} g^{2} h^{2} - 5 \, b^{5} c^{4} d g h^{3} + b^{5} c^{5} h^{4}\right )} q r\right )} \log \left (d x + c\right ) + 60 \, {\left (b^{5} d^{5} h^{4} r x^{5} + 5 \, b^{5} d^{5} g h^{3} r x^{4} + 10 \, b^{5} d^{5} g^{2} h^{2} r x^{3} + 10 \, b^{5} d^{5} g^{3} h r x^{2} + 5 \, b^{5} d^{5} g^{4} r x\right )} \log \left (f\right )}{300 \, b^{5} d^{5}} \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 [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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Mupad [B]
time = 1.03, size = 1128, normalized size = 3.38 \begin {gather*} \ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )\,\left (g^4\,x+2\,g^3\,h\,x^2+2\,g^2\,h^2\,x^3+g\,h^3\,x^4+\frac {h^4\,x^5}{5}\right )-x^2\,\left (\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {h^3\,r\,\left (b\,c\,h\,p+5\,b\,d\,g\,p+a\,d\,h\,q+5\,b\,d\,g\,q\right )}{5\,b\,d}-\frac {h^4\,r\,\left (p+q\right )\,\left (5\,a\,d+5\,b\,c\right )}{25\,b\,d}\right )}{5\,b\,d}-\frac {g\,h^2\,r\,\left (b\,c\,h\,p+2\,b\,d\,g\,p+a\,d\,h\,q+2\,b\,d\,g\,q\right )}{b\,d}+\frac {a\,c\,h^4\,r\,\left (p+q\right )}{5\,b\,d}\right )}{10\,b\,d}-\frac {a\,c\,\left (\frac {h^3\,r\,\left (b\,c\,h\,p+5\,b\,d\,g\,p+a\,d\,h\,q+5\,b\,d\,g\,q\right )}{5\,b\,d}-\frac {h^4\,r\,\left (p+q\right )\,\left (5\,a\,d+5\,b\,c\right )}{25\,b\,d}\right )}{2\,b\,d}+\frac {g^2\,h\,r\,\left (b\,c\,h\,p+b\,d\,g\,p+a\,d\,h\,q+b\,d\,g\,q\right )}{b\,d}\right )-x^4\,\left (\frac {h^3\,r\,\left (b\,c\,h\,p+5\,b\,d\,g\,p+a\,d\,h\,q+5\,b\,d\,g\,q\right )}{20\,b\,d}-\frac {h^4\,r\,\left (p+q\right )\,\left (5\,a\,d+5\,b\,c\right )}{100\,b\,d}\right )-x\,\left (\frac {a\,c\,\left (\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {h^3\,r\,\left (b\,c\,h\,p+5\,b\,d\,g\,p+a\,d\,h\,q+5\,b\,d\,g\,q\right )}{5\,b\,d}-\frac {h^4\,r\,\left (p+q\right )\,\left (5\,a\,d+5\,b\,c\right )}{25\,b\,d}\right )}{5\,b\,d}-\frac {g\,h^2\,r\,\left (b\,c\,h\,p+2\,b\,d\,g\,p+a\,d\,h\,q+2\,b\,d\,g\,q\right )}{b\,d}+\frac {a\,c\,h^4\,r\,\left (p+q\right )}{5\,b\,d}\right )}{b\,d}-\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {h^3\,r\,\left (b\,c\,h\,p+5\,b\,d\,g\,p+a\,d\,h\,q+5\,b\,d\,g\,q\right )}{5\,b\,d}-\frac {h^4\,r\,\left (p+q\right )\,\left (5\,a\,d+5\,b\,c\right )}{25\,b\,d}\right )}{5\,b\,d}-\frac {g\,h^2\,r\,\left (b\,c\,h\,p+2\,b\,d\,g\,p+a\,d\,h\,q+2\,b\,d\,g\,q\right )}{b\,d}+\frac {a\,c\,h^4\,r\,\left (p+q\right )}{5\,b\,d}\right )}{5\,b\,d}-\frac {a\,c\,\left (\frac {h^3\,r\,\left (b\,c\,h\,p+5\,b\,d\,g\,p+a\,d\,h\,q+5\,b\,d\,g\,q\right )}{5\,b\,d}-\frac {h^4\,r\,\left (p+q\right )\,\left (5\,a\,d+5\,b\,c\right )}{25\,b\,d}\right )}{b\,d}+\frac {2\,g^2\,h\,r\,\left (b\,c\,h\,p+b\,d\,g\,p+a\,d\,h\,q+b\,d\,g\,q\right )}{b\,d}\right )}{5\,b\,d}+\frac {g^3\,r\,\left (2\,b\,c\,h\,p+b\,d\,g\,p+2\,a\,d\,h\,q+b\,d\,g\,q\right )}{b\,d}\right )+x^3\,\left (\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {h^3\,r\,\left (b\,c\,h\,p+5\,b\,d\,g\,p+a\,d\,h\,q+5\,b\,d\,g\,q\right )}{5\,b\,d}-\frac {h^4\,r\,\left (p+q\right )\,\left (5\,a\,d+5\,b\,c\right )}{25\,b\,d}\right )}{15\,b\,d}-\frac {g\,h^2\,r\,\left (b\,c\,h\,p+2\,b\,d\,g\,p+a\,d\,h\,q+2\,b\,d\,g\,q\right )}{3\,b\,d}+\frac {a\,c\,h^4\,r\,\left (p+q\right )}{15\,b\,d}\right )+\frac {\ln \left (a+b\,x\right )\,\left (\frac {p\,r\,a^5\,h^4}{5}-p\,r\,a^4\,b\,g\,h^3+2\,p\,r\,a^3\,b^2\,g^2\,h^2-2\,p\,r\,a^2\,b^3\,g^3\,h+p\,r\,a\,b^4\,g^4\right )}{b^5}+\frac {\ln \left (c+d\,x\right )\,\left (\frac {q\,r\,c^5\,h^4}{5}-q\,r\,c^4\,d\,g\,h^3+2\,q\,r\,c^3\,d^2\,g^2\,h^2-2\,q\,r\,c^2\,d^3\,g^3\,h+q\,r\,c\,d^4\,g^4\right )}{d^5}-\frac {h^4\,r\,x^5\,\left (p+q\right )}{25} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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