Optimal. Leaf size=218 \[ -\frac {p r (b g-a h)^3 \log (a+b x)}{3 b^3 h}-\frac {p r x (b g-a h)^2}{3 b^2}+\frac {(g+h x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}-\frac {p r (g+h x)^2 (b g-a h)}{6 b h}-\frac {q r (d g-c h)^3 \log (c+d x)}{3 d^3 h}-\frac {q r x (d g-c h)^2}{3 d^2}-\frac {q r (g+h x)^2 (d g-c h)}{6 d h}-\frac {p r (g+h x)^3}{9 h}-\frac {q r (g+h x)^3}{9 h} \]
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Rubi [A] time = 0.10, antiderivative size = 218, 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 = {2495, 43} \[ -\frac {p r x (b g-a h)^2}{3 b^2}-\frac {p r (b g-a h)^3 \log (a+b x)}{3 b^3 h}+\frac {(g+h x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}-\frac {p r (g+h x)^2 (b g-a h)}{6 b h}-\frac {q r x (d g-c h)^2}{3 d^2}-\frac {q r (d g-c h)^3 \log (c+d x)}{3 d^3 h}-\frac {q r (g+h x)^2 (d g-c h)}{6 d h}-\frac {p r (g+h x)^3}{9 h}-\frac {q r (g+h x)^3}{9 h} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2495
Rubi steps
\begin {align*} \int (g+h x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx &=\frac {(g+h x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}-\frac {(b p r) \int \frac {(g+h x)^3}{a+b x} \, dx}{3 h}-\frac {(d q r) \int \frac {(g+h x)^3}{c+d x} \, dx}{3 h}\\ &=\frac {(g+h x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}-\frac {(b p r) \int \left (\frac {h (b g-a h)^2}{b^3}+\frac {(b g-a h)^3}{b^3 (a+b x)}+\frac {h (b g-a h) (g+h x)}{b^2}+\frac {h (g+h x)^2}{b}\right ) \, dx}{3 h}-\frac {(d q r) \int \left (\frac {h (d g-c h)^2}{d^3}+\frac {(d g-c h)^3}{d^3 (c+d x)}+\frac {h (d g-c h) (g+h x)}{d^2}+\frac {h (g+h x)^2}{d}\right ) \, dx}{3 h}\\ &=-\frac {(b g-a h)^2 p r x}{3 b^2}-\frac {(d g-c h)^2 q r x}{3 d^2}-\frac {(b g-a h) p r (g+h x)^2}{6 b h}-\frac {(d g-c h) q r (g+h x)^2}{6 d h}-\frac {p r (g+h x)^3}{9 h}-\frac {q r (g+h x)^3}{9 h}-\frac {(b g-a h)^3 p r \log (a+b x)}{3 b^3 h}-\frac {(d g-c h)^3 q r \log (c+d x)}{3 d^3 h}+\frac {(g+h x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 209, normalized size = 0.96 \[ \frac {(g+h x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-\frac {r \left (b \left (6 a^2 d^3 h^3 p x-3 a b d^3 h p \left (g^2+6 g h x+h^2 x^2\right )+b^2 d \left (6 c^2 h^3 q x-3 c d h q \left (g^2+6 g h x+h^2 x^2\right )+d^2 (p+q) \left (5 g^3+18 g^2 h x+9 g h^2 x^2+2 h^3 x^3\right )\right )+6 b^2 q (d g-c h)^3 \log (c+d x)\right )+6 d^3 p (b g-a h)^3 \log (a+b x)\right )}{6 b^3 d^3}}{3 h} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 441, normalized size = 2.02 \[ -\frac {2 \, {\left (b^{3} d^{3} h^{2} p + b^{3} d^{3} h^{2} q\right )} r x^{3} + 3 \, {\left ({\left (3 \, b^{3} d^{3} g h - a b^{2} d^{3} h^{2}\right )} p + {\left (3 \, b^{3} d^{3} g h - b^{3} c d^{2} h^{2}\right )} q\right )} r x^{2} + 6 \, {\left ({\left (3 \, b^{3} d^{3} g^{2} - 3 \, a b^{2} d^{3} g h + a^{2} b d^{3} h^{2}\right )} p + {\left (3 \, b^{3} d^{3} g^{2} - 3 \, b^{3} c d^{2} g h + b^{3} c^{2} d h^{2}\right )} q\right )} r x - 6 \, {\left (b^{3} d^{3} h^{2} p r x^{3} + 3 \, b^{3} d^{3} g h p r x^{2} + 3 \, b^{3} d^{3} g^{2} p r x + {\left (3 \, a b^{2} d^{3} g^{2} - 3 \, a^{2} b d^{3} g h + a^{3} d^{3} h^{2}\right )} p r\right )} \log \left (b x + a\right ) - 6 \, {\left (b^{3} d^{3} h^{2} q r x^{3} + 3 \, b^{3} d^{3} g h q r x^{2} + 3 \, b^{3} d^{3} g^{2} q r x + {\left (3 \, b^{3} c d^{2} g^{2} - 3 \, b^{3} c^{2} d g h + b^{3} c^{3} h^{2}\right )} q r\right )} \log \left (d x + c\right ) - 6 \, {\left (b^{3} d^{3} h^{2} x^{3} + 3 \, b^{3} d^{3} g h x^{2} + 3 \, b^{3} d^{3} g^{2} x\right )} \log \relax (e) - 6 \, {\left (b^{3} d^{3} h^{2} r x^{3} + 3 \, b^{3} d^{3} g h r x^{2} + 3 \, b^{3} d^{3} g^{2} r x\right )} \log \relax (f)}{18 \, b^{3} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 100.51, size = 357, normalized size = 1.64 \[ -\frac {1}{9} \, {\left (h^{2} p r + h^{2} q r - 3 \, h^{2} r \log \relax (f) - 3 \, h^{2}\right )} x^{3} + \frac {1}{3} \, {\left (h^{2} p r x^{3} + 3 \, g h p r x^{2} + 3 \, g^{2} p r x\right )} \log \left (b x + a\right ) + \frac {1}{3} \, {\left (h^{2} q r x^{3} + 3 \, g h q r x^{2} + 3 \, g^{2} q r x\right )} \log \left (d x + c\right ) - \frac {{\left (3 \, b d g h p r - a d h^{2} p r + 3 \, b d g h q r - b c h^{2} q r - 6 \, b d g h r \log \relax (f) - 6 \, b d g h\right )} x^{2}}{6 \, b d} + \frac {{\left (3 \, a b^{2} g^{2} p r - 3 \, a^{2} b g h p r + a^{3} h^{2} p r\right )} \log \left (b x + a\right )}{3 \, b^{3}} + \frac {{\left (3 \, c d^{2} g^{2} q r - 3 \, c^{2} d g h q r + c^{3} h^{2} q r\right )} \log \left (-d x - c\right )}{3 \, d^{3}} - \frac {{\left (3 \, b^{2} d^{2} g^{2} p r - 3 \, a b d^{2} g h p r + a^{2} d^{2} h^{2} p r + 3 \, b^{2} d^{2} g^{2} q r - 3 \, b^{2} c d g h q r + b^{2} c^{2} h^{2} q r - 3 \, b^{2} d^{2} g^{2} r \log \relax (f) - 3 \, b^{2} d^{2} g^{2}\right )} x}{3 \, b^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.29, size = 0, normalized size = 0.00 \[ \int \left (h x +g \right )^{2} \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.92, size = 269, normalized size = 1.23 \[ \frac {1}{3} \, {\left (h^{2} x^{3} + 3 \, g h x^{2} + 3 \, g^{2} x\right )} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) + \frac {r {\left (\frac {6 \, {\left (3 \, a b^{2} f g^{2} p - 3 \, a^{2} b f g h p + a^{3} f h^{2} p\right )} \log \left (b x + a\right )}{b^{3}} + \frac {6 \, {\left (3 \, c d^{2} f g^{2} q - 3 \, c^{2} d f g h q + c^{3} f h^{2} q\right )} \log \left (d x + c\right )}{d^{3}} - \frac {2 \, b^{2} d^{2} f h^{2} {\left (p + q\right )} x^{3} - 3 \, {\left (a b d^{2} f h^{2} p - {\left (3 \, d^{2} f g h {\left (p + q\right )} - c d f h^{2} q\right )} b^{2}\right )} x^{2} - 6 \, {\left (3 \, a b d^{2} f g h p - a^{2} d^{2} f h^{2} p - {\left (3 \, d^{2} f g^{2} {\left (p + q\right )} - 3 \, c d f g h q + c^{2} f h^{2} q\right )} b^{2}\right )} x}{b^{2} d^{2}}\right )}}{18 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.65, size = 328, normalized size = 1.50 \[ x\,\left (\frac {\left (\frac {h\,r\,\left (b\,c\,h\,p+3\,b\,d\,g\,p+a\,d\,h\,q+3\,b\,d\,g\,q\right )}{3\,b\,d}-\frac {h^2\,r\,\left (p+q\right )\,\left (3\,a\,d+3\,b\,c\right )}{9\,b\,d}\right )\,\left (3\,a\,d+3\,b\,c\right )}{3\,b\,d}-\frac {g\,r\,\left (b\,c\,h\,p+b\,d\,g\,p+a\,d\,h\,q+b\,d\,g\,q\right )}{b\,d}+\frac {a\,c\,h^2\,r\,\left (p+q\right )}{3\,b\,d}\right )-x^2\,\left (\frac {h\,r\,\left (b\,c\,h\,p+3\,b\,d\,g\,p+a\,d\,h\,q+3\,b\,d\,g\,q\right )}{6\,b\,d}-\frac {h^2\,r\,\left (p+q\right )\,\left (3\,a\,d+3\,b\,c\right )}{18\,b\,d}\right )+\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )\,\left (g^2\,x+g\,h\,x^2+\frac {h^2\,x^3}{3}\right )+\frac {\ln \left (a+b\,x\right )\,\left (p\,r\,a^3\,h^2-3\,p\,r\,a^2\,b\,g\,h+3\,p\,r\,a\,b^2\,g^2\right )}{3\,b^3}+\frac {\ln \left (c+d\,x\right )\,\left (q\,r\,c^3\,h^2-3\,q\,r\,c^2\,d\,g\,h+3\,q\,r\,c\,d^2\,g^2\right )}{3\,d^3}-\frac {h^2\,r\,x^3\,\left (p+q\right )}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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