Integrand size = 23, antiderivative size = 126 \[ \int \frac {(a+b x) (c+d x) (e+f x)}{g+h x} \, dx=\frac {(b (d g-c h) (f g-e h)-a h (d f g-d e h-c f h)) x}{h^3}+\frac {(a d f h-b (d f g-d e h-c f h)) x^2}{2 h^2}+\frac {b d f x^3}{3 h}-\frac {(b g-a h) (d g-c h) (f g-e h) \log (g+h x)}{h^4} \]
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Time = 0.14 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {147} \[ \int \frac {(a+b x) (c+d x) (e+f x)}{g+h x} \, dx=-\frac {(b g-a h) (d g-c h) (f g-e h) \log (g+h x)}{h^4}+\frac {x (b (d g-c h) (f g-e h)-a h (-c f h-d e h+d f g))}{h^3}+\frac {x^2 (a d f h-b (-c f h-d e h+d f g))}{2 h^2}+\frac {b d f x^3}{3 h} \]
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Rule 147
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {b (d g-c h) (f g-e h)-a h (d f g-d e h-c f h)}{h^3}+\frac {(a d f h-b (d f g-d e h-c f h)) x}{h^2}+\frac {b d f x^2}{h}+\frac {(-b g+a h) (-d g+c h) (-f g+e h)}{h^3 (g+h x)}\right ) \, dx \\ & = \frac {(b (d g-c h) (f g-e h)-a h (d f g-d e h-c f h)) x}{h^3}+\frac {(a d f h-b (d f g-d e h-c f h)) x^2}{2 h^2}+\frac {b d f x^3}{3 h}-\frac {(b g-a h) (d g-c h) (f g-e h) \log (g+h x)}{h^4} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.98 \[ \int \frac {(a+b x) (c+d x) (e+f x)}{g+h x} \, dx=\frac {h x \left (3 a h (2 c f h+d (-2 f g+2 e h+f h x))+b \left (3 d e h (-2 g+h x)+3 c h (-2 f g+2 e h+f h x)+d f \left (6 g^2-3 g h x+2 h^2 x^2\right )\right )\right )-6 (b g-a h) (d g-c h) (f g-e h) \log (g+h x)}{6 h^4} \]
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Time = 1.52 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.39
method | result | size |
norman | \(\frac {\left (a c f \,h^{2}+a d e \,h^{2}-a d f g h +b c e \,h^{2}-b c f g h -b d e g h +b d f \,g^{2}\right ) x}{h^{3}}+\frac {\left (a d f h +b c f h +b d e h -b d f g \right ) x^{2}}{2 h^{2}}+\frac {b d f \,x^{3}}{3 h}+\frac {\left (a c e \,h^{3}-a c f g \,h^{2}-a d e g \,h^{2}+a d f \,g^{2} h -b c e g \,h^{2}+b c f \,g^{2} h +b d e \,g^{2} h -b d f \,g^{3}\right ) \ln \left (h x +g \right )}{h^{4}}\) | \(175\) |
default | \(\frac {\frac {1}{3} b d f \,x^{3} h^{2}+\frac {1}{2} a d f \,h^{2} x^{2}+\frac {1}{2} b c f \,h^{2} x^{2}+\frac {1}{2} b d e \,h^{2} x^{2}-\frac {1}{2} b d f g h \,x^{2}+a c f \,h^{2} x +a d e \,h^{2} x -a d f g h x +b c e \,h^{2} x -b c f g h x -b d e g h x +b d f \,g^{2} x}{h^{3}}+\frac {\left (a c e \,h^{3}-a c f g \,h^{2}-a d e g \,h^{2}+a d f \,g^{2} h -b c e g \,h^{2}+b c f \,g^{2} h +b d e \,g^{2} h -b d f \,g^{3}\right ) \ln \left (h x +g \right )}{h^{4}}\) | \(194\) |
risch | \(\frac {b d f \,x^{3}}{3 h}+\frac {a d f \,x^{2}}{2 h}+\frac {b c f \,x^{2}}{2 h}+\frac {b d e \,x^{2}}{2 h}-\frac {b d f g \,x^{2}}{2 h^{2}}+\frac {a c f x}{h}+\frac {a d e x}{h}-\frac {a d f g x}{h^{2}}+\frac {b c e x}{h}-\frac {b c f g x}{h^{2}}-\frac {b d e g x}{h^{2}}+\frac {b d f \,g^{2} x}{h^{3}}+\frac {\ln \left (h x +g \right ) a c e}{h}-\frac {\ln \left (h x +g \right ) a c f g}{h^{2}}-\frac {\ln \left (h x +g \right ) a d e g}{h^{2}}+\frac {\ln \left (h x +g \right ) a d f \,g^{2}}{h^{3}}-\frac {\ln \left (h x +g \right ) b c e g}{h^{2}}+\frac {\ln \left (h x +g \right ) b c f \,g^{2}}{h^{3}}+\frac {\ln \left (h x +g \right ) b d e \,g^{2}}{h^{3}}-\frac {\ln \left (h x +g \right ) b d f \,g^{3}}{h^{4}}\) | \(246\) |
parallelrisch | \(\frac {2 b d f \,x^{3} h^{3}+3 x^{2} a d f \,h^{3}+3 x^{2} b c f \,h^{3}+3 x^{2} b d e \,h^{3}-3 x^{2} b d f g \,h^{2}+6 \ln \left (h x +g \right ) a c e \,h^{3}-6 \ln \left (h x +g \right ) a c f g \,h^{2}-6 \ln \left (h x +g \right ) a d e g \,h^{2}+6 \ln \left (h x +g \right ) a d f \,g^{2} h -6 \ln \left (h x +g \right ) b c e g \,h^{2}+6 \ln \left (h x +g \right ) b c f \,g^{2} h +6 \ln \left (h x +g \right ) b d e \,g^{2} h -6 \ln \left (h x +g \right ) b d f \,g^{3}+6 x a c f \,h^{3}+6 x a d e \,h^{3}-6 x a d f g \,h^{2}+6 x b c e \,h^{3}-6 x b c f g \,h^{2}-6 x b d e g \,h^{2}+6 x b d f \,g^{2} h}{6 h^{4}}\) | \(248\) |
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Time = 0.24 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.29 \[ \int \frac {(a+b x) (c+d x) (e+f x)}{g+h x} \, dx=\frac {2 \, b d f h^{3} x^{3} - 3 \, {\left (b d f g h^{2} - {\left (b d e + {\left (b c + a d\right )} f\right )} h^{3}\right )} x^{2} + 6 \, {\left (b d f g^{2} h - {\left (b d e + {\left (b c + a d\right )} f\right )} g h^{2} + {\left (a c f + {\left (b c + a d\right )} e\right )} h^{3}\right )} x - 6 \, {\left (b d f g^{3} - a c e h^{3} - {\left (b d e + {\left (b c + a d\right )} f\right )} g^{2} h + {\left (a c f + {\left (b c + a d\right )} e\right )} g h^{2}\right )} \log \left (h x + g\right )}{6 \, h^{4}} \]
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Time = 0.31 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.16 \[ \int \frac {(a+b x) (c+d x) (e+f x)}{g+h x} \, dx=\frac {b d f x^{3}}{3 h} + x^{2} \left (\frac {a d f}{2 h} + \frac {b c f}{2 h} + \frac {b d e}{2 h} - \frac {b d f g}{2 h^{2}}\right ) + x \left (\frac {a c f}{h} + \frac {a d e}{h} - \frac {a d f g}{h^{2}} + \frac {b c e}{h} - \frac {b c f g}{h^{2}} - \frac {b d e g}{h^{2}} + \frac {b d f g^{2}}{h^{3}}\right ) + \frac {\left (a h - b g\right ) \left (c h - d g\right ) \left (e h - f g\right ) \log {\left (g + h x \right )}}{h^{4}} \]
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Time = 0.19 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.29 \[ \int \frac {(a+b x) (c+d x) (e+f x)}{g+h x} \, dx=\frac {2 \, b d f h^{2} x^{3} - 3 \, {\left (b d f g h - {\left (b d e + {\left (b c + a d\right )} f\right )} h^{2}\right )} x^{2} + 6 \, {\left (b d f g^{2} - {\left (b d e + {\left (b c + a d\right )} f\right )} g h + {\left (a c f + {\left (b c + a d\right )} e\right )} h^{2}\right )} x}{6 \, h^{3}} - \frac {{\left (b d f g^{3} - a c e h^{3} - {\left (b d e + {\left (b c + a d\right )} f\right )} g^{2} h + {\left (a c f + {\left (b c + a d\right )} e\right )} g h^{2}\right )} \log \left (h x + g\right )}{h^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.59 \[ \int \frac {(a+b x) (c+d x) (e+f x)}{g+h x} \, dx=\frac {2 \, b d f h^{2} x^{3} - 3 \, b d f g h x^{2} + 3 \, b d e h^{2} x^{2} + 3 \, b c f h^{2} x^{2} + 3 \, a d f h^{2} x^{2} + 6 \, b d f g^{2} x - 6 \, b d e g h x - 6 \, b c f g h x - 6 \, a d f g h x + 6 \, b c e h^{2} x + 6 \, a d e h^{2} x + 6 \, a c f h^{2} x}{6 \, h^{3}} - \frac {{\left (b d f g^{3} - b d e g^{2} h - b c f g^{2} h - a d f g^{2} h + b c e g h^{2} + a d e g h^{2} + a c f g h^{2} - a c e h^{3}\right )} \log \left ({\left | h x + g \right |}\right )}{h^{4}} \]
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Time = 2.82 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.38 \[ \int \frac {(a+b x) (c+d x) (e+f x)}{g+h x} \, dx=x\,\left (\frac {a\,c\,f+a\,d\,e+b\,c\,e}{h}-\frac {g\,\left (\frac {a\,d\,f+b\,c\,f+b\,d\,e}{h}-\frac {b\,d\,f\,g}{h^2}\right )}{h}\right )+x^2\,\left (\frac {a\,d\,f+b\,c\,f+b\,d\,e}{2\,h}-\frac {b\,d\,f\,g}{2\,h^2}\right )+\frac {\ln \left (g+h\,x\right )\,\left (a\,c\,e\,h^3-b\,d\,f\,g^3-a\,c\,f\,g\,h^2-a\,d\,e\,g\,h^2-b\,c\,e\,g\,h^2+a\,d\,f\,g^2\,h+b\,c\,f\,g^2\,h+b\,d\,e\,g^2\,h\right )}{h^4}+\frac {b\,d\,f\,x^3}{3\,h} \]
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