Integrand size = 20, antiderivative size = 155 \[ \int \frac {(A+B x) (d+e x)^4}{a+b x} \, dx=\frac {(A b-a B) e (b d-a e)^3 x}{b^5}+\frac {(A b-a B) (b d-a e)^2 (d+e x)^2}{2 b^4}+\frac {(A b-a B) (b d-a e) (d+e x)^3}{3 b^3}+\frac {(A b-a B) (d+e x)^4}{4 b^2}+\frac {B (d+e x)^5}{5 b e}+\frac {(A b-a B) (b d-a e)^4 \log (a+b x)}{b^6} \]
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Time = 0.07 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \[ \int \frac {(A+B x) (d+e x)^4}{a+b x} \, dx=\frac {(A b-a B) (b d-a e)^4 \log (a+b x)}{b^6}+\frac {e x (A b-a B) (b d-a e)^3}{b^5}+\frac {(d+e x)^2 (A b-a B) (b d-a e)^2}{2 b^4}+\frac {(d+e x)^3 (A b-a B) (b d-a e)}{3 b^3}+\frac {(d+e x)^4 (A b-a B)}{4 b^2}+\frac {B (d+e x)^5}{5 b e} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(A b-a B) e (b d-a e)^3}{b^5}+\frac {(A b-a B) (b d-a e)^4}{b^5 (a+b x)}+\frac {(A b-a B) e (b d-a e)^2 (d+e x)}{b^4}+\frac {(A b-a B) e (b d-a e) (d+e x)^2}{b^3}+\frac {(A b-a B) e (d+e x)^3}{b^2}+\frac {B (d+e x)^4}{b}\right ) \, dx \\ & = \frac {(A b-a B) e (b d-a e)^3 x}{b^5}+\frac {(A b-a B) (b d-a e)^2 (d+e x)^2}{2 b^4}+\frac {(A b-a B) (b d-a e) (d+e x)^3}{3 b^3}+\frac {(A b-a B) (d+e x)^4}{4 b^2}+\frac {B (d+e x)^5}{5 b e}+\frac {(A b-a B) (b d-a e)^4 \log (a+b x)}{b^6} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.66 \[ \int \frac {(A+B x) (d+e x)^4}{a+b x} \, dx=\frac {b x \left (60 a^4 B e^4-30 a^3 b e^3 (8 B d+2 A e+B e x)+10 a^2 b^2 e^2 \left (3 A e (8 d+e x)+2 B \left (18 d^2+6 d e x+e^2 x^2\right )\right )-5 a b^3 e \left (4 A e \left (18 d^2+6 d e x+e^2 x^2\right )+B \left (48 d^3+36 d^2 e x+16 d e^2 x^2+3 e^3 x^3\right )\right )+b^4 \left (5 A e \left (48 d^3+36 d^2 e x+16 d e^2 x^2+3 e^3 x^3\right )+12 B \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )\right )\right )+60 (A b-a B) (b d-a e)^4 \log (a+b x)}{60 b^6} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(403\) vs. \(2(147)=294\).
Time = 0.71 (sec) , antiderivative size = 404, normalized size of antiderivative = 2.61
method | result | size |
norman | \(-\frac {\left (A \,a^{3} b \,e^{4}-4 A \,a^{2} b^{2} d \,e^{3}+6 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e -B \,a^{4} e^{4}+4 B \,a^{3} b d \,e^{3}-6 B \,a^{2} b^{2} d^{2} e^{2}+4 B a \,b^{3} d^{3} e -B \,b^{4} d^{4}\right ) x}{b^{5}}+\frac {B \,e^{4} x^{5}}{5 b}+\frac {e \left (A \,a^{2} b \,e^{3}-4 A a \,b^{2} d \,e^{2}+6 A \,b^{3} d^{2} e -B \,a^{3} e^{3}+4 B \,a^{2} b d \,e^{2}-6 B a \,b^{2} d^{2} e +4 b^{3} B \,d^{3}\right ) x^{2}}{2 b^{4}}-\frac {e^{2} \left (A a b \,e^{2}-4 A \,b^{2} d e -B \,a^{2} e^{2}+4 B a b d e -6 b^{2} B \,d^{2}\right ) x^{3}}{3 b^{3}}+\frac {e^{3} \left (A b e -B a e +4 B b d \right ) x^{4}}{4 b^{2}}+\frac {\left (A \,a^{4} b \,e^{4}-4 A \,a^{3} b^{2} d \,e^{3}+6 A \,a^{2} b^{3} d^{2} e^{2}-4 A a \,b^{4} d^{3} e +A \,b^{5} d^{4}-B \,a^{5} e^{4}+4 B \,a^{4} b d \,e^{3}-6 B \,a^{3} b^{2} d^{2} e^{2}+4 B \,a^{2} b^{3} d^{3} e -B a \,b^{4} d^{4}\right ) \ln \left (b x +a \right )}{b^{6}}\) | \(404\) |
default | \(-\frac {3 B a \,b^{3} d^{2} e^{2} x^{2}-4 A \,a^{2} b^{2} d \,e^{3} x +6 A a \,b^{3} d^{2} e^{2} x +2 A a \,b^{3} d \,e^{3} x^{2}-2 B \,a^{2} b^{2} d \,e^{3} x^{2}-\frac {1}{4} A \,b^{4} e^{4} x^{4}-\frac {1}{2} A \,a^{2} b^{2} e^{4} x^{2}-3 A \,b^{4} d^{2} e^{2} x^{2}+\frac {1}{2} B \,a^{3} b \,e^{4} x^{2}-2 B \,b^{4} d^{3} e \,x^{2}+A \,a^{3} b \,e^{4} x -4 A \,b^{4} d^{3} e x +\frac {1}{3} A a \,b^{3} e^{4} x^{3}-\frac {4}{3} A \,b^{4} d \,e^{3} x^{3}-\frac {1}{3} B \,a^{2} b^{2} e^{4} x^{3}-2 B \,b^{4} d^{2} e^{2} x^{3}+\frac {1}{4} B a \,b^{3} e^{4} x^{4}-B \,b^{4} d \,e^{3} x^{4}-\frac {1}{5} b^{4} B \,x^{5} e^{4}-B \,a^{4} e^{4} x -B \,b^{4} d^{4} x +\frac {4}{3} B a \,b^{3} d \,e^{3} x^{3}+4 B \,a^{3} b d \,e^{3} x -6 B \,a^{2} b^{2} d^{2} e^{2} x +4 B a \,b^{3} d^{3} e x}{b^{5}}+\frac {\left (A \,a^{4} b \,e^{4}-4 A \,a^{3} b^{2} d \,e^{3}+6 A \,a^{2} b^{3} d^{2} e^{2}-4 A a \,b^{4} d^{3} e +A \,b^{5} d^{4}-B \,a^{5} e^{4}+4 B \,a^{4} b d \,e^{3}-6 B \,a^{3} b^{2} d^{2} e^{2}+4 B \,a^{2} b^{3} d^{3} e -B a \,b^{4} d^{4}\right ) \ln \left (b x +a \right )}{b^{6}}\) | \(464\) |
risch | \(-\frac {4 \ln \left (b x +a \right ) A \,a^{3} d \,e^{3}}{b^{4}}-\frac {4 B \,a^{3} d \,e^{3} x}{b^{4}}+\frac {6 \ln \left (b x +a \right ) A \,a^{2} d^{2} e^{2}}{b^{3}}-\frac {4 \ln \left (b x +a \right ) A a \,d^{3} e}{b^{2}}+\frac {4 \ln \left (b x +a \right ) B \,a^{4} d \,e^{3}}{b^{5}}-\frac {6 \ln \left (b x +a \right ) B \,a^{3} d^{2} e^{2}}{b^{4}}+\frac {4 \ln \left (b x +a \right ) B \,a^{2} d^{3} e}{b^{3}}+\frac {A \,e^{4} x^{4}}{4 b}+\frac {B \,d^{4} x}{b}+\frac {\ln \left (b x +a \right ) A \,d^{4}}{b}-\frac {B \,a^{3} e^{4} x^{2}}{2 b^{4}}+\frac {2 B \,d^{3} e \,x^{2}}{b}-\frac {A \,a^{3} e^{4} x}{b^{4}}+\frac {4 A \,d^{3} e x}{b}-\frac {A a \,e^{4} x^{3}}{3 b^{2}}+\frac {4 A d \,e^{3} x^{3}}{3 b}+\frac {B \,a^{2} e^{4} x^{3}}{3 b^{3}}+\frac {2 B \,d^{2} e^{2} x^{3}}{b}-\frac {B a \,e^{4} x^{4}}{4 b^{2}}+\frac {B d \,e^{3} x^{4}}{b}-\frac {2 A a d \,e^{3} x^{2}}{b^{2}}+\frac {2 B \,a^{2} d \,e^{3} x^{2}}{b^{3}}-\frac {4 B a d \,e^{3} x^{3}}{3 b^{2}}-\frac {3 B a \,d^{2} e^{2} x^{2}}{b^{2}}+\frac {4 A \,a^{2} d \,e^{3} x}{b^{3}}-\frac {6 A a \,d^{2} e^{2} x}{b^{2}}+\frac {A \,a^{2} e^{4} x^{2}}{2 b^{3}}+\frac {3 A \,d^{2} e^{2} x^{2}}{b}+\frac {B \,a^{4} e^{4} x}{b^{5}}+\frac {\ln \left (b x +a \right ) A \,a^{4} e^{4}}{b^{5}}-\frac {\ln \left (b x +a \right ) B \,a^{5} e^{4}}{b^{6}}-\frac {\ln \left (b x +a \right ) B a \,d^{4}}{b^{2}}+\frac {B \,e^{4} x^{5}}{5 b}+\frac {6 B \,a^{2} d^{2} e^{2} x}{b^{3}}-\frac {4 B a \,d^{3} e x}{b^{2}}\) | \(521\) |
parallelrisch | \(\frac {60 A \ln \left (b x +a \right ) a^{4} b \,e^{4}-60 B \ln \left (b x +a \right ) a \,b^{4} d^{4}-15 B \,x^{4} a \,b^{4} e^{4}+60 B \,x^{4} b^{5} d \,e^{3}-20 A \,x^{3} a \,b^{4} e^{4}+80 A \,x^{3} b^{5} d \,e^{3}+20 B \,x^{3} a^{2} b^{3} e^{4}+120 B \,x^{3} b^{5} d^{2} e^{2}+30 A \,x^{2} a^{2} b^{3} e^{4}+180 A \,x^{2} b^{5} d^{2} e^{2}-30 B \,x^{2} a^{3} b^{2} e^{4}+120 B \,x^{2} b^{5} d^{3} e -60 A x \,a^{3} b^{2} e^{4}+240 A x \,b^{5} d^{3} e +60 B x \,a^{4} b \,e^{4}+360 B x \,a^{2} b^{3} d^{2} e^{2}-240 A \ln \left (b x +a \right ) a^{3} b^{2} d \,e^{3}+360 A \ln \left (b x +a \right ) a^{2} b^{3} d^{2} e^{2}-240 A \ln \left (b x +a \right ) a \,b^{4} d^{3} e +240 B \ln \left (b x +a \right ) a^{4} b d \,e^{3}-360 B \ln \left (b x +a \right ) a^{3} b^{2} d^{2} e^{2}+15 A \,x^{4} b^{5} e^{4}+60 B x \,b^{5} d^{4}+60 A \ln \left (b x +a \right ) b^{5} d^{4}-60 B \ln \left (b x +a \right ) a^{5} e^{4}+240 B \ln \left (b x +a \right ) a^{2} b^{3} d^{3} e -240 B x a \,b^{4} d^{3} e -80 B \,x^{3} a \,b^{4} d \,e^{3}-120 A \,x^{2} a \,b^{4} d \,e^{3}+120 B \,x^{2} a^{2} b^{3} d \,e^{3}-180 B \,x^{2} a \,b^{4} d^{2} e^{2}+240 A x \,a^{2} b^{3} d \,e^{3}-360 A x a \,b^{4} d^{2} e^{2}-240 B x \,a^{3} b^{2} d \,e^{3}+12 B \,x^{5} e^{4} b^{5}}{60 b^{6}}\) | \(522\) |
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Leaf count of result is larger than twice the leaf count of optimal. 403 vs. \(2 (149) = 298\).
Time = 0.22 (sec) , antiderivative size = 403, normalized size of antiderivative = 2.60 \[ \int \frac {(A+B x) (d+e x)^4}{a+b x} \, dx=\frac {12 \, B b^{5} e^{4} x^{5} + 15 \, {\left (4 \, B b^{5} d e^{3} - {\left (B a b^{4} - A b^{5}\right )} e^{4}\right )} x^{4} + 20 \, {\left (6 \, B b^{5} d^{2} e^{2} - 4 \, {\left (B a b^{4} - A b^{5}\right )} d e^{3} + {\left (B a^{2} b^{3} - A a b^{4}\right )} e^{4}\right )} x^{3} + 30 \, {\left (4 \, B b^{5} d^{3} e - 6 \, {\left (B a b^{4} - A b^{5}\right )} d^{2} e^{2} + 4 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} d e^{3} - {\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} e^{4}\right )} x^{2} + 60 \, {\left (B b^{5} d^{4} - 4 \, {\left (B a b^{4} - A b^{5}\right )} d^{3} e + 6 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} d^{2} e^{2} - 4 \, {\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} d e^{3} + {\left (B a^{4} b - A a^{3} b^{2}\right )} e^{4}\right )} x - 60 \, {\left ({\left (B a b^{4} - A b^{5}\right )} d^{4} - 4 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} d^{3} e + 6 \, {\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} d^{2} e^{2} - 4 \, {\left (B a^{4} b - A a^{3} b^{2}\right )} d e^{3} + {\left (B a^{5} - A a^{4} b\right )} e^{4}\right )} \log \left (b x + a\right )}{60 \, b^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (136) = 272\).
Time = 0.55 (sec) , antiderivative size = 352, normalized size of antiderivative = 2.27 \[ \int \frac {(A+B x) (d+e x)^4}{a+b x} \, dx=\frac {B e^{4} x^{5}}{5 b} + x^{4} \left (\frac {A e^{4}}{4 b} - \frac {B a e^{4}}{4 b^{2}} + \frac {B d e^{3}}{b}\right ) + x^{3} \left (- \frac {A a e^{4}}{3 b^{2}} + \frac {4 A d e^{3}}{3 b} + \frac {B a^{2} e^{4}}{3 b^{3}} - \frac {4 B a d e^{3}}{3 b^{2}} + \frac {2 B d^{2} e^{2}}{b}\right ) + x^{2} \left (\frac {A a^{2} e^{4}}{2 b^{3}} - \frac {2 A a d e^{3}}{b^{2}} + \frac {3 A d^{2} e^{2}}{b} - \frac {B a^{3} e^{4}}{2 b^{4}} + \frac {2 B a^{2} d e^{3}}{b^{3}} - \frac {3 B a d^{2} e^{2}}{b^{2}} + \frac {2 B d^{3} e}{b}\right ) + x \left (- \frac {A a^{3} e^{4}}{b^{4}} + \frac {4 A a^{2} d e^{3}}{b^{3}} - \frac {6 A a d^{2} e^{2}}{b^{2}} + \frac {4 A d^{3} e}{b} + \frac {B a^{4} e^{4}}{b^{5}} - \frac {4 B a^{3} d e^{3}}{b^{4}} + \frac {6 B a^{2} d^{2} e^{2}}{b^{3}} - \frac {4 B a d^{3} e}{b^{2}} + \frac {B d^{4}}{b}\right ) - \frac {\left (- A b + B a\right ) \left (a e - b d\right )^{4} \log {\left (a + b x \right )}}{b^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 400 vs. \(2 (149) = 298\).
Time = 0.20 (sec) , antiderivative size = 400, normalized size of antiderivative = 2.58 \[ \int \frac {(A+B x) (d+e x)^4}{a+b x} \, dx=\frac {12 \, B b^{4} e^{4} x^{5} + 15 \, {\left (4 \, B b^{4} d e^{3} - {\left (B a b^{3} - A b^{4}\right )} e^{4}\right )} x^{4} + 20 \, {\left (6 \, B b^{4} d^{2} e^{2} - 4 \, {\left (B a b^{3} - A b^{4}\right )} d e^{3} + {\left (B a^{2} b^{2} - A a b^{3}\right )} e^{4}\right )} x^{3} + 30 \, {\left (4 \, B b^{4} d^{3} e - 6 \, {\left (B a b^{3} - A b^{4}\right )} d^{2} e^{2} + 4 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d e^{3} - {\left (B a^{3} b - A a^{2} b^{2}\right )} e^{4}\right )} x^{2} + 60 \, {\left (B b^{4} d^{4} - 4 \, {\left (B a b^{3} - A b^{4}\right )} d^{3} e + 6 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2} e^{2} - 4 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} d e^{3} + {\left (B a^{4} - A a^{3} b\right )} e^{4}\right )} x}{60 \, b^{5}} - \frac {{\left ({\left (B a b^{4} - A b^{5}\right )} d^{4} - 4 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} d^{3} e + 6 \, {\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} d^{2} e^{2} - 4 \, {\left (B a^{4} b - A a^{3} b^{2}\right )} d e^{3} + {\left (B a^{5} - A a^{4} b\right )} e^{4}\right )} \log \left (b x + a\right )}{b^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (149) = 298\).
Time = 0.29 (sec) , antiderivative size = 466, normalized size of antiderivative = 3.01 \[ \int \frac {(A+B x) (d+e x)^4}{a+b x} \, dx=\frac {12 \, B b^{4} e^{4} x^{5} + 60 \, B b^{4} d e^{3} x^{4} - 15 \, B a b^{3} e^{4} x^{4} + 15 \, A b^{4} e^{4} x^{4} + 120 \, B b^{4} d^{2} e^{2} x^{3} - 80 \, B a b^{3} d e^{3} x^{3} + 80 \, A b^{4} d e^{3} x^{3} + 20 \, B a^{2} b^{2} e^{4} x^{3} - 20 \, A a b^{3} e^{4} x^{3} + 120 \, B b^{4} d^{3} e x^{2} - 180 \, B a b^{3} d^{2} e^{2} x^{2} + 180 \, A b^{4} d^{2} e^{2} x^{2} + 120 \, B a^{2} b^{2} d e^{3} x^{2} - 120 \, A a b^{3} d e^{3} x^{2} - 30 \, B a^{3} b e^{4} x^{2} + 30 \, A a^{2} b^{2} e^{4} x^{2} + 60 \, B b^{4} d^{4} x - 240 \, B a b^{3} d^{3} e x + 240 \, A b^{4} d^{3} e x + 360 \, B a^{2} b^{2} d^{2} e^{2} x - 360 \, A a b^{3} d^{2} e^{2} x - 240 \, B a^{3} b d e^{3} x + 240 \, A a^{2} b^{2} d e^{3} x + 60 \, B a^{4} e^{4} x - 60 \, A a^{3} b e^{4} x}{60 \, b^{5}} - \frac {{\left (B a b^{4} d^{4} - A b^{5} d^{4} - 4 \, B a^{2} b^{3} d^{3} e + 4 \, A a b^{4} d^{3} e + 6 \, B a^{3} b^{2} d^{2} e^{2} - 6 \, A a^{2} b^{3} d^{2} e^{2} - 4 \, B a^{4} b d e^{3} + 4 \, A a^{3} b^{2} d e^{3} + B a^{5} e^{4} - A a^{4} b e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} \]
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Time = 1.21 (sec) , antiderivative size = 411, normalized size of antiderivative = 2.65 \[ \int \frac {(A+B x) (d+e x)^4}{a+b x} \, dx=x\,\left (\frac {B\,d^4+4\,A\,e\,d^3}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{b}-\frac {B\,a\,e^4}{b^2}\right )}{b}-\frac {2\,d\,e^2\,\left (2\,A\,e+3\,B\,d\right )}{b}\right )}{b}+\frac {2\,d^2\,e\,\left (3\,A\,e+2\,B\,d\right )}{b}\right )}{b}\right )-x^3\,\left (\frac {a\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{b}-\frac {B\,a\,e^4}{b^2}\right )}{3\,b}-\frac {2\,d\,e^2\,\left (2\,A\,e+3\,B\,d\right )}{3\,b}\right )+x^4\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{4\,b}-\frac {B\,a\,e^4}{4\,b^2}\right )+x^2\,\left (\frac {a\,\left (\frac {a\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{b}-\frac {B\,a\,e^4}{b^2}\right )}{b}-\frac {2\,d\,e^2\,\left (2\,A\,e+3\,B\,d\right )}{b}\right )}{2\,b}+\frac {d^2\,e\,\left (3\,A\,e+2\,B\,d\right )}{b}\right )+\frac {\ln \left (a+b\,x\right )\,\left (-B\,a^5\,e^4+4\,B\,a^4\,b\,d\,e^3+A\,a^4\,b\,e^4-6\,B\,a^3\,b^2\,d^2\,e^2-4\,A\,a^3\,b^2\,d\,e^3+4\,B\,a^2\,b^3\,d^3\,e+6\,A\,a^2\,b^3\,d^2\,e^2-B\,a\,b^4\,d^4-4\,A\,a\,b^4\,d^3\,e+A\,b^5\,d^4\right )}{b^6}+\frac {B\,e^4\,x^5}{5\,b} \]
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