Integrand size = 29, antiderivative size = 92 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{d+e x} \, dx=\frac {b (b d-a e) (B d-A e) x}{e^3}-\frac {(B d-A e) (a+b x)^2}{2 e^2}+\frac {B (a+b x)^3}{3 b e}-\frac {(b d-a e)^2 (B d-A e) \log (d+e x)}{e^4} \]
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Time = 0.04 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {27, 78} \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{d+e x} \, dx=-\frac {(b d-a e)^2 (B d-A e) \log (d+e x)}{e^4}+\frac {b x (b d-a e) (B d-A e)}{e^3}-\frac {(a+b x)^2 (B d-A e)}{2 e^2}+\frac {B (a+b x)^3}{3 b e} \]
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Rule 27
Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b x)^2 (A+B x)}{d+e x} \, dx \\ & = \int \left (-\frac {b (b d-a e) (-B d+A e)}{e^3}+\frac {b (-B d+A e) (a+b x)}{e^2}+\frac {B (a+b x)^2}{e}+\frac {(-b d+a e)^2 (-B d+A e)}{e^3 (d+e x)}\right ) \, dx \\ & = \frac {b (b d-a e) (B d-A e) x}{e^3}-\frac {(B d-A e) (a+b x)^2}{2 e^2}+\frac {B (a+b x)^3}{3 b e}-\frac {(b d-a e)^2 (B d-A e) \log (d+e x)}{e^4} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.11 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{d+e x} \, dx=\frac {e x \left (6 a^2 B e^2+6 a b e (-2 B d+2 A e+B e x)+b^2 \left (3 A e (-2 d+e x)+B \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )\right )-6 (b d-a e)^2 (B d-A e) \log (d+e x)}{6 e^4} \]
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Time = 0.20 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.61
method | result | size |
norman | \(\frac {\left (2 A a b \,e^{2}-A \,b^{2} d e +a^{2} B \,e^{2}-2 B a b d e +B \,b^{2} d^{2}\right ) x}{e^{3}}+\frac {B \,b^{2} x^{3}}{3 e}+\frac {b \left (A b e +2 B a e -B b d \right ) x^{2}}{2 e^{2}}+\frac {\left (A \,a^{2} e^{3}-2 A a b d \,e^{2}+A \,b^{2} d^{2} e -B \,a^{2} d \,e^{2}+2 B a b \,d^{2} e -B \,b^{2} d^{3}\right ) \ln \left (e x +d \right )}{e^{4}}\) | \(148\) |
default | \(\frac {\frac {1}{3} b^{2} B \,x^{3} e^{2}+\frac {1}{2} A \,b^{2} e^{2} x^{2}+B a b \,e^{2} x^{2}-\frac {1}{2} B \,b^{2} d e \,x^{2}+2 A a b \,e^{2} x -A \,b^{2} d e x +B \,a^{2} e^{2} x -2 B a b d e x +B \,b^{2} d^{2} x}{e^{3}}+\frac {\left (A \,a^{2} e^{3}-2 A a b d \,e^{2}+A \,b^{2} d^{2} e -B \,a^{2} d \,e^{2}+2 B a b \,d^{2} e -B \,b^{2} d^{3}\right ) \ln \left (e x +d \right )}{e^{4}}\) | \(161\) |
risch | \(\frac {B \,b^{2} x^{3}}{3 e}+\frac {A \,b^{2} x^{2}}{2 e}+\frac {B a b \,x^{2}}{e}-\frac {B \,b^{2} d \,x^{2}}{2 e^{2}}+\frac {2 A a b x}{e}-\frac {A \,b^{2} d x}{e^{2}}+\frac {B \,a^{2} x}{e}-\frac {2 B a b d x}{e^{2}}+\frac {B \,b^{2} d^{2} x}{e^{3}}+\frac {\ln \left (e x +d \right ) A \,a^{2}}{e}-\frac {2 \ln \left (e x +d \right ) A a b d}{e^{2}}+\frac {\ln \left (e x +d \right ) A \,b^{2} d^{2}}{e^{3}}-\frac {\ln \left (e x +d \right ) B \,a^{2} d}{e^{2}}+\frac {2 \ln \left (e x +d \right ) B a b \,d^{2}}{e^{3}}-\frac {\ln \left (e x +d \right ) B \,b^{2} d^{3}}{e^{4}}\) | \(197\) |
parallelrisch | \(\frac {2 B \,b^{2} x^{3} e^{3}+3 A \,b^{2} e^{3} x^{2}+6 B \,x^{2} a b \,e^{3}-3 B \,x^{2} b^{2} d \,e^{2}+6 A \ln \left (e x +d \right ) a^{2} e^{3}-12 A \ln \left (e x +d \right ) a b d \,e^{2}+6 A \ln \left (e x +d \right ) b^{2} d^{2} e +12 A x a b \,e^{3}-6 A x \,b^{2} d \,e^{2}-6 B \ln \left (e x +d \right ) a^{2} d \,e^{2}+12 B \ln \left (e x +d \right ) a b \,d^{2} e -6 B \ln \left (e x +d \right ) b^{2} d^{3}+6 B x \,a^{2} e^{3}-12 B x a b d \,e^{2}+6 B x \,b^{2} d^{2} e}{6 e^{4}}\) | \(198\) |
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Time = 0.31 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.66 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{d+e x} \, dx=\frac {2 \, B b^{2} e^{3} x^{3} - 3 \, {\left (B b^{2} d e^{2} - {\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 6 \, {\left (B b^{2} d^{2} e - {\left (2 \, B a b + A b^{2}\right )} d e^{2} + {\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x - 6 \, {\left (B b^{2} d^{3} - A a^{2} e^{3} - {\left (2 \, B a b + A b^{2}\right )} d^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )} \log \left (e x + d\right )}{6 \, e^{4}} \]
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Time = 0.23 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.27 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{d+e x} \, dx=\frac {B b^{2} x^{3}}{3 e} + x^{2} \left (\frac {A b^{2}}{2 e} + \frac {B a b}{e} - \frac {B b^{2} d}{2 e^{2}}\right ) + x \left (\frac {2 A a b}{e} - \frac {A b^{2} d}{e^{2}} + \frac {B a^{2}}{e} - \frac {2 B a b d}{e^{2}} + \frac {B b^{2} d^{2}}{e^{3}}\right ) - \frac {\left (- A e + B d\right ) \left (a e - b d\right )^{2} \log {\left (d + e x \right )}}{e^{4}} \]
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Time = 0.20 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.65 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{d+e x} \, dx=\frac {2 \, B b^{2} e^{2} x^{3} - 3 \, {\left (B b^{2} d e - {\left (2 \, B a b + A b^{2}\right )} e^{2}\right )} x^{2} + 6 \, {\left (B b^{2} d^{2} - {\left (2 \, B a b + A b^{2}\right )} d e + {\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )} x}{6 \, e^{3}} - \frac {{\left (B b^{2} d^{3} - A a^{2} e^{3} - {\left (2 \, B a b + A b^{2}\right )} d^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )} \log \left (e x + d\right )}{e^{4}} \]
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Time = 0.27 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.80 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{d+e x} \, dx=\frac {2 \, B b^{2} e^{2} x^{3} - 3 \, B b^{2} d e x^{2} + 6 \, B a b e^{2} x^{2} + 3 \, A b^{2} e^{2} x^{2} + 6 \, B b^{2} d^{2} x - 12 \, B a b d e x - 6 \, A b^{2} d e x + 6 \, B a^{2} e^{2} x + 12 \, A a b e^{2} x}{6 \, e^{3}} - \frac {{\left (B b^{2} d^{3} - 2 \, B a b d^{2} e - A b^{2} d^{2} e + B a^{2} d e^{2} + 2 \, A a b d e^{2} - A a^{2} e^{3}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{4}} \]
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Time = 10.75 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.73 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{d+e x} \, dx=x\,\left (\frac {B\,a^2+2\,A\,b\,a}{e}-\frac {d\,\left (\frac {A\,b^2+2\,B\,a\,b}{e}-\frac {B\,b^2\,d}{e^2}\right )}{e}\right )+x^2\,\left (\frac {A\,b^2+2\,B\,a\,b}{2\,e}-\frac {B\,b^2\,d}{2\,e^2}\right )+\frac {\ln \left (d+e\,x\right )\,\left (-B\,a^2\,d\,e^2+A\,a^2\,e^3+2\,B\,a\,b\,d^2\,e-2\,A\,a\,b\,d\,e^2-B\,b^2\,d^3+A\,b^2\,d^2\,e\right )}{e^4}+\frac {B\,b^2\,x^3}{3\,e} \]
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