Integrand size = 29, antiderivative size = 59 \[ \int \frac {(a+b x)^5}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=-\frac {(b c-a d)^2}{2 d^3 (c+d x)^2}+\frac {2 b (b c-a d)}{d^3 (c+d x)}+\frac {b^2 \log (c+d x)}{d^3} \]
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Time = 0.03 (sec) , antiderivative size = 59, 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 = {640, 45} \[ \int \frac {(a+b x)^5}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\frac {2 b (b c-a d)}{d^3 (c+d x)}-\frac {(b c-a d)^2}{2 d^3 (c+d x)^2}+\frac {b^2 \log (c+d x)}{d^3} \]
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Rule 45
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b x)^2}{(c+d x)^3} \, dx \\ & = \int \left (\frac {(-b c+a d)^2}{d^2 (c+d x)^3}-\frac {2 b (b c-a d)}{d^2 (c+d x)^2}+\frac {b^2}{d^2 (c+d x)}\right ) \, dx \\ & = -\frac {(b c-a d)^2}{2 d^3 (c+d x)^2}+\frac {2 b (b c-a d)}{d^3 (c+d x)}+\frac {b^2 \log (c+d x)}{d^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x)^5}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\frac {\frac {(b c-a d) (3 b c+a d+4 b d x)}{(c+d x)^2}+2 b^2 \log (c+d x)}{2 d^3} \]
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Time = 2.68 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.12
method | result | size |
risch | \(\frac {-\frac {2 b \left (a d -b c \right ) x}{d^{2}}-\frac {a^{2} d^{2}+2 a b c d -3 b^{2} c^{2}}{2 d^{3}}}{\left (d x +c \right )^{2}}+\frac {b^{2} \ln \left (d x +c \right )}{d^{3}}\) | \(66\) |
default | \(-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{2 d^{3} \left (d x +c \right )^{2}}+\frac {b^{2} \ln \left (d x +c \right )}{d^{3}}-\frac {2 b \left (a d -b c \right )}{d^{3} \left (d x +c \right )}\) | \(69\) |
parallelrisch | \(\frac {2 \ln \left (d x +c \right ) x^{2} b^{2} d^{2}+4 \ln \left (d x +c \right ) x \,b^{2} c d +2 \ln \left (d x +c \right ) b^{2} c^{2}-4 x a b \,d^{2}+4 x \,b^{2} c d -a^{2} d^{2}-2 a b c d +3 b^{2} c^{2}}{2 d^{3} \left (d x +c \right )^{2}}\) | \(97\) |
norman | \(\frac {-\frac {\left (9 a^{2} b^{4} d^{2}-6 a c d \,b^{5}-3 c^{2} b^{6}\right ) x^{2}}{2 d^{3} b^{2}}-\frac {2 \left (a d \,b^{4}-c \,b^{5}\right ) x^{3}}{d^{2} b}-\frac {\left (b^{2} d^{2} a^{2}+2 a \,b^{3} c d -3 b^{4} c^{2}\right ) a^{2}}{2 d^{3} b^{2}}-\frac {a \left (3 a^{2} b^{3} d^{2}-3 b^{5} c^{2}\right ) x}{d^{3} b^{2}}}{\left (b x +a \right )^{2} \left (d x +c \right )^{2}}+\frac {b^{2} \ln \left (d x +c \right )}{d^{3}}\) | \(162\) |
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Time = 0.27 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.69 \[ \int \frac {(a+b x)^5}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\frac {3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2} + 4 \, {\left (b^{2} c d - a b d^{2}\right )} x + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (d x + c\right )}{2 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]
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Time = 0.27 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.36 \[ \int \frac {(a+b x)^5}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\frac {b^{2} \log {\left (c + d x \right )}}{d^{3}} + \frac {- a^{2} d^{2} - 2 a b c d + 3 b^{2} c^{2} + x \left (- 4 a b d^{2} + 4 b^{2} c d\right )}{2 c^{2} d^{3} + 4 c d^{4} x + 2 d^{5} x^{2}} \]
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Time = 0.20 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.36 \[ \int \frac {(a+b x)^5}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\frac {3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2} + 4 \, {\left (b^{2} c d - a b d^{2}\right )} x}{2 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} + \frac {b^{2} \log \left (d x + c\right )}{d^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.17 \[ \int \frac {(a+b x)^5}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\frac {b^{2} \log \left ({\left | d x + c \right |}\right )}{d^{3}} + \frac {4 \, {\left (b^{2} c - a b d\right )} x + \frac {3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}}{d}}{2 \, {\left (d x + c\right )}^{2} d^{2}} \]
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Time = 9.91 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.31 \[ \int \frac {(a+b x)^5}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\frac {b^2\,\ln \left (c+d\,x\right )}{d^3}-\frac {\frac {a^2\,d^2+2\,a\,b\,c\,d-3\,b^2\,c^2}{2\,d^3}+\frac {2\,b\,x\,\left (a\,d-b\,c\right )}{d^2}}{c^2+2\,c\,d\,x+d^2\,x^2} \]
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