Integrand size = 33, antiderivative size = 132 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^2} \, dx=\frac {b^2 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^2 (a+b x)}-\frac {(b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x) (d+e x)}-\frac {2 b (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^3 (a+b x)} \]
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Time = 0.05 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {784, 21, 45} \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^2} \, dx=-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^3 (a+b x) (d+e x)}-\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^3 (a+b x)}+\frac {b^2 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^2 (a+b x)} \]
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Rule 21
Rule 45
Rule 784
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x) \left (a b+b^2 x\right )}{(d+e x)^2} \, dx}{a b+b^2 x} \\ & = \frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^2}{(d+e x)^2} \, dx}{a b+b^2 x} \\ & = \frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {b^2}{e^2}+\frac {(-b d+a e)^2}{e^2 (d+e x)^2}-\frac {2 b (b d-a e)}{e^2 (d+e x)}\right ) \, dx}{a b+b^2 x} \\ & = \frac {b^2 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^2 (a+b x)}-\frac {(b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x) (d+e x)}-\frac {2 b (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^3 (a+b x)} \\ \end{align*}
Time = 1.04 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.66 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^2} \, dx=\frac {\sqrt {(a+b x)^2} \left (2 a b d e-a^2 e^2+b^2 \left (-d^2+d e x+e^2 x^2\right )-2 b (b d-a e) (d+e x) \log (d+e x)\right )}{e^3 (a+b x) (d+e x)} \]
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Time = 0.32 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.84
method | result | size |
risch | \(\frac {b^{2} x \sqrt {\left (b x +a \right )^{2}}}{e^{2} \left (b x +a \right )}-\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )}{\left (b x +a \right ) e^{3} \left (e x +d \right )}+\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (a e -b d \right ) b \ln \left (e x +d \right )}{\left (b x +a \right ) e^{3}}\) | \(111\) |
default | \(\frac {\operatorname {csgn}\left (b x +a \right ) \left (2 \ln \left (-b e x -b d \right ) a b \,e^{2} x -2 \ln \left (-b e x -b d \right ) b^{2} d e x +b^{2} e^{2} x^{2}+2 \ln \left (-b e x -b d \right ) a b d e -2 \ln \left (-b e x -b d \right ) b^{2} d^{2}+a b \,e^{2} x +b^{2} d e x -e^{2} a^{2}+3 a b d e -b^{2} d^{2}\right )}{e^{3} \left (e x +d \right )}\) | \(139\) |
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Time = 0.31 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.70 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^2} \, dx=\frac {b^{2} e^{2} x^{2} + b^{2} d e x - b^{2} d^{2} + 2 \, a b d e - a^{2} e^{2} - 2 \, {\left (b^{2} d^{2} - a b d e + {\left (b^{2} d e - a b e^{2}\right )} x\right )} \log \left (e x + d\right )}{e^{4} x + d e^{3}} \]
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\[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^2} \, dx=\int \frac {\left (a + b x\right ) \sqrt {\left (a + b x\right )^{2}}}{\left (d + e x\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.26 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.77 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^2} \, dx=\frac {b^{2} x \mathrm {sgn}\left (b x + a\right )}{e^{2}} - \frac {2 \, {\left (b^{2} d \mathrm {sgn}\left (b x + a\right ) - a b e \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | e x + d \right |}\right )}{e^{3}} - \frac {b^{2} d^{2} \mathrm {sgn}\left (b x + a\right ) - 2 \, a b d e \mathrm {sgn}\left (b x + a\right ) + a^{2} e^{2} \mathrm {sgn}\left (b x + a\right )}{{\left (e x + d\right )} e^{3}} \]
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Timed out. \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^2} \, dx=\int \frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (a+b\,x\right )}{{\left (d+e\,x\right )}^2} \,d x \]
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