Integrand size = 28, antiderivative size = 166 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{d+e x} \, dx=\frac {b (b d-a e)^2 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x)}-\frac {(b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^2}+\frac {(a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e}-\frac {(b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^4 (a+b x)} \]
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Time = 0.05 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {660, 45} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{d+e x} \, dx=-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 \log (d+e x)}{e^4 (a+b x)}+\frac {b x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^3 (a+b x)}-\frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{2 e^2}+\frac {(a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e} \]
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Rule 45
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^3}{d+e x} \, dx}{b^2 \left (a b+b^2 x\right )} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^4 (b d-a e)^2}{e^3}-\frac {b^3 (b d-a e) \left (a b+b^2 x\right )}{e^2}+\frac {b^2 \left (a b+b^2 x\right )^2}{e}-\frac {b^3 (b d-a e)^3}{e^3 (d+e x)}\right ) \, dx}{b^2 \left (a b+b^2 x\right )} \\ & = \frac {b (b d-a e)^2 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x)}-\frac {(b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^2}+\frac {(a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e}-\frac {(b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^4 (a+b x)} \\ \end{align*}
Time = 1.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.55 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{d+e x} \, dx=\frac {\sqrt {(a+b x)^2} \left (b e x \left (18 a^2 e^2+9 a b e (-2 d+e x)+b^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )-6 (b d-a e)^3 \log (d+e x)\right )}{6 e^4 (a+b x)} \]
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Time = 2.80 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.85
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b \left (\frac {1}{3} b^{2} e^{2} x^{3}+\frac {3}{2} x^{2} a b \,e^{2}-\frac {1}{2} b^{2} d e \,x^{2}+3 a^{2} e^{2} x -3 a b d e x +b^{2} d^{2} x \right )}{\left (b x +a \right ) e^{3}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \ln \left (e x +d \right )}{\left (b x +a \right ) e^{4}}\) | \(141\) |
default | \(\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} \left (2 e^{3} x^{3} b^{3}+9 x^{2} a \,b^{2} e^{3}-3 x^{2} b^{3} d \,e^{2}+6 \ln \left (e x +d \right ) a^{3} e^{3}-18 \ln \left (e x +d \right ) a^{2} b d \,e^{2}+18 \ln \left (e x +d \right ) a \,b^{2} d^{2} e -6 \ln \left (e x +d \right ) b^{3} d^{3}+18 a^{2} b \,e^{3} x -18 x a \,b^{2} d \,e^{2}+6 b^{3} d^{2} e x \right )}{6 \left (b x +a \right )^{3} e^{4}}\) | \(149\) |
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Time = 0.29 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.69 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{d+e x} \, dx=\frac {2 \, b^{3} e^{3} x^{3} - 3 \, {\left (b^{3} d e^{2} - 3 \, a b^{2} e^{3}\right )} x^{2} + 6 \, {\left (b^{3} d^{2} e - 3 \, a b^{2} d e^{2} + 3 \, a^{2} b e^{3}\right )} x - 6 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \log \left (e x + d\right )}{6 \, e^{4}} \]
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\[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{d+e x} \, dx=\int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{d + e x}\, dx \]
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Exception generated. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{d+e x} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.27 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{d+e x} \, dx=\frac {2 \, b^{3} e^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, b^{3} d e x^{2} \mathrm {sgn}\left (b x + a\right ) + 9 \, a b^{2} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, b^{3} d^{2} x \mathrm {sgn}\left (b x + a\right ) - 18 \, a b^{2} d e x \mathrm {sgn}\left (b x + a\right ) + 18 \, a^{2} b e^{2} x \mathrm {sgn}\left (b x + a\right )}{6 \, e^{3}} - \frac {{\left (b^{3} d^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a b^{2} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b d e^{2} \mathrm {sgn}\left (b x + a\right ) - a^{3} e^{3} \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | e x + d \right |}\right )}{e^{4}} \]
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Timed out. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{d+e x} \, dx=\int \frac {{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{d+e\,x} \,d x \]
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