Integrand size = 35, antiderivative size = 148 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{3/2}} \, dx=-\frac {2 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x) \sqrt {d+e x}}-\frac {4 b (b d-a e) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x)}+\frac {2 b^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^3 (a+b x)} \]
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Time = 0.05 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {784, 21, 45} \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{3/2}} \, dx=\frac {2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^3 (a+b x)}-\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)}{e^3 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^3 (a+b x) \sqrt {d+e x}} \]
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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)^{3/2}} \, 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)^{3/2}} \, 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 d+a e)^2}{e^2 (d+e x)^{3/2}}-\frac {2 b (b d-a e)}{e^2 \sqrt {d+e x}}+\frac {b^2 \sqrt {d+e x}}{e^2}\right ) \, dx}{a b+b^2 x} \\ & = -\frac {2 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x) \sqrt {d+e x}}-\frac {4 b (b d-a e) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x)}+\frac {2 b^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^3 (a+b x)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.53 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{3/2}} \, dx=-\frac {2 \sqrt {(a+b x)^2} \left (3 a^2 e^2-6 a b e (2 d+e x)+b^2 \left (8 d^2+4 d e x-e^2 x^2\right )\right )}{3 e^3 (a+b x) \sqrt {d+e x}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.26 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.47
method | result | size |
default | \(-\frac {2 \,\operatorname {csgn}\left (b x +a \right ) \left (-b^{2} e^{2} x^{2}-6 a b \,e^{2} x +4 b^{2} d e x +3 e^{2} a^{2}-12 a b d e +8 b^{2} d^{2}\right )}{3 e^{3} \sqrt {e x +d}}\) | \(69\) |
gosper | \(-\frac {2 \left (-b^{2} e^{2} x^{2}-6 a b \,e^{2} x +4 b^{2} d e x +3 e^{2} a^{2}-12 a b d e +8 b^{2} d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{3 \sqrt {e x +d}\, e^{3} \left (b x +a \right )}\) | \(79\) |
risch | \(\frac {2 b \left (b e x +6 a e -5 b d \right ) \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{3 e^{3} \left (b x +a \right )}-\frac {2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{e^{3} \sqrt {e x +d}\, \left (b x +a \right )}\) | \(93\) |
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Time = 0.34 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.49 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (b^{2} e^{2} x^{2} - 8 \, b^{2} d^{2} + 12 \, a b d e - 3 \, a^{2} e^{2} - 2 \, {\left (2 \, b^{2} d e - 3 \, a b e^{2}\right )} x\right )} \sqrt {e x + d}}{3 \, {\left (e^{4} x + d e^{3}\right )}} \]
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\[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{3/2}} \, dx=\int \frac {\left (a + b x\right ) \sqrt {\left (a + b x\right )^{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.51 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (b e x + 2 \, b d - a e\right )} a}{\sqrt {e x + d} e^{2}} + \frac {2 \, {\left (b e^{2} x^{2} - 8 \, b d^{2} + 6 \, a d e - {\left (4 \, b d e - 3 \, a e^{2}\right )} x\right )} b}{3 \, \sqrt {e x + d} e^{3}} \]
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Time = 0.26 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{3/2}} \, dx=-\frac {2 \, {\left (b^{2} d^{2} \mathrm {sgn}\left (b x + a\right ) - 2 \, a b d e \mathrm {sgn}\left (b x + a\right ) + a^{2} e^{2} \mathrm {sgn}\left (b x + a\right )\right )}}{\sqrt {e x + d} e^{3}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} b^{2} e^{6} \mathrm {sgn}\left (b x + a\right ) - 6 \, \sqrt {e x + d} b^{2} d e^{6} \mathrm {sgn}\left (b x + a\right ) + 6 \, \sqrt {e x + d} a b e^{7} \mathrm {sgn}\left (b x + a\right )\right )}}{3 \, e^{9}} \]
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Time = 11.50 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.61 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{3/2}} \, dx=\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (\frac {4\,x\,\left (3\,a\,e-2\,b\,d\right )}{3\,e^2}+\frac {2\,b\,x^2}{3\,e}-\frac {6\,a^2\,e^2-24\,a\,b\,d\,e+16\,b^2\,d^2}{3\,b\,e^3}\right )}{x\,\sqrt {d+e\,x}+\frac {a\,\sqrt {d+e\,x}}{b}} \]
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