Integrand size = 35, antiderivative size = 148 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{5/2}} \, dx=-\frac {2 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^3 (a+b x) (d+e x)^{3/2}}+\frac {4 b (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x) \sqrt {d+e x}}+\frac {2 b^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x)} \]
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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} \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{5/2}} \, dx=\frac {2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x}}{e^3 (a+b x)}+\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^3 (a+b x) \sqrt {d+e x}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{3 e^3 (a+b x) (d+e x)^{3/2}} \]
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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)^{5/2}} \, 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)^{5/2}} \, 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)^{5/2}}-\frac {2 b (b d-a e)}{e^2 (d+e x)^{3/2}}+\frac {b^2}{e^2 \sqrt {d+e x}}\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}}{3 e^3 (a+b x) (d+e x)^{3/2}}+\frac {4 b (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x) \sqrt {d+e x}}+\frac {2 b^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.53 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{5/2}} \, dx=-\frac {2 \sqrt {(a+b x)^2} \left (a^2 e^2+2 a b e (2 d+3 e x)-b^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )\right )}{3 e^3 (a+b x) (d+e x)^{3/2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.26 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.46
method | result | size |
default | \(-\frac {2 \,\operatorname {csgn}\left (b x +a \right ) \left (-3 b^{2} e^{2} x^{2}+6 a b \,e^{2} x -12 b^{2} d e x +e^{2} a^{2}+4 a b d e -8 b^{2} d^{2}\right )}{3 e^{3} \left (e x +d \right )^{\frac {3}{2}}}\) | \(68\) |
gosper | \(-\frac {2 \left (-3 b^{2} e^{2} x^{2}+6 a b \,e^{2} x -12 b^{2} d e x +e^{2} a^{2}+4 a b d e -8 b^{2} d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{3 \left (e x +d \right )^{\frac {3}{2}} e^{3} \left (b x +a \right )}\) | \(78\) |
risch | \(\frac {2 b^{2} \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{e^{3} \left (b x +a \right )}-\frac {2 \left (6 b e x +a e +5 b d \right ) \left (a e -b d \right ) \sqrt {\left (b x +a \right )^{2}}}{3 e^{3} \left (e x +d \right )^{\frac {3}{2}} \left (b x +a \right )}\) | \(82\) |
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Time = 0.31 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.57 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (3 \, b^{2} e^{2} x^{2} + 8 \, b^{2} d^{2} - 4 \, a b d e - a^{2} e^{2} + 6 \, {\left (2 \, b^{2} d e - a b e^{2}\right )} x\right )} \sqrt {e x + d}}{3 \, {\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} \]
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\[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{5/2}} \, dx=\int \frac {\left (a + b x\right ) \sqrt {\left (a + b x\right )^{2}}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.65 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{5/2}} \, dx=-\frac {2 \, {\left (3 \, b e x + 2 \, b d + a e\right )} a}{3 \, {\left (e^{3} x + d e^{2}\right )} \sqrt {e x + d}} + \frac {2 \, {\left (3 \, b e^{2} x^{2} + 8 \, b d^{2} - 2 \, a d e + 3 \, {\left (4 \, b d e - a e^{2}\right )} x\right )} b}{3 \, {\left (e^{4} x + d e^{3}\right )} \sqrt {e x + d}} \]
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Time = 0.36 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.73 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{5/2}} \, dx=\frac {2 \, \sqrt {e x + d} b^{2} \mathrm {sgn}\left (b x + a\right )}{e^{3}} + \frac {2 \, {\left (6 \, {\left (e x + d\right )} b^{2} d \mathrm {sgn}\left (b x + a\right ) - b^{2} d^{2} \mathrm {sgn}\left (b x + a\right ) - 6 \, {\left (e x + d\right )} a b e \mathrm {sgn}\left (b x + a\right ) + 2 \, a b d e \mathrm {sgn}\left (b x + a\right ) - a^{2} e^{2} \mathrm {sgn}\left (b x + a\right )\right )}}{3 \, {\left (e x + d\right )}^{\frac {3}{2}} e^{3}} \]
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Time = 11.55 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{5/2}} \, dx=-\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (\frac {4\,x\,\left (a\,e-2\,b\,d\right )}{e^3}-\frac {2\,b\,x^2}{e^2}+\frac {2\,a^2\,e^2+8\,a\,b\,d\,e-16\,b^2\,d^2}{3\,b\,e^4}\right )}{x^2\,\sqrt {d+e\,x}+\frac {a\,d\,\sqrt {d+e\,x}}{b\,e}+\frac {x\,\left (3\,a\,e^4+3\,b\,d\,e^3\right )\,\sqrt {d+e\,x}}{3\,b\,e^4}} \]
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