Integrand size = 35, antiderivative size = 160 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{3/2}} \, dx=-\frac {2 (b d-a e) (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x) \sqrt {d+e x}}-\frac {2 (2 b B d-A b e-a B e) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x)}+\frac {2 b B (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^3 (a+b x)} \]
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Time = 0.06 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {784, 78} \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{3/2}} \, dx=-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (-a B e-A b e+2 b B d)}{e^3 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{e^3 (a+b x) \sqrt {d+e x}}+\frac {2 b B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^3 (a+b x)} \]
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Rule 78
Rule 784
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 ) (A+B x)}{(d+e x)^{3/2}} \, dx}{a b+b^2 x} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b (b d-a e) (-B d+A e)}{e^2 (d+e x)^{3/2}}+\frac {b (-2 b B d+A b e+a B e)}{e^2 \sqrt {d+e x}}+\frac {b^2 B \sqrt {d+e x}}{e^2}\right ) \, dx}{a b+b^2 x} \\ & = -\frac {2 (b d-a e) (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 (2 b B d-A b e-a B e) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x)}+\frac {2 b B (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 = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.53 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{3/2}} \, dx=\frac {2 \sqrt {(a+b x)^2} \left (3 A b e (2 d+e x)+3 a e (2 B d-A e+B e x)+b B \left (-8 d^2-4 d e x+e^2 x^2\right )\right )}{3 e^3 (a+b x) \sqrt {d+e x}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.23 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.49
method | result | size |
default | \(-\frac {2 \,\operatorname {csgn}\left (b x +a \right ) \left (-B b \,e^{2} x^{2}-3 A b \,e^{2} x -3 B a \,e^{2} x +4 B b d e x +3 A a \,e^{2}-6 A b d e -6 B a d e +8 B b \,d^{2}\right )}{3 e^{3} \sqrt {e x +d}}\) | \(79\) |
gosper | \(-\frac {2 \left (-B b \,e^{2} x^{2}-3 A b \,e^{2} x -3 B a \,e^{2} x +4 B b d e x +3 A a \,e^{2}-6 A b d e -6 B a d e +8 B b \,d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{3 \sqrt {e x +d}\, e^{3} \left (b x +a \right )}\) | \(89\) |
risch | \(\frac {2 \left (B b e x +3 A b e +3 B a e -5 B b d \right ) \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{3 e^{3} \left (b x +a \right )}-\frac {2 \left (A a \,e^{2}-A b d e -B a d e +B b \,d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{e^{3} \sqrt {e x +d}\, \left (b x +a \right )}\) | \(104\) |
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Time = 0.28 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.49 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (B b e^{2} x^{2} - 8 \, B b d^{2} - 3 \, A a e^{2} + 6 \, {\left (B a + A b\right )} d e - {\left (4 \, B b d e - 3 \, {\left (B a + A b\right )} e^{2}\right )} x\right )} \sqrt {e x + d}}{3 \, {\left (e^{4} x + d e^{3}\right )}} \]
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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 = 0.23 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.47 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (b e x + 2 \, b d - a e\right )} A}{\sqrt {e x + d} e^{2}} + \frac {2 \, {\left (b e^{2} x^{2} - 8 \, b d^{2} + 6 \, a d e - {\left (4 \, b d e - 3 \, a e^{2}\right )} x\right )} B}{3 \, \sqrt {e x + d} e^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.92 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{3/2}} \, dx=-\frac {2 \, {\left (B b d^{2} \mathrm {sgn}\left (b x + a\right ) - B a d e \mathrm {sgn}\left (b x + a\right ) - A b d e \mathrm {sgn}\left (b x + a\right ) + A a e^{2} \mathrm {sgn}\left (b x + a\right )\right )}}{\sqrt {e x + d} e^{3}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} B b e^{6} \mathrm {sgn}\left (b x + a\right ) - 6 \, \sqrt {e x + d} B b d e^{6} \mathrm {sgn}\left (b x + a\right ) + 3 \, \sqrt {e x + d} B a e^{7} \mathrm {sgn}\left (b x + a\right ) + 3 \, \sqrt {e x + d} A b e^{7} \mathrm {sgn}\left (b x + a\right )\right )}}{3 \, e^{9}} \]
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Time = 11.18 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.68 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{3/2}} \, dx=\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (\frac {2\,B\,x^2}{3\,e}-\frac {6\,A\,a\,e^2+16\,B\,b\,d^2-12\,A\,b\,d\,e-12\,B\,a\,d\,e}{3\,b\,e^3}+\frac {x\,\left (6\,A\,b\,e^2+6\,B\,a\,e^2-8\,B\,b\,d\,e\right )}{3\,b\,e^3}\right )}{x\,\sqrt {d+e\,x}+\frac {a\,\sqrt {d+e\,x}}{b}} \]
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