Integrand size = 33, antiderivative size = 104 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx=\frac {(A b-a B) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 (b d-a e)^2 (d+e x)^2}+\frac {(B d-A e) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 (b d-a e)^2 (d+e x)^3} \]
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Time = 0.04 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {783, 660, 37} \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx=\frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2} (B d-A e)}{3 (d+e x)^3 (b d-a e)^2}+\frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (A b-a B)}{2 (d+e x)^2 (b d-a e)^2} \]
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Rule 37
Rule 660
Rule 783
Rubi steps \begin{align*} \text {integral}& = \frac {(B d-A e) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 (b d-a e)^2 (d+e x)^3}+\frac {(A b-a B) \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^3} \, dx}{b d-a e} \\ & = \frac {(B d-A e) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 (b d-a e)^2 (d+e x)^3}+\frac {\left ((A b-a B) \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {a b+b^2 x}{(d+e x)^3} \, dx}{(b d-a e) \left (a b+b^2 x\right )} \\ & = \frac {(A b-a B) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 (b d-a e)^2 (d+e x)^2}+\frac {(B d-A e) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 (b d-a e)^2 (d+e x)^3} \\ \end{align*}
Time = 1.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.78 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx=-\frac {\sqrt {(a+b x)^2} \left (a e (2 A e+B (d+3 e x))+b \left (A e (d+3 e x)+2 B \left (d^2+3 d e x+3 e^2 x^2\right )\right )\right )}{6 e^3 (a+b x) (d+e x)^3} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.61 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.74
method | result | size |
default | \(-\frac {\operatorname {csgn}\left (b x +a \right ) \left (6 B b \,e^{2} x^{2}+3 A b \,e^{2} x +3 B a \,e^{2} x +6 B b d e x +2 A a \,e^{2}+A b d e +B a d e +2 B b \,d^{2}\right )}{6 e^{3} \left (e x +d \right )^{3}}\) | \(77\) |
risch | \(\frac {\left (-\frac {B b \,x^{2}}{e}-\frac {\left (A b e +B a e +2 B b d \right ) x}{2 e^{2}}-\frac {2 A a \,e^{2}+A b d e +B a d e +2 B b \,d^{2}}{6 e^{3}}\right ) \sqrt {\left (b x +a \right )^{2}}}{\left (e x +d \right )^{3} \left (b x +a \right )}\) | \(86\) |
gosper | \(-\frac {\left (6 B b \,e^{2} x^{2}+3 A b \,e^{2} x +3 B a \,e^{2} x +6 B b d e x +2 A a \,e^{2}+A b d e +B a d e +2 B b \,d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{6 \left (e x +d \right )^{3} e^{3} \left (b x +a \right )}\) | \(87\) |
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none
Time = 0.32 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.89 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx=-\frac {6 \, B b e^{2} x^{2} + 2 \, B b d^{2} + 2 \, A a e^{2} + {\left (B a + A b\right )} d e + 3 \, {\left (2 \, B b d e + {\left (B a + A b\right )} e^{2}\right )} x}{6 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]
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Timed out. \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 177 vs. \(2 (87) = 174\).
Time = 0.26 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.70 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx=\frac {{\left (2 \, B b^{3} d - 3 \, B a b^{2} e + A b^{3} e\right )} \mathrm {sgn}\left (b x + a\right )}{6 \, {\left (b^{2} d^{2} e^{3} - 2 \, a b d e^{4} + a^{2} e^{5}\right )}} - \frac {6 \, B b e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, B b d e x \mathrm {sgn}\left (b x + a\right ) + 3 \, B a e^{2} x \mathrm {sgn}\left (b x + a\right ) + 3 \, A b e^{2} x \mathrm {sgn}\left (b x + a\right ) + 2 \, B b d^{2} \mathrm {sgn}\left (b x + a\right ) + B a d e \mathrm {sgn}\left (b x + a\right ) + A b d e \mathrm {sgn}\left (b x + a\right ) + 2 \, A a e^{2} \mathrm {sgn}\left (b x + a\right )}{6 \, {\left (e x + d\right )}^{3} e^{3}} \]
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Time = 10.70 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.83 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx=-\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (2\,A\,a\,e^2+2\,B\,b\,d^2+3\,A\,b\,e^2\,x+3\,B\,a\,e^2\,x+6\,B\,b\,e^2\,x^2+A\,b\,d\,e+B\,a\,d\,e+6\,B\,b\,d\,e\,x\right )}{6\,e^3\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^3} \]
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