Integrand size = 33, antiderivative size = 158 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^7} \, dx=-\frac {(b d-a e) (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^3 (a+b x) (d+e x)^6}+\frac {(2 b B d-A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^3 (a+b x) (d+e x)^5}-\frac {b B \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^3 (a+b x) (d+e x)^4} \]
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Time = 0.06 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {784, 78} \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^7} \, dx=\frac {\sqrt {a^2+2 a b x+b^2 x^2} (-a B e-A b e+2 b B d)}{5 e^3 (a+b x) (d+e x)^5}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{6 e^3 (a+b x) (d+e x)^6}-\frac {b B \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^3 (a+b x) (d+e x)^4} \]
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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)^7} \, 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)^7}+\frac {b (-2 b B d+A b e+a B e)}{e^2 (d+e x)^6}+\frac {b^2 B}{e^2 (d+e x)^5}\right ) \, dx}{a b+b^2 x} \\ & = -\frac {(b d-a e) (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^3 (a+b x) (d+e x)^6}+\frac {(2 b B d-A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^3 (a+b x) (d+e x)^5}-\frac {b B \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^3 (a+b x) (d+e x)^4} \\ \end{align*}
Time = 1.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.52 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^7} \, dx=-\frac {\sqrt {(a+b x)^2} \left (2 a e (5 A e+B (d+6 e x))+b \left (2 A e (d+6 e x)+B \left (d^2+6 d e x+15 e^2 x^2\right )\right )\right )}{60 e^3 (a+b x) (d+e x)^6} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 1.19 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.49
method | result | size |
default | \(-\frac {\operatorname {csgn}\left (b x +a \right ) \left (15 B b \,e^{2} x^{2}+12 A b \,e^{2} x +12 B a \,e^{2} x +6 B b d e x +10 A a \,e^{2}+2 A b d e +2 B a d e +B b \,d^{2}\right )}{60 e^{3} \left (e x +d \right )^{6}}\) | \(78\) |
gosper | \(-\frac {\left (15 B b \,e^{2} x^{2}+12 A b \,e^{2} x +12 B a \,e^{2} x +6 B b d e x +10 A a \,e^{2}+2 A b d e +2 B a d e +B b \,d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{60 e^{3} \left (e x +d \right )^{6} \left (b x +a \right )}\) | \(88\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {B b \,x^{2}}{4 e}-\frac {\left (2 A b e +2 B a e +B b d \right ) x}{10 e^{2}}-\frac {10 A a \,e^{2}+2 A b d e +2 B a d e +B b \,d^{2}}{60 e^{3}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{6}}\) | \(88\) |
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Time = 0.30 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.80 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^7} \, dx=-\frac {15 \, B b e^{2} x^{2} + B b d^{2} + 10 \, A a e^{2} + 2 \, {\left (B a + A b\right )} d e + 6 \, {\left (B b d e + 2 \, {\left (B a + A b\right )} e^{2}\right )} x}{60 \, {\left (e^{9} x^{6} + 6 \, d e^{8} x^{5} + 15 \, d^{2} e^{7} x^{4} + 20 \, d^{3} e^{6} x^{3} + 15 \, d^{4} e^{5} x^{2} + 6 \, d^{5} e^{4} x + d^{6} e^{3}\right )}} \]
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Timed out. \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^7} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^7} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.28 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.40 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^7} \, dx=\frac {{\left (B b^{6} d - 3 \, B a b^{5} e + 2 \, A b^{6} e\right )} \mathrm {sgn}\left (b x + a\right )}{60 \, {\left (b^{5} d^{5} e^{3} - 5 \, a b^{4} d^{4} e^{4} + 10 \, a^{2} b^{3} d^{3} e^{5} - 10 \, a^{3} b^{2} d^{2} e^{6} + 5 \, a^{4} b d e^{7} - a^{5} e^{8}\right )}} - \frac {15 \, B b e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, B b d e x \mathrm {sgn}\left (b x + a\right ) + 12 \, B a e^{2} x \mathrm {sgn}\left (b x + a\right ) + 12 \, A b e^{2} x \mathrm {sgn}\left (b x + a\right ) + B b d^{2} \mathrm {sgn}\left (b x + a\right ) + 2 \, B a d e \mathrm {sgn}\left (b x + a\right ) + 2 \, A b d e \mathrm {sgn}\left (b x + a\right ) + 10 \, A a e^{2} \mathrm {sgn}\left (b x + a\right )}{60 \, {\left (e x + d\right )}^{6} e^{3}} \]
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Time = 10.86 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.55 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^7} \, dx=-\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (10\,A\,a\,e^2+B\,b\,d^2+12\,A\,b\,e^2\,x+12\,B\,a\,e^2\,x+15\,B\,b\,e^2\,x^2+2\,A\,b\,d\,e+2\,B\,a\,d\,e+6\,B\,b\,d\,e\,x\right )}{60\,e^3\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6} \]
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