Integrand size = 35, antiderivative size = 194 \[ \int \frac {A+B x}{(d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {2 (B d-A e) (a+b x)}{3 e (b d-a e) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x)}{(b d-a e)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 \sqrt {b} (A b-a B) (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.09 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {784, 79, 53, 65, 214} \[ \int \frac {A+B x}{(d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {2 \sqrt {b} (a+b x) (A b-a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{5/2}}+\frac {2 (a+b x) (A b-a B)}{\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^2}-\frac {2 (a+b x) (B d-A e)}{3 e \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)} \]
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Rule 53
Rule 65
Rule 79
Rule 214
Rule 784
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a b+b^2 x\right ) \int \frac {A+B x}{\left (a b+b^2 x\right ) (d+e x)^{5/2}} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {2 (B d-A e) (a+b x)}{3 e (b d-a e) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left ((A b-a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) (d+e x)^{3/2}} \, dx}{(b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {2 (B d-A e) (a+b x)}{3 e (b d-a e) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x)}{(b d-a e)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (b (A b-a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{(b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {2 (B d-A e) (a+b x)}{3 e (b d-a e) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x)}{(b d-a e)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 b (A b-a B) \left (a b+b^2 x\right )\right ) \text {Subst}\left (\int \frac {1}{a b-\frac {b^2 d}{e}+\frac {b^2 x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{e (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {2 (B d-A e) (a+b x)}{3 e (b d-a e) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x)}{(b d-a e)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 \sqrt {b} (A b-a B) (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.70 \[ \int \frac {A+B x}{(d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {2 (a+b x) \left (\frac {-b B d^2+A b e (4 d+3 e x)-a e (2 B d+A e+3 B e x)}{e (b d-a e)^2 (d+e x)^{3/2}}+\frac {3 \sqrt {b} (A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{5/2}}\right )}{3 \sqrt {(a+b x)^2}} \]
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Time = 0.25 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.21
method | result | size |
default | \(\frac {2 \left (b x +a \right ) \left (3 A \left (e x +d \right )^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) b^{2} e -3 B \left (e x +d \right )^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a b e +3 A \sqrt {\left (a e -b d \right ) b}\, b \,e^{2} x -3 B \sqrt {\left (a e -b d \right ) b}\, a \,e^{2} x -A \sqrt {\left (a e -b d \right ) b}\, a \,e^{2}+4 A \sqrt {\left (a e -b d \right ) b}\, b d e -2 B \sqrt {\left (a e -b d \right ) b}\, a d e -B \sqrt {\left (a e -b d \right ) b}\, b \,d^{2}\right )}{3 \sqrt {\left (b x +a \right )^{2}}\, e \left (a e -b d \right )^{2} \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}}\) | \(235\) |
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Time = 0.40 (sec) , antiderivative size = 506, normalized size of antiderivative = 2.61 \[ \int \frac {A+B x}{(d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\left [-\frac {3 \, {\left ({\left (B a - A b\right )} e^{3} x^{2} + 2 \, {\left (B a - A b\right )} d e^{2} x + {\left (B a - A b\right )} d^{2} e\right )} \sqrt {\frac {b}{b d - a e}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, {\left (b d - a e\right )} \sqrt {e x + d} \sqrt {\frac {b}{b d - a e}}}{b x + a}\right ) + 2 \, {\left (B b d^{2} + A a e^{2} + 3 \, {\left (B a - A b\right )} e^{2} x + 2 \, {\left (B a - 2 \, A b\right )} d e\right )} \sqrt {e x + d}}{3 \, {\left (b^{2} d^{4} e - 2 \, a b d^{3} e^{2} + a^{2} d^{2} e^{3} + {\left (b^{2} d^{2} e^{3} - 2 \, a b d e^{4} + a^{2} e^{5}\right )} x^{2} + 2 \, {\left (b^{2} d^{3} e^{2} - 2 \, a b d^{2} e^{3} + a^{2} d e^{4}\right )} x\right )}}, \frac {2 \, {\left (3 \, {\left ({\left (B a - A b\right )} e^{3} x^{2} + 2 \, {\left (B a - A b\right )} d e^{2} x + {\left (B a - A b\right )} d^{2} e\right )} \sqrt {-\frac {b}{b d - a e}} \arctan \left (-\frac {{\left (b d - a e\right )} \sqrt {e x + d} \sqrt {-\frac {b}{b d - a e}}}{b e x + b d}\right ) - {\left (B b d^{2} + A a e^{2} + 3 \, {\left (B a - A b\right )} e^{2} x + 2 \, {\left (B a - 2 \, A b\right )} d e\right )} \sqrt {e x + d}\right )}}{3 \, {\left (b^{2} d^{4} e - 2 \, a b d^{3} e^{2} + a^{2} d^{2} e^{3} + {\left (b^{2} d^{2} e^{3} - 2 \, a b d e^{4} + a^{2} e^{5}\right )} x^{2} + 2 \, {\left (b^{2} d^{3} e^{2} - 2 \, a b d^{2} e^{3} + a^{2} d e^{4}\right )} x\right )}}\right ] \]
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\[ \int \frac {A+B x}{(d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {A + B x}{\left (d + e x\right )^{\frac {5}{2}} \sqrt {\left (a + b x\right )^{2}}}\, dx \]
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\[ \int \frac {A+B x}{(d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int { \frac {B x + A}{\sqrt {{\left (b x + a\right )}^{2}} {\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 0.30 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.04 \[ \int \frac {A+B x}{(d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {2 \, {\left (B a b \mathrm {sgn}\left (b x + a\right ) - A b^{2} \mathrm {sgn}\left (b x + a\right )\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \sqrt {-b^{2} d + a b e}} - \frac {2 \, {\left (B b d^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, {\left (e x + d\right )} B a e \mathrm {sgn}\left (b x + a\right ) - 3 \, {\left (e x + d\right )} A b e \mathrm {sgn}\left (b x + a\right ) - B a d e \mathrm {sgn}\left (b x + a\right ) - A b d e \mathrm {sgn}\left (b x + a\right ) + A a e^{2} \mathrm {sgn}\left (b x + a\right )\right )}}{3 \, {\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\right )} {\left (e x + d\right )}^{\frac {3}{2}}} \]
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Timed out. \[ \int \frac {A+B x}{(d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {A+B\,x}{\sqrt {{\left (a+b\,x\right )}^2}\,{\left (d+e\,x\right )}^{5/2}} \,d x \]
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