Integrand size = 33, antiderivative size = 174 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{a^2+2 a b x+b^2 x^2} \, dx=\frac {(2 b B d+3 A b e-5 a B e) \sqrt {d+e x}}{b^3}+\frac {(2 b B d+3 A b e-5 a B e) (d+e x)^{3/2}}{3 b^2 (b d-a e)}-\frac {(A b-a B) (d+e x)^{5/2}}{b (b d-a e) (a+b x)}-\frac {\sqrt {b d-a e} (2 b B d+3 A b e-5 a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{7/2}} \]
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Time = 0.10 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {27, 79, 52, 65, 214} \[ \int \frac {(A+B x) (d+e x)^{3/2}}{a^2+2 a b x+b^2 x^2} \, dx=-\frac {\sqrt {b d-a e} (-5 a B e+3 A b e+2 b B d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{7/2}}+\frac {\sqrt {d+e x} (-5 a B e+3 A b e+2 b B d)}{b^3}+\frac {(d+e x)^{3/2} (-5 a B e+3 A b e+2 b B d)}{3 b^2 (b d-a e)}-\frac {(d+e x)^{5/2} (A b-a B)}{b (a+b x) (b d-a e)} \]
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Rule 27
Rule 52
Rule 65
Rule 79
Rule 214
Rubi steps \begin{align*} \text {integral}& = \int \frac {(A+B x) (d+e x)^{3/2}}{(a+b x)^2} \, dx \\ & = -\frac {(A b-a B) (d+e x)^{5/2}}{b (b d-a e) (a+b x)}+\frac {(2 b B d+3 A b e-5 a B e) \int \frac {(d+e x)^{3/2}}{a+b x} \, dx}{2 b (b d-a e)} \\ & = \frac {(2 b B d+3 A b e-5 a B e) (d+e x)^{3/2}}{3 b^2 (b d-a e)}-\frac {(A b-a B) (d+e x)^{5/2}}{b (b d-a e) (a+b x)}+\frac {(2 b B d+3 A b e-5 a B e) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{2 b^2} \\ & = \frac {(2 b B d+3 A b e-5 a B e) \sqrt {d+e x}}{b^3}+\frac {(2 b B d+3 A b e-5 a B e) (d+e x)^{3/2}}{3 b^2 (b d-a e)}-\frac {(A b-a B) (d+e x)^{5/2}}{b (b d-a e) (a+b x)}+\frac {((b d-a e) (2 b B d+3 A b e-5 a B e)) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{2 b^3} \\ & = \frac {(2 b B d+3 A b e-5 a B e) \sqrt {d+e x}}{b^3}+\frac {(2 b B d+3 A b e-5 a B e) (d+e x)^{3/2}}{3 b^2 (b d-a e)}-\frac {(A b-a B) (d+e x)^{5/2}}{b (b d-a e) (a+b x)}+\frac {((b d-a e) (2 b B d+3 A b e-5 a B e)) \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{b^3 e} \\ & = \frac {(2 b B d+3 A b e-5 a B e) \sqrt {d+e x}}{b^3}+\frac {(2 b B d+3 A b e-5 a B e) (d+e x)^{3/2}}{3 b^2 (b d-a e)}-\frac {(A b-a B) (d+e x)^{5/2}}{b (b d-a e) (a+b x)}-\frac {\sqrt {b d-a e} (2 b B d+3 A b e-5 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{7/2}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.80 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{a^2+2 a b x+b^2 x^2} \, dx=\frac {\sqrt {d+e x} \left (3 A b (-b d+3 a e+2 b e x)+B \left (-15 a^2 e+a b (11 d-10 e x)+2 b^2 x (4 d+e x)\right )\right )}{3 b^3 (a+b x)}-\frac {\sqrt {-b d+a e} (2 b B d+3 A b e-5 a B e) \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{b^{7/2}} \]
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Time = 0.34 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.79
method | result | size |
risch | \(\frac {2 \left (B b e x +3 A b e -6 B a e +4 B b d \right ) \sqrt {e x +d}}{3 b^{3}}-\frac {\left (2 a e -2 b d \right ) \left (\frac {\left (-\frac {1}{2} A b e +\frac {1}{2} B a e \right ) \sqrt {e x +d}}{b \left (e x +d \right )+a e -b d}+\frac {\left (3 A b e -5 B a e +2 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 \sqrt {\left (a e -b d \right ) b}}\right )}{b^{3}}\) | \(138\) |
pseudoelliptic | \(\frac {-3 \left (a e -b d \right ) \left (\left (A e +\frac {2 B d}{3}\right ) b -\frac {5 B a e}{3}\right ) \left (b x +a \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )+3 \left (\frac {\left (2 \left (\frac {B x}{3}+A \right ) x e -d \left (-\frac {8 B x}{3}+A \right )\right ) b^{2}}{3}+\left (e \left (-\frac {10 B x}{9}+A \right )+\frac {11 B d}{9}\right ) a b -\frac {5 B e \,a^{2}}{3}\right ) \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}}{b^{3} \left (b x +a \right ) \sqrt {\left (a e -b d \right ) b}}\) | \(148\) |
derivativedivides | \(\frac {\frac {2 B \left (e x +d \right )^{\frac {3}{2}} b}{3}+2 A b e \sqrt {e x +d}-4 B a e \sqrt {e x +d}+2 B b d \sqrt {e x +d}}{b^{3}}-\frac {2 \left (\frac {\left (-\frac {1}{2} A a b \,e^{2}+\frac {1}{2} A \,b^{2} d e +\frac {1}{2} a^{2} B \,e^{2}-\frac {1}{2} B a b d e \right ) \sqrt {e x +d}}{b \left (e x +d \right )+a e -b d}+\frac {\left (3 A a b \,e^{2}-3 A \,b^{2} d e -5 a^{2} B \,e^{2}+7 B a b d e -2 B \,b^{2} d^{2}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 \sqrt {\left (a e -b d \right ) b}}\right )}{b^{3}}\) | \(195\) |
default | \(\frac {\frac {2 B \left (e x +d \right )^{\frac {3}{2}} b}{3}+2 A b e \sqrt {e x +d}-4 B a e \sqrt {e x +d}+2 B b d \sqrt {e x +d}}{b^{3}}-\frac {2 \left (\frac {\left (-\frac {1}{2} A a b \,e^{2}+\frac {1}{2} A \,b^{2} d e +\frac {1}{2} a^{2} B \,e^{2}-\frac {1}{2} B a b d e \right ) \sqrt {e x +d}}{b \left (e x +d \right )+a e -b d}+\frac {\left (3 A a b \,e^{2}-3 A \,b^{2} d e -5 a^{2} B \,e^{2}+7 B a b d e -2 B \,b^{2} d^{2}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 \sqrt {\left (a e -b d \right ) b}}\right )}{b^{3}}\) | \(195\) |
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Time = 0.69 (sec) , antiderivative size = 392, normalized size of antiderivative = 2.25 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{a^2+2 a b x+b^2 x^2} \, dx=\left [\frac {3 \, {\left (2 \, B a b d - {\left (5 \, B a^{2} - 3 \, A a b\right )} e + {\left (2 \, B b^{2} d - {\left (5 \, B a b - 3 \, A b^{2}\right )} e\right )} x\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) + 2 \, {\left (2 \, B b^{2} e x^{2} + {\left (11 \, B a b - 3 \, A b^{2}\right )} d - 3 \, {\left (5 \, B a^{2} - 3 \, A a b\right )} e + 2 \, {\left (4 \, B b^{2} d - {\left (5 \, B a b - 3 \, A b^{2}\right )} e\right )} x\right )} \sqrt {e x + d}}{6 \, {\left (b^{4} x + a b^{3}\right )}}, -\frac {3 \, {\left (2 \, B a b d - {\left (5 \, B a^{2} - 3 \, A a b\right )} e + {\left (2 \, B b^{2} d - {\left (5 \, B a b - 3 \, A b^{2}\right )} e\right )} x\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (2 \, B b^{2} e x^{2} + {\left (11 \, B a b - 3 \, A b^{2}\right )} d - 3 \, {\left (5 \, B a^{2} - 3 \, A a b\right )} e + 2 \, {\left (4 \, B b^{2} d - {\left (5 \, B a b - 3 \, A b^{2}\right )} e\right )} x\right )} \sqrt {e x + d}}{3 \, {\left (b^{4} x + a b^{3}\right )}}\right ] \]
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Timed out. \[ \int \frac {(A+B x) (d+e x)^{3/2}}{a^2+2 a b x+b^2 x^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(A+B x) (d+e x)^{3/2}}{a^2+2 a b x+b^2 x^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.28 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.29 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{a^2+2 a b x+b^2 x^2} \, dx=\frac {{\left (2 \, B b^{2} d^{2} - 7 \, B a b d e + 3 \, A b^{2} d e + 5 \, B a^{2} e^{2} - 3 \, A a b e^{2}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b^{3}} + \frac {\sqrt {e x + d} B a b d e - \sqrt {e x + d} A b^{2} d e - \sqrt {e x + d} B a^{2} e^{2} + \sqrt {e x + d} A a b e^{2}}{{\left ({\left (e x + d\right )} b - b d + a e\right )} b^{3}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} B b^{4} + 3 \, \sqrt {e x + d} B b^{4} d - 6 \, \sqrt {e x + d} B a b^{3} e + 3 \, \sqrt {e x + d} A b^{4} e\right )}}{3 \, b^{6}} \]
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Time = 10.66 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{a^2+2 a b x+b^2 x^2} \, dx=\left (\frac {2\,A\,e-2\,B\,d}{b^2}+\frac {2\,B\,\left (2\,b^2\,d-2\,a\,b\,e\right )}{b^4}\right )\,\sqrt {d+e\,x}-\frac {\sqrt {d+e\,x}\,\left (B\,a^2\,e^2-A\,a\,b\,e^2-B\,d\,a\,b\,e+A\,d\,b^2\,e\right )}{b^4\,\left (d+e\,x\right )-b^4\,d+a\,b^3\,e}+\frac {2\,B\,{\left (d+e\,x\right )}^{3/2}}{3\,b^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,1{}\mathrm {i}}{\sqrt {b\,d-a\,e}}\right )\,\sqrt {b\,d-a\,e}\,\left (3\,A\,b\,e-5\,B\,a\,e+2\,B\,b\,d\right )\,1{}\mathrm {i}}{b^{7/2}} \]
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