Integrand size = 35, antiderivative size = 230 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {2 (A b-a B) (b d-a e) (a+b x) \sqrt {d+e x}}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x) (d+e x)^{3/2}}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B (a+b x) (d+e x)^{5/2}}{5 b e \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 (A b-a B) (b d-a e)^{3/2} (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.10 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {784, 81, 52, 65, 214} \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {2 (a+b x) (A b-a B) (b d-a e)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{3/2} (A b-a B)}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) \sqrt {d+e x} (A b-a B) (b d-a e)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B (a+b x) (d+e x)^{5/2}}{5 b e \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rule 52
Rule 65
Rule 81
Rule 214
Rule 784
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a b+b^2 x\right ) \int \frac {(A+B x) (d+e x)^{3/2}}{a b+b^2 x} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {2 B (a+b x) (d+e x)^{5/2}}{5 b e \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 \left (\frac {5}{2} A b^2 e-\frac {5}{2} a b B e\right ) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{3/2}}{a b+b^2 x} \, dx}{5 b^2 e \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {2 (A b-a B) (a+b x) (d+e x)^{3/2}}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B (a+b x) (d+e x)^{5/2}}{5 b e \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 \left (b^2 d-a b e\right ) \left (\frac {5}{2} A b^2 e-\frac {5}{2} a b B e\right ) \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{a b+b^2 x} \, dx}{5 b^4 e \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {2 (A b-a B) (b d-a e) (a+b x) \sqrt {d+e x}}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x) (d+e x)^{3/2}}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B (a+b x) (d+e x)^{5/2}}{5 b e \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 \left (b^2 d-a b e\right )^2 \left (\frac {5}{2} A b^2 e-\frac {5}{2} a b B e\right ) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{5 b^6 e \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {2 (A b-a B) (b d-a e) (a+b x) \sqrt {d+e x}}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x) (d+e x)^{3/2}}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B (a+b x) (d+e x)^{5/2}}{5 b e \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (4 \left (b^2 d-a b e\right )^2 \left (\frac {5}{2} A b^2 e-\frac {5}{2} a b B e\right ) \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 )}{5 b^6 e^2 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {2 (A b-a B) (b d-a e) (a+b x) \sqrt {d+e x}}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x) (d+e x)^{3/2}}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B (a+b x) (d+e x)^{5/2}}{5 b e \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 (A b-a B) (b d-a e)^{3/2} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.64 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {2 (a+b x) \left (\frac {\sqrt {b} \sqrt {d+e x} \left (15 a^2 B e^2-5 a b e (4 B d+3 A e+B e x)+b^2 \left (3 B (d+e x)^2+5 A e (4 d+e x)\right )\right )}{e}+15 (A b-a B) (-b d+a e)^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )\right )}{15 b^{7/2} \sqrt {(a+b x)^2}} \]
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Time = 0.24 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.98
method | result | size |
risch | \(-\frac {2 \left (-3 B \,b^{2} e^{2} x^{2}-5 A \,b^{2} e^{2} x +5 B a b \,e^{2} x -6 B \,b^{2} d e x +15 A a b \,e^{2}-20 A \,b^{2} d e -15 a^{2} B \,e^{2}+20 B a b d e -3 B \,b^{2} d^{2}\right ) \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{15 e \,b^{3} \left (b x +a \right )}+\frac {2 \left (A \,a^{2} b \,e^{2}-2 A \,b^{2} d e a +A \,d^{2} b^{3}-B \,e^{2} a^{3}+2 B \,a^{2} b d e -B \,b^{2} d^{2} a \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) \sqrt {\left (b x +a \right )^{2}}}{b^{3} \sqrt {\left (a e -b d \right ) b}\, \left (b x +a \right )}\) | \(225\) |
default | \(\frac {2 \left (b x +a \right ) \left (3 B \left (e x +d \right )^{\frac {5}{2}} \sqrt {\left (a e -b d \right ) b}\, b^{2}+5 A \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, b^{2} e +15 A \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{2} b \,e^{3}-30 A \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a \,b^{2} d \,e^{2}+15 A \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) b^{3} d^{2} e -5 B \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, a b e -15 B \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{3} e^{3}+30 B \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{2} b d \,e^{2}-15 B \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a \,b^{2} d^{2} e -15 A \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a b \,e^{2}+15 A \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, b^{2} d e +15 B \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a^{2} e^{2}-15 B \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a b d e \right )}{15 \sqrt {\left (b x +a \right )^{2}}\, e \,b^{3} \sqrt {\left (a e -b d \right ) b}}\) | \(414\) |
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Time = 0.35 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.62 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\left [-\frac {15 \, {\left ({\left (B a b - A b^{2}\right )} d e - {\left (B a^{2} - A a b\right )} e^{2}\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) - 2 \, {\left (3 \, B b^{2} e^{2} x^{2} + 3 \, B b^{2} d^{2} - 20 \, {\left (B a b - A b^{2}\right )} d e + 15 \, {\left (B a^{2} - A a b\right )} e^{2} + {\left (6 \, B b^{2} d e - 5 \, {\left (B a b - A b^{2}\right )} e^{2}\right )} x\right )} \sqrt {e x + d}}{15 \, b^{3} e}, \frac {2 \, {\left (15 \, {\left ({\left (B a b - A b^{2}\right )} d e - {\left (B a^{2} - A a b\right )} e^{2}\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) + {\left (3 \, B b^{2} e^{2} x^{2} + 3 \, B b^{2} d^{2} - 20 \, {\left (B a b - A b^{2}\right )} d e + 15 \, {\left (B a^{2} - A a b\right )} e^{2} + {\left (6 \, B b^{2} d e - 5 \, {\left (B a b - A b^{2}\right )} e^{2}\right )} x\right )} \sqrt {e x + d}\right )}}{15 \, b^{3} e}\right ] \]
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\[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {\left (A + B x\right ) \left (d + e x\right )^{\frac {3}{2}}}{\sqrt {\left (a + b x\right )^{2}}}\, dx \]
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\[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{\sqrt {{\left (b x + a\right )}^{2}}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.32 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {2 \, {\left (B a b^{2} d^{2} \mathrm {sgn}\left (b x + a\right ) - A b^{3} d^{2} \mathrm {sgn}\left (b x + a\right ) - 2 \, B a^{2} b d e \mathrm {sgn}\left (b x + a\right ) + 2 \, A a b^{2} d e \mathrm {sgn}\left (b x + a\right ) + B a^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - A a^{2} b e^{2} \mathrm {sgn}\left (b x + a\right )\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b^{3}} + \frac {2 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} B b^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) - 5 \, {\left (e x + d\right )}^{\frac {3}{2}} B a b^{3} e^{5} \mathrm {sgn}\left (b x + a\right ) + 5 \, {\left (e x + d\right )}^{\frac {3}{2}} A b^{4} e^{5} \mathrm {sgn}\left (b x + a\right ) - 15 \, \sqrt {e x + d} B a b^{3} d e^{5} \mathrm {sgn}\left (b x + a\right ) + 15 \, \sqrt {e x + d} A b^{4} d e^{5} \mathrm {sgn}\left (b x + a\right ) + 15 \, \sqrt {e x + d} B a^{2} b^{2} e^{6} \mathrm {sgn}\left (b x + a\right ) - 15 \, \sqrt {e x + d} A a b^{3} e^{6} \mathrm {sgn}\left (b x + a\right )\right )}}{15 \, b^{5} e^{5}} \]
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Timed out. \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{3/2}}{\sqrt {{\left (a+b\,x\right )}^2}} \,d x \]
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