Integrand size = 22, antiderivative size = 124 \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^{5/2}} \, dx=\frac {2 (b d-a e)^2 (B d-A e)}{3 e^4 (d+e x)^{3/2}}-\frac {2 (b d-a e) (3 b B d-2 A b e-a B e)}{e^4 \sqrt {d+e x}}-\frac {2 b (3 b B d-A b e-2 a B e) \sqrt {d+e x}}{e^4}+\frac {2 b^2 B (d+e x)^{3/2}}{3 e^4} \]
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Time = 0.04 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {78} \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^{5/2}} \, dx=-\frac {2 b \sqrt {d+e x} (-2 a B e-A b e+3 b B d)}{e^4}-\frac {2 (b d-a e) (-a B e-2 A b e+3 b B d)}{e^4 \sqrt {d+e x}}+\frac {2 (b d-a e)^2 (B d-A e)}{3 e^4 (d+e x)^{3/2}}+\frac {2 b^2 B (d+e x)^{3/2}}{3 e^4} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b d+a e)^2 (-B d+A e)}{e^3 (d+e x)^{5/2}}+\frac {(-b d+a e) (-3 b B d+2 A b e+a B e)}{e^3 (d+e x)^{3/2}}+\frac {b (-3 b B d+A b e+2 a B e)}{e^3 \sqrt {d+e x}}+\frac {b^2 B \sqrt {d+e x}}{e^3}\right ) \, dx \\ & = \frac {2 (b d-a e)^2 (B d-A e)}{3 e^4 (d+e x)^{3/2}}-\frac {2 (b d-a e) (3 b B d-2 A b e-a B e)}{e^4 \sqrt {d+e x}}-\frac {2 b (3 b B d-A b e-2 a B e) \sqrt {d+e x}}{e^4}+\frac {2 b^2 B (d+e x)^{3/2}}{3 e^4} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^{5/2}} \, dx=-\frac {2 \left (a^2 e^2 (2 B d+A e+3 B e x)-2 a b e \left (-A e (2 d+3 e x)+B \left (8 d^2+12 d e x+3 e^2 x^2\right )\right )+b^2 \left (-A e \left (8 d^2+12 d e x+3 e^2 x^2\right )+B \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )\right )\right )}{3 e^4 (d+e x)^{3/2}} \]
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Time = 0.97 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.85
method | result | size |
risch | \(\frac {2 b \left (b B x e +3 A b e +6 B a e -8 B b d \right ) \sqrt {e x +d}}{3 e^{4}}-\frac {2 \left (6 A b \,e^{2} x +3 B a \,e^{2} x -9 B b d e x +A a \,e^{2}+5 A b d e +2 B a d e -8 b B \,d^{2}\right ) \left (a e -b d \right )}{3 e^{4} \left (e x +d \right )^{\frac {3}{2}}}\) | \(105\) |
pseudoelliptic | \(-\frac {2 \left (\left (\left (-x^{3} B -3 A \,x^{2}\right ) b^{2}+6 a x \left (-B x +A \right ) b +a^{2} \left (3 B x +A \right )\right ) e^{3}+4 d \left (\left (\frac {3}{2} x^{2} B -3 A x \right ) b^{2}+a \left (-6 B x +A \right ) b +\frac {a^{2} B}{2}\right ) e^{2}-8 \left (\left (-3 B x +A \right ) b +2 B a \right ) d^{2} b e +16 b^{2} B \,d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{4}}\) | \(123\) |
gosper | \(-\frac {2 \left (-b^{2} B \,x^{3} e^{3}-3 A \,b^{2} e^{3} x^{2}-6 B a b \,e^{3} x^{2}+6 B \,b^{2} d \,e^{2} x^{2}+6 A a b \,e^{3} x -12 A \,b^{2} d \,e^{2} x +3 B \,a^{2} e^{3} x -24 B a b d \,e^{2} x +24 B \,b^{2} d^{2} e x +a^{2} A \,e^{3}+4 A a b d \,e^{2}-8 A \,b^{2} d^{2} e +2 B \,a^{2} d \,e^{2}-16 B a b \,d^{2} e +16 b^{2} B \,d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{4}}\) | \(168\) |
trager | \(-\frac {2 \left (-b^{2} B \,x^{3} e^{3}-3 A \,b^{2} e^{3} x^{2}-6 B a b \,e^{3} x^{2}+6 B \,b^{2} d \,e^{2} x^{2}+6 A a b \,e^{3} x -12 A \,b^{2} d \,e^{2} x +3 B \,a^{2} e^{3} x -24 B a b d \,e^{2} x +24 B \,b^{2} d^{2} e x +a^{2} A \,e^{3}+4 A a b d \,e^{2}-8 A \,b^{2} d^{2} e +2 B \,a^{2} d \,e^{2}-16 B a b \,d^{2} e +16 b^{2} B \,d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{4}}\) | \(168\) |
derivativedivides | \(\frac {\frac {2 b^{2} B \left (e x +d \right )^{\frac {3}{2}}}{3}+2 A \,b^{2} e \sqrt {e x +d}+4 B a b e \sqrt {e x +d}-6 B \,b^{2} d \sqrt {e x +d}-\frac {2 \left (2 A a b \,e^{2}-2 A \,b^{2} d e +B \,a^{2} e^{2}-4 B a b d e +3 b^{2} B \,d^{2}\right )}{\sqrt {e x +d}}-\frac {2 \left (a^{2} A \,e^{3}-2 A a b d \,e^{2}+A \,b^{2} d^{2} e -B \,a^{2} d \,e^{2}+2 B a b \,d^{2} e -b^{2} B \,d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{4}}\) | \(174\) |
default | \(\frac {\frac {2 b^{2} B \left (e x +d \right )^{\frac {3}{2}}}{3}+2 A \,b^{2} e \sqrt {e x +d}+4 B a b e \sqrt {e x +d}-6 B \,b^{2} d \sqrt {e x +d}-\frac {2 \left (2 A a b \,e^{2}-2 A \,b^{2} d e +B \,a^{2} e^{2}-4 B a b d e +3 b^{2} B \,d^{2}\right )}{\sqrt {e x +d}}-\frac {2 \left (a^{2} A \,e^{3}-2 A a b d \,e^{2}+A \,b^{2} d^{2} e -B \,a^{2} d \,e^{2}+2 B a b \,d^{2} e -b^{2} B \,d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{4}}\) | \(174\) |
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Time = 0.22 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.41 \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (B b^{2} e^{3} x^{3} - 16 \, B b^{2} d^{3} - A a^{2} e^{3} + 8 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e - 2 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{2} - 3 \, {\left (2 \, B b^{2} d e^{2} - {\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} - 3 \, {\left (8 \, B b^{2} d^{2} e - 4 \, {\left (2 \, B a b + A b^{2}\right )} d e^{2} + {\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x\right )} \sqrt {e x + d}}{3 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 709 vs. \(2 (124) = 248\).
Time = 0.40 (sec) , antiderivative size = 709, normalized size of antiderivative = 5.72 \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^{5/2}} \, dx=\begin {cases} - \frac {2 A a^{2} e^{3}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {8 A a b d e^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {12 A a b e^{3} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {16 A b^{2} d^{2} e}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {24 A b^{2} d e^{2} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {6 A b^{2} e^{3} x^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {4 B a^{2} d e^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {6 B a^{2} e^{3} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {32 B a b d^{2} e}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {48 B a b d e^{2} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {12 B a b e^{3} x^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {32 B b^{2} d^{3}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {48 B b^{2} d^{2} e x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {12 B b^{2} d e^{2} x^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {2 B b^{2} e^{3} x^{3}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {A a^{2} x + A a b x^{2} + \frac {A b^{2} x^{3}}{3} + \frac {B a^{2} x^{2}}{2} + \frac {2 B a b x^{3}}{3} + \frac {B b^{2} x^{4}}{4}}{d^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.31 \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {{\left (e x + d\right )}^{\frac {3}{2}} B b^{2} - 3 \, {\left (3 \, B b^{2} d - {\left (2 \, B a b + A b^{2}\right )} e\right )} \sqrt {e x + d}}{e^{3}} + \frac {B b^{2} d^{3} - A a^{2} e^{3} - {\left (2 \, B a b + A b^{2}\right )} d^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d e^{2} - 3 \, {\left (3 \, B b^{2} d^{2} - 2 \, {\left (2 \, B a b + A b^{2}\right )} d e + {\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{3}}\right )}}{3 \, e} \]
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Time = 0.28 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.64 \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^{5/2}} \, dx=-\frac {2 \, {\left (9 \, {\left (e x + d\right )} B b^{2} d^{2} - B b^{2} d^{3} - 12 \, {\left (e x + d\right )} B a b d e - 6 \, {\left (e x + d\right )} A b^{2} d e + 2 \, B a b d^{2} e + A b^{2} d^{2} e + 3 \, {\left (e x + d\right )} B a^{2} e^{2} + 6 \, {\left (e x + d\right )} A a b e^{2} - B a^{2} d e^{2} - 2 \, A a b d e^{2} + A a^{2} e^{3}\right )}}{3 \, {\left (e x + d\right )}^{\frac {3}{2}} e^{4}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} B b^{2} e^{8} - 9 \, \sqrt {e x + d} B b^{2} d e^{8} + 6 \, \sqrt {e x + d} B a b e^{9} + 3 \, \sqrt {e x + d} A b^{2} e^{9}\right )}}{3 \, e^{12}} \]
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Time = 1.52 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.52 \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^{5/2}} \, dx=\frac {2\,B\,b^2\,d^3-2\,A\,a^2\,e^3+2\,B\,b^2\,{\left (d+e\,x\right )}^3+6\,A\,b^2\,e\,{\left (d+e\,x\right )}^2-6\,B\,a^2\,e^2\,\left (d+e\,x\right )-18\,B\,b^2\,d\,{\left (d+e\,x\right )}^2-18\,B\,b^2\,d^2\,\left (d+e\,x\right )-2\,A\,b^2\,d^2\,e+2\,B\,a^2\,d\,e^2-12\,A\,a\,b\,e^2\,\left (d+e\,x\right )+12\,B\,a\,b\,e\,{\left (d+e\,x\right )}^2+12\,A\,b^2\,d\,e\,\left (d+e\,x\right )+4\,A\,a\,b\,d\,e^2-4\,B\,a\,b\,d^2\,e+24\,B\,a\,b\,d\,e\,\left (d+e\,x\right )}{3\,e^4\,{\left (d+e\,x\right )}^{3/2}} \]
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