Integrand size = 33, antiderivative size = 125 \[ \int (a+b x) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {(b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 b^3}+\frac {e (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{2 b^3}+\frac {e^2 (a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{5 b^3} \]
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Time = 0.06 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {784, 21, 45} \[ \int (a+b x) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^3 (b d-a e)}{2 b^3}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^2 (b d-a e)^2}{3 b^3}+\frac {e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^4}{5 b^3} \]
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Rule 21
Rule 45
Rule 784
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x) \left (a b+b^2 x\right ) (d+e x)^2 \, dx}{a b+b^2 x} \\ & = \frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^2 (d+e x)^2 \, dx}{a b+b^2 x} \\ & = \frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(b d-a e)^2 (a+b x)^2}{b^2}+\frac {2 e (b d-a e) (a+b x)^3}{b^2}+\frac {e^2 (a+b x)^4}{b^2}\right ) \, dx}{a b+b^2 x} \\ & = \frac {(b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 b^3}+\frac {e (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{2 b^3}+\frac {e^2 (a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{5 b^3} \\ \end{align*}
Time = 1.03 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.78 \[ \int (a+b x) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (10 a^2 \left (3 d^2+3 d e x+e^2 x^2\right )+5 a b x \left (6 d^2+8 d e x+3 e^2 x^2\right )+b^2 x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )\right )}{30 (a+b x)} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.37 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.54
method | result | size |
default | \(\frac {\operatorname {csgn}\left (b x +a \right ) \left (b x +a \right )^{3} \left (6 b^{2} e^{2} x^{2}-3 a b \,e^{2} x +15 b^{2} d e x +e^{2} a^{2}-5 a b d e +10 b^{2} d^{2}\right )}{30 b^{3}}\) | \(68\) |
gosper | \(\frac {x \left (6 b^{2} e^{2} x^{4}+15 x^{3} a b \,e^{2}+15 x^{3} b^{2} d e +10 x^{2} e^{2} a^{2}+40 x^{2} a b d e +10 x^{2} b^{2} d^{2}+30 a^{2} d e x +30 b \,d^{2} a x +30 a^{2} d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{30 b x +30 a}\) | \(107\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{2} e^{2} x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (2 a b \,e^{2}+2 b^{2} d e \right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (e^{2} a^{2}+4 a b d e +b^{2} d^{2}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (2 a^{2} d e +2 b a \,d^{2}\right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, a^{2} d^{2} x}{b x +a}\) | \(167\) |
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Time = 0.29 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.65 \[ \int (a+b x) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {1}{5} \, b^{2} e^{2} x^{5} + a^{2} d^{2} x + \frac {1}{2} \, {\left (b^{2} d e + a b e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (b^{2} d^{2} + 4 \, a b d e + a^{2} e^{2}\right )} x^{3} + {\left (a b d^{2} + a^{2} d e\right )} x^{2} \]
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Time = 2.10 (sec) , antiderivative size = 704, normalized size of antiderivative = 5.63 \[ \int (a+b x) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=a d^{2} \left (\begin {cases} \left (\frac {a}{2 b} + \frac {x}{2}\right ) \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} & \text {for}\: b^{2} \neq 0 \\\frac {\left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3 a b} & \text {for}\: a b \neq 0 \\x \sqrt {a^{2}} & \text {otherwise} \end {cases}\right ) + 2 a d e \left (\begin {cases} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} \left (- \frac {a^{2}}{6 b^{2}} + \frac {a x}{6 b} + \frac {x^{2}}{3}\right ) & \text {for}\: b^{2} \neq 0 \\\frac {- \frac {a^{2} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5}}{2 a^{2} b^{2}} & \text {for}\: a b \neq 0 \\\frac {x^{2} \sqrt {a^{2}}}{2} & \text {otherwise} \end {cases}\right ) + a e^{2} \left (\begin {cases} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} \left (\frac {a^{3}}{12 b^{3}} - \frac {a^{2} x}{12 b^{2}} + \frac {a x^{2}}{12 b} + \frac {x^{3}}{4}\right ) & \text {for}\: b^{2} \neq 0 \\\frac {\frac {a^{4} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3} - \frac {2 a^{2} \left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {7}{2}}}{7}}{4 a^{3} b^{3}} & \text {for}\: a b \neq 0 \\\frac {x^{3} \sqrt {a^{2}}}{3} & \text {otherwise} \end {cases}\right ) + b d^{2} \left (\begin {cases} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} \left (- \frac {a^{2}}{6 b^{2}} + \frac {a x}{6 b} + \frac {x^{2}}{3}\right ) & \text {for}\: b^{2} \neq 0 \\\frac {- \frac {a^{2} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5}}{2 a^{2} b^{2}} & \text {for}\: a b \neq 0 \\\frac {x^{2} \sqrt {a^{2}}}{2} & \text {otherwise} \end {cases}\right ) + 2 b d e \left (\begin {cases} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} \left (\frac {a^{3}}{12 b^{3}} - \frac {a^{2} x}{12 b^{2}} + \frac {a x^{2}}{12 b} + \frac {x^{3}}{4}\right ) & \text {for}\: b^{2} \neq 0 \\\frac {\frac {a^{4} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3} - \frac {2 a^{2} \left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {7}{2}}}{7}}{4 a^{3} b^{3}} & \text {for}\: a b \neq 0 \\\frac {x^{3} \sqrt {a^{2}}}{3} & \text {otherwise} \end {cases}\right ) + b e^{2} \left (\begin {cases} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} \left (- \frac {a^{4}}{20 b^{4}} + \frac {a^{3} x}{20 b^{3}} - \frac {a^{2} x^{2}}{20 b^{2}} + \frac {a x^{3}}{20 b} + \frac {x^{4}}{5}\right ) & \text {for}\: b^{2} \neq 0 \\\frac {- \frac {a^{6} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3} + \frac {3 a^{4} \left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5} - \frac {3 a^{2} \left (a^{2} + 2 a b x\right )^{\frac {7}{2}}}{7} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {9}{2}}}{9}}{8 a^{4} b^{4}} & \text {for}\: a b \neq 0 \\\frac {x^{4} \sqrt {a^{2}}}{4} & \text {otherwise} \end {cases}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 452 vs. \(2 (86) = 172\).
Time = 0.25 (sec) , antiderivative size = 452, normalized size of antiderivative = 3.62 \[ \int (a+b x) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {1}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a d^{2} x - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3} e^{2} x}{2 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} e^{2} x^{2}}{5 \, b} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} d^{2}}{2 \, b} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{4} e^{2}}{2 \, b^{3}} - \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a e^{2} x}{20 \, b^{2}} + \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} e^{2}}{20 \, b^{3}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} {\left (2 \, b d e + a e^{2}\right )} a^{2} x}{2 \, b^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} {\left (b d^{2} + 2 \, a d e\right )} a x}{2 \, b} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} {\left (2 \, b d e + a e^{2}\right )} a^{3}}{2 \, b^{3}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} {\left (b d^{2} + 2 \, a d e\right )} a^{2}}{2 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (2 \, b d e + a e^{2}\right )} x}{4 \, b^{2}} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (2 \, b d e + a e^{2}\right )} a}{12 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (b d^{2} + 2 \, a d e\right )}}{3 \, b^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 181 vs. \(2 (86) = 172\).
Time = 0.27 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.45 \[ \int (a+b x) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {1}{5} \, b^{2} e^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, b^{2} d e x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, a b e^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, b^{2} d^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{3} \, a b d e x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, a^{2} e^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + a b d^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + a^{2} d e x^{2} \mathrm {sgn}\left (b x + a\right ) + a^{2} d^{2} x \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (10 \, a^{3} b^{2} d^{2} - 5 \, a^{4} b d e + a^{5} e^{2}\right )} \mathrm {sgn}\left (b x + a\right )}{30 \, b^{3}} \]
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Time = 11.21 (sec) , antiderivative size = 438, normalized size of antiderivative = 3.50 \[ \int (a+b x) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=a\,d^2\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}+\frac {d^2\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{24\,b^3}+\frac {e^2\,x^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{5\,b}-\frac {11\,a^2\,e^2\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{160\,b^5}-\frac {a^3\,e^2\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,b^2}+\frac {d\,e\,x\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{2\,b}-\frac {7\,a\,e^2\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (a^3-5\,a\,b^2\,x^2+3\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-4\,a^2\,b\,x\right )}{60\,b^3}+\frac {a\,e^2\,x\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{4\,b^2}-\frac {a^2\,d\,e\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,b}-\frac {a\,d\,e\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{48\,b^4} \]
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