Integrand size = 31, antiderivative size = 78 \[ \int (a+b x) (d+e x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {(b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 b^2}+\frac {e (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 b^2} \]
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Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {784, 21, 45} \[ \int (a+b x) (d+e x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^2 (b d-a e)}{3 b^2}+\frac {e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^3}{4 b^2} \]
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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) \, 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) \, 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) (a+b x)^2}{b}+\frac {e (a+b x)^3}{b}\right ) \, dx}{a b+b^2 x} \\ & = \frac {(b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 b^2}+\frac {e (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 b^2} \\ \end{align*}
Time = 0.74 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.44 \[ \int (a+b x) (d+e x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {x \left (6 a^2 (2 d+e x)+4 a b x (3 d+2 e x)+b^2 x^2 (4 d+3 e x)\right ) \left (\sqrt {a^2} b x+a \left (\sqrt {a^2}-\sqrt {(a+b x)^2}\right )\right )}{12 \left (-a^2-a b x+\sqrt {a^2} \sqrt {(a+b x)^2}\right )} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.07 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.41
method | result | size |
default | \(-\frac {\operatorname {csgn}\left (b x +a \right ) \left (b x +a \right )^{3} \left (-3 b e x +a e -4 b d \right )}{12 b^{2}}\) | \(32\) |
gosper | \(\frac {x \left (3 b^{2} e \,x^{3}+8 x^{2} a b e +4 x^{2} b^{2} d +6 a^{2} e x +12 a b d x +12 a^{2} d \right ) \sqrt {\left (b x +a \right )^{2}}}{12 b x +12 a}\) | \(66\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{2} e \,x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (2 a b e +b^{2} d \right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (a^{2} e +2 b d a \right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, a^{2} d x}{b x +a}\) | \(113\) |
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none
Time = 0.29 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.62 \[ \int (a+b x) (d+e x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {1}{4} \, b^{2} e x^{4} + a^{2} d x + \frac {1}{3} \, {\left (b^{2} d + 2 \, a b e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b d + a^{2} e\right )} x^{2} \]
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Time = 1.56 (sec) , antiderivative size = 393, normalized size of antiderivative = 5.04 \[ \int (a+b x) (d+e x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=a d \left (\begin {cases} \left (\frac {a}{2 b} + \frac {x}{2}\right ) \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} & \text {for}\: b^{2} \neq 0 \\\frac {\left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3 a b} & \text {for}\: a b \neq 0 \\x \sqrt {a^{2}} & \text {otherwise} \end {cases}\right ) + a e \left (\begin {cases} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} \left (- \frac {a^{2}}{6 b^{2}} + \frac {a x}{6 b} + \frac {x^{2}}{3}\right ) & \text {for}\: b^{2} \neq 0 \\\frac {- \frac {a^{2} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5}}{2 a^{2} b^{2}} & \text {for}\: a b \neq 0 \\\frac {x^{2} \sqrt {a^{2}}}{2} & \text {otherwise} \end {cases}\right ) + b d \left (\begin {cases} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} \left (- \frac {a^{2}}{6 b^{2}} + \frac {a x}{6 b} + \frac {x^{2}}{3}\right ) & \text {for}\: b^{2} \neq 0 \\\frac {- \frac {a^{2} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5}}{2 a^{2} b^{2}} & \text {for}\: a b \neq 0 \\\frac {x^{2} \sqrt {a^{2}}}{2} & \text {otherwise} \end {cases}\right ) + b e \left (\begin {cases} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} \left (\frac {a^{3}}{12 b^{3}} - \frac {a^{2} x}{12 b^{2}} + \frac {a x^{2}}{12 b} + \frac {x^{3}}{4}\right ) & \text {for}\: b^{2} \neq 0 \\\frac {\frac {a^{4} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3} - \frac {2 a^{2} \left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {7}{2}}}{7}}{4 a^{3} b^{3}} & \text {for}\: a b \neq 0 \\\frac {x^{3} \sqrt {a^{2}}}{3} & \text {otherwise} \end {cases}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 251 vs. \(2 (52) = 104\).
Time = 0.25 (sec) , antiderivative size = 251, normalized size of antiderivative = 3.22 \[ \int (a+b x) (d+e x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {1}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a d x + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} e x}{2 \, b} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} d}{2 \, b} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3} e}{2 \, b^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} {\left (b d + a e\right )} a x}{2 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} e x}{4 \, b} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} {\left (b d + a e\right )} a^{2}}{2 \, b^{2}} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a e}{12 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (b d + a e\right )}}{3 \, b^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (52) = 104\).
Time = 0.26 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.41 \[ \int (a+b x) (d+e x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {1}{4} \, b^{2} e x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, b^{2} d x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, a b e x^{3} \mathrm {sgn}\left (b x + a\right ) + a b d x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, a^{2} e x^{2} \mathrm {sgn}\left (b x + a\right ) + a^{2} d x \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (4 \, a^{3} b d - a^{4} e\right )} \mathrm {sgn}\left (b x + a\right )}{12 \, b^{2}} \]
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Time = 11.17 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.81 \[ \int (a+b x) (d+e x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {d\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{24\,b^3}+\frac {e\,x\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{4\,b}-\frac {a^2\,e\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,b}-\frac {5\,a\,e\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{96\,b^4}+\frac {a\,\left (a+b\,x\right )\,\left (3\,b\,d-a\,e+2\,b\,e\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{6\,b^2} \]
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