Integrand size = 33, antiderivative size = 146 \[ \int (a+b x) (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {(b d-a e)^2 (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^3 (a+b x)}-\frac {2 b (b d-a e) (d+e x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^3 (a+b x)}+\frac {b^2 (d+e x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^3 (a+b x)} \]
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Time = 0.07 (sec) , antiderivative size = 146, 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} \[ \int (a+b x) (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^6}{6 e^3 (a+b x)}-\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^5 (b d-a e)}{5 e^3 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^4 (b d-a e)^2}{4 e^3 (a+b x)} \]
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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)^3 \, 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)^3 \, 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 (d+e x)^3}{e^2}-\frac {2 b (b d-a e) (d+e x)^4}{e^2}+\frac {b^2 (d+e x)^5}{e^2}\right ) \, dx}{a b+b^2 x} \\ & = \frac {(b d-a e)^2 (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^3 (a+b x)}-\frac {2 b (b d-a e) (d+e x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^3 (a+b x)}+\frac {b^2 (d+e x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^3 (a+b x)} \\ \end{align*}
Time = 1.03 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.89 \[ \int (a+b x) (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (15 a^2 \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+6 a b x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )+b^2 x^2 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )\right )}{60 (a+b x)} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.37 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.83
method | result | size |
default | \(-\frac {\operatorname {csgn}\left (b x +a \right ) \left (b x +a \right )^{3} \left (-10 b^{3} x^{3} e^{3}+6 x^{2} a \,b^{2} e^{3}-36 x^{2} b^{3} d \,e^{2}-3 x \,a^{2} b \,e^{3}+18 x a \,b^{2} d \,e^{2}-45 x \,b^{3} d^{2} e +a^{3} e^{3}-6 a^{2} b d \,e^{2}+15 a \,b^{2} d^{2} e -20 b^{3} d^{3}\right )}{60 b^{4}}\) | \(121\) |
gosper | \(\frac {x \left (10 x^{5} b^{2} e^{3}+24 x^{4} a b \,e^{3}+36 x^{4} b^{2} d \,e^{2}+15 x^{3} a^{2} e^{3}+90 a b d \,e^{2} x^{3}+45 b^{2} d^{2} e \,x^{3}+60 x^{2} a^{2} d \,e^{2}+120 x^{2} a b \,d^{2} e +20 x^{2} b^{2} d^{3}+90 x \,a^{2} d^{2} e +60 b \,d^{3} a x +60 a^{2} d^{3}\right ) \sqrt {\left (b x +a \right )^{2}}}{60 b x +60 a}\) | \(148\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{2} e^{3} x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (2 a b \,e^{3}+3 b^{2} d \,e^{2}\right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (a^{2} e^{3}+6 a b d \,e^{2}+3 b^{2} d^{2} e \right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (3 a^{2} d \,e^{2}+6 a b \,d^{2} e +b^{2} d^{3}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (3 a^{2} d^{2} e +2 b a \,d^{3}\right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, a^{2} d^{3} x}{b x +a}\) | \(221\) |
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Time = 0.29 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.85 \[ \int (a+b x) (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {1}{6} \, b^{2} e^{3} x^{6} + a^{2} d^{3} x + \frac {1}{5} \, {\left (3 \, b^{2} d e^{2} + 2 \, a b e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, b^{2} d^{2} e + 6 \, a b d e^{2} + a^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (b^{2} d^{3} + 6 \, a b d^{2} e + 3 \, a^{2} d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e\right )} x^{2} \]
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Time = 2.88 (sec) , antiderivative size = 1080, normalized size of antiderivative = 7.40 \[ \int (a+b x) (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 693 vs. \(2 (107) = 214\).
Time = 0.20 (sec) , antiderivative size = 693, normalized size of antiderivative = 4.75 \[ \int (a+b x) (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} e^{3} x^{3}}{6 \, b} + \frac {1}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a d^{3} x + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{4} e^{3} x}{2 \, b^{3}} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a e^{3} x^{2}}{10 \, b^{2}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} d^{3}}{2 \, b} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{5} e^{3}}{2 \, b^{4}} + \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} e^{3} x}{5 \, b^{3}} - \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3} e^{3}}{15 \, b^{4}} - \frac {{\left (3 \, b d e^{2} + a e^{3}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3} x}{2 \, b^{3}} + \frac {3 \, {\left (b d^{2} e + a d e^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} x}{2 \, b^{2}} - \frac {{\left (b d^{3} + 3 \, a d^{2} e\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a x}{2 \, b} + \frac {{\left (3 \, b d e^{2} + a e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} x^{2}}{5 \, b^{2}} - \frac {{\left (3 \, b d e^{2} + a e^{3}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{4}}{2 \, b^{4}} + \frac {3 \, {\left (b d^{2} e + a d e^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3}}{2 \, b^{3}} - \frac {{\left (b d^{3} + 3 \, a d^{2} e\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2}}{2 \, b^{2}} - \frac {7 \, {\left (3 \, b d e^{2} + a e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a x}{20 \, b^{3}} + \frac {3 \, {\left (b d^{2} e + a d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} x}{4 \, b^{2}} + \frac {9 \, {\left (3 \, b d e^{2} + a e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2}}{20 \, b^{4}} - \frac {5 \, {\left (b d^{2} e + a d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a}{4 \, b^{3}} + \frac {{\left (b d^{3} + 3 \, a d^{2} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}}}{3 \, b^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (107) = 214\).
Time = 0.26 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.75 \[ \int (a+b x) (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {1}{6} \, b^{2} e^{3} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{5} \, b^{2} d e^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{5} \, a b e^{3} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{4} \, b^{2} d^{2} e x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, a b d e^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, a^{2} e^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, b^{2} d^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, a b d^{2} e x^{3} \mathrm {sgn}\left (b x + a\right ) + a^{2} d e^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + a b d^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, a^{2} d^{2} e x^{2} \mathrm {sgn}\left (b x + a\right ) + a^{2} d^{3} x \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (20 \, a^{3} b^{3} d^{3} - 15 \, a^{4} b^{2} d^{2} e + 6 \, a^{5} b d e^{2} - a^{6} e^{3}\right )} \mathrm {sgn}\left (b x + a\right )}{60 \, b^{4}} \]
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Time = 11.46 (sec) , antiderivative size = 734, normalized size of antiderivative = 5.03 \[ \int (a+b x) (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=a\,d^3\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}+\frac {d^3\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{24\,b^3}+\frac {e^3\,x^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{6\,b}-\frac {19\,a^2\,e^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (a^3-5\,a\,b^2\,x^2+3\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-4\,a^2\,b\,x\right )}{120\,b^4}-\frac {a^3\,e^3\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{60\,b^6}+\frac {a\,e^3\,x^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{5\,b^2}+\frac {3\,d\,e^2\,x^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{5\,b}-\frac {3\,a\,e^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (4\,b^2\,x^2\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-a^4+9\,a^2\,b^2\,x^2+8\,a^3\,b\,x-7\,a\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )\right )}{40\,b^4}+\frac {3\,d^2\,e\,x\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{4\,b}-\frac {7\,a\,d\,e^2\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (a^3-5\,a\,b^2\,x^2+3\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-4\,a^2\,b\,x\right )}{20\,b^3}-\frac {a\,d^2\,e\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{32\,b^4}+\frac {3\,a\,d\,e^2\,x\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{4\,b^2}-\frac {33\,a^2\,d\,e^2\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{160\,b^5}-\frac {3\,a^2\,d^2\,e\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,b}-\frac {3\,a^3\,d\,e^2\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,b^2} \]
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