Integrand size = 28, antiderivative size = 92 \[ \int (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=-\frac {(b d-a e) (d+e x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^2 (a+b x)}+\frac {b (d+e x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^2 (a+b x)} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {660, 45} \[ \int (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^6}{6 e^2 (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^5 (b d-a e)}{5 e^2 (a+b x)} \]
[In]
[Out]
Rule 45
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right ) (d+e x)^4 \, dx}{a b+b^2 x} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b (b d-a e) (d+e x)^4}{e}+\frac {b^2 (d+e x)^5}{e}\right ) \, dx}{a b+b^2 x} \\ & = -\frac {(b d-a e) (d+e x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^2 (a+b x)}+\frac {b (d+e x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^2 (a+b x)} \\ \end{align*}
Time = 1.03 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.21 \[ \int (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (6 a \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )+b x \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )\right )}{30 (a+b x)} \]
[In]
[Out]
Time = 2.20 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.24
method | result | size |
gosper | \(\frac {x \left (5 b \,e^{4} x^{5}+6 x^{4} e^{4} a +24 x^{4} b d \,e^{3}+30 x^{3} a d \,e^{3}+45 x^{3} b \,d^{2} e^{2}+60 x^{2} a \,d^{2} e^{2}+40 x^{2} b \,d^{3} e +60 a \,d^{3} e x +15 x b \,d^{4}+30 a \,d^{4}\right ) \sqrt {\left (b x +a \right )^{2}}}{30 b x +30 a}\) | \(114\) |
default | \(\frac {\operatorname {csgn}\left (b x +a \right ) \left (b x +a \right )^{2} \left (5 b^{4} x^{4} e^{4}-4 x^{3} a \,b^{3} e^{4}+24 x^{3} b^{4} d \,e^{3}+3 x^{2} a^{2} b^{2} e^{4}-18 x^{2} a \,b^{3} d \,e^{3}+45 x^{2} b^{4} d^{2} e^{2}-2 x \,a^{3} b \,e^{4}+12 x \,a^{2} b^{2} d \,e^{3}-30 x a \,b^{3} d^{2} e^{2}+40 x \,b^{4} d^{3} e +e^{4} a^{4}-6 b \,e^{3} d \,a^{3}+15 b^{2} e^{2} d^{2} a^{2}-20 a \,b^{3} d^{3} e +15 b^{4} d^{4}\right )}{30 b^{5}}\) | \(191\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b \,e^{4} x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (e^{4} a +4 b d \,e^{3}\right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (4 a d \,e^{3}+6 b \,d^{2} e^{2}\right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (6 a \,d^{2} e^{2}+4 b \,d^{3} e \right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (4 a \,d^{3} e +b \,d^{4}\right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, a \,d^{4} x}{b x +a}\) | \(193\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.04 \[ \int (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {1}{6} \, b e^{4} x^{6} + a d^{4} x + \frac {1}{5} \, {\left (4 \, b d e^{3} + a e^{4}\right )} x^{5} + \frac {1}{2} \, {\left (3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} x^{4} + \frac {2}{3} \, {\left (2 \, b d^{3} e + 3 \, a d^{2} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (b d^{4} + 4 \, a d^{3} e\right )} x^{2} \]
[In]
[Out]
Time = 2.20 (sec) , antiderivative size = 666, normalized size of antiderivative = 7.24 \[ \int (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=d^{4} \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 ) + 4 d^{3} 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 ) + 6 d^{2} 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 ) + 4 d e^{3} \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 ) + e^{4} \left (\begin {cases} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} \left (\frac {a^{5}}{30 b^{5}} - \frac {a^{4} x}{30 b^{4}} + \frac {a^{3} x^{2}}{30 b^{3}} - \frac {a^{2} x^{3}}{30 b^{2}} + \frac {a x^{4}}{30 b} + \frac {x^{5}}{6}\right ) & \text {for}\: b^{2} \neq 0 \\\frac {\frac {a^{8} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3} - \frac {4 a^{6} \left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5} + \frac {6 a^{4} \left (a^{2} + 2 a b x\right )^{\frac {7}{2}}}{7} - \frac {4 a^{2} \left (a^{2} + 2 a b x\right )^{\frac {9}{2}}}{9} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {11}{2}}}{11}}{16 a^{5} b^{5}} & \text {for}\: a b \neq 0 \\\frac {x^{5} \sqrt {a^{2}}}{5} & \text {otherwise} \end {cases}\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 590 vs. \(2 (66) = 132\).
Time = 0.20 (sec) , antiderivative size = 590, normalized size of antiderivative = 6.41 \[ \int (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} e^{4} x^{3}}{6 \, b^{2}} + \frac {1}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} d^{4} x - \frac {2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a d^{3} e x}{b} + \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} d^{2} e^{2} x}{b^{2}} - \frac {2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3} d e^{3} x}{b^{3}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{4} e^{4} x}{2 \, b^{4}} + \frac {4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} d e^{3} x^{2}}{5 \, b^{2}} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a e^{4} x^{2}}{10 \, b^{3}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a d^{4}}{2 \, b} - \frac {2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} d^{3} e}{b^{2}} + \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3} d^{2} e^{2}}{b^{3}} - \frac {2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{4} d e^{3}}{b^{4}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{5} e^{4}}{2 \, b^{5}} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} d^{2} e^{2} x}{2 \, b^{2}} - \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a d e^{3} x}{5 \, b^{3}} + \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} e^{4} x}{5 \, b^{4}} + \frac {4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} d^{3} e}{3 \, b^{2}} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a d^{2} e^{2}}{2 \, b^{3}} + \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} d e^{3}}{5 \, b^{4}} - \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3} e^{4}}{15 \, b^{5}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (66) = 132\).
Time = 0.28 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.42 \[ \int (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {1}{6} \, b e^{4} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{5} \, b d e^{3} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, a e^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, b d^{2} e^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + a d e^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{3} \, b d^{3} e x^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, a d^{2} e^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, b d^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 2 \, a d^{3} e x^{2} \mathrm {sgn}\left (b x + a\right ) + a d^{4} x \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (15 \, a^{2} b^{4} d^{4} - 20 \, a^{3} b^{3} d^{3} e + 15 \, a^{4} b^{2} d^{2} e^{2} - 6 \, a^{5} b d e^{3} + a^{6} e^{4}\right )} \mathrm {sgn}\left (b x + a\right )}{30 \, b^{5}} \]
[In]
[Out]
Time = 10.55 (sec) , antiderivative size = 580, normalized size of antiderivative = 6.30 \[ \int (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=d^4\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}+\frac {e^4\,x^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{6\,b^2}-\frac {a^2\,e^4\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (a^3-5\,a\,b^2\,x^2+3\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-4\,a^2\,b\,x\right )}{24\,b^5}+\frac {3\,d^2\,e^2\,x\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{2\,b^2}+\frac {4\,d\,e^3\,x^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{5\,b^2}-\frac {3\,a\,e^4\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (4\,b^2\,x^2\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-a^4+9\,a^2\,b^2\,x^2+8\,a^3\,b\,x-7\,a\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )\right )}{40\,b^5}+\frac {d^3\,e\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{6\,b^4}-\frac {7\,a\,d\,e^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (a^3-5\,a\,b^2\,x^2+3\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-4\,a^2\,b\,x\right )}{15\,b^4}-\frac {3\,a^2\,d^2\,e^2\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,b^2}-\frac {5\,a\,d^2\,e^2\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{16\,b^5}-\frac {a^2\,d\,e^3\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{15\,b^6} \]
[In]
[Out]