Integrand size = 31, antiderivative size = 320 \[ \int x^{7/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 a^5 A x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {2 a^4 (5 A b+a B) x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac {10 a^3 b (2 A b+a B) x^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 (a+b x)}+\frac {4 a^2 b^2 (A b+a B) x^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {10 a b^3 (A b+2 a B) x^{17/2} \sqrt {a^2+2 a b x+b^2 x^2}}{17 (a+b x)}+\frac {2 b^4 (A b+5 a B) x^{19/2} \sqrt {a^2+2 a b x+b^2 x^2}}{19 (a+b x)}+\frac {2 b^5 B x^{21/2} \sqrt {a^2+2 a b x+b^2 x^2}}{21 (a+b x)} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {784, 77} \[ \int x^{7/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {4 a^2 b^2 x^{15/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{3 (a+b x)}+\frac {2 b^4 x^{19/2} \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{19 (a+b x)}+\frac {10 a b^3 x^{17/2} \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{17 (a+b x)}+\frac {2 b^5 B x^{21/2} \sqrt {a^2+2 a b x+b^2 x^2}}{21 (a+b x)}+\frac {2 a^5 A x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {2 a^4 x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{11 (a+b x)}+\frac {10 a^3 b x^{13/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{13 (a+b x)} \]
[In]
[Out]
Rule 77
Rule 784
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int x^{7/2} \left (a b+b^2 x\right )^5 (A+B x) \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a^5 A b^5 x^{7/2}+a^4 b^5 (5 A b+a B) x^{9/2}+5 a^3 b^6 (2 A b+a B) x^{11/2}+10 a^2 b^7 (A b+a B) x^{13/2}+5 a b^8 (A b+2 a B) x^{15/2}+b^9 (A b+5 a B) x^{17/2}+b^{10} B x^{19/2}\right ) \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = \frac {2 a^5 A x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {2 a^4 (5 A b+a B) x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac {10 a^3 b (2 A b+a B) x^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 (a+b x)}+\frac {4 a^2 b^2 (A b+a B) x^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {10 a b^3 (A b+2 a B) x^{17/2} \sqrt {a^2+2 a b x+b^2 x^2}}{17 (a+b x)}+\frac {2 b^4 (A b+5 a B) x^{19/2} \sqrt {a^2+2 a b x+b^2 x^2}}{19 (a+b x)}+\frac {2 b^5 B x^{21/2} \sqrt {a^2+2 a b x+b^2 x^2}}{21 (a+b x)} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.40 \[ \int x^{7/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 x^{9/2} \sqrt {(a+b x)^2} \left (29393 a^5 (11 A+9 B x)+101745 a^4 b x (13 A+11 B x)+149226 a^3 b^2 x^2 (15 A+13 B x)+114114 a^2 b^3 x^3 (17 A+15 B x)+45045 a b^4 x^4 (19 A+17 B x)+7293 b^5 x^5 (21 A+19 B x)\right )}{2909907 (a+b x)} \]
[In]
[Out]
Time = 0.16 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.44
method | result | size |
gosper | \(\frac {2 x^{\frac {9}{2}} \left (138567 B \,b^{5} x^{6}+153153 A \,b^{5} x^{5}+765765 B a \,b^{4} x^{5}+855855 A a \,b^{4} x^{4}+1711710 B \,a^{2} b^{3} x^{4}+1939938 A \,a^{2} b^{3} x^{3}+1939938 B \,a^{3} b^{2} x^{3}+2238390 A \,a^{3} b^{2} x^{2}+1119195 B \,a^{4} b \,x^{2}+1322685 A \,a^{4} b x +264537 a^{5} B x +323323 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{2909907 \left (b x +a \right )^{5}}\) | \(140\) |
default | \(\frac {2 x^{\frac {9}{2}} \left (138567 B \,b^{5} x^{6}+153153 A \,b^{5} x^{5}+765765 B a \,b^{4} x^{5}+855855 A a \,b^{4} x^{4}+1711710 B \,a^{2} b^{3} x^{4}+1939938 A \,a^{2} b^{3} x^{3}+1939938 B \,a^{3} b^{2} x^{3}+2238390 A \,a^{3} b^{2} x^{2}+1119195 B \,a^{4} b \,x^{2}+1322685 A \,a^{4} b x +264537 a^{5} B x +323323 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{2909907 \left (b x +a \right )^{5}}\) | \(140\) |
risch | \(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, x^{\frac {9}{2}} \left (138567 B \,b^{5} x^{6}+153153 A \,b^{5} x^{5}+765765 B a \,b^{4} x^{5}+855855 A a \,b^{4} x^{4}+1711710 B \,a^{2} b^{3} x^{4}+1939938 A \,a^{2} b^{3} x^{3}+1939938 B \,a^{3} b^{2} x^{3}+2238390 A \,a^{3} b^{2} x^{2}+1119195 B \,a^{4} b \,x^{2}+1322685 A \,a^{4} b x +264537 a^{5} B x +323323 A \,a^{5}\right )}{2909907 \left (b x +a \right )}\) | \(140\) |
[In]
[Out]
none
Time = 0.35 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.39 \[ \int x^{7/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2}{2909907} \, {\left (138567 \, B b^{5} x^{10} + 323323 \, A a^{5} x^{4} + 153153 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{9} + 855855 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 1939938 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{7} + 1119195 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 264537 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{5}\right )} \sqrt {x} \]
[In]
[Out]
Timed out. \[ \int x^{7/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.75 \[ \int x^{7/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2}{2078505} \, {\left (6435 \, {\left (17 \, b^{5} x^{2} + 19 \, a b^{4} x\right )} x^{\frac {15}{2}} + 32604 \, {\left (15 \, a b^{4} x^{2} + 17 \, a^{2} b^{3} x\right )} x^{\frac {13}{2}} + 63954 \, {\left (13 \, a^{2} b^{3} x^{2} + 15 \, a^{3} b^{2} x\right )} x^{\frac {11}{2}} + 58140 \, {\left (11 \, a^{3} b^{2} x^{2} + 13 \, a^{4} b x\right )} x^{\frac {9}{2}} + 20995 \, {\left (9 \, a^{4} b x^{2} + 11 \, a^{5} x\right )} x^{\frac {7}{2}}\right )} A + \frac {2}{4849845} \, {\left (12155 \, {\left (19 \, b^{5} x^{2} + 21 \, a b^{4} x\right )} x^{\frac {17}{2}} + 60060 \, {\left (17 \, a b^{4} x^{2} + 19 \, a^{2} b^{3} x\right )} x^{\frac {15}{2}} + 114114 \, {\left (15 \, a^{2} b^{3} x^{2} + 17 \, a^{3} b^{2} x\right )} x^{\frac {13}{2}} + 99484 \, {\left (13 \, a^{3} b^{2} x^{2} + 15 \, a^{4} b x\right )} x^{\frac {11}{2}} + 33915 \, {\left (11 \, a^{4} b x^{2} + 13 \, a^{5} x\right )} x^{\frac {9}{2}}\right )} B \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.62 \[ \int x^{7/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2}{21} \, B b^{5} x^{\frac {21}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{19} \, B a b^{4} x^{\frac {19}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{19} \, A b^{5} x^{\frac {19}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{17} \, B a^{2} b^{3} x^{\frac {17}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{17} \, A a b^{4} x^{\frac {17}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{3} \, B a^{3} b^{2} x^{\frac {15}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{3} \, A a^{2} b^{3} x^{\frac {15}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{13} \, B a^{4} b x^{\frac {13}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{13} \, A a^{3} b^{2} x^{\frac {13}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{11} \, B a^{5} x^{\frac {11}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{11} \, A a^{4} b x^{\frac {11}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{9} \, A a^{5} x^{\frac {9}{2}} \mathrm {sgn}\left (b x + a\right ) \]
[In]
[Out]
Timed out. \[ \int x^{7/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int x^{7/2}\,\left (A+B\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \]
[In]
[Out]