Integrand size = 29, antiderivative size = 111 \[ \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2}{7} a^4 A x^{7/2}+\frac {2}{9} a^3 (4 A b+a B) x^{9/2}+\frac {4}{11} a^2 b (3 A b+2 a B) x^{11/2}+\frac {4}{13} a b^2 (2 A b+3 a B) x^{13/2}+\frac {2}{15} b^3 (A b+4 a B) x^{15/2}+\frac {2}{17} b^4 B x^{17/2} \]
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Time = 0.04 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {27, 77} \[ \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2}{7} a^4 A x^{7/2}+\frac {2}{9} a^3 x^{9/2} (a B+4 A b)+\frac {4}{11} a^2 b x^{11/2} (2 a B+3 A b)+\frac {2}{15} b^3 x^{15/2} (4 a B+A b)+\frac {4}{13} a b^2 x^{13/2} (3 a B+2 A b)+\frac {2}{17} b^4 B x^{17/2} \]
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Rule 27
Rule 77
Rubi steps \begin{align*} \text {integral}& = \int x^{5/2} (a+b x)^4 (A+B x) \, dx \\ & = \int \left (a^4 A x^{5/2}+a^3 (4 A b+a B) x^{7/2}+2 a^2 b (3 A b+2 a B) x^{9/2}+2 a b^2 (2 A b+3 a B) x^{11/2}+b^3 (A b+4 a B) x^{13/2}+b^4 B x^{15/2}\right ) \, dx \\ & = \frac {2}{7} a^4 A x^{7/2}+\frac {2}{9} a^3 (4 A b+a B) x^{9/2}+\frac {4}{11} a^2 b (3 A b+2 a B) x^{11/2}+\frac {4}{13} a b^2 (2 A b+3 a B) x^{13/2}+\frac {2}{15} b^3 (A b+4 a B) x^{15/2}+\frac {2}{17} b^4 B x^{17/2} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.81 \[ \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 x^{7/2} \left (12155 a^4 (9 A+7 B x)+30940 a^3 b x (11 A+9 B x)+32130 a^2 b^2 x^2 (13 A+11 B x)+15708 a b^3 x^3 (15 A+13 B x)+3003 b^4 x^4 (17 A+15 B x)\right )}{765765} \]
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Time = 0.22 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.90
method | result | size |
gosper | \(\frac {2 x^{\frac {7}{2}} \left (45045 b^{4} B \,x^{5}+51051 A \,b^{4} x^{4}+204204 x^{4} B \,b^{3} a +235620 a A \,b^{3} x^{3}+353430 x^{3} B \,a^{2} b^{2}+417690 a^{2} A \,b^{2} x^{2}+278460 x^{2} B \,a^{3} b +340340 a^{3} A b x +85085 a^{4} B x +109395 A \,a^{4}\right )}{765765}\) | \(100\) |
derivativedivides | \(\frac {2 b^{4} B \,x^{\frac {17}{2}}}{17}+\frac {2 \left (A \,b^{4}+4 B \,b^{3} a \right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (4 A \,b^{3} a +6 B \,a^{2} b^{2}\right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (6 A \,a^{2} b^{2}+4 B \,a^{3} b \right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (4 A \,a^{3} b +B \,a^{4}\right ) x^{\frac {9}{2}}}{9}+\frac {2 a^{4} A \,x^{\frac {7}{2}}}{7}\) | \(100\) |
default | \(\frac {2 b^{4} B \,x^{\frac {17}{2}}}{17}+\frac {2 \left (A \,b^{4}+4 B \,b^{3} a \right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (4 A \,b^{3} a +6 B \,a^{2} b^{2}\right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (6 A \,a^{2} b^{2}+4 B \,a^{3} b \right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (4 A \,a^{3} b +B \,a^{4}\right ) x^{\frac {9}{2}}}{9}+\frac {2 a^{4} A \,x^{\frac {7}{2}}}{7}\) | \(100\) |
trager | \(\frac {2 x^{\frac {7}{2}} \left (45045 b^{4} B \,x^{5}+51051 A \,b^{4} x^{4}+204204 x^{4} B \,b^{3} a +235620 a A \,b^{3} x^{3}+353430 x^{3} B \,a^{2} b^{2}+417690 a^{2} A \,b^{2} x^{2}+278460 x^{2} B \,a^{3} b +340340 a^{3} A b x +85085 a^{4} B x +109395 A \,a^{4}\right )}{765765}\) | \(100\) |
risch | \(\frac {2 x^{\frac {7}{2}} \left (45045 b^{4} B \,x^{5}+51051 A \,b^{4} x^{4}+204204 x^{4} B \,b^{3} a +235620 a A \,b^{3} x^{3}+353430 x^{3} B \,a^{2} b^{2}+417690 a^{2} A \,b^{2} x^{2}+278460 x^{2} B \,a^{3} b +340340 a^{3} A b x +85085 a^{4} B x +109395 A \,a^{4}\right )}{765765}\) | \(100\) |
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Time = 0.29 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.94 \[ \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2}{765765} \, {\left (45045 \, B b^{4} x^{8} + 109395 \, A a^{4} x^{3} + 51051 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{7} + 117810 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{6} + 139230 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{5} + 85085 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x^{4}\right )} \sqrt {x} \]
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Time = 0.53 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.33 \[ \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 A a^{4} x^{\frac {7}{2}}}{7} + \frac {8 A a^{3} b x^{\frac {9}{2}}}{9} + \frac {12 A a^{2} b^{2} x^{\frac {11}{2}}}{11} + \frac {8 A a b^{3} x^{\frac {13}{2}}}{13} + \frac {2 A b^{4} x^{\frac {15}{2}}}{15} + \frac {2 B a^{4} x^{\frac {9}{2}}}{9} + \frac {8 B a^{3} b x^{\frac {11}{2}}}{11} + \frac {12 B a^{2} b^{2} x^{\frac {13}{2}}}{13} + \frac {8 B a b^{3} x^{\frac {15}{2}}}{15} + \frac {2 B b^{4} x^{\frac {17}{2}}}{17} \]
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Time = 0.19 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.89 \[ \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2}{17} \, B b^{4} x^{\frac {17}{2}} + \frac {2}{7} \, A a^{4} x^{\frac {7}{2}} + \frac {2}{15} \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{\frac {15}{2}} + \frac {4}{13} \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{\frac {13}{2}} + \frac {4}{11} \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{\frac {11}{2}} + \frac {2}{9} \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x^{\frac {9}{2}} \]
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Time = 0.39 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.91 \[ \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2}{17} \, B b^{4} x^{\frac {17}{2}} + \frac {8}{15} \, B a b^{3} x^{\frac {15}{2}} + \frac {2}{15} \, A b^{4} x^{\frac {15}{2}} + \frac {12}{13} \, B a^{2} b^{2} x^{\frac {13}{2}} + \frac {8}{13} \, A a b^{3} x^{\frac {13}{2}} + \frac {8}{11} \, B a^{3} b x^{\frac {11}{2}} + \frac {12}{11} \, A a^{2} b^{2} x^{\frac {11}{2}} + \frac {2}{9} \, B a^{4} x^{\frac {9}{2}} + \frac {8}{9} \, A a^{3} b x^{\frac {9}{2}} + \frac {2}{7} \, A a^{4} x^{\frac {7}{2}} \]
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Time = 0.04 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.82 \[ \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=x^{9/2}\,\left (\frac {2\,B\,a^4}{9}+\frac {8\,A\,b\,a^3}{9}\right )+x^{15/2}\,\left (\frac {2\,A\,b^4}{15}+\frac {8\,B\,a\,b^3}{15}\right )+\frac {2\,A\,a^4\,x^{7/2}}{7}+\frac {2\,B\,b^4\,x^{17/2}}{17}+\frac {4\,a^2\,b\,x^{11/2}\,\left (3\,A\,b+2\,B\,a\right )}{11}+\frac {4\,a\,b^2\,x^{13/2}\,\left (2\,A\,b+3\,B\,a\right )}{13} \]
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