Integrand size = 29, antiderivative size = 157 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {x}} \, dx=2 a^6 A \sqrt {x}+\frac {2}{3} a^5 (6 A b+a B) x^{3/2}+\frac {6}{5} a^4 b (5 A b+2 a B) x^{5/2}+\frac {10}{7} a^3 b^2 (4 A b+3 a B) x^{7/2}+\frac {10}{9} a^2 b^3 (3 A b+4 a B) x^{9/2}+\frac {6}{11} a b^4 (2 A b+5 a B) x^{11/2}+\frac {2}{13} b^5 (A b+6 a B) x^{13/2}+\frac {2}{15} b^6 B x^{15/2} \]
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Time = 0.06 (sec) , antiderivative size = 157, 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} \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {x}} \, dx=2 a^6 A \sqrt {x}+\frac {2}{3} a^5 x^{3/2} (a B+6 A b)+\frac {6}{5} a^4 b x^{5/2} (2 a B+5 A b)+\frac {10}{7} a^3 b^2 x^{7/2} (3 a B+4 A b)+\frac {10}{9} a^2 b^3 x^{9/2} (4 a B+3 A b)+\frac {2}{13} b^5 x^{13/2} (6 a B+A b)+\frac {6}{11} a b^4 x^{11/2} (5 a B+2 A b)+\frac {2}{15} b^6 B x^{15/2} \]
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Rule 27
Rule 77
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b x)^6 (A+B x)}{\sqrt {x}} \, dx \\ & = \int \left (\frac {a^6 A}{\sqrt {x}}+a^5 (6 A b+a B) \sqrt {x}+3 a^4 b (5 A b+2 a B) x^{3/2}+5 a^3 b^2 (4 A b+3 a B) x^{5/2}+5 a^2 b^3 (3 A b+4 a B) x^{7/2}+3 a b^4 (2 A b+5 a B) x^{9/2}+b^5 (A b+6 a B) x^{11/2}+b^6 B x^{13/2}\right ) \, dx \\ & = 2 a^6 A \sqrt {x}+\frac {2}{3} a^5 (6 A b+a B) x^{3/2}+\frac {6}{5} a^4 b (5 A b+2 a B) x^{5/2}+\frac {10}{7} a^3 b^2 (4 A b+3 a B) x^{7/2}+\frac {10}{9} a^2 b^3 (3 A b+4 a B) x^{9/2}+\frac {6}{11} a b^4 (2 A b+5 a B) x^{11/2}+\frac {2}{13} b^5 (A b+6 a B) x^{13/2}+\frac {2}{15} b^6 B x^{15/2} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.81 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {x}} \, dx=\frac {2 \sqrt {x} \left (15015 a^6 (3 A+B x)+18018 a^5 b x (5 A+3 B x)+19305 a^4 b^2 x^2 (7 A+5 B x)+14300 a^3 b^3 x^3 (9 A+7 B x)+6825 a^2 b^4 x^4 (11 A+9 B x)+1890 a b^5 x^5 (13 A+11 B x)+231 b^6 x^6 (15 A+13 B x)\right )}{45045} \]
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Time = 0.20 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.94
method | result | size |
trager | \(\left (\frac {2}{15} b^{6} B \,x^{7}+\frac {2}{13} A \,b^{6} x^{6}+\frac {12}{13} x^{6} B a \,b^{5}+\frac {12}{11} a A \,b^{5} x^{5}+\frac {30}{11} x^{5} B \,b^{4} a^{2}+\frac {10}{3} a^{2} A \,b^{4} x^{4}+\frac {40}{9} x^{4} B \,a^{3} b^{3}+\frac {40}{7} a^{3} A \,b^{3} x^{3}+\frac {30}{7} x^{3} B \,a^{4} b^{2}+6 a^{4} A \,b^{2} x^{2}+\frac {12}{5} x^{2} B \,a^{5} b +4 a^{5} A b x +\frac {2}{3} x B \,a^{6}+2 A \,a^{6}\right ) \sqrt {x}\) | \(147\) |
gosper | \(\frac {2 \sqrt {x}\, \left (3003 b^{6} B \,x^{7}+3465 A \,b^{6} x^{6}+20790 x^{6} B a \,b^{5}+24570 a A \,b^{5} x^{5}+61425 x^{5} B \,b^{4} a^{2}+75075 a^{2} A \,b^{4} x^{4}+100100 x^{4} B \,a^{3} b^{3}+128700 a^{3} A \,b^{3} x^{3}+96525 x^{3} B \,a^{4} b^{2}+135135 a^{4} A \,b^{2} x^{2}+54054 x^{2} B \,a^{5} b +90090 a^{5} A b x +15015 x B \,a^{6}+45045 A \,a^{6}\right )}{45045}\) | \(148\) |
derivativedivides | \(\frac {2 b^{6} B \,x^{\frac {15}{2}}}{15}+\frac {2 \left (A \,b^{6}+6 B a \,b^{5}\right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (6 A a \,b^{5}+15 B \,b^{4} a^{2}\right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (15 A \,b^{4} a^{2}+20 B \,a^{3} b^{3}\right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (20 A \,a^{3} b^{3}+15 B \,a^{4} b^{2}\right ) x^{\frac {7}{2}}}{7}+\frac {2 \left (15 A \,a^{4} b^{2}+6 B \,a^{5} b \right ) x^{\frac {5}{2}}}{5}+\frac {2 \left (6 A \,a^{5} b +B \,a^{6}\right ) x^{\frac {3}{2}}}{3}+2 a^{6} A \sqrt {x}\) | \(148\) |
default | \(\frac {2 b^{6} B \,x^{\frac {15}{2}}}{15}+\frac {2 \left (A \,b^{6}+6 B a \,b^{5}\right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (6 A a \,b^{5}+15 B \,b^{4} a^{2}\right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (15 A \,b^{4} a^{2}+20 B \,a^{3} b^{3}\right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (20 A \,a^{3} b^{3}+15 B \,a^{4} b^{2}\right ) x^{\frac {7}{2}}}{7}+\frac {2 \left (15 A \,a^{4} b^{2}+6 B \,a^{5} b \right ) x^{\frac {5}{2}}}{5}+\frac {2 \left (6 A \,a^{5} b +B \,a^{6}\right ) x^{\frac {3}{2}}}{3}+2 a^{6} A \sqrt {x}\) | \(148\) |
risch | \(\frac {2 \sqrt {x}\, \left (3003 b^{6} B \,x^{7}+3465 A \,b^{6} x^{6}+20790 x^{6} B a \,b^{5}+24570 a A \,b^{5} x^{5}+61425 x^{5} B \,b^{4} a^{2}+75075 a^{2} A \,b^{4} x^{4}+100100 x^{4} B \,a^{3} b^{3}+128700 a^{3} A \,b^{3} x^{3}+96525 x^{3} B \,a^{4} b^{2}+135135 a^{4} A \,b^{2} x^{2}+54054 x^{2} B \,a^{5} b +90090 a^{5} A b x +15015 x B \,a^{6}+45045 A \,a^{6}\right )}{45045}\) | \(148\) |
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Time = 0.27 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.94 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {x}} \, dx=\frac {2}{45045} \, {\left (3003 \, B b^{6} x^{7} + 45045 \, A a^{6} + 3465 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 12285 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 25025 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 32175 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 27027 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 15015 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )} \sqrt {x} \]
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Time = 0.33 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.34 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {x}} \, dx=2 A a^{6} \sqrt {x} + 4 A a^{5} b x^{\frac {3}{2}} + 6 A a^{4} b^{2} x^{\frac {5}{2}} + \frac {40 A a^{3} b^{3} x^{\frac {7}{2}}}{7} + \frac {10 A a^{2} b^{4} x^{\frac {9}{2}}}{3} + \frac {12 A a b^{5} x^{\frac {11}{2}}}{11} + \frac {2 A b^{6} x^{\frac {13}{2}}}{13} + \frac {2 B a^{6} x^{\frac {3}{2}}}{3} + \frac {12 B a^{5} b x^{\frac {5}{2}}}{5} + \frac {30 B a^{4} b^{2} x^{\frac {7}{2}}}{7} + \frac {40 B a^{3} b^{3} x^{\frac {9}{2}}}{9} + \frac {30 B a^{2} b^{4} x^{\frac {11}{2}}}{11} + \frac {12 B a b^{5} x^{\frac {13}{2}}}{13} + \frac {2 B b^{6} x^{\frac {15}{2}}}{15} \]
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Time = 0.21 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.94 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {x}} \, dx=\frac {2}{15} \, B b^{6} x^{\frac {15}{2}} + 2 \, A a^{6} \sqrt {x} + \frac {2}{13} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{\frac {13}{2}} + \frac {6}{11} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{\frac {11}{2}} + \frac {10}{9} \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{\frac {9}{2}} + \frac {10}{7} \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{\frac {7}{2}} + \frac {6}{5} \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{\frac {5}{2}} + \frac {2}{3} \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x^{\frac {3}{2}} \]
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Time = 0.30 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.95 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {x}} \, dx=\frac {2}{15} \, B b^{6} x^{\frac {15}{2}} + \frac {12}{13} \, B a b^{5} x^{\frac {13}{2}} + \frac {2}{13} \, A b^{6} x^{\frac {13}{2}} + \frac {30}{11} \, B a^{2} b^{4} x^{\frac {11}{2}} + \frac {12}{11} \, A a b^{5} x^{\frac {11}{2}} + \frac {40}{9} \, B a^{3} b^{3} x^{\frac {9}{2}} + \frac {10}{3} \, A a^{2} b^{4} x^{\frac {9}{2}} + \frac {30}{7} \, B a^{4} b^{2} x^{\frac {7}{2}} + \frac {40}{7} \, A a^{3} b^{3} x^{\frac {7}{2}} + \frac {12}{5} \, B a^{5} b x^{\frac {5}{2}} + 6 \, A a^{4} b^{2} x^{\frac {5}{2}} + \frac {2}{3} \, B a^{6} x^{\frac {3}{2}} + 4 \, A a^{5} b x^{\frac {3}{2}} + 2 \, A a^{6} \sqrt {x} \]
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Time = 0.05 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.83 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {x}} \, dx=x^{3/2}\,\left (\frac {2\,B\,a^6}{3}+4\,A\,b\,a^5\right )+x^{13/2}\,\left (\frac {2\,A\,b^6}{13}+\frac {12\,B\,a\,b^5}{13}\right )+2\,A\,a^6\,\sqrt {x}+\frac {2\,B\,b^6\,x^{15/2}}{15}+\frac {10\,a^3\,b^2\,x^{7/2}\,\left (4\,A\,b+3\,B\,a\right )}{7}+\frac {10\,a^2\,b^3\,x^{9/2}\,\left (3\,A\,b+4\,B\,a\right )}{9}+\frac {6\,a^4\,b\,x^{5/2}\,\left (5\,A\,b+2\,B\,a\right )}{5}+\frac {6\,a\,b^4\,x^{11/2}\,\left (2\,A\,b+5\,B\,a\right )}{11} \]
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