Integrand size = 24, antiderivative size = 172 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{11/2}} \, dx=\frac {5 c (4 b B+3 A c) \sqrt {b x+c x^2}}{4 \sqrt {x}}+\frac {5 c (4 b B+3 A c) \left (b x+c x^2\right )^{3/2}}{12 b x^{3/2}}-\frac {(4 b B+3 A c) \left (b x+c x^2\right )^{5/2}}{4 b x^{7/2}}-\frac {A \left (b x+c x^2\right )^{7/2}}{2 b x^{11/2}}-\frac {5}{4} \sqrt {b} c (4 b B+3 A c) \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right ) \]
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Time = 0.10 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {806, 676, 678, 674, 213} \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{11/2}} \, dx=-\frac {5}{4} \sqrt {b} c (3 A c+4 b B) \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )+\frac {5 c \sqrt {b x+c x^2} (3 A c+4 b B)}{4 \sqrt {x}}-\frac {\left (b x+c x^2\right )^{5/2} (3 A c+4 b B)}{4 b x^{7/2}}+\frac {5 c \left (b x+c x^2\right )^{3/2} (3 A c+4 b B)}{12 b x^{3/2}}-\frac {A \left (b x+c x^2\right )^{7/2}}{2 b x^{11/2}} \]
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Rule 213
Rule 674
Rule 676
Rule 678
Rule 806
Rubi steps \begin{align*} \text {integral}& = -\frac {A \left (b x+c x^2\right )^{7/2}}{2 b x^{11/2}}+\frac {\left (-\frac {11}{2} (-b B+A c)+\frac {7}{2} (-b B+2 A c)\right ) \int \frac {\left (b x+c x^2\right )^{5/2}}{x^{9/2}} \, dx}{2 b} \\ & = -\frac {(4 b B+3 A c) \left (b x+c x^2\right )^{5/2}}{4 b x^{7/2}}-\frac {A \left (b x+c x^2\right )^{7/2}}{2 b x^{11/2}}+\frac {(5 c (4 b B+3 A c)) \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{5/2}} \, dx}{8 b} \\ & = \frac {5 c (4 b B+3 A c) \left (b x+c x^2\right )^{3/2}}{12 b x^{3/2}}-\frac {(4 b B+3 A c) \left (b x+c x^2\right )^{5/2}}{4 b x^{7/2}}-\frac {A \left (b x+c x^2\right )^{7/2}}{2 b x^{11/2}}+\frac {1}{8} (5 c (4 b B+3 A c)) \int \frac {\sqrt {b x+c x^2}}{x^{3/2}} \, dx \\ & = \frac {5 c (4 b B+3 A c) \sqrt {b x+c x^2}}{4 \sqrt {x}}+\frac {5 c (4 b B+3 A c) \left (b x+c x^2\right )^{3/2}}{12 b x^{3/2}}-\frac {(4 b B+3 A c) \left (b x+c x^2\right )^{5/2}}{4 b x^{7/2}}-\frac {A \left (b x+c x^2\right )^{7/2}}{2 b x^{11/2}}+\frac {1}{8} (5 b c (4 b B+3 A c)) \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx \\ & = \frac {5 c (4 b B+3 A c) \sqrt {b x+c x^2}}{4 \sqrt {x}}+\frac {5 c (4 b B+3 A c) \left (b x+c x^2\right )^{3/2}}{12 b x^{3/2}}-\frac {(4 b B+3 A c) \left (b x+c x^2\right )^{5/2}}{4 b x^{7/2}}-\frac {A \left (b x+c x^2\right )^{7/2}}{2 b x^{11/2}}+\frac {1}{4} (5 b c (4 b B+3 A c)) \text {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right ) \\ & = \frac {5 c (4 b B+3 A c) \sqrt {b x+c x^2}}{4 \sqrt {x}}+\frac {5 c (4 b B+3 A c) \left (b x+c x^2\right )^{3/2}}{12 b x^{3/2}}-\frac {(4 b B+3 A c) \left (b x+c x^2\right )^{5/2}}{4 b x^{7/2}}-\frac {A \left (b x+c x^2\right )^{7/2}}{2 b x^{11/2}}-\frac {5}{4} \sqrt {b} c (4 b B+3 A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right ) \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.71 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{11/2}} \, dx=\frac {\sqrt {x (b+c x)} \left (\sqrt {b+c x} \left (-3 A \left (2 b^2+9 b c x-8 c^2 x^2\right )+4 B x \left (-3 b^2+14 b c x+2 c^2 x^2\right )\right )-15 \sqrt {b} c (4 b B+3 A c) x^2 \text {arctanh}\left (\frac {\sqrt {b+c x}}{\sqrt {b}}\right )\right )}{12 x^{5/2} \sqrt {b+c x}} \]
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Time = 0.08 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.69
method | result | size |
risch | \(-\frac {b \left (c x +b \right ) \left (9 A c x +4 B b x +2 A b \right )}{4 x^{\frac {3}{2}} \sqrt {x \left (c x +b \right )}}+\frac {c \left (\frac {16 B \left (c x +b \right )^{\frac {3}{2}}}{3}+16 A c \sqrt {c x +b}+32 B b \sqrt {c x +b}-10 \sqrt {b}\, \left (3 A c +4 B b \right ) \operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )\right ) \sqrt {c x +b}\, \sqrt {x}}{8 \sqrt {x \left (c x +b \right )}}\) | \(118\) |
default | \(-\frac {\sqrt {x \left (c x +b \right )}\, \left (-8 B \,c^{2} x^{3} \sqrt {b}\, \sqrt {c x +b}+45 A \,\operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right ) b \,c^{2} x^{2}-24 A \,c^{2} x^{2} \sqrt {c x +b}\, \sqrt {b}+60 B \,\operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right ) b^{2} c \,x^{2}-56 B \,b^{\frac {3}{2}} c \,x^{2} \sqrt {c x +b}+27 A \,b^{\frac {3}{2}} c x \sqrt {c x +b}+12 B \,b^{\frac {5}{2}} x \sqrt {c x +b}+6 A \,b^{\frac {5}{2}} \sqrt {c x +b}\right )}{12 x^{\frac {5}{2}} \sqrt {c x +b}\, \sqrt {b}}\) | \(167\) |
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Time = 0.30 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.38 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{11/2}} \, dx=\left [\frac {15 \, {\left (4 \, B b c + 3 \, A c^{2}\right )} \sqrt {b} x^{3} \log \left (-\frac {c x^{2} + 2 \, b x - 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) + 2 \, {\left (8 \, B c^{2} x^{3} - 6 \, A b^{2} + 8 \, {\left (7 \, B b c + 3 \, A c^{2}\right )} x^{2} - 3 \, {\left (4 \, B b^{2} + 9 \, A b c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{24 \, x^{3}}, \frac {15 \, {\left (4 \, B b c + 3 \, A c^{2}\right )} \sqrt {-b} x^{3} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) + {\left (8 \, B c^{2} x^{3} - 6 \, A b^{2} + 8 \, {\left (7 \, B b c + 3 \, A c^{2}\right )} x^{2} - 3 \, {\left (4 \, B b^{2} + 9 \, A b c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{12 \, x^{3}}\right ] \]
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\[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{11/2}} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (A + B x\right )}{x^{\frac {11}{2}}}\, dx \]
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\[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{11/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} {\left (B x + A\right )}}{x^{\frac {11}{2}}} \,d x } \]
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Time = 0.33 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.90 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{11/2}} \, dx=\frac {8 \, {\left (c x + b\right )}^{\frac {3}{2}} B c^{2} + 48 \, \sqrt {c x + b} B b c^{2} + 24 \, \sqrt {c x + b} A c^{3} + \frac {15 \, {\left (4 \, B b^{2} c^{2} + 3 \, A b c^{3}\right )} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} - \frac {3 \, {\left (4 \, {\left (c x + b\right )}^{\frac {3}{2}} B b^{2} c^{2} - 4 \, \sqrt {c x + b} B b^{3} c^{2} + 9 \, {\left (c x + b\right )}^{\frac {3}{2}} A b c^{3} - 7 \, \sqrt {c x + b} A b^{2} c^{3}\right )}}{c^{2} x^{2}}}{12 \, c} \]
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Timed out. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{11/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{5/2}\,\left (A+B\,x\right )}{x^{11/2}} \,d x \]
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