Integrand size = 19, antiderivative size = 171 \[ \int (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=-\frac {5 b^4 (b B-2 A c) (b+2 c x) \sqrt {b x+c x^2}}{1024 c^4}+\frac {5 b^2 (b B-2 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^3}-\frac {(b B-2 A c) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{24 c^2}+\frac {B \left (b x+c x^2\right )^{7/2}}{7 c}+\frac {5 b^6 (b B-2 A c) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{1024 c^{9/2}} \]
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Time = 0.04 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {654, 626, 634, 212} \[ \int (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\frac {5 b^6 (b B-2 A c) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{1024 c^{9/2}}-\frac {5 b^4 (b+2 c x) \sqrt {b x+c x^2} (b B-2 A c)}{1024 c^4}+\frac {5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} (b B-2 A c)}{384 c^3}-\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2} (b B-2 A c)}{24 c^2}+\frac {B \left (b x+c x^2\right )^{7/2}}{7 c} \]
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Rule 212
Rule 626
Rule 634
Rule 654
Rubi steps \begin{align*} \text {integral}& = \frac {B \left (b x+c x^2\right )^{7/2}}{7 c}+\frac {(-b B+2 A c) \int \left (b x+c x^2\right )^{5/2} \, dx}{2 c} \\ & = -\frac {(b B-2 A c) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{24 c^2}+\frac {B \left (b x+c x^2\right )^{7/2}}{7 c}+\frac {\left (5 b^2 (b B-2 A c)\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{48 c^2} \\ & = \frac {5 b^2 (b B-2 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^3}-\frac {(b B-2 A c) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{24 c^2}+\frac {B \left (b x+c x^2\right )^{7/2}}{7 c}-\frac {\left (5 b^4 (b B-2 A c)\right ) \int \sqrt {b x+c x^2} \, dx}{256 c^3} \\ & = -\frac {5 b^4 (b B-2 A c) (b+2 c x) \sqrt {b x+c x^2}}{1024 c^4}+\frac {5 b^2 (b B-2 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^3}-\frac {(b B-2 A c) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{24 c^2}+\frac {B \left (b x+c x^2\right )^{7/2}}{7 c}+\frac {\left (5 b^6 (b B-2 A c)\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{2048 c^4} \\ & = -\frac {5 b^4 (b B-2 A c) (b+2 c x) \sqrt {b x+c x^2}}{1024 c^4}+\frac {5 b^2 (b B-2 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^3}-\frac {(b B-2 A c) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{24 c^2}+\frac {B \left (b x+c x^2\right )^{7/2}}{7 c}+\frac {\left (5 b^6 (b B-2 A c)\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{1024 c^4} \\ & = -\frac {5 b^4 (b B-2 A c) (b+2 c x) \sqrt {b x+c x^2}}{1024 c^4}+\frac {5 b^2 (b B-2 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^3}-\frac {(b B-2 A c) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{24 c^2}+\frac {B \left (b x+c x^2\right )^{7/2}}{7 c}+\frac {5 b^6 (b B-2 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{1024 c^{9/2}} \\ \end{align*}
Time = 1.07 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.39 \[ \int (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\frac {\sqrt {x} \sqrt {b+c x} \left (\sqrt {c} \sqrt {x} \sqrt {b+c x} \left (-105 b^6 B+70 b^5 c (3 A+B x)-28 b^4 c^2 x (5 A+2 B x)+16 b^3 c^3 x^2 (7 A+3 B x)+512 c^6 x^5 (7 A+6 B x)+256 b c^5 x^4 (35 A+29 B x)+32 b^2 c^4 x^3 (189 A+148 B x)\right )+420 A b^6 c \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}-\sqrt {b+c x}}\right )+210 b^7 B \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{-\sqrt {b}+\sqrt {b+c x}}\right )\right )}{21504 c^{9/2} \sqrt {x (b+c x)}} \]
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Time = 0.11 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.13
method | result | size |
risch | \(\frac {\left (3072 B \,c^{6} x^{6}+3584 A \,c^{6} x^{5}+7424 B b \,c^{5} x^{5}+8960 A b \,c^{5} x^{4}+4736 B \,b^{2} c^{4} x^{4}+6048 A \,b^{2} c^{4} x^{3}+48 B \,b^{3} c^{3} x^{3}+112 A \,b^{3} c^{3} x^{2}-56 B \,b^{4} c^{2} x^{2}-140 A \,b^{4} c^{2} x +70 B \,b^{5} c x +210 A \,b^{5} c -105 B \,b^{6}\right ) x \left (c x +b \right )}{21504 c^{4} \sqrt {x \left (c x +b \right )}}-\frac {5 b^{6} \left (2 A c -B b \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2048 c^{\frac {9}{2}}}\) | \(193\) |
default | \(A \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{12 c}-\frac {5 b^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )+B \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{12 c}-\frac {5 b^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )\) | \(263\) |
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Time = 0.27 (sec) , antiderivative size = 392, normalized size of antiderivative = 2.29 \[ \int (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\left [-\frac {105 \, {\left (B b^{7} - 2 \, A b^{6} c\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (3072 \, B c^{7} x^{6} - 105 \, B b^{6} c + 210 \, A b^{5} c^{2} + 256 \, {\left (29 \, B b c^{6} + 14 \, A c^{7}\right )} x^{5} + 128 \, {\left (37 \, B b^{2} c^{5} + 70 \, A b c^{6}\right )} x^{4} + 48 \, {\left (B b^{3} c^{4} + 126 \, A b^{2} c^{5}\right )} x^{3} - 56 \, {\left (B b^{4} c^{3} - 2 \, A b^{3} c^{4}\right )} x^{2} + 70 \, {\left (B b^{5} c^{2} - 2 \, A b^{4} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{43008 \, c^{5}}, -\frac {105 \, {\left (B b^{7} - 2 \, A b^{6} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (3072 \, B c^{7} x^{6} - 105 \, B b^{6} c + 210 \, A b^{5} c^{2} + 256 \, {\left (29 \, B b c^{6} + 14 \, A c^{7}\right )} x^{5} + 128 \, {\left (37 \, B b^{2} c^{5} + 70 \, A b c^{6}\right )} x^{4} + 48 \, {\left (B b^{3} c^{4} + 126 \, A b^{2} c^{5}\right )} x^{3} - 56 \, {\left (B b^{4} c^{3} - 2 \, A b^{3} c^{4}\right )} x^{2} + 70 \, {\left (B b^{5} c^{2} - 2 \, A b^{4} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{21504 \, c^{5}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 570 vs. \(2 (163) = 326\).
Time = 0.55 (sec) , antiderivative size = 570, normalized size of antiderivative = 3.33 \[ \int (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\begin {cases} - \frac {5 b^{3} \left (A b^{3} - \frac {7 b \left (3 A b^{2} c + B b^{3} - \frac {9 b \left (3 A b c^{2} + 3 B b^{2} c - \frac {11 b \left (A c^{3} + \frac {29 B b c^{2}}{14}\right )}{12 c}\right )}{10 c}\right )}{8 c}\right ) \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {b x + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: \frac {b^{2}}{c} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {c \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right )}{16 c^{3}} + \sqrt {b x + c x^{2}} \left (\frac {B c^{2} x^{6}}{7} + \frac {5 b^{2} \left (A b^{3} - \frac {7 b \left (3 A b^{2} c + B b^{3} - \frac {9 b \left (3 A b c^{2} + 3 B b^{2} c - \frac {11 b \left (A c^{3} + \frac {29 B b c^{2}}{14}\right )}{12 c}\right )}{10 c}\right )}{8 c}\right )}{8 c^{3}} - \frac {5 b x \left (A b^{3} - \frac {7 b \left (3 A b^{2} c + B b^{3} - \frac {9 b \left (3 A b c^{2} + 3 B b^{2} c - \frac {11 b \left (A c^{3} + \frac {29 B b c^{2}}{14}\right )}{12 c}\right )}{10 c}\right )}{8 c}\right )}{12 c^{2}} + \frac {x^{5} \left (A c^{3} + \frac {29 B b c^{2}}{14}\right )}{6 c} + \frac {x^{4} \cdot \left (3 A b c^{2} + 3 B b^{2} c - \frac {11 b \left (A c^{3} + \frac {29 B b c^{2}}{14}\right )}{12 c}\right )}{5 c} + \frac {x^{3} \cdot \left (3 A b^{2} c + B b^{3} - \frac {9 b \left (3 A b c^{2} + 3 B b^{2} c - \frac {11 b \left (A c^{3} + \frac {29 B b c^{2}}{14}\right )}{12 c}\right )}{10 c}\right )}{4 c} + \frac {x^{2} \left (A b^{3} - \frac {7 b \left (3 A b^{2} c + B b^{3} - \frac {9 b \left (3 A b c^{2} + 3 B b^{2} c - \frac {11 b \left (A c^{3} + \frac {29 B b c^{2}}{14}\right )}{12 c}\right )}{10 c}\right )}{8 c}\right )}{3 c}\right ) & \text {for}\: c \neq 0 \\\frac {2 \left (\frac {A \left (b x\right )^{\frac {7}{2}}}{7} + \frac {B \left (b x\right )^{\frac {9}{2}}}{9 b}\right )}{b} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (147) = 294\).
Time = 0.19 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.86 \[ \int (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\frac {1}{6} \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} A x - \frac {5 \, \sqrt {c x^{2} + b x} B b^{5} x}{512 \, c^{3}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{3} x}{192 \, c^{2}} + \frac {5 \, \sqrt {c x^{2} + b x} A b^{4} x}{256 \, c^{2}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b x}{12 \, c} - \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b^{2} x}{96 \, c} + \frac {5 \, B b^{7} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2048 \, c^{\frac {9}{2}}} - \frac {5 \, A b^{6} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{1024 \, c^{\frac {7}{2}}} - \frac {5 \, \sqrt {c x^{2} + b x} B b^{6}}{1024 \, c^{4}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{4}}{384 \, c^{3}} + \frac {5 \, \sqrt {c x^{2} + b x} A b^{5}}{512 \, c^{3}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b^{2}}{24 \, c^{2}} - \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b^{3}}{192 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {7}{2}} B}{7 \, c} + \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} A b}{12 \, c} \]
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Time = 0.29 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.28 \[ \int (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\frac {1}{21504} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (12 \, B c^{2} x + \frac {29 \, B b c^{7} + 14 \, A c^{8}}{c^{6}}\right )} x + \frac {37 \, B b^{2} c^{6} + 70 \, A b c^{7}}{c^{6}}\right )} x + \frac {3 \, {\left (B b^{3} c^{5} + 126 \, A b^{2} c^{6}\right )}}{c^{6}}\right )} x - \frac {7 \, {\left (B b^{4} c^{4} - 2 \, A b^{3} c^{5}\right )}}{c^{6}}\right )} x + \frac {35 \, {\left (B b^{5} c^{3} - 2 \, A b^{4} c^{4}\right )}}{c^{6}}\right )} x - \frac {105 \, {\left (B b^{6} c^{2} - 2 \, A b^{5} c^{3}\right )}}{c^{6}}\right )} - \frac {5 \, {\left (B b^{7} - 2 \, A b^{6} c\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{2048 \, c^{\frac {9}{2}}} \]
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Timed out. \[ \int (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\int {\left (c\,x^2+b\,x\right )}^{5/2}\,\left (A+B\,x\right ) \,d x \]
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