Integrand size = 22, antiderivative size = 162 \[ \int \frac {x^3 (A+B x)}{\sqrt {b x+c x^2}} \, dx=-\frac {5 b^2 (7 b B-8 A c) \sqrt {b x+c x^2}}{64 c^4}+\frac {5 b (7 b B-8 A c) x \sqrt {b x+c x^2}}{96 c^3}-\frac {(7 b B-8 A c) x^2 \sqrt {b x+c x^2}}{24 c^2}+\frac {B x^3 \sqrt {b x+c x^2}}{4 c}+\frac {5 b^3 (7 b B-8 A c) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{64 c^{9/2}} \]
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Time = 0.09 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {808, 684, 654, 634, 212} \[ \int \frac {x^3 (A+B x)}{\sqrt {b x+c x^2}} \, dx=\frac {5 b^3 (7 b B-8 A c) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{64 c^{9/2}}-\frac {5 b^2 \sqrt {b x+c x^2} (7 b B-8 A c)}{64 c^4}+\frac {5 b x \sqrt {b x+c x^2} (7 b B-8 A c)}{96 c^3}-\frac {x^2 \sqrt {b x+c x^2} (7 b B-8 A c)}{24 c^2}+\frac {B x^3 \sqrt {b x+c x^2}}{4 c} \]
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Rule 212
Rule 634
Rule 654
Rule 684
Rule 808
Rubi steps \begin{align*} \text {integral}& = \frac {B x^3 \sqrt {b x+c x^2}}{4 c}+\frac {\left (3 (-b B+A c)+\frac {1}{2} (-b B+2 A c)\right ) \int \frac {x^3}{\sqrt {b x+c x^2}} \, dx}{4 c} \\ & = -\frac {(7 b B-8 A c) x^2 \sqrt {b x+c x^2}}{24 c^2}+\frac {B x^3 \sqrt {b x+c x^2}}{4 c}+\frac {(5 b (7 b B-8 A c)) \int \frac {x^2}{\sqrt {b x+c x^2}} \, dx}{48 c^2} \\ & = \frac {5 b (7 b B-8 A c) x \sqrt {b x+c x^2}}{96 c^3}-\frac {(7 b B-8 A c) x^2 \sqrt {b x+c x^2}}{24 c^2}+\frac {B x^3 \sqrt {b x+c x^2}}{4 c}-\frac {\left (5 b^2 (7 b B-8 A c)\right ) \int \frac {x}{\sqrt {b x+c x^2}} \, dx}{64 c^3} \\ & = -\frac {5 b^2 (7 b B-8 A c) \sqrt {b x+c x^2}}{64 c^4}+\frac {5 b (7 b B-8 A c) x \sqrt {b x+c x^2}}{96 c^3}-\frac {(7 b B-8 A c) x^2 \sqrt {b x+c x^2}}{24 c^2}+\frac {B x^3 \sqrt {b x+c x^2}}{4 c}+\frac {\left (5 b^3 (7 b B-8 A c)\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{128 c^4} \\ & = -\frac {5 b^2 (7 b B-8 A c) \sqrt {b x+c x^2}}{64 c^4}+\frac {5 b (7 b B-8 A c) x \sqrt {b x+c x^2}}{96 c^3}-\frac {(7 b B-8 A c) x^2 \sqrt {b x+c x^2}}{24 c^2}+\frac {B x^3 \sqrt {b x+c x^2}}{4 c}+\frac {\left (5 b^3 (7 b B-8 A c)\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{64 c^4} \\ & = -\frac {5 b^2 (7 b B-8 A c) \sqrt {b x+c x^2}}{64 c^4}+\frac {5 b (7 b B-8 A c) x \sqrt {b x+c x^2}}{96 c^3}-\frac {(7 b B-8 A c) x^2 \sqrt {b x+c x^2}}{24 c^2}+\frac {B x^3 \sqrt {b x+c x^2}}{4 c}+\frac {5 b^3 (7 b B-8 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{64 c^{9/2}} \\ \end{align*}
Time = 0.47 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.16 \[ \int \frac {x^3 (A+B x)}{\sqrt {b x+c x^2}} \, dx=\frac {\sqrt {x} (b+c x) \left (-105 b^3 B \sqrt {x}+120 A b^2 c \sqrt {x}+70 b^2 B c x^{3/2}-80 A b c^2 x^{3/2}-56 b B c^2 x^{5/2}+64 A c^3 x^{5/2}+48 B c^3 x^{7/2}\right )}{192 c^4 \sqrt {x (b+c x)}}+\frac {5 b^3 (7 b B-8 A c) \sqrt {x} \sqrt {b+c x} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{-\sqrt {b}+\sqrt {b+c x}}\right )}{32 c^{9/2} \sqrt {x (b+c x)}} \]
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Time = 0.17 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.61
method | result | size |
pseudoelliptic | \(-\frac {5 \left (\frac {3 \left (A \,b^{3} c -\frac {7}{8} b^{4} B \right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )}{2}+\sqrt {x \left (c x +b \right )}\, \left (-\frac {3 \left (\frac {7 B x}{12}+A \right ) b^{2} c^{\frac {3}{2}}}{2}+b x \left (\frac {7 B x}{10}+A \right ) c^{\frac {5}{2}}-\frac {4 x^{2} \left (\frac {3 B x}{4}+A \right ) c^{\frac {7}{2}}}{5}+\frac {21 B \sqrt {c}\, b^{3}}{16}\right )\right )}{12 c^{\frac {9}{2}}}\) | \(99\) |
risch | \(\frac {\left (48 B \,c^{3} x^{3}+64 A \,c^{3} x^{2}-56 B b \,c^{2} x^{2}-80 A b \,c^{2} x +70 B \,b^{2} c x +120 A \,b^{2} c -105 B \,b^{3}\right ) x \left (c x +b \right )}{192 c^{4} \sqrt {x \left (c x +b \right )}}-\frac {5 b^{3} \left (8 A c -7 B b \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{128 c^{\frac {9}{2}}}\) | \(121\) |
default | \(B \left (\frac {x^{3} \sqrt {c \,x^{2}+b x}}{4 c}-\frac {7 b \left (\frac {x^{2} \sqrt {c \,x^{2}+b x}}{3 c}-\frac {5 b \left (\frac {x \sqrt {c \,x^{2}+b x}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}\right )}{6 c}\right )}{8 c}\right )+A \left (\frac {x^{2} \sqrt {c \,x^{2}+b x}}{3 c}-\frac {5 b \left (\frac {x \sqrt {c \,x^{2}+b x}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}\right )}{6 c}\right )\) | \(224\) |
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Time = 0.29 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.58 \[ \int \frac {x^3 (A+B x)}{\sqrt {b x+c x^2}} \, dx=\left [-\frac {15 \, {\left (7 \, B b^{4} - 8 \, A b^{3} c\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (48 \, B c^{4} x^{3} - 105 \, B b^{3} c + 120 \, A b^{2} c^{2} - 8 \, {\left (7 \, B b c^{3} - 8 \, A c^{4}\right )} x^{2} + 10 \, {\left (7 \, B b^{2} c^{2} - 8 \, A b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{384 \, c^{5}}, -\frac {15 \, {\left (7 \, B b^{4} - 8 \, A b^{3} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (48 \, B c^{4} x^{3} - 105 \, B b^{3} c + 120 \, A b^{2} c^{2} - 8 \, {\left (7 \, B b c^{3} - 8 \, A c^{4}\right )} x^{2} + 10 \, {\left (7 \, B b^{2} c^{2} - 8 \, A b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{192 \, c^{5}}\right ] \]
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Time = 0.50 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.26 \[ \int \frac {x^3 (A+B x)}{\sqrt {b x+c x^2}} \, dx=\begin {cases} - \frac {5 b^{3} \left (A - \frac {7 B b}{8 c}\right ) \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {b x + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: \frac {b^{2}}{c} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {c \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right )}{16 c^{3}} + \sqrt {b x + c x^{2}} \left (\frac {B x^{3}}{4 c} + \frac {5 b^{2} \left (A - \frac {7 B b}{8 c}\right )}{8 c^{3}} - \frac {5 b x \left (A - \frac {7 B b}{8 c}\right )}{12 c^{2}} + \frac {x^{2} \left (A - \frac {7 B b}{8 c}\right )}{3 c}\right ) & \text {for}\: c \neq 0 \\\frac {2 \left (\frac {A \left (b x\right )^{\frac {7}{2}}}{7} + \frac {B \left (b x\right )^{\frac {9}{2}}}{9 b}\right )}{b^{4}} & \text {for}\: b \neq 0 \\\tilde {\infty } \left (\frac {A x^{4}}{4} + \frac {B x^{5}}{5}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.18 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.27 \[ \int \frac {x^3 (A+B x)}{\sqrt {b x+c x^2}} \, dx=\frac {\sqrt {c x^{2} + b x} B x^{3}}{4 \, c} - \frac {7 \, \sqrt {c x^{2} + b x} B b x^{2}}{24 \, c^{2}} + \frac {\sqrt {c x^{2} + b x} A x^{2}}{3 \, c} + \frac {35 \, \sqrt {c x^{2} + b x} B b^{2} x}{96 \, c^{3}} - \frac {5 \, \sqrt {c x^{2} + b x} A b x}{12 \, c^{2}} + \frac {35 \, B b^{4} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{128 \, c^{\frac {9}{2}}} - \frac {5 \, A b^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{16 \, c^{\frac {7}{2}}} - \frac {35 \, \sqrt {c x^{2} + b x} B b^{3}}{64 \, c^{4}} + \frac {5 \, \sqrt {c x^{2} + b x} A b^{2}}{8 \, c^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.83 \[ \int \frac {x^3 (A+B x)}{\sqrt {b x+c x^2}} \, dx=\frac {1}{192} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (\frac {6 \, B x}{c} - \frac {7 \, B b c^{2} - 8 \, A c^{3}}{c^{4}}\right )} x + \frac {5 \, {\left (7 \, B b^{2} c - 8 \, A b c^{2}\right )}}{c^{4}}\right )} x - \frac {15 \, {\left (7 \, B b^{3} - 8 \, A b^{2} c\right )}}{c^{4}}\right )} - \frac {5 \, {\left (7 \, B b^{4} - 8 \, A b^{3} c\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{128 \, c^{\frac {9}{2}}} \]
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Timed out. \[ \int \frac {x^3 (A+B x)}{\sqrt {b x+c x^2}} \, dx=\int \frac {x^3\,\left (A+B\,x\right )}{\sqrt {c\,x^2+b\,x}} \,d x \]
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