Integrand size = 19, antiderivative size = 111 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{11/2}} \, dx=-\frac {c \sqrt {b x+c x^2}}{4 x^{5/2}}-\frac {c^2 \sqrt {b x+c x^2}}{8 b x^{3/2}}-\frac {\left (b x+c x^2\right )^{3/2}}{3 x^{9/2}}+\frac {c^3 \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{8 b^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {676, 686, 674, 213} \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{11/2}} \, dx=\frac {c^3 \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{8 b^{3/2}}-\frac {c^2 \sqrt {b x+c x^2}}{8 b x^{3/2}}-\frac {c \sqrt {b x+c x^2}}{4 x^{5/2}}-\frac {\left (b x+c x^2\right )^{3/2}}{3 x^{9/2}} \]
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Rule 213
Rule 674
Rule 676
Rule 686
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (b x+c x^2\right )^{3/2}}{3 x^{9/2}}+\frac {1}{2} c \int \frac {\sqrt {b x+c x^2}}{x^{7/2}} \, dx \\ & = -\frac {c \sqrt {b x+c x^2}}{4 x^{5/2}}-\frac {\left (b x+c x^2\right )^{3/2}}{3 x^{9/2}}+\frac {1}{8} c^2 \int \frac {1}{x^{3/2} \sqrt {b x+c x^2}} \, dx \\ & = -\frac {c \sqrt {b x+c x^2}}{4 x^{5/2}}-\frac {c^2 \sqrt {b x+c x^2}}{8 b x^{3/2}}-\frac {\left (b x+c x^2\right )^{3/2}}{3 x^{9/2}}-\frac {c^3 \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx}{16 b} \\ & = -\frac {c \sqrt {b x+c x^2}}{4 x^{5/2}}-\frac {c^2 \sqrt {b x+c x^2}}{8 b x^{3/2}}-\frac {\left (b x+c x^2\right )^{3/2}}{3 x^{9/2}}-\frac {c^3 \text {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )}{8 b} \\ & = -\frac {c \sqrt {b x+c x^2}}{4 x^{5/2}}-\frac {c^2 \sqrt {b x+c x^2}}{8 b x^{3/2}}-\frac {\left (b x+c x^2\right )^{3/2}}{3 x^{9/2}}+\frac {c^3 \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{8 b^{3/2}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.85 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{11/2}} \, dx=\frac {\sqrt {x (b+c x)} \left (-\sqrt {b} \sqrt {b+c x} \left (8 b^2+14 b c x+3 c^2 x^2\right )+3 c^3 x^3 \text {arctanh}\left (\frac {\sqrt {b+c x}}{\sqrt {b}}\right )\right )}{24 b^{3/2} x^{7/2} \sqrt {b+c x}} \]
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Time = 1.96 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.74
method | result | size |
risch | \(-\frac {\left (c x +b \right ) \left (3 c^{2} x^{2}+14 b c x +8 b^{2}\right )}{24 x^{\frac {5}{2}} b \sqrt {x \left (c x +b \right )}}+\frac {c^{3} \operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right ) \sqrt {c x +b}\, \sqrt {x}}{8 b^{\frac {3}{2}} \sqrt {x \left (c x +b \right )}}\) | \(82\) |
default | \(\frac {\sqrt {x \left (c x +b \right )}\, \left (3 \,\operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right ) c^{3} x^{3}-3 c^{2} x^{2} \sqrt {b}\, \sqrt {c x +b}-14 b^{\frac {3}{2}} c x \sqrt {c x +b}-8 b^{\frac {5}{2}} \sqrt {c x +b}\right )}{24 b^{\frac {3}{2}} x^{\frac {7}{2}} \sqrt {c x +b}}\) | \(90\) |
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Time = 0.26 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.57 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{11/2}} \, dx=\left [\frac {3 \, \sqrt {b} c^{3} x^{4} \log \left (-\frac {c x^{2} + 2 \, b x + 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) - 2 \, {\left (3 \, b c^{2} x^{2} + 14 \, b^{2} c x + 8 \, b^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{48 \, b^{2} x^{4}}, -\frac {3 \, \sqrt {-b} c^{3} x^{4} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) + {\left (3 \, b c^{2} x^{2} + 14 \, b^{2} c x + 8 \, b^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{24 \, b^{2} x^{4}}\right ] \]
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\[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{11/2}} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}{x^{\frac {11}{2}}}\, dx \]
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\[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{11/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}}}{x^{\frac {11}{2}}} \,d x } \]
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Time = 0.29 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.76 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{11/2}} \, dx=-\frac {\frac {3 \, c^{4} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b} + \frac {3 \, {\left (c x + b\right )}^{\frac {5}{2}} c^{4} + 8 \, {\left (c x + b\right )}^{\frac {3}{2}} b c^{4} - 3 \, \sqrt {c x + b} b^{2} c^{4}}{b c^{3} x^{3}}}{24 \, c} \]
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Timed out. \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{11/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}}{x^{11/2}} \,d x \]
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