Integrand size = 27, antiderivative size = 167 \[ \int (a+b x)^3 \sqrt {-\frac {a^2 c}{b^2}+c x^2} \, dx=\frac {7}{8} a^3 x \sqrt {-\frac {a^2 c}{b^2}+c x^2}+\frac {7 a^2 b \left (-\frac {a^2 c}{b^2}+c x^2\right )^{3/2}}{12 c}+\frac {7 a b (a+b x) \left (-\frac {a^2 c}{b^2}+c x^2\right )^{3/2}}{20 c}+\frac {b (a+b x)^2 \left (-\frac {a^2 c}{b^2}+c x^2\right )^{3/2}}{5 c}-\frac {7 a^5 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {-\frac {a^2 c}{b^2}+c x^2}}\right )}{8 b^2} \]
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Time = 0.06 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {685, 655, 201, 223, 212} \[ \int (a+b x)^3 \sqrt {-\frac {a^2 c}{b^2}+c x^2} \, dx=\frac {7 a^2 b \left (c x^2-\frac {a^2 c}{b^2}\right )^{3/2}}{12 c}+\frac {7 a b (a+b x) \left (c x^2-\frac {a^2 c}{b^2}\right )^{3/2}}{20 c}+\frac {b (a+b x)^2 \left (c x^2-\frac {a^2 c}{b^2}\right )^{3/2}}{5 c}-\frac {7 a^5 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {c x^2-\frac {a^2 c}{b^2}}}\right )}{8 b^2}+\frac {7}{8} a^3 x \sqrt {c x^2-\frac {a^2 c}{b^2}} \]
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Rule 201
Rule 212
Rule 223
Rule 655
Rule 685
Rubi steps \begin{align*} \text {integral}& = \frac {b (a+b x)^2 \left (-\frac {a^2 c}{b^2}+c x^2\right )^{3/2}}{5 c}+\frac {1}{5} (7 a) \int (a+b x)^2 \sqrt {-\frac {a^2 c}{b^2}+c x^2} \, dx \\ & = \frac {7 a b (a+b x) \left (-\frac {a^2 c}{b^2}+c x^2\right )^{3/2}}{20 c}+\frac {b (a+b x)^2 \left (-\frac {a^2 c}{b^2}+c x^2\right )^{3/2}}{5 c}+\frac {1}{4} \left (7 a^2\right ) \int (a+b x) \sqrt {-\frac {a^2 c}{b^2}+c x^2} \, dx \\ & = \frac {7 a^2 b \left (-\frac {a^2 c}{b^2}+c x^2\right )^{3/2}}{12 c}+\frac {7 a b (a+b x) \left (-\frac {a^2 c}{b^2}+c x^2\right )^{3/2}}{20 c}+\frac {b (a+b x)^2 \left (-\frac {a^2 c}{b^2}+c x^2\right )^{3/2}}{5 c}+\frac {1}{4} \left (7 a^3\right ) \int \sqrt {-\frac {a^2 c}{b^2}+c x^2} \, dx \\ & = \frac {7}{8} a^3 x \sqrt {-\frac {a^2 c}{b^2}+c x^2}+\frac {7 a^2 b \left (-\frac {a^2 c}{b^2}+c x^2\right )^{3/2}}{12 c}+\frac {7 a b (a+b x) \left (-\frac {a^2 c}{b^2}+c x^2\right )^{3/2}}{20 c}+\frac {b (a+b x)^2 \left (-\frac {a^2 c}{b^2}+c x^2\right )^{3/2}}{5 c}-\frac {\left (7 a^5 c\right ) \int \frac {1}{\sqrt {-\frac {a^2 c}{b^2}+c x^2}} \, dx}{8 b^2} \\ & = \frac {7}{8} a^3 x \sqrt {-\frac {a^2 c}{b^2}+c x^2}+\frac {7 a^2 b \left (-\frac {a^2 c}{b^2}+c x^2\right )^{3/2}}{12 c}+\frac {7 a b (a+b x) \left (-\frac {a^2 c}{b^2}+c x^2\right )^{3/2}}{20 c}+\frac {b (a+b x)^2 \left (-\frac {a^2 c}{b^2}+c x^2\right )^{3/2}}{5 c}-\frac {\left (7 a^5 c\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {-\frac {a^2 c}{b^2}+c x^2}}\right )}{8 b^2} \\ & = \frac {7}{8} a^3 x \sqrt {-\frac {a^2 c}{b^2}+c x^2}+\frac {7 a^2 b \left (-\frac {a^2 c}{b^2}+c x^2\right )^{3/2}}{12 c}+\frac {7 a b (a+b x) \left (-\frac {a^2 c}{b^2}+c x^2\right )^{3/2}}{20 c}+\frac {b (a+b x)^2 \left (-\frac {a^2 c}{b^2}+c x^2\right )^{3/2}}{5 c}-\frac {7 a^5 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {-\frac {a^2 c}{b^2}+c x^2}}\right )}{8 b^2} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.76 \[ \int (a+b x)^3 \sqrt {-\frac {a^2 c}{b^2}+c x^2} \, dx=\frac {\sqrt {c \left (-\frac {a^2}{b^2}+x^2\right )} \left (b \sqrt {-\frac {a^2}{b^2}+x^2} \left (-136 a^4+15 a^3 b x+112 a^2 b^2 x^2+90 a b^3 x^3+24 b^4 x^4\right )+105 a^5 \log \left (-x+\sqrt {-\frac {a^2}{b^2}+x^2}\right )\right )}{120 b^2 \sqrt {-\frac {a^2}{b^2}+x^2}} \]
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Time = 2.36 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.18
method | result | size |
risch | \(-\frac {\left (-24 b^{4} x^{4}-90 a \,b^{3} x^{3}-112 a^{2} b^{2} x^{2}-15 a^{3} b x +136 a^{4}\right ) \sqrt {-\frac {c \left (-b^{2} x^{2}+a^{2}\right )}{b^{2}}}\, \sqrt {-c \left (-b^{2} x^{2}+a^{2}\right )}}{120 b \sqrt {c \left (b^{2} x^{2}-a^{2}\right )}}+\frac {7 a^{5} \ln \left (\frac {b^{2} c x}{\sqrt {b^{2} c}}+\sqrt {b^{2} c \,x^{2}-a^{2} c}\right ) \sqrt {-\frac {c \left (-b^{2} x^{2}+a^{2}\right )}{b^{2}}}\, \sqrt {-c \left (-b^{2} x^{2}+a^{2}\right )}}{8 \sqrt {b^{2} c}\, \left (-b^{2} x^{2}+a^{2}\right )}\) | \(197\) |
default | \(a^{3} \left (\frac {x \sqrt {-\frac {a^{2} c}{b^{2}}+c \,x^{2}}}{2}-\frac {\sqrt {c}\, a^{2} \ln \left (\sqrt {c}\, x +\sqrt {-\frac {a^{2} c}{b^{2}}+c \,x^{2}}\right )}{2 b^{2}}\right )+b^{3} \left (\frac {x^{2} \left (-\frac {a^{2} c}{b^{2}}+c \,x^{2}\right )^{\frac {3}{2}}}{5 c}+\frac {2 a^{2} \left (-\frac {a^{2} c}{b^{2}}+c \,x^{2}\right )^{\frac {3}{2}}}{15 c \,b^{2}}\right )+3 a \,b^{2} \left (\frac {x \left (-\frac {a^{2} c}{b^{2}}+c \,x^{2}\right )^{\frac {3}{2}}}{4 c}+\frac {a^{2} \left (\frac {x \sqrt {-\frac {a^{2} c}{b^{2}}+c \,x^{2}}}{2}-\frac {\sqrt {c}\, a^{2} \ln \left (\sqrt {c}\, x +\sqrt {-\frac {a^{2} c}{b^{2}}+c \,x^{2}}\right )}{2 b^{2}}\right )}{4 b^{2}}\right )+\frac {a^{2} b {\left (-\frac {c \left (-b^{2} x^{2}+a^{2}\right )}{b^{2}}\right )}^{\frac {3}{2}}}{c}\) | \(242\) |
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Time = 0.28 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.56 \[ \int (a+b x)^3 \sqrt {-\frac {a^2 c}{b^2}+c x^2} \, dx=\left [\frac {105 \, a^{5} \sqrt {c} \log \left (2 \, b^{2} c x^{2} - 2 \, b^{2} \sqrt {c} x \sqrt {\frac {b^{2} c x^{2} - a^{2} c}{b^{2}}} - a^{2} c\right ) + 2 \, {\left (24 \, b^{5} x^{4} + 90 \, a b^{4} x^{3} + 112 \, a^{2} b^{3} x^{2} + 15 \, a^{3} b^{2} x - 136 \, a^{4} b\right )} \sqrt {\frac {b^{2} c x^{2} - a^{2} c}{b^{2}}}}{240 \, b^{2}}, \frac {105 \, a^{5} \sqrt {-c} \arctan \left (\frac {b^{2} \sqrt {-c} x \sqrt {\frac {b^{2} c x^{2} - a^{2} c}{b^{2}}}}{b^{2} c x^{2} - a^{2} c}\right ) + {\left (24 \, b^{5} x^{4} + 90 \, a b^{4} x^{3} + 112 \, a^{2} b^{3} x^{2} + 15 \, a^{3} b^{2} x - 136 \, a^{4} b\right )} \sqrt {\frac {b^{2} c x^{2} - a^{2} c}{b^{2}}}}{120 \, b^{2}}\right ] \]
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Time = 0.58 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.96 \[ \int (a+b x)^3 \sqrt {-\frac {a^2 c}{b^2}+c x^2} \, dx=\begin {cases} - \frac {7 a^{5} c \left (\begin {cases} \frac {\log {\left (2 \sqrt {c} \sqrt {- \frac {a^{2} c}{b^{2}} + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: \frac {a^{2} c}{b^{2}} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {c x^{2}}} & \text {otherwise} \end {cases}\right )}{8 b^{2}} + \sqrt {- \frac {a^{2} c}{b^{2}} + c x^{2}} \left (- \frac {17 a^{4}}{15 b} + \frac {a^{3} x}{8} + \frac {14 a^{2} b x^{2}}{15} + \frac {3 a b^{2} x^{3}}{4} + \frac {b^{3} x^{4}}{5}\right ) & \text {for}\: c \neq 0 \\\sqrt {- \frac {a^{2} c}{b^{2}}} \left (\begin {cases} a^{3} x & \text {for}\: b = 0 \\\frac {\left (a + b x\right )^{4}}{4 b} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.86 \[ \int (a+b x)^3 \sqrt {-\frac {a^2 c}{b^2}+c x^2} \, dx=\frac {{\left (c x^{2} - \frac {a^{2} c}{b^{2}}\right )}^{\frac {3}{2}} b^{3} x^{2}}{5 \, c} + \frac {7}{8} \, \sqrt {c x^{2} - \frac {a^{2} c}{b^{2}}} a^{3} x + \frac {3 \, {\left (c x^{2} - \frac {a^{2} c}{b^{2}}\right )}^{\frac {3}{2}} a b^{2} x}{4 \, c} - \frac {7 \, a^{5} \sqrt {c} \log \left (2 \, c x + 2 \, \sqrt {c x^{2} - \frac {a^{2} c}{b^{2}}} \sqrt {c}\right )}{8 \, b^{2}} + \frac {17 \, {\left (c x^{2} - \frac {a^{2} c}{b^{2}}\right )}^{\frac {3}{2}} a^{2} b}{15 \, c} \]
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Time = 0.30 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.68 \[ \int (a+b x)^3 \sqrt {-\frac {a^2 c}{b^2}+c x^2} \, dx=\frac {{\left (\frac {105 \, a^{5} \sqrt {c} \log \left ({\left | -\sqrt {b^{2} c} x + \sqrt {b^{2} c x^{2} - a^{2} c} \right |}\right )}{{\left | b \right |}} - \sqrt {b^{2} c x^{2} - a^{2} c} {\left (\frac {136 \, a^{4}}{b} - {\left (15 \, a^{3} + 2 \, {\left (56 \, a^{2} b + 3 \, {\left (4 \, b^{3} x + 15 \, a b^{2}\right )} x\right )} x\right )} x\right )}\right )} {\left | b \right |}}{120 \, b^{2}} \]
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Timed out. \[ \int (a+b x)^3 \sqrt {-\frac {a^2 c}{b^2}+c x^2} \, dx=\int \sqrt {c\,x^2-\frac {a^2\,c}{b^2}}\,{\left (a+b\,x\right )}^3 \,d x \]
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