Integrand size = 24, antiderivative size = 144 \[ \int \frac {x^3 \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=-\frac {a (b c-a d)^2 \sqrt {c+d x^2}}{b^4}-\frac {a (b c-a d) \left (c+d x^2\right )^{3/2}}{3 b^3}-\frac {a \left (c+d x^2\right )^{5/2}}{5 b^2}+\frac {\left (c+d x^2\right )^{7/2}}{7 b d}+\frac {a (b c-a d)^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{b^{9/2}} \]
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Time = 0.11 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {457, 81, 52, 65, 214} \[ \int \frac {x^3 \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\frac {a (b c-a d)^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{b^{9/2}}-\frac {a \sqrt {c+d x^2} (b c-a d)^2}{b^4}-\frac {a \left (c+d x^2\right )^{3/2} (b c-a d)}{3 b^3}-\frac {a \left (c+d x^2\right )^{5/2}}{5 b^2}+\frac {\left (c+d x^2\right )^{7/2}}{7 b d} \]
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Rule 52
Rule 65
Rule 81
Rule 214
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x (c+d x)^{5/2}}{a+b x} \, dx,x,x^2\right ) \\ & = \frac {\left (c+d x^2\right )^{7/2}}{7 b d}-\frac {a \text {Subst}\left (\int \frac {(c+d x)^{5/2}}{a+b x} \, dx,x,x^2\right )}{2 b} \\ & = -\frac {a \left (c+d x^2\right )^{5/2}}{5 b^2}+\frac {\left (c+d x^2\right )^{7/2}}{7 b d}-\frac {(a (b c-a d)) \text {Subst}\left (\int \frac {(c+d x)^{3/2}}{a+b x} \, dx,x,x^2\right )}{2 b^2} \\ & = -\frac {a (b c-a d) \left (c+d x^2\right )^{3/2}}{3 b^3}-\frac {a \left (c+d x^2\right )^{5/2}}{5 b^2}+\frac {\left (c+d x^2\right )^{7/2}}{7 b d}-\frac {\left (a (b c-a d)^2\right ) \text {Subst}\left (\int \frac {\sqrt {c+d x}}{a+b x} \, dx,x,x^2\right )}{2 b^3} \\ & = -\frac {a (b c-a d)^2 \sqrt {c+d x^2}}{b^4}-\frac {a (b c-a d) \left (c+d x^2\right )^{3/2}}{3 b^3}-\frac {a \left (c+d x^2\right )^{5/2}}{5 b^2}+\frac {\left (c+d x^2\right )^{7/2}}{7 b d}-\frac {\left (a (b c-a d)^3\right ) \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{2 b^4} \\ & = -\frac {a (b c-a d)^2 \sqrt {c+d x^2}}{b^4}-\frac {a (b c-a d) \left (c+d x^2\right )^{3/2}}{3 b^3}-\frac {a \left (c+d x^2\right )^{5/2}}{5 b^2}+\frac {\left (c+d x^2\right )^{7/2}}{7 b d}-\frac {\left (a (b c-a d)^3\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{b^4 d} \\ & = -\frac {a (b c-a d)^2 \sqrt {c+d x^2}}{b^4}-\frac {a (b c-a d) \left (c+d x^2\right )^{3/2}}{3 b^3}-\frac {a \left (c+d x^2\right )^{5/2}}{5 b^2}+\frac {\left (c+d x^2\right )^{7/2}}{7 b d}+\frac {a (b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{b^{9/2}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.97 \[ \int \frac {x^3 \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\frac {\sqrt {c+d x^2} \left (-105 a^3 d^3+15 b^3 \left (c+d x^2\right )^3+35 a^2 b d^2 \left (7 c+d x^2\right )-7 a b^2 d \left (23 c^2+11 c d x^2+3 d^2 x^4\right )\right )}{105 b^4 d}+\frac {a (-b c+a d)^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{b^{9/2}} \]
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Time = 3.04 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.02
method | result | size |
pseudoelliptic | \(-\frac {\sqrt {\left (a d -b c \right ) b}\, \left (-\frac {\left (d \,x^{2}+c \right )^{3} b^{3}}{7}+\frac {23 d \left (\frac {3}{23} d^{2} x^{4}+\frac {11}{23} c d \,x^{2}+c^{2}\right ) a \,b^{2}}{15}-\frac {7 \left (\frac {d \,x^{2}}{7}+c \right ) d^{2} a^{2} b}{3}+a^{3} d^{3}\right ) \sqrt {d \,x^{2}+c}-a d \left (a d -b c \right )^{3} \arctan \left (\frac {b \sqrt {d \,x^{2}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}\, d \,b^{4}}\) | \(147\) |
risch | \(-\frac {\left (-15 b^{3} d^{3} x^{6}+21 a \,b^{2} d^{3} x^{4}-45 b^{3} c \,d^{2} x^{4}-35 x^{2} a^{2} b \,d^{3}+77 x^{2} a \,b^{2} c \,d^{2}-45 x^{2} b^{3} c^{2} d +105 a^{3} d^{3}-245 a^{2} b c \,d^{2}+161 a \,b^{2} c^{2} d -15 b^{3} c^{3}\right ) \sqrt {d \,x^{2}+c}}{105 d \,b^{4}}+\frac {a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (-\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 b \sqrt {-\frac {a d -b c}{b}}}-\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 b \sqrt {-\frac {a d -b c}{b}}}\right )}{b^{4}}\) | \(468\) |
default | \(\text {Expression too large to display}\) | \(2087\) |
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Time = 0.29 (sec) , antiderivative size = 527, normalized size of antiderivative = 3.66 \[ \int \frac {x^3 \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\left [\frac {105 \, {\left (a b^{2} c^{2} d - 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} + 4 \, {\left (b^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, {\left (15 \, b^{3} d^{3} x^{6} + 15 \, b^{3} c^{3} - 161 \, a b^{2} c^{2} d + 245 \, a^{2} b c d^{2} - 105 \, a^{3} d^{3} + 3 \, {\left (15 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{4} + {\left (45 \, b^{3} c^{2} d - 77 \, a b^{2} c d^{2} + 35 \, a^{2} b d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{420 \, b^{4} d}, \frac {105 \, {\left (a b^{2} c^{2} d - 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{b}}}{2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}}\right ) + 2 \, {\left (15 \, b^{3} d^{3} x^{6} + 15 \, b^{3} c^{3} - 161 \, a b^{2} c^{2} d + 245 \, a^{2} b c d^{2} - 105 \, a^{3} d^{3} + 3 \, {\left (15 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{4} + {\left (45 \, b^{3} c^{2} d - 77 \, a b^{2} c d^{2} + 35 \, a^{2} b d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{210 \, b^{4} d}\right ] \]
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Time = 12.50 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.30 \[ \int \frac {x^3 \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\begin {cases} \frac {2 \left (- \frac {a d \left (c + d x^{2}\right )^{\frac {5}{2}}}{10 b^{2}} + \frac {a d \left (a d - b c\right )^{3} \operatorname {atan}{\left (\frac {\sqrt {c + d x^{2}}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{2 b^{5} \sqrt {\frac {a d - b c}{b}}} + \frac {\left (c + d x^{2}\right )^{\frac {7}{2}}}{14 b} + \frac {\left (c + d x^{2}\right )^{\frac {3}{2}} \left (a^{2} d^{2} - a b c d\right )}{6 b^{3}} + \frac {\sqrt {c + d x^{2}} \left (- a^{3} d^{3} + 2 a^{2} b c d^{2} - a b^{2} c^{2} d\right )}{2 b^{4}}\right )}{d} & \text {for}\: d \neq 0 \\c^{\frac {5}{2}} \left (- \frac {a \left (\begin {cases} \frac {x^{2}}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + b x^{2} \right )}}{b} & \text {otherwise} \end {cases}\right )}{2 b} + \frac {x^{2}}{2 b}\right ) & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {x^3 \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.29 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.58 \[ \int \frac {x^3 \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=-\frac {{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b^{4}} + \frac {15 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{6} d^{6} - 21 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b^{5} d^{7} - 35 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b^{5} c d^{7} - 105 \, \sqrt {d x^{2} + c} a b^{5} c^{2} d^{7} + 35 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} b^{4} d^{8} + 210 \, \sqrt {d x^{2} + c} a^{2} b^{4} c d^{8} - 105 \, \sqrt {d x^{2} + c} a^{3} b^{3} d^{9}}{105 \, b^{7} d^{7}} \]
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Time = 5.29 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.74 \[ \int \frac {x^3 \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\frac {{\left (d\,x^2+c\right )}^{7/2}}{7\,b\,d}-{\left (d\,x^2+c\right )}^{5/2}\,\left (\frac {c}{5\,b\,d}+\frac {a\,d^2-b\,c\,d}{5\,b^2\,d^2}\right )+\frac {a\,\mathrm {atan}\left (\frac {a\,\sqrt {b}\,\sqrt {d\,x^2+c}\,{\left (a\,d-b\,c\right )}^{5/2}}{a^4\,d^3-3\,a^3\,b\,c\,d^2+3\,a^2\,b^2\,c^2\,d-a\,b^3\,c^3}\right )\,{\left (a\,d-b\,c\right )}^{5/2}}{b^{9/2}}+\frac {{\left (d\,x^2+c\right )}^{3/2}\,\left (a\,d^2-b\,c\,d\right )\,\left (\frac {c}{b\,d}+\frac {a\,d^2-b\,c\,d}{b^2\,d^2}\right )}{3\,b\,d}-\frac {\sqrt {d\,x^2+c}\,{\left (a\,d^2-b\,c\,d\right )}^2\,\left (\frac {c}{b\,d}+\frac {a\,d^2-b\,c\,d}{b^2\,d^2}\right )}{b^2\,d^2} \]
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