Integrand size = 22, antiderivative size = 119 \[ \int \frac {x \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\frac {(b c-a d)^2 \sqrt {c+d x^2}}{b^3}+\frac {(b c-a d) \left (c+d x^2\right )^{3/2}}{3 b^2}+\frac {\left (c+d x^2\right )^{5/2}}{5 b}-\frac {(b c-a d)^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{b^{7/2}} \]
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Time = 0.07 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {455, 52, 65, 214} \[ \int \frac {x \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=-\frac {(b c-a d)^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{b^{7/2}}+\frac {\sqrt {c+d x^2} (b c-a d)^2}{b^3}+\frac {\left (c+d x^2\right )^{3/2} (b c-a d)}{3 b^2}+\frac {\left (c+d x^2\right )^{5/2}}{5 b} \]
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Rule 52
Rule 65
Rule 214
Rule 455
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {(c+d x)^{5/2}}{a+b x} \, dx,x,x^2\right ) \\ & = \frac {\left (c+d x^2\right )^{5/2}}{5 b}+\frac {(b c-a d) \text {Subst}\left (\int \frac {(c+d x)^{3/2}}{a+b x} \, dx,x,x^2\right )}{2 b} \\ & = \frac {(b c-a d) \left (c+d x^2\right )^{3/2}}{3 b^2}+\frac {\left (c+d x^2\right )^{5/2}}{5 b}+\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {\sqrt {c+d x}}{a+b x} \, dx,x,x^2\right )}{2 b^2} \\ & = \frac {(b c-a d)^2 \sqrt {c+d x^2}}{b^3}+\frac {(b c-a d) \left (c+d x^2\right )^{3/2}}{3 b^2}+\frac {\left (c+d x^2\right )^{5/2}}{5 b}+\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{2 b^3} \\ & = \frac {(b c-a d)^2 \sqrt {c+d x^2}}{b^3}+\frac {(b c-a d) \left (c+d x^2\right )^{3/2}}{3 b^2}+\frac {\left (c+d x^2\right )^{5/2}}{5 b}+\frac {(b c-a d)^3 \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^3 d} \\ & = \frac {(b c-a d)^2 \sqrt {c+d x^2}}{b^3}+\frac {(b c-a d) \left (c+d x^2\right )^{3/2}}{3 b^2}+\frac {\left (c+d x^2\right )^{5/2}}{5 b}-\frac {(b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{b^{7/2}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.97 \[ \int \frac {x \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\frac {\sqrt {c+d x^2} \left (15 a^2 d^2-5 a b d \left (7 c+d x^2\right )+b^2 \left (23 c^2+11 c d x^2+3 d^2 x^4\right )\right )}{15 b^3}-\frac {(-b c+a d)^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{b^{7/2}} \]
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Time = 3.11 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.03
method | result | size |
pseudoelliptic | \(-\frac {\left (a d -b c \right )^{3} \arctan \left (\frac {b \sqrt {d \,x^{2}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )-\sqrt {d \,x^{2}+c}\, \left (\frac {\left (d^{2} x^{4}+\frac {11}{3} c d \,x^{2}+\frac {23}{3} c^{2}\right ) b^{2}}{5}-\frac {7 \left (\frac {d \,x^{2}}{7}+c \right ) d a b}{3}+a^{2} d^{2}\right ) \sqrt {\left (a d -b c \right ) b}}{\sqrt {\left (a d -b c \right ) b}\, b^{3}}\) | \(123\) |
risch | \(\frac {\left (3 b^{2} d^{2} x^{4}-5 x^{2} a b \,d^{2}+11 x^{2} b^{2} c d +15 a^{2} d^{2}-35 a b c d +23 b^{2} c^{2}\right ) \sqrt {d \,x^{2}+c}}{15 b^{3}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (-\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 b \sqrt {-\frac {a d -b c}{b}}}-\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 b \sqrt {-\frac {a d -b c}{b}}}\right )}{b^{3}}\) | \(410\) |
default | \(\text {Expression too large to display}\) | \(2068\) |
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Time = 0.27 (sec) , antiderivative size = 405, normalized size of antiderivative = 3.40 \[ \int \frac {x \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\left [\frac {15 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \, {\left (b^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, {\left (3 \, b^{2} d^{2} x^{4} + 23 \, b^{2} c^{2} - 35 \, a b c d + 15 \, a^{2} d^{2} + {\left (11 \, b^{2} c d - 5 \, a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{60 \, b^{3}}, -\frac {15 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{b}}}{2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}}\right ) - 2 \, {\left (3 \, b^{2} d^{2} x^{4} + 23 \, b^{2} c^{2} - 35 \, a b c d + 15 \, a^{2} d^{2} + {\left (11 \, b^{2} c d - 5 \, a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{30 \, b^{3}}\right ] \]
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Time = 9.03 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.31 \[ \int \frac {x \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\begin {cases} \frac {2 \left (\frac {d \left (c + d x^{2}\right )^{\frac {5}{2}}}{10 b} + \frac {\left (c + d x^{2}\right )^{\frac {3}{2}} \left (- a d^{2} + b c d\right )}{6 b^{2}} + \frac {\sqrt {c + d x^{2}} \left (a^{2} d^{3} - 2 a b c d^{2} + b^{2} c^{2} d\right )}{2 b^{3}} - \frac {d \left (a d - b c\right )^{3} \operatorname {atan}{\left (\frac {\sqrt {c + d x^{2}}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{2 b^{4} \sqrt {\frac {a d - b c}{b}}}\right )}{d} & \text {for}\: d \neq 0 \\c^{\frac {5}{2}} \left (\begin {cases} \frac {x^{2}}{2 a} & \text {for}\: b = 0 \\\frac {\log {\left (2 a + 2 b x^{2} \right )}}{2 b} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {x \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.29 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.55 \[ \int \frac {x \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b^{3}} + \frac {3 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{4} + 5 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{4} c + 15 \, \sqrt {d x^{2} + c} b^{4} c^{2} - 5 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b^{3} d - 30 \, \sqrt {d x^{2} + c} a b^{3} c d + 15 \, \sqrt {d x^{2} + c} a^{2} b^{2} d^{2}}{15 \, b^{5}} \]
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Time = 5.25 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.15 \[ \int \frac {x \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\frac {{\left (d\,x^2+c\right )}^{5/2}}{5\,b}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d\,x^2+c}\,{\left (a\,d-b\,c\right )}^{5/2}}{a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}\right )\,{\left (a\,d-b\,c\right )}^{5/2}}{b^{7/2}}-\frac {{\left (d\,x^2+c\right )}^{3/2}\,\left (a\,d-b\,c\right )}{3\,b^2}+\frac {\sqrt {d\,x^2+c}\,{\left (a\,d-b\,c\right )}^2}{b^3} \]
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