Integrand size = 24, antiderivative size = 163 \[ \int \frac {x^3 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\frac {(2 b c-5 a d) \sqrt {c+d x^2}}{2 b^3}+\frac {(2 b c-5 a d) \left (c+d x^2\right )^{3/2}}{6 b^2 (b c-a d)}+\frac {a \left (c+d x^2\right )^{5/2}}{2 b (b c-a d) \left (a+b x^2\right )}-\frac {(2 b c-5 a d) \sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 b^{7/2}} \]
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Time = 0.10 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {457, 79, 52, 65, 214} \[ \int \frac {x^3 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=-\frac {(2 b c-5 a d) \sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 b^{7/2}}+\frac {\sqrt {c+d x^2} (2 b c-5 a d)}{2 b^3}+\frac {\left (c+d x^2\right )^{3/2} (2 b c-5 a d)}{6 b^2 (b c-a d)}+\frac {a \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right ) (b c-a d)} \]
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Rule 52
Rule 65
Rule 79
Rule 214
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x (c+d x)^{3/2}}{(a+b x)^2} \, dx,x,x^2\right ) \\ & = \frac {a \left (c+d x^2\right )^{5/2}}{2 b (b c-a d) \left (a+b x^2\right )}+\frac {(2 b c-5 a d) \text {Subst}\left (\int \frac {(c+d x)^{3/2}}{a+b x} \, dx,x,x^2\right )}{4 b (b c-a d)} \\ & = \frac {(2 b c-5 a d) \left (c+d x^2\right )^{3/2}}{6 b^2 (b c-a d)}+\frac {a \left (c+d x^2\right )^{5/2}}{2 b (b c-a d) \left (a+b x^2\right )}+\frac {(2 b c-5 a d) \text {Subst}\left (\int \frac {\sqrt {c+d x}}{a+b x} \, dx,x,x^2\right )}{4 b^2} \\ & = \frac {(2 b c-5 a d) \sqrt {c+d x^2}}{2 b^3}+\frac {(2 b c-5 a d) \left (c+d x^2\right )^{3/2}}{6 b^2 (b c-a d)}+\frac {a \left (c+d x^2\right )^{5/2}}{2 b (b c-a d) \left (a+b x^2\right )}+\frac {((2 b c-5 a d) (b c-a d)) \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{4 b^3} \\ & = \frac {(2 b c-5 a d) \sqrt {c+d x^2}}{2 b^3}+\frac {(2 b c-5 a d) \left (c+d x^2\right )^{3/2}}{6 b^2 (b c-a d)}+\frac {a \left (c+d x^2\right )^{5/2}}{2 b (b c-a d) \left (a+b x^2\right )}+\frac {((2 b c-5 a d) (b c-a d)) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{2 b^3 d} \\ & = \frac {(2 b c-5 a d) \sqrt {c+d x^2}}{2 b^3}+\frac {(2 b c-5 a d) \left (c+d x^2\right )^{3/2}}{6 b^2 (b c-a d)}+\frac {a \left (c+d x^2\right )^{5/2}}{2 b (b c-a d) \left (a+b x^2\right )}-\frac {(2 b c-5 a d) \sqrt {b c-a d} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 b^{7/2}} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.77 \[ \int \frac {x^3 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\frac {\sqrt {c+d x^2} \left (-15 a^2 d+a b \left (11 c-10 d x^2\right )+2 b^2 x^2 \left (4 c+d x^2\right )\right )}{6 b^3 \left (a+b x^2\right )}-\frac {(2 b c-5 a d) \sqrt {-b c+a d} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{2 b^{7/2}} \]
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Time = 3.10 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.82
method | result | size |
pseudoelliptic | \(-\frac {5 \left (-\left (b \,x^{2}+a \right ) \left (a d -\frac {2 b c}{5}\right ) \left (a d -b c \right ) \arctan \left (\frac {b \sqrt {d \,x^{2}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )+\sqrt {\left (a d -b c \right ) b}\, \left (-\frac {8 x^{2} \left (\frac {d \,x^{2}}{4}+c \right ) b^{2}}{15}-\frac {11 \left (-\frac {10 d \,x^{2}}{11}+c \right ) a b}{15}+a^{2} d \right ) \sqrt {d \,x^{2}+c}\right )}{2 \sqrt {\left (a d -b c \right ) b}\, b^{3} \left (b \,x^{2}+a \right )}\) | \(133\) |
risch | \(-\frac {\left (-b d \,x^{2}+6 a d -4 b c \right ) \sqrt {d \,x^{2}+c}}{3 b^{3}}+\frac {-\frac {\left (\frac {3}{2} a^{2} d^{2}-2 a b c d +\frac {1}{2} b^{2} c^{2}\right ) \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 )}{b \sqrt {-\frac {a d -b c}{b}}}-\frac {\left (\frac {3}{2} a^{2} d^{2}-2 a b c d +\frac {1}{2} b^{2} c^{2}\right ) \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 )}{b \sqrt {-\frac {a d -b c}{b}}}-\frac {\sqrt {-a b}\, \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {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}}}{\left (a d -b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {d \sqrt {-a b}\, \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 )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{4 b^{2}}+\frac {\sqrt {-a b}\, \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {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}}}{\left (a d -b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {d \sqrt {-a b}\, \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 )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{4 b^{2}}}{b^{3}}\) | \(939\) |
default | \(\text {Expression too large to display}\) | \(3381\) |
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Time = 0.30 (sec) , antiderivative size = 413, normalized size of antiderivative = 2.53 \[ \int \frac {x^3 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\left [-\frac {3 \, {\left (2 \, a b c - 5 \, a^{2} d + {\left (2 \, b^{2} c - 5 \, a b d\right )} x^{2}\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} + 4 \, {\left (b^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left (2 \, b^{2} d x^{4} + 11 \, a b c - 15 \, a^{2} d + 2 \, {\left (4 \, b^{2} c - 5 \, a b d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{24 \, {\left (b^{4} x^{2} + a b^{3}\right )}}, -\frac {3 \, {\left (2 \, a b c - 5 \, a^{2} d + {\left (2 \, b^{2} c - 5 \, a b d\right )} x^{2}\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{b}}}{2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}}\right ) - 2 \, {\left (2 \, b^{2} d x^{4} + 11 \, a b c - 15 \, a^{2} d + 2 \, {\left (4 \, b^{2} c - 5 \, a b d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{12 \, {\left (b^{4} x^{2} + a b^{3}\right )}}\right ] \]
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\[ \int \frac {x^3 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\int \frac {x^{3} \left (c + d x^{2}\right )^{\frac {3}{2}}}{\left (a + b x^{2}\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {x^3 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.28 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.06 \[ \int \frac {x^3 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\frac {{\left (2 \, b^{2} c^{2} - 7 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, \sqrt {-b^{2} c + a b d} b^{3}} + \frac {\sqrt {d x^{2} + c} a b c d - \sqrt {d x^{2} + c} a^{2} d^{2}}{2 \, {\left ({\left (d x^{2} + c\right )} b - b c + a d\right )} b^{3}} + \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{4} + 3 \, \sqrt {d x^{2} + c} b^{4} c - 6 \, \sqrt {d x^{2} + c} a b^{3} d}{3 \, b^{6}} \]
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Time = 5.59 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.12 \[ \int \frac {x^3 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\frac {{\left (d\,x^2+c\right )}^{3/2}}{3\,b^2}-\sqrt {d\,x^2+c}\,\left (\frac {c}{b^2}-\frac {2\,b^2\,c-2\,a\,b\,d}{b^4}\right )-\frac {\left (\frac {a^2\,d^2}{2}-\frac {a\,b\,c\,d}{2}\right )\,\sqrt {d\,x^2+c}}{b^4\,\left (d\,x^2+c\right )-b^4\,c+a\,b^3\,d}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d\,x^2+c}\,\sqrt {a\,d-b\,c}\,\left (5\,a\,d-2\,b\,c\right )}{5\,a^2\,d^2-7\,a\,b\,c\,d+2\,b^2\,c^2}\right )\,\sqrt {a\,d-b\,c}\,\left (5\,a\,d-2\,b\,c\right )}{2\,b^{7/2}} \]
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