Integrand size = 24, antiderivative size = 108 \[ \int \frac {x^3 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {c (b c-a d)^2}{d^4 \sqrt {c+d x^2}}+\frac {(b c-a d) (3 b c-a d) \sqrt {c+d x^2}}{d^4}-\frac {b (3 b c-2 a d) \left (c+d x^2\right )^{3/2}}{3 d^4}+\frac {b^2 \left (c+d x^2\right )^{5/2}}{5 d^4} \]
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Time = 0.06 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {457, 78} \[ \int \frac {x^3 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=-\frac {b \left (c+d x^2\right )^{3/2} (3 b c-2 a d)}{3 d^4}+\frac {\sqrt {c+d x^2} (b c-a d) (3 b c-a d)}{d^4}+\frac {c (b c-a d)^2}{d^4 \sqrt {c+d x^2}}+\frac {b^2 \left (c+d x^2\right )^{5/2}}{5 d^4} \]
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Rule 78
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x (a+b x)^2}{(c+d x)^{3/2}} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (-\frac {c (b c-a d)^2}{d^3 (c+d x)^{3/2}}+\frac {(b c-a d) (3 b c-a d)}{d^3 \sqrt {c+d x}}-\frac {b (3 b c-2 a d) \sqrt {c+d x}}{d^3}+\frac {b^2 (c+d x)^{3/2}}{d^3}\right ) \, dx,x,x^2\right ) \\ & = \frac {c (b c-a d)^2}{d^4 \sqrt {c+d x^2}}+\frac {(b c-a d) (3 b c-a d) \sqrt {c+d x^2}}{d^4}-\frac {b (3 b c-2 a d) \left (c+d x^2\right )^{3/2}}{3 d^4}+\frac {b^2 \left (c+d x^2\right )^{5/2}}{5 d^4} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.90 \[ \int \frac {x^3 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {15 a^2 d^2 \left (2 c+d x^2\right )+10 a b d \left (-8 c^2-4 c d x^2+d^2 x^4\right )+3 b^2 \left (16 c^3+8 c^2 d x^2-2 c d^2 x^4+d^3 x^6\right )}{15 d^4 \sqrt {c+d x^2}} \]
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Time = 2.91 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.85
method | result | size |
pseudoelliptic | \(\frac {3 \left (d^{3} x^{6}-2 c \,d^{2} x^{4}+8 c^{2} d \,x^{2}+16 c^{3}\right ) b^{2}-80 d a \left (-\frac {1}{8} d^{2} x^{4}+\frac {1}{2} c d \,x^{2}+c^{2}\right ) b +30 d^{2} a^{2} \left (\frac {d \,x^{2}}{2}+c \right )}{15 \sqrt {d \,x^{2}+c}\, d^{4}}\) | \(92\) |
risch | \(\frac {\left (3 b^{2} d^{2} x^{4}+10 x^{2} a b \,d^{2}-9 x^{2} b^{2} c d +15 a^{2} d^{2}-50 a b c d +33 b^{2} c^{2}\right ) \sqrt {d \,x^{2}+c}}{15 d^{4}}+\frac {c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{\sqrt {d \,x^{2}+c}\, d^{4}}\) | \(105\) |
gosper | \(\frac {3 b^{2} d^{3} x^{6}+10 a b \,d^{3} x^{4}-6 b^{2} c \,d^{2} x^{4}+15 a^{2} d^{3} x^{2}-40 a b c \,d^{2} x^{2}+24 b^{2} c^{2} d \,x^{2}+30 c \,a^{2} d^{2}-80 a b \,c^{2} d +48 b^{2} c^{3}}{15 \sqrt {d \,x^{2}+c}\, d^{4}}\) | \(108\) |
trager | \(\frac {3 b^{2} d^{3} x^{6}+10 a b \,d^{3} x^{4}-6 b^{2} c \,d^{2} x^{4}+15 a^{2} d^{3} x^{2}-40 a b c \,d^{2} x^{2}+24 b^{2} c^{2} d \,x^{2}+30 c \,a^{2} d^{2}-80 a b \,c^{2} d +48 b^{2} c^{3}}{15 \sqrt {d \,x^{2}+c}\, d^{4}}\) | \(108\) |
default | \(b^{2} \left (\frac {x^{6}}{5 d \sqrt {d \,x^{2}+c}}-\frac {6 c \left (\frac {x^{4}}{3 d \sqrt {d \,x^{2}+c}}-\frac {4 c \left (\frac {x^{2}}{d \sqrt {d \,x^{2}+c}}+\frac {2 c}{d^{2} \sqrt {d \,x^{2}+c}}\right )}{3 d}\right )}{5 d}\right )+a^{2} \left (\frac {x^{2}}{d \sqrt {d \,x^{2}+c}}+\frac {2 c}{d^{2} \sqrt {d \,x^{2}+c}}\right )+2 a b \left (\frac {x^{4}}{3 d \sqrt {d \,x^{2}+c}}-\frac {4 c \left (\frac {x^{2}}{d \sqrt {d \,x^{2}+c}}+\frac {2 c}{d^{2} \sqrt {d \,x^{2}+c}}\right )}{3 d}\right )\) | \(182\) |
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Time = 0.26 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.06 \[ \int \frac {x^3 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {{\left (3 \, b^{2} d^{3} x^{6} + 48 \, b^{2} c^{3} - 80 \, a b c^{2} d + 30 \, a^{2} c d^{2} - 2 \, {\left (3 \, b^{2} c d^{2} - 5 \, a b d^{3}\right )} x^{4} + {\left (24 \, b^{2} c^{2} d - 40 \, a b c d^{2} + 15 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{15 \, {\left (d^{5} x^{2} + c d^{4}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (97) = 194\).
Time = 0.32 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.19 \[ \int \frac {x^3 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\begin {cases} \frac {2 a^{2} c}{d^{2} \sqrt {c + d x^{2}}} + \frac {a^{2} x^{2}}{d \sqrt {c + d x^{2}}} - \frac {16 a b c^{2}}{3 d^{3} \sqrt {c + d x^{2}}} - \frac {8 a b c x^{2}}{3 d^{2} \sqrt {c + d x^{2}}} + \frac {2 a b x^{4}}{3 d \sqrt {c + d x^{2}}} + \frac {16 b^{2} c^{3}}{5 d^{4} \sqrt {c + d x^{2}}} + \frac {8 b^{2} c^{2} x^{2}}{5 d^{3} \sqrt {c + d x^{2}}} - \frac {2 b^{2} c x^{4}}{5 d^{2} \sqrt {c + d x^{2}}} + \frac {b^{2} x^{6}}{5 d \sqrt {c + d x^{2}}} & \text {for}\: d \neq 0 \\\frac {\frac {a^{2} x^{4}}{4} + \frac {a b x^{6}}{3} + \frac {b^{2} x^{8}}{8}}{c^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.67 \[ \int \frac {x^3 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {b^{2} x^{6}}{5 \, \sqrt {d x^{2} + c} d} - \frac {2 \, b^{2} c x^{4}}{5 \, \sqrt {d x^{2} + c} d^{2}} + \frac {2 \, a b x^{4}}{3 \, \sqrt {d x^{2} + c} d} + \frac {8 \, b^{2} c^{2} x^{2}}{5 \, \sqrt {d x^{2} + c} d^{3}} - \frac {8 \, a b c x^{2}}{3 \, \sqrt {d x^{2} + c} d^{2}} + \frac {a^{2} x^{2}}{\sqrt {d x^{2} + c} d} + \frac {16 \, b^{2} c^{3}}{5 \, \sqrt {d x^{2} + c} d^{4}} - \frac {16 \, a b c^{2}}{3 \, \sqrt {d x^{2} + c} d^{3}} + \frac {2 \, a^{2} c}{\sqrt {d x^{2} + c} d^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.38 \[ \int \frac {x^3 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}}{\sqrt {d x^{2} + c} d^{4}} + \frac {3 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} d^{16} - 15 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c d^{16} + 45 \, \sqrt {d x^{2} + c} b^{2} c^{2} d^{16} + 10 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d^{17} - 60 \, \sqrt {d x^{2} + c} a b c d^{17} + 15 \, \sqrt {d x^{2} + c} a^{2} d^{18}}{15 \, d^{20}} \]
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Time = 5.64 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.99 \[ \int \frac {x^3 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {30\,a^2\,c\,d^2+15\,a^2\,d^3\,x^2-80\,a\,b\,c^2\,d-40\,a\,b\,c\,d^2\,x^2+10\,a\,b\,d^3\,x^4+48\,b^2\,c^3+24\,b^2\,c^2\,d\,x^2-6\,b^2\,c\,d^2\,x^4+3\,b^2\,d^3\,x^6}{15\,d^4\,\sqrt {d\,x^2+c}} \]
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