Integrand size = 26, antiderivative size = 138 \[ \int \frac {x^3}{\left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}} \, dx=\frac {a \sqrt {c+d x^2}}{5 b (b c-a d) \left (a+b x^2\right )^{5/2}}-\frac {(5 b c-a d) \sqrt {c+d x^2}}{15 b (b c-a d)^2 \left (a+b x^2\right )^{3/2}}+\frac {2 d (5 b c-a d) \sqrt {c+d x^2}}{15 b (b c-a d)^3 \sqrt {a+b x^2}} \]
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Time = 0.08 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {457, 79, 47, 37} \[ \int \frac {x^3}{\left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}} \, dx=\frac {a \sqrt {c+d x^2}}{5 b \left (a+b x^2\right )^{5/2} (b c-a d)}+\frac {2 d \sqrt {c+d x^2} (5 b c-a d)}{15 b \sqrt {a+b x^2} (b c-a d)^3}-\frac {\sqrt {c+d x^2} (5 b c-a d)}{15 b \left (a+b x^2\right )^{3/2} (b c-a d)^2} \]
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Rule 37
Rule 47
Rule 79
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x}{(a+b x)^{7/2} \sqrt {c+d x}} \, dx,x,x^2\right ) \\ & = \frac {a \sqrt {c+d x^2}}{5 b (b c-a d) \left (a+b x^2\right )^{5/2}}+\frac {(5 b c-a d) \text {Subst}\left (\int \frac {1}{(a+b x)^{5/2} \sqrt {c+d x}} \, dx,x,x^2\right )}{10 b (b c-a d)} \\ & = \frac {a \sqrt {c+d x^2}}{5 b (b c-a d) \left (a+b x^2\right )^{5/2}}-\frac {(5 b c-a d) \sqrt {c+d x^2}}{15 b (b c-a d)^2 \left (a+b x^2\right )^{3/2}}-\frac {(d (5 b c-a d)) \text {Subst}\left (\int \frac {1}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx,x,x^2\right )}{15 b (b c-a d)^2} \\ & = \frac {a \sqrt {c+d x^2}}{5 b (b c-a d) \left (a+b x^2\right )^{5/2}}-\frac {(5 b c-a d) \sqrt {c+d x^2}}{15 b (b c-a d)^2 \left (a+b x^2\right )^{3/2}}+\frac {2 d (5 b c-a d) \sqrt {c+d x^2}}{15 b (b c-a d)^3 \sqrt {a+b x^2}} \\ \end{align*}
Time = 2.00 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.66 \[ \int \frac {x^3}{\left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {c+d x^2} \left (-5 b^2 c x^2 \left (c-2 d x^2\right )-5 a^2 d \left (-2 c+d x^2\right )-2 a b \left (c^2-13 c d x^2+d^2 x^4\right )\right )}{15 (b c-a d)^3 \left (a+b x^2\right )^{5/2}} \]
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Time = 3.15 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.87
method | result | size |
default | \(-\frac {\sqrt {d \,x^{2}+c}\, \left (-2 a b \,d^{2} x^{4}+10 b^{2} c d \,x^{4}-5 a^{2} d^{2} x^{2}+26 a b c d \,x^{2}-5 b^{2} c^{2} x^{2}+10 a^{2} c d -2 b \,c^{2} a \right )}{15 \left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (a d -b c \right ) \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(120\) |
gosper | \(-\frac {\sqrt {d \,x^{2}+c}\, \left (-2 a b \,d^{2} x^{4}+10 b^{2} c d \,x^{4}-5 a^{2} d^{2} x^{2}+26 a b c d \,x^{2}-5 b^{2} c^{2} x^{2}+10 a^{2} c d -2 b \,c^{2} a \right )}{15 \left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(125\) |
elliptic | \(-\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {d \,x^{2}+c}\, \left (-2 a b \,d^{2} x^{4}+10 b^{2} c d \,x^{4}-5 a^{2} d^{2} x^{2}+26 a b c d \,x^{2}-5 b^{2} c^{2} x^{2}+10 a^{2} c d -2 b \,c^{2} a \right )}{15 \sqrt {b \,x^{2}+a}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right ) \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(186\) |
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Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (120) = 240\).
Time = 0.44 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.95 \[ \int \frac {x^3}{\left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}} \, dx=\frac {{\left (2 \, {\left (5 \, b^{2} c d - a b d^{2}\right )} x^{4} - 2 \, a b c^{2} + 10 \, a^{2} c d - {\left (5 \, b^{2} c^{2} - 26 \, a b c d + 5 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{15 \, {\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3} + {\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{6} + 3 \, {\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{4} + 3 \, {\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\right )} x^{2}\right )}} \]
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\[ \int \frac {x^3}{\left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}} \, dx=\int \frac {x^{3}}{\left (a + b x^{2}\right )^{\frac {7}{2}} \sqrt {c + d x^{2}}}\, dx \]
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Exception generated. \[ \int \frac {x^3}{\left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 472 vs. \(2 (120) = 240\).
Time = 0.36 (sec) , antiderivative size = 472, normalized size of antiderivative = 3.42 \[ \int \frac {x^3}{\left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}} \, dx=\frac {4 \, {\left (5 \, \sqrt {b d} b^{8} c^{3} d - 11 \, \sqrt {b d} a b^{7} c^{2} d^{2} + 7 \, \sqrt {b d} a^{2} b^{6} c d^{3} - \sqrt {b d} a^{3} b^{5} d^{4} - 25 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} b^{6} c^{2} d + 30 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a b^{5} c d^{2} - 5 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a^{2} b^{4} d^{3} + 35 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{4} b^{4} c d + 5 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{4} a b^{3} d^{2} - 15 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{6} b^{2} d\right )}}{15 \, {\left (b^{2} c - a b d - {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}\right )}^{5} b {\left | b \right |}} \]
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Time = 6.11 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.64 \[ \int \frac {x^3}{\left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}} \, dx=-\frac {\sqrt {b\,x^2+a}\,\left (\frac {x^2\,\left (5\,a^2\,c\,d^2+24\,a\,b\,c^2\,d-5\,b^2\,c^3\right )}{15\,b^3\,{\left (a\,d-b\,c\right )}^3}+\frac {x^4\,\left (-5\,a^2\,d^3+24\,a\,b\,c\,d^2+5\,b^2\,c^2\,d\right )}{15\,b^3\,{\left (a\,d-b\,c\right )}^3}-\frac {2\,d^2\,x^6\,\left (a\,d-5\,b\,c\right )}{15\,b^2\,{\left (a\,d-b\,c\right )}^3}+\frac {2\,a\,c^2\,\left (5\,a\,d-b\,c\right )}{15\,b^3\,{\left (a\,d-b\,c\right )}^3}\right )}{x^6\,\sqrt {d\,x^2+c}+\frac {a^3\,\sqrt {d\,x^2+c}}{b^3}+\frac {3\,a\,x^4\,\sqrt {d\,x^2+c}}{b}+\frac {3\,a^2\,x^2\,\sqrt {d\,x^2+c}}{b^2}} \]
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