Integrand size = 26, antiderivative size = 89 \[ \int \frac {x^3}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=\frac {a \sqrt {c+d x^2}}{3 b (b c-a d) \left (a+b x^2\right )^{3/2}}-\frac {(3 b c-a d) \sqrt {c+d x^2}}{3 b (b c-a d)^2 \sqrt {a+b x^2}} \]
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Time = 0.06 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {457, 79, 37} \[ \int \frac {x^3}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=\frac {a \sqrt {c+d x^2}}{3 b \left (a+b x^2\right )^{3/2} (b c-a d)}-\frac {\sqrt {c+d x^2} (3 b c-a d)}{3 b \sqrt {a+b x^2} (b c-a d)^2} \]
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Rule 37
Rule 79
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x}{(a+b x)^{5/2} \sqrt {c+d x}} \, dx,x,x^2\right ) \\ & = \frac {a \sqrt {c+d x^2}}{3 b (b c-a d) \left (a+b x^2\right )^{3/2}}+\frac {(3 b c-a d) \text {Subst}\left (\int \frac {1}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx,x,x^2\right )}{6 b (b c-a d)} \\ & = \frac {a \sqrt {c+d x^2}}{3 b (b c-a d) \left (a+b x^2\right )^{3/2}}-\frac {(3 b c-a d) \sqrt {c+d x^2}}{3 b (b c-a d)^2 \sqrt {a+b x^2}} \\ \end{align*}
Time = 1.85 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.61 \[ \int \frac {x^3}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {c+d x^2} \left (-2 a c-3 b c x^2+a d x^2\right )}{3 (b c-a d)^2 \left (a+b x^2\right )^{3/2}} \]
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Time = 3.06 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.56
method | result | size |
default | \(-\frac {\sqrt {d \,x^{2}+c}\, \left (-a d \,x^{2}+3 c b \,x^{2}+2 a c \right )}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (a d -b c \right )^{2}}\) | \(50\) |
gosper | \(-\frac {\sqrt {d \,x^{2}+c}\, \left (-a d \,x^{2}+3 c b \,x^{2}+2 a c \right )}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(63\) |
elliptic | \(-\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {d \,x^{2}+c}\, \left (-a d \,x^{2}+3 c b \,x^{2}+2 a c \right )}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(104\) |
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Time = 0.30 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.44 \[ \int \frac {x^3}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=-\frac {{\left ({\left (3 \, b c - a d\right )} x^{2} + 2 \, a c\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4} + 2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2}\right )}} \]
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\[ \int \frac {x^3}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=\int \frac {x^{3}}{\left (a + b x^{2}\right )^{\frac {5}{2}} \sqrt {c + d x^{2}}}\, dx \]
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Exception generated. \[ \int \frac {x^3}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (77) = 154\).
Time = 0.33 (sec) , antiderivative size = 214, normalized size of antiderivative = 2.40 \[ \int \frac {x^3}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {b d} b^{5} c^{2} - 4 \, \sqrt {b d} a b^{4} c d + \sqrt {b d} a^{2} b^{3} d^{2} - 6 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} b^{3} c + 3 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{4} b\right )}}{3 \, {\left (b^{2} c - a b d - {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}\right )}^{3} b {\left | b \right |}} \]
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Time = 6.16 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.56 \[ \int \frac {x^3}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=-\frac {\sqrt {b\,x^2+a}\,\left (\frac {2\,a\,c^2}{3\,b^2\,{\left (a\,d-b\,c\right )}^2}+\frac {x^2\,\left (3\,b\,c^2+a\,d\,c\right )}{3\,b^2\,{\left (a\,d-b\,c\right )}^2}-\frac {x^4\,\left (a\,d^2-3\,b\,c\,d\right )}{3\,b^2\,{\left (a\,d-b\,c\right )}^2}\right )}{x^4\,\sqrt {d\,x^2+c}+\frac {a^2\,\sqrt {d\,x^2+c}}{b^2}+\frac {2\,a\,x^2\,\sqrt {d\,x^2+c}}{b}} \]
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