Integrand size = 26, antiderivative size = 192 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^5 \sqrt {c+d x^2}} \, dx=-\frac {a (7 b c-3 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 c^2 x^2}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 c x^4}-\frac {\sqrt {a} \left (15 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 c^{5/2}}+\frac {b^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{\sqrt {d}} \]
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Time = 0.15 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {457, 100, 154, 163, 65, 223, 212, 95, 214} \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^5 \sqrt {c+d x^2}} \, dx=-\frac {\sqrt {a} \left (3 a^2 d^2-10 a b c d+15 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 c^{5/2}}+\frac {b^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{\sqrt {d}}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2} (7 b c-3 a d)}{8 c^2 x^2}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 c x^4} \]
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Rule 65
Rule 95
Rule 100
Rule 154
Rule 163
Rule 212
Rule 214
Rule 223
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^{5/2}}{x^3 \sqrt {c+d x}} \, dx,x,x^2\right ) \\ & = -\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 c x^4}-\frac {\text {Subst}\left (\int \frac {\sqrt {a+b x} \left (-\frac {1}{2} a (7 b c-3 a d)-2 b^2 c x\right )}{x^2 \sqrt {c+d x}} \, dx,x,x^2\right )}{4 c} \\ & = -\frac {a (7 b c-3 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 c^2 x^2}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 c x^4}-\frac {\text {Subst}\left (\int \frac {-\frac {1}{4} a \left (15 b^2 c^2-10 a b c d+3 a^2 d^2\right )-2 b^3 c^2 x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )}{4 c^2} \\ & = -\frac {a (7 b c-3 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 c^2 x^2}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 c x^4}+\frac {1}{2} b^3 \text {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )+\frac {\left (a \left (15 b^2 c^2-10 a b c d+3 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )}{16 c^2} \\ & = -\frac {a (7 b c-3 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 c^2 x^2}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 c x^4}+b^2 \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x^2}\right )+\frac {\left (a \left (15 b^2 c^2-10 a b c d+3 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2}}\right )}{8 c^2} \\ & = -\frac {a (7 b c-3 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 c^2 x^2}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 c x^4}-\frac {\sqrt {a} \left (15 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 c^{5/2}}+b^2 \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2}}\right ) \\ & = -\frac {a (7 b c-3 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 c^2 x^2}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 c x^4}-\frac {\sqrt {a} \left (15 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 c^{5/2}}+\frac {b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{\sqrt {d}} \\ \end{align*}
Time = 3.62 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^5 \sqrt {c+d x^2}} \, dx=\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-2 a c-9 b c x^2+3 a d x^2\right )}{8 c^2 x^4}+\frac {(b c-a d)^{5/2} \left (\frac {b \left (c+d x^2\right )}{b c-a d}\right )^{5/2} \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b c-a d}}\right )}{\sqrt {d} \left (c+d x^2\right )^{5/2}}-\frac {\sqrt {a} \left (15 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 c^{5/2}} \]
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Time = 3.21 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.20
method | result | size |
risch | \(-\frac {a \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (-3 a d \,x^{2}+9 c b \,x^{2}+2 a c \right )}{8 c^{2} x^{4}}+\frac {\left (\frac {4 c^{2} b^{3} \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right )}{\sqrt {b d}}-\frac {a \left (3 a^{2} d^{2}-10 a b c d +15 b^{2} c^{2}\right ) \ln \left (\frac {2 a c +\left (a d +b c \right ) x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{x^{2}}\right )}{2 \sqrt {a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{8 c^{2} \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(230\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {b^{3} \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right )}{2 \sqrt {b d}}-\frac {a^{2} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{4 c \,x^{4}}+\frac {3 a^{2} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, d}{8 c^{2} x^{2}}-\frac {9 a \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, b}{8 c \,x^{2}}-\frac {3 a^{3} \ln \left (\frac {2 a c +\left (a d +b c \right ) x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{x^{2}}\right ) d^{2}}{16 c^{2} \sqrt {a c}}+\frac {5 a^{2} \ln \left (\frac {2 a c +\left (a d +b c \right ) x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{x^{2}}\right ) d b}{8 c \sqrt {a c}}-\frac {15 a \,b^{2} \ln \left (\frac {2 a c +\left (a d +b c \right ) x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{x^{2}}\right )}{16 \sqrt {a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(391\) |
default | \(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (3 \ln \left (\frac {a d \,x^{2}+c b \,x^{2}+2 \sqrt {a c}\, \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}+2 a c}{x^{2}}\right ) a^{3} d^{2} x^{4} \sqrt {b d}-10 \ln \left (\frac {a d \,x^{2}+c b \,x^{2}+2 \sqrt {a c}\, \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}+2 a c}{x^{2}}\right ) a^{2} b c d \,x^{4} \sqrt {b d}+15 \ln \left (\frac {a d \,x^{2}+c b \,x^{2}+2 \sqrt {a c}\, \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}+2 a c}{x^{2}}\right ) a \,b^{2} c^{2} x^{4} \sqrt {b d}-8 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{3} c^{2} x^{4} \sqrt {a c}-6 a^{2} d \,x^{2} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, \sqrt {a c}+18 a b c \,x^{2} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, \sqrt {a c}+4 a^{2} c \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, \sqrt {a c}\right )}{16 c^{2} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, x^{4} \sqrt {a c}\, \sqrt {b d}}\) | \(408\) |
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Time = 1.33 (sec) , antiderivative size = 1123, normalized size of antiderivative = 5.85 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^5 \sqrt {c+d x^2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^5 \sqrt {c+d x^2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {5}{2}}}{x^{5} \sqrt {c + d x^{2}}}\, dx \]
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Exception generated. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^5 \sqrt {c+d x^2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1175 vs. \(2 (154) = 308\).
Time = 0.40 (sec) , antiderivative size = 1175, normalized size of antiderivative = 6.12 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^5 \sqrt {c+d x^2}} \, dx=-\frac {{\left (\frac {4 \, \sqrt {b d} b^{2} \log \left ({\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 )}{d} + \frac {{\left (15 \, \sqrt {b d} a b^{3} c^{2} - 10 \, \sqrt {b d} a^{2} b^{2} c d + 3 \, \sqrt {b d} a^{3} b d^{2}\right )} \arctan \left (-\frac {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}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} b c^{2}} + \frac {2 \, {\left (9 \, \sqrt {b d} a b^{9} c^{5} - 39 \, \sqrt {b d} a^{2} b^{8} c^{4} d + 66 \, \sqrt {b d} a^{3} b^{7} c^{3} d^{2} - 54 \, \sqrt {b d} a^{4} b^{6} c^{2} d^{3} + 21 \, \sqrt {b d} a^{5} b^{5} c d^{4} - 3 \, \sqrt {b d} a^{6} b^{4} d^{5} - 27 \, \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^{7} c^{4} + 40 \, \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^{6} c^{3} d + 10 \, \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^{3} b^{5} c^{2} d^{2} - 32 \, \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^{4} b^{4} c d^{3} + 9 \, \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^{5} b^{3} d^{4} + 27 \, \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^{5} c^{3} + 9 \, \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^{2} b^{4} c^{2} d + 21 \, \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^{3} b^{3} c d^{2} - 9 \, \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^{4} b^{2} d^{3} - 9 \, \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} a b^{3} c^{2} - 10 \, \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} a^{2} b^{2} c d + 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 )}^{6} a^{3} b d^{2}\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\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^{2} c - 2 \, {\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 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}\right )}^{2} c^{2}}\right )} b}{8 \, {\left | b \right |}} \]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^5 \sqrt {c+d x^2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}}{x^5\,\sqrt {d\,x^2+c}} \,d x \]
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