Integrand size = 26, antiderivative size = 187 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x \sqrt {c+d x^2}} \, dx=-\frac {b (3 b c-7 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 d^2}+\frac {b \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 d}-\frac {a^{5/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {c}}+\frac {\sqrt {b} \left (3 b^2 c^2-10 a b c d+15 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{8 d^{5/2}} \]
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Time = 0.18 (sec) , antiderivative size = 187, 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, 104, 159, 163, 65, 223, 212, 95, 214} \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x \sqrt {c+d x^2}} \, dx=-\frac {a^{5/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {c}}+\frac {\sqrt {b} \left (15 a^2 d^2-10 a b c d+3 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{8 d^{5/2}}-\frac {b \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-7 a d)}{8 d^2}+\frac {b \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 d} \]
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Rule 65
Rule 95
Rule 104
Rule 159
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 \sqrt {c+d x}} \, dx,x,x^2\right ) \\ & = \frac {b \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 d}+\frac {\text {Subst}\left (\int \frac {\sqrt {a+b x} \left (2 a^2 d-\frac {1}{2} b (3 b c-7 a d) x\right )}{x \sqrt {c+d x}} \, dx,x,x^2\right )}{4 d} \\ & = -\frac {b (3 b c-7 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 d^2}+\frac {b \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 d}+\frac {\text {Subst}\left (\int \frac {2 a^3 d^2+\frac {1}{4} b \left (3 b^2 c^2-10 a b c d+15 a^2 d^2\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )}{4 d^2} \\ & = -\frac {b (3 b c-7 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 d^2}+\frac {b \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 d}+\frac {1}{2} a^3 \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )+\frac {\left (b \left (3 b^2 c^2-10 a b c d+15 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )}{16 d^2} \\ & = -\frac {b (3 b c-7 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 d^2}+\frac {b \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 d}+a^3 \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2}}\right )+\frac {\left (3 b^2 c^2-10 a b c d+15 a^2 d^2\right ) \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 )}{8 d^2} \\ & = -\frac {b (3 b c-7 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 d^2}+\frac {b \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 d}-\frac {a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {c}}+\frac {\left (3 b^2 c^2-10 a b c d+15 a^2 d^2\right ) \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 )}{8 d^2} \\ & = -\frac {b (3 b c-7 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 d^2}+\frac {b \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 d}-\frac {a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {c}}+\frac {\sqrt {b} \left (3 b^2 c^2-10 a b c d+15 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{8 d^{5/2}} \\ \end{align*}
Time = 1.94 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x \sqrt {c+d x^2}} \, dx=\frac {1}{8} \left (\frac {b \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-3 b c+9 a d+2 b d x^2\right )}{d^2}-\frac {8 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x^2}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c}}+\frac {\sqrt {b} \left (3 b^2 c^2-10 a b c d+15 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {d} \sqrt {a+b x^2}}\right )}{d^{5/2}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(374\) vs. \(2(149)=298\).
Time = 3.04 (sec) , antiderivative size = 375, normalized size of antiderivative = 2.01
method | result | size |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {b^{2} x^{2} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{4 d}+\frac {9 b \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a}{8 d}-\frac {3 b^{2} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, c}{8 d^{2}}+\frac {15 a^{2} b \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right )}{16 \sqrt {b d}}-\frac {5 b^{2} \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) a c}{8 d \sqrt {b d}}+\frac {3 b^{3} \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) c^{2}}{16 d^{2} \sqrt {b d}}-\frac {a^{3} \ln \left (\frac {2 a c +\left (a d +b c \right ) x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{x^{2}}\right )}{2 \sqrt {a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(375\) |
default | \(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (-4 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, b^{2} d \,x^{2}+8 \sqrt {b d}\, \ln \left (\frac {a d \,x^{2}+c b \,x^{2}+2 \sqrt {a c}\, \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}+2 a c}{x^{2}}\right ) a^{3} d^{2}-15 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a^{2} b \,d^{2}+10 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a \,b^{2} c d -3 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, b^{3} c^{2}-18 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, a b d +6 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, b^{2} c \right )}{16 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, d^{2} \sqrt {b d}\, \sqrt {a c}}\) | \(390\) |
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Time = 1.68 (sec) , antiderivative size = 1075, normalized size of antiderivative = 5.75 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x \sqrt {c+d x^2}} \, dx=\left [\frac {8 \, a^{2} d^{2} \sqrt {\frac {a}{c}} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2} - 4 \, {\left (2 \, a c^{2} + {\left (b c^{2} + a c d\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {\frac {a}{c}}}{x^{4}}\right ) + {\left (3 \, b^{2} c^{2} - 10 \, a b c d + 15 \, a^{2} d^{2}\right )} \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, b d^{2} x^{2} + b c d + a d^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {\frac {b}{d}}\right ) + 4 \, {\left (2 \, b^{2} d x^{2} - 3 \, b^{2} c + 9 \, a b d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{32 \, d^{2}}, \frac {4 \, a^{2} d^{2} \sqrt {\frac {a}{c}} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2} - 4 \, {\left (2 \, a c^{2} + {\left (b c^{2} + a c d\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {\frac {a}{c}}}{x^{4}}\right ) - {\left (3 \, b^{2} c^{2} - 10 \, a b c d + 15 \, a^{2} d^{2}\right )} \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{4} + a b c + {\left (b^{2} c + a b d\right )} x^{2}\right )}}\right ) + 2 \, {\left (2 \, b^{2} d x^{2} - 3 \, b^{2} c + 9 \, a b d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{16 \, d^{2}}, \frac {16 \, a^{2} d^{2} \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{4} + a^{2} c + {\left (a b c + a^{2} d\right )} x^{2}\right )}}\right ) + {\left (3 \, b^{2} c^{2} - 10 \, a b c d + 15 \, a^{2} d^{2}\right )} \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, b d^{2} x^{2} + b c d + a d^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {\frac {b}{d}}\right ) + 4 \, {\left (2 \, b^{2} d x^{2} - 3 \, b^{2} c + 9 \, a b d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{32 \, d^{2}}, \frac {8 \, a^{2} d^{2} \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{4} + a^{2} c + {\left (a b c + a^{2} d\right )} x^{2}\right )}}\right ) - {\left (3 \, b^{2} c^{2} - 10 \, a b c d + 15 \, a^{2} d^{2}\right )} \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{4} + a b c + {\left (b^{2} c + a b d\right )} x^{2}\right )}}\right ) + 2 \, {\left (2 \, b^{2} d x^{2} - 3 \, b^{2} c + 9 \, a b d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{16 \, d^{2}}\right ] \]
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\[ \int \frac {\left (a+b x^2\right )^{5/2}}{x \sqrt {c+d x^2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {5}{2}}}{x \sqrt {c + d x^{2}}}\, dx \]
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Exception generated. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x \sqrt {c+d x^2}} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x \sqrt {c+d x^2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x \sqrt {c+d x^2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}}{x\,\sqrt {d\,x^2+c}} \,d x \]
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