Integrand size = 19, antiderivative size = 219 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^2} \, dx=-\frac {5 c \left (8 d \left (c d^2+a e^2\right )-e \left (4 c d^2+3 a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 e^5}-\frac {5 c (4 d-3 e x) \left (a+c x^2\right )^{3/2}}{12 e^3}-\frac {\left (a+c x^2\right )^{5/2}}{e (d+e x)}+\frac {5 \sqrt {c} \left (8 c^2 d^4+12 a c d^2 e^2+3 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 e^6}+\frac {5 c d \left (c d^2+a e^2\right )^{3/2} \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^6} \]
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Time = 0.19 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {747, 829, 858, 223, 212, 739} \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {5 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (3 a^2 e^4+12 a c d^2 e^2+8 c^2 d^4\right )}{8 e^6}+\frac {5 c d \left (a e^2+c d^2\right )^{3/2} \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^6}-\frac {5 c \sqrt {a+c x^2} \left (8 d \left (a e^2+c d^2\right )-e x \left (3 a e^2+4 c d^2\right )\right )}{8 e^5}-\frac {5 c \left (a+c x^2\right )^{3/2} (4 d-3 e x)}{12 e^3}-\frac {\left (a+c x^2\right )^{5/2}}{e (d+e x)} \]
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Rule 212
Rule 223
Rule 739
Rule 747
Rule 829
Rule 858
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+c x^2\right )^{5/2}}{e (d+e x)}+\frac {(5 c) \int \frac {x \left (a+c x^2\right )^{3/2}}{d+e x} \, dx}{e} \\ & = -\frac {5 c (4 d-3 e x) \left (a+c x^2\right )^{3/2}}{12 e^3}-\frac {\left (a+c x^2\right )^{5/2}}{e (d+e x)}+\frac {5 \int \frac {\left (-a c d e+c \left (4 c d^2+3 a e^2\right ) x\right ) \sqrt {a+c x^2}}{d+e x} \, dx}{4 e^3} \\ & = -\frac {5 c \left (8 d \left (c d^2+a e^2\right )-e \left (4 c d^2+3 a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 e^5}-\frac {5 c (4 d-3 e x) \left (a+c x^2\right )^{3/2}}{12 e^3}-\frac {\left (a+c x^2\right )^{5/2}}{e (d+e x)}+\frac {5 \int \frac {-a c^2 d e \left (4 c d^2+5 a e^2\right )+c^2 \left (8 c^2 d^4+12 a c d^2 e^2+3 a^2 e^4\right ) x}{(d+e x) \sqrt {a+c x^2}} \, dx}{8 c e^5} \\ & = -\frac {5 c \left (8 d \left (c d^2+a e^2\right )-e \left (4 c d^2+3 a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 e^5}-\frac {5 c (4 d-3 e x) \left (a+c x^2\right )^{3/2}}{12 e^3}-\frac {\left (a+c x^2\right )^{5/2}}{e (d+e x)}-\frac {\left (5 c d \left (c d^2+a e^2\right )^2\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{e^6}+\frac {\left (5 c \left (8 c^2 d^4+12 a c d^2 e^2+3 a^2 e^4\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{8 e^6} \\ & = -\frac {5 c \left (8 d \left (c d^2+a e^2\right )-e \left (4 c d^2+3 a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 e^5}-\frac {5 c (4 d-3 e x) \left (a+c x^2\right )^{3/2}}{12 e^3}-\frac {\left (a+c x^2\right )^{5/2}}{e (d+e x)}+\frac {\left (5 c d \left (c d^2+a e^2\right )^2\right ) \text {Subst}\left (\int \frac {1}{c d^2+a e^2-x^2} \, dx,x,\frac {a e-c d x}{\sqrt {a+c x^2}}\right )}{e^6}+\frac {\left (5 c \left (8 c^2 d^4+12 a c d^2 e^2+3 a^2 e^4\right )\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{8 e^6} \\ & = -\frac {5 c \left (8 d \left (c d^2+a e^2\right )-e \left (4 c d^2+3 a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 e^5}-\frac {5 c (4 d-3 e x) \left (a+c x^2\right )^{3/2}}{12 e^3}-\frac {\left (a+c x^2\right )^{5/2}}{e (d+e x)}+\frac {5 \sqrt {c} \left (8 c^2 d^4+12 a c d^2 e^2+3 a^2 e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 e^6}+\frac {5 c d \left (c d^2+a e^2\right )^{3/2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^6} \\ \end{align*}
Time = 1.14 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^2} \, dx=-\frac {\frac {e \sqrt {a+c x^2} \left (24 a^2 e^4+a c e^2 \left (160 d^2+85 d e x-27 e^2 x^2\right )+2 c^2 \left (60 d^4+30 d^3 e x-10 d^2 e^2 x^2+5 d e^3 x^3-3 e^4 x^4\right )\right )}{d+e x}-240 c d \left (-c d^2-a e^2\right )^{3/2} \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )+15 \sqrt {c} \left (8 c^2 d^4+12 a c d^2 e^2+3 a^2 e^4\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{24 e^6} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(529\) vs. \(2(195)=390\).
Time = 2.01 (sec) , antiderivative size = 530, normalized size of antiderivative = 2.42
method | result | size |
risch | \(-\frac {c \left (-6 c \,x^{3} e^{3}+16 c d \,x^{2} e^{2}-27 a \,e^{3} x -36 c \,d^{2} e x +112 a d \,e^{2}+96 d^{3} c \right ) \sqrt {c \,x^{2}+a}}{24 e^{5}}+\frac {\frac {5 \sqrt {c}\, \left (3 a^{2} e^{4}+12 a c \,d^{2} e^{2}+8 c^{2} d^{4}\right ) \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{e}+\frac {48 c d \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}+\frac {\left (8 e^{6} a^{3}+24 d^{2} e^{4} a^{2} c +24 d^{4} e^{2} c^{2} a +8 c^{3} d^{6}\right ) \left (-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{\left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {c d e \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{e^{3}}}{8 e^{5}}\) | \(530\) |
default | \(\text {Expression too large to display}\) | \(1309\) |
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Time = 8.79 (sec) , antiderivative size = 1372, normalized size of antiderivative = 6.26 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^2} \, dx=\text {Too large to display} \]
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\[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^2} \, dx=\int \frac {\left (a + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^2} \, dx=\text {Exception raised: ValueError} \]
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Timed out. \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^2} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{5/2}}{{\left (d+e\,x\right )}^2} \,d x \]
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