Integrand size = 19, antiderivative size = 811 \[ \int \frac {(d+e x)^{5/2}}{\left (a+c x^2\right )^2} \, dx=-\frac {d e \sqrt {d+e x}}{2 a c}-\frac {(a e-c d x) (d+e x)^{3/2}}{2 a c \left (a+c x^2\right )}+\frac {e \left (c^{3/2} d^3+a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c^{3/2} d^3+a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c^{3/2} d^3+a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (c^{3/2} d^3+a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]
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Time = 1.98 (sec) , antiderivative size = 811, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {753, 839, 841, 1183, 648, 632, 212, 642} \[ \int \frac {(d+e x)^{5/2}}{\left (a+c x^2\right )^2} \, dx=-\frac {(a e-c d x) (d+e x)^{3/2}}{2 a c \left (c x^2+a\right )}-\frac {d e \sqrt {d+e x}}{2 a c}+\frac {e \left (c^{3/2} d^3+a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c^{3/2} d^3+a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c^{3/2} d^3+a \sqrt {c} e^2 d-\sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \log \left (\sqrt {c} (d+e x)-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{8 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (c^{3/2} d^3+a \sqrt {c} e^2 d-\sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \log \left (\sqrt {c} (d+e x)+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{8 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 753
Rule 839
Rule 841
Rule 1183
Rubi steps \begin{align*} \text {integral}& = -\frac {(a e-c d x) (d+e x)^{3/2}}{2 a c \left (a+c x^2\right )}+\frac {\int \frac {\sqrt {d+e x} \left (\frac {1}{2} \left (2 c d^2+3 a e^2\right )-\frac {1}{2} c d e x\right )}{a+c x^2} \, dx}{2 a c} \\ & = -\frac {d e \sqrt {d+e x}}{2 a c}-\frac {(a e-c d x) (d+e x)^{3/2}}{2 a c \left (a+c x^2\right )}+\frac {\int \frac {c d \left (c d^2+2 a e^2\right )+\frac {1}{2} c e \left (c d^2+3 a e^2\right ) x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx}{2 a c^2} \\ & = -\frac {d e \sqrt {d+e x}}{2 a c}-\frac {(a e-c d x) (d+e x)^{3/2}}{2 a c \left (a+c x^2\right )}+\frac {\text {Subst}\left (\int \frac {c d e \left (c d^2+2 a e^2\right )-\frac {1}{2} c d e \left (c d^2+3 a e^2\right )+\frac {1}{2} c e \left (c d^2+3 a e^2\right ) x^2}{c d^2+a e^2-2 c d x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{a c^2} \\ & = -\frac {d e \sqrt {d+e x}}{2 a c}-\frac {(a e-c d x) (d+e x)^{3/2}}{2 a c \left (a+c x^2\right )}+\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c d e \left (c d^2+2 a e^2\right )-\frac {1}{2} c d e \left (c d^2+3 a e^2\right )\right )}{\sqrt [4]{c}}-\left (c d e \left (c d^2+2 a e^2\right )-\frac {1}{2} c d e \left (c d^2+3 a e^2\right )-\frac {1}{2} \sqrt {c} e \sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \sqrt {2} a c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c d e \left (c d^2+2 a e^2\right )-\frac {1}{2} c d e \left (c d^2+3 a e^2\right )\right )}{\sqrt [4]{c}}+\left (c d e \left (c d^2+2 a e^2\right )-\frac {1}{2} c d e \left (c d^2+3 a e^2\right )-\frac {1}{2} \sqrt {c} e \sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \sqrt {2} a c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \\ & = -\frac {d e \sqrt {d+e x}}{2 a c}-\frac {(a e-c d x) (d+e x)^{3/2}}{2 a c \left (a+c x^2\right )}-\frac {\left (e \left (c^{3/2} d^3+a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right )\right ) \text {Subst}\left (\int \frac {-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{8 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (e \left (c^{3/2} d^3+a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right )\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{8 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (e \left (c^{3/2} d^3+a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{8 a c^2 \sqrt {c d^2+a e^2}}+\frac {\left (e \left (c^{3/2} d^3+a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{8 a c^2 \sqrt {c d^2+a e^2}} \\ & = -\frac {d e \sqrt {d+e x}}{2 a c}-\frac {(a e-c d x) (d+e x)^{3/2}}{2 a c \left (a+c x^2\right )}-\frac {e \left (c^{3/2} d^3+a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (c^{3/2} d^3+a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\left (e \left (c^{3/2} d^3+a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-x^2} \, dx,x,-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{4 a c^2 \sqrt {c d^2+a e^2}}-\frac {\left (e \left (c^{3/2} d^3+a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-x^2} \, dx,x,\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{4 a c^2 \sqrt {c d^2+a e^2}} \\ & = -\frac {d e \sqrt {d+e x}}{2 a c}-\frac {(a e-c d x) (d+e x)^{3/2}}{2 a c \left (a+c x^2\right )}+\frac {e \left (c^{3/2} d^3+a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}-\sqrt {2} \sqrt {d+e x}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c^{3/2} d^3+a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+\sqrt {2} \sqrt {d+e x}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c^{3/2} d^3+a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (c^{3/2} d^3+a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.78 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.35 \[ \int \frac {(d+e x)^{5/2}}{\left (a+c x^2\right )^2} \, dx=\frac {\frac {2 \sqrt {a} c \sqrt {d+e x} \left (c d^2 x-a e (2 d+e x)\right )}{a+c x^2}-i \sqrt {-c d-i \sqrt {a} \sqrt {c} e} \left (2 c d^2-i \sqrt {a} \sqrt {c} d e+3 a e^2\right ) \arctan \left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )+i \sqrt {-c d+i \sqrt {a} \sqrt {c} e} \left (2 c d^2+i \sqrt {a} \sqrt {c} d e+3 a e^2\right ) \arctan \left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{4 a^{3/2} c^2} \]
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Time = 2.75 (sec) , antiderivative size = 862, normalized size of antiderivative = 1.06
method | result | size |
pseudoelliptic | \(-\frac {-\frac {\sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}\, \left (\left (-d \left (c^{\frac {7}{2}} x^{2}+a \,c^{\frac {5}{2}}\right ) \sqrt {e^{2} a +c \,d^{2}}+3 c^{2} \left (e^{2} a +\frac {c \,d^{2}}{3}\right ) \left (c \,x^{2}+a \right )\right ) \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+\left (\left (c^{\frac {9}{2}} x^{2}+a \,c^{\frac {7}{2}}\right ) d \sqrt {e^{2} a +c \,d^{2}}-3 c^{3} \left (e^{2} a +\frac {c \,d^{2}}{3}\right ) \left (c \,x^{2}+a \right )\right ) d \right ) \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}\, \ln \left (\left (e x +d \right ) \sqrt {c}-\sqrt {e x +d}\, \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}+\sqrt {e^{2} a +c \,d^{2}}\right )}{4}+\frac {\sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}\, \left (\left (-d \left (c^{\frac {7}{2}} x^{2}+a \,c^{\frac {5}{2}}\right ) \sqrt {e^{2} a +c \,d^{2}}+3 c^{2} \left (e^{2} a +\frac {c \,d^{2}}{3}\right ) \left (c \,x^{2}+a \right )\right ) \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+\left (\left (c^{\frac {9}{2}} x^{2}+a \,c^{\frac {7}{2}}\right ) d \sqrt {e^{2} a +c \,d^{2}}-3 c^{3} \left (e^{2} a +\frac {c \,d^{2}}{3}\right ) \left (c \,x^{2}+a \right )\right ) d \right ) \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}\, \ln \left (\left (e x +d \right ) \sqrt {c}+\sqrt {e x +d}\, \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}+\sqrt {e^{2} a +c \,d^{2}}\right )}{4}+\left (-2 \sqrt {e x +d}\, \left (-2 a e \left (\frac {e x}{2}+d \right ) c^{\frac {7}{2}}+x \,c^{\frac {9}{2}} d^{2}\right ) \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}+\left (\left (c^{\frac {9}{2}} x^{2}+a \,c^{\frac {7}{2}}\right ) d \sqrt {e^{2} a +c \,d^{2}}+3 c^{3} \left (e^{2} a +\frac {c \,d^{2}}{3}\right ) \left (c \,x^{2}+a \right )\right ) \left (\arctan \left (\frac {-2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}}{\sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}\right )-\arctan \left (\frac {2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}}{\sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}\right )\right ) e \right ) e a}{4 c^{\frac {9}{2}} \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}\, e \left (c \,x^{2}+a \right ) a^{2}}\) | \(862\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1673\) |
default | \(\text {Expression too large to display}\) | \(1673\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1383 vs. \(2 (657) = 1314\).
Time = 0.34 (sec) , antiderivative size = 1383, normalized size of antiderivative = 1.71 \[ \int \frac {(d+e x)^{5/2}}{\left (a+c x^2\right )^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(d+e x)^{5/2}}{\left (a+c x^2\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {(d+e x)^{5/2}}{\left (a+c x^2\right )^2} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {5}{2}}}{{\left (c x^{2} + a\right )}^{2}} \,d x } \]
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Time = 0.39 (sec) , antiderivative size = 492, normalized size of antiderivative = 0.61 \[ \int \frac {(d+e x)^{5/2}}{\left (a+c x^2\right )^2} \, dx=\frac {{\left (2 \, a c^{4} d^{4} e + 4 \, a^{2} c^{3} d^{2} e^{3} + {\left (c d^{2} e + 3 \, a e^{3}\right )} a^{2} c^{2} e^{2} - {\left (\sqrt {-a c} c^{2} d^{3} e + \sqrt {-a c} a c d e^{3}\right )} {\left | a \right |} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a c^{2} d + \sqrt {a^{2} c^{4} d^{2} - {\left (a c^{2} d^{2} + a^{2} c e^{2}\right )} a c^{2}}}{a c^{2}}}}\right )}{4 \, {\left (a^{2} c^{3} e + \sqrt {-a c} a c^{3} d\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e} {\left | a \right |} {\left | e \right |}} + \frac {{\left (2 \, a c^{4} d^{4} e + 4 \, a^{2} c^{3} d^{2} e^{3} + {\left (c d^{2} e + 3 \, a e^{3}\right )} a^{2} c^{2} e^{2} + {\left (\sqrt {-a c} c^{2} d^{3} e + \sqrt {-a c} a c d e^{3}\right )} {\left | a \right |} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a c^{2} d - \sqrt {a^{2} c^{4} d^{2} - {\left (a c^{2} d^{2} + a^{2} c e^{2}\right )} a c^{2}}}{a c^{2}}}}\right )}{4 \, {\left (a^{2} c^{3} e - \sqrt {-a c} a c^{3} d\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e} {\left | a \right |} {\left | e \right |}} + \frac {{\left (e x + d\right )}^{\frac {3}{2}} c d^{2} e - \sqrt {e x + d} c d^{3} e - {\left (e x + d\right )}^{\frac {3}{2}} a e^{3} - \sqrt {e x + d} a d e^{3}}{2 \, {\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} + a e^{2}\right )} a c} \]
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Time = 0.54 (sec) , antiderivative size = 2031, normalized size of antiderivative = 2.50 \[ \int \frac {(d+e x)^{5/2}}{\left (a+c x^2\right )^2} \, dx=\text {Too large to display} \]
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