Integrand size = 19, antiderivative size = 689 \[ \int \frac {(d+e x)^{3/2}}{a+c x^2} \, dx=\frac {2 e \sqrt {d+e x}}{c}-\frac {e \left (c d^2+a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}+\frac {e \left (c d^2+a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}+\frac {e \left (c d^2+a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {e \left (c d^2+a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]
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Time = 1.00 (sec) , antiderivative size = 689, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {718, 841, 1183, 648, 632, 212, 642} \[ \int \frac {(d+e x)^{3/2}}{a+c x^2} \, dx=-\frac {e \left (-2 \sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+c d^2\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {2} c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}+\frac {e \left (-2 \sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+c d^2\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {2} c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}+\frac {e \left (2 \sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+c d^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}-\frac {e \left (2 \sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+c d^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {2 e \sqrt {d+e x}}{c} \]
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 718
Rule 841
Rule 1183
Rubi steps \begin{align*} \text {integral}& = \frac {2 e \sqrt {d+e x}}{c}+\frac {\int \frac {c d^2-a e^2+2 c d e x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx}{c} \\ & = \frac {2 e \sqrt {d+e x}}{c}+\frac {2 \text {Subst}\left (\int \frac {-2 c d^2 e+e \left (c d^2-a e^2\right )+2 c d e x^2}{c d^2+a e^2-2 c d x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{c} \\ & = \frac {2 e \sqrt {d+e x}}{c}+\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \left (-2 c d^2 e+e \left (c d^2-a e^2\right )\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}-\left (-2 c d^2 e+e \left (c d^2-a e^2\right )-2 \sqrt {c} d e \sqrt {c d^2+a e^2}\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 )}{\sqrt {2} c^{5/4} \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} \left (-2 c d^2 e+e \left (c d^2-a e^2\right )\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+\left (-2 c d^2 e+e \left (c d^2-a e^2\right )-2 \sqrt {c} d e \sqrt {c d^2+a e^2}\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 )}{\sqrt {2} c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \\ & = \frac {2 e \sqrt {d+e x}}{c}-\frac {\left (e \left (c d^2+a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\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 )}{2 c^{3/2} \sqrt {c d^2+a e^2}}-\frac {\left (e \left (c d^2+a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\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 )}{2 c^{3/2} \sqrt {c d^2+a e^2}}+\frac {\left (e \left (c d^2+a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\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 )}{2 \sqrt {2} c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\left (e \left (c d^2+a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\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 )}{2 \sqrt {2} c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \\ & = \frac {2 e \sqrt {d+e x}}{c}+\frac {e \left (c d^2+a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\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 )}{2 \sqrt {2} c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {e \left (c d^2+a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\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 )}{2 \sqrt {2} c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (e \left (c d^2+a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\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 )}{c^{3/2} \sqrt {c d^2+a e^2}}+\frac {\left (e \left (c d^2+a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\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 )}{c^{3/2} \sqrt {c d^2+a e^2}} \\ & = \frac {2 e \sqrt {d+e x}}{c}-\frac {e \left (c d^2+a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\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 )}{\sqrt {2} c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}+\frac {e \left (c d^2+a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\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 )}{\sqrt {2} c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}+\frac {e \left (c d^2+a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\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 )}{2 \sqrt {2} c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {e \left (c d^2+a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\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 )}{2 \sqrt {2} c^{5/4} \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 = 0.61 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.34 \[ \int \frac {(d+e x)^{3/2}}{a+c x^2} \, dx=\frac {2 e \sqrt {d+e x}+\frac {i \left (\sqrt {c} d+i \sqrt {a} e\right )^2 \arctan \left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {a} \sqrt {-c d-i \sqrt {a} \sqrt {c} e}}-\frac {i \left (\sqrt {c} d-i \sqrt {a} e\right )^2 \arctan \left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{\sqrt {a} \sqrt {-c d+i \sqrt {a} \sqrt {c} e}}}{c} \]
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Time = 2.75 (sec) , antiderivative size = 685, normalized size of antiderivative = 0.99
method | result | size |
pseudoelliptic | \(\frac {\frac {\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}\, \left (\left (\sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}+2 c d \right ) \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c^{2} d^{2}-c^{\frac {3}{2}} \sqrt {e^{2} a +c \,d^{2}}\, d \right ) \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}\, \ln \left (\left (e x +d \right ) \sqrt {c}-\sqrt {e x +d}\, \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}+\sqrt {e^{2} a +c \,d^{2}}\right )}{4}-\frac {\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}\, \left (\left (\sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}+2 c d \right ) \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c^{2} d^{2}-c^{\frac {3}{2}} \sqrt {e^{2} a +c \,d^{2}}\, d \right ) \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}\, \ln \left (\left (e x +d \right ) \sqrt {c}+\sqrt {e x +d}\, \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}+\sqrt {e^{2} a +c \,d^{2}}\right )}{4}+e^{2} \left (2 \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}\, c^{\frac {3}{2}} \sqrt {e x +d}+\left (\arctan \left (\frac {-2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}}{\sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}\right )-\arctan \left (\frac {2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}}{\sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}\right )\right ) \left (-2 c^{2} d +\sqrt {e^{2} a +c \,d^{2}}\, c^{\frac {3}{2}}\right )\right ) a}{c^{\frac {5}{2}} \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}\, a e}\) | \(685\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1244\) |
default | \(\text {Expression too large to display}\) | \(1244\) |
risch | \(\text {Expression too large to display}\) | \(1245\) |
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Time = 0.49 (sec) , antiderivative size = 998, normalized size of antiderivative = 1.45 \[ \int \frac {(d+e x)^{3/2}}{a+c x^2} \, dx=\frac {c \sqrt {-\frac {c d^{3} - 3 \, a d e^{2} + a c^{2} \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}} \log \left (-{\left (3 \, c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} \sqrt {e x + d} + {\left (3 \, a c^{2} d^{2} e^{2} - a^{2} c e^{4} + a c^{4} d \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}\right )} \sqrt {-\frac {c d^{3} - 3 \, a d e^{2} + a c^{2} \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}}\right ) - c \sqrt {-\frac {c d^{3} - 3 \, a d e^{2} + a c^{2} \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}} \log \left (-{\left (3 \, c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} \sqrt {e x + d} - {\left (3 \, a c^{2} d^{2} e^{2} - a^{2} c e^{4} + a c^{4} d \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}\right )} \sqrt {-\frac {c d^{3} - 3 \, a d e^{2} + a c^{2} \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}}\right ) + c \sqrt {-\frac {c d^{3} - 3 \, a d e^{2} - a c^{2} \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}} \log \left (-{\left (3 \, c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} \sqrt {e x + d} + {\left (3 \, a c^{2} d^{2} e^{2} - a^{2} c e^{4} - a c^{4} d \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}\right )} \sqrt {-\frac {c d^{3} - 3 \, a d e^{2} - a c^{2} \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}}\right ) - c \sqrt {-\frac {c d^{3} - 3 \, a d e^{2} - a c^{2} \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}} \log \left (-{\left (3 \, c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} \sqrt {e x + d} - {\left (3 \, a c^{2} d^{2} e^{2} - a^{2} c e^{4} - a c^{4} d \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}\right )} \sqrt {-\frac {c d^{3} - 3 \, a d e^{2} - a c^{2} \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}}\right ) + 4 \, \sqrt {e x + d} e}{2 \, c} \]
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\[ \int \frac {(d+e x)^{3/2}}{a+c x^2} \, dx=\int \frac {\left (d + e x\right )^{\frac {3}{2}}}{a + c x^{2}}\, dx \]
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\[ \int \frac {(d+e x)^{3/2}}{a+c x^2} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{c x^{2} + a} \,d x } \]
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Time = 0.32 (sec) , antiderivative size = 316, normalized size of antiderivative = 0.46 \[ \int \frac {(d+e x)^{3/2}}{a+c x^2} \, dx=\frac {2 \, \sqrt {e x + d} e}{c} - \frac {{\left (\sqrt {-a c} c^{3} d^{3} e + \sqrt {-a c} a c^{2} d e^{3} - {\left (a c^{2} d^{2} e + a^{2} c e^{3}\right )} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{2} d + \sqrt {c^{4} d^{2} - {\left (c^{2} d^{2} + a c e^{2}\right )} c^{2}}}{c^{2}}}}\right )}{{\left (a c^{3} d - \sqrt {-a c} a c^{2} e\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e} {\left | e \right |}} + \frac {{\left (\sqrt {-a c} c^{3} d^{3} e + \sqrt {-a c} a c^{2} d e^{3} + {\left (a c^{2} d^{2} e + a^{2} c e^{3}\right )} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{2} d - \sqrt {c^{4} d^{2} - {\left (c^{2} d^{2} + a c e^{2}\right )} c^{2}}}{c^{2}}}}\right )}{{\left (a c^{3} d + \sqrt {-a c} a c^{2} e\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e} {\left | e \right |}} \]
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Time = 9.92 (sec) , antiderivative size = 1625, normalized size of antiderivative = 2.36 \[ \int \frac {(d+e x)^{3/2}}{a+c x^2} \, dx=\text {Too large to display} \]
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