Integrand size = 19, antiderivative size = 846 \[ \int \frac {(d+e x)^{5/2}}{\left (a+c x^2\right )^3} \, dx=-\frac {(a e-c d x) (d+e x)^{3/2}}{4 a c \left (a+c x^2\right )^2}-\frac {3 \sqrt {d+e x} \left (a d e-\left (2 c d^2+a e^2\right ) x\right )}{16 a^2 c \left (a+c x^2\right )}+\frac {3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (2 c d^2+a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{32 \sqrt {2} a^2 c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (2 c d^2+a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{32 \sqrt {2} a^2 c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (2 c d^2+a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{64 \sqrt {2} a^2 c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (2 c d^2+a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{64 \sqrt {2} a^2 c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]
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Time = 2.07 (sec) , antiderivative size = 846, 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, 835, 841, 1183, 648, 632, 212, 642} \[ \int \frac {(d+e x)^{5/2}}{\left (a+c x^2\right )^3} \, dx=-\frac {(a e-c d x) (d+e x)^{3/2}}{4 a c \left (c x^2+a\right )^2}-\frac {3 \left (a d e-\left (2 c d^2+a e^2\right ) x\right ) \sqrt {d+e x}}{16 a^2 c \left (c x^2+a\right )}+\frac {3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (2 c d^2+a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{32 \sqrt {2} a^2 c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (2 c d^2+a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{32 \sqrt {2} a^2 c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} e^2 d-\sqrt {c d^2+a e^2} \left (2 c d^2+a e^2\right )\right ) \log \left (\sqrt {c} (d+e x)-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{64 \sqrt {2} a^2 c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} e^2 d-\sqrt {c d^2+a e^2} \left (2 c d^2+a e^2\right )\right ) \log \left (\sqrt {c} (d+e x)+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{64 \sqrt {2} a^2 c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 753
Rule 835
Rule 841
Rule 1183
Rubi steps \begin{align*} \text {integral}& = -\frac {(a e-c d x) (d+e x)^{3/2}}{4 a c \left (a+c x^2\right )^2}+\frac {\int \frac {\sqrt {d+e x} \left (\frac {3}{2} \left (2 c d^2+a e^2\right )+\frac {3}{2} c d e x\right )}{\left (a+c x^2\right )^2} \, dx}{4 a c} \\ & = -\frac {(a e-c d x) (d+e x)^{3/2}}{4 a c \left (a+c x^2\right )^2}-\frac {3 \sqrt {d+e x} \left (a d e-\left (2 c d^2+a e^2\right ) x\right )}{16 a^2 c \left (a+c x^2\right )}+\frac {\int \frac {\frac {3}{4} c d \left (4 c d^2+3 a e^2\right )+\frac {3}{4} c e \left (2 c d^2+a e^2\right ) x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx}{8 a^2 c^2} \\ & = -\frac {(a e-c d x) (d+e x)^{3/2}}{4 a c \left (a+c x^2\right )^2}-\frac {3 \sqrt {d+e x} \left (a d e-\left (2 c d^2+a e^2\right ) x\right )}{16 a^2 c \left (a+c x^2\right )}+\frac {\text {Subst}\left (\int \frac {-\frac {3}{4} c d e \left (2 c d^2+a e^2\right )+\frac {3}{4} c d e \left (4 c d^2+3 a e^2\right )+\frac {3}{4} c e \left (2 c d^2+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 )}{4 a^2 c^2} \\ & = -\frac {(a e-c d x) (d+e x)^{3/2}}{4 a c \left (a+c x^2\right )^2}-\frac {3 \sqrt {d+e x} \left (a d e-\left (2 c d^2+a e^2\right ) x\right )}{16 a^2 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 (-\frac {3}{4} c d e \left (2 c d^2+a e^2\right )+\frac {3}{4} c d e \left (4 c d^2+3 a e^2\right )\right )}{\sqrt [4]{c}}-\left (-\frac {3}{4} c d e \left (2 c d^2+a e^2\right )-\frac {3}{4} \sqrt {c} e \sqrt {c d^2+a e^2} \left (2 c d^2+a e^2\right )+\frac {3}{4} c d e \left (4 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 )}{8 \sqrt {2} a^2 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 (-\frac {3}{4} c d e \left (2 c d^2+a e^2\right )+\frac {3}{4} c d e \left (4 c d^2+3 a e^2\right )\right )}{\sqrt [4]{c}}+\left (-\frac {3}{4} c d e \left (2 c d^2+a e^2\right )-\frac {3}{4} \sqrt {c} e \sqrt {c d^2+a e^2} \left (2 c d^2+a e^2\right )+\frac {3}{4} c d e \left (4 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 )}{8 \sqrt {2} a^2 c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \\ & = -\frac {(a e-c d x) (d+e x)^{3/2}}{4 a c \left (a+c x^2\right )^2}-\frac {3 \sqrt {d+e x} \left (a d e-\left (2 c d^2+a e^2\right ) x\right )}{16 a^2 c \left (a+c x^2\right )}-\frac {\left (3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (2 c d^2+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 )}{64 \sqrt {2} a^2 c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (2 c d^2+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 )}{64 \sqrt {2} a^2 c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (2 c d^2+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 )}{64 a^2 c^2 \sqrt {c d^2+a e^2}}+\frac {\left (3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (2 c d^2+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 )}{64 a^2 c^2 \sqrt {c d^2+a e^2}} \\ & = -\frac {(a e-c d x) (d+e x)^{3/2}}{4 a c \left (a+c x^2\right )^2}-\frac {3 \sqrt {d+e x} \left (a d e-\left (2 c d^2+a e^2\right ) x\right )}{16 a^2 c \left (a+c x^2\right )}-\frac {3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (2 c d^2+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 )}{64 \sqrt {2} a^2 c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (2 c d^2+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 )}{64 \sqrt {2} a^2 c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\left (3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (2 c d^2+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 )}{32 a^2 c^2 \sqrt {c d^2+a e^2}}-\frac {\left (3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (2 c d^2+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 )}{32 a^2 c^2 \sqrt {c d^2+a e^2}} \\ & = -\frac {(a e-c d x) (d+e x)^{3/2}}{4 a c \left (a+c x^2\right )^2}-\frac {3 \sqrt {d+e x} \left (a d e-\left (2 c d^2+a e^2\right ) x\right )}{16 a^2 c \left (a+c x^2\right )}+\frac {3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (2 c d^2+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 )}{32 \sqrt {2} a^2 c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (2 c d^2+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 )}{32 \sqrt {2} a^2 c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (2 c d^2+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 )}{64 \sqrt {2} a^2 c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (2 c d^2+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 )}{64 \sqrt {2} a^2 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 = 2.51 (sec) , antiderivative size = 308, normalized size of antiderivative = 0.36 \[ \int \frac {(d+e x)^{5/2}}{\left (a+c x^2\right )^3} \, dx=\frac {\frac {2 \sqrt {a} c \sqrt {d+e x} \left (6 c^2 d^2 x^3-a^2 e (7 d+e x)+a c x \left (10 d^2+d e x+3 e^2 x^2\right )\right )}{\left (a+c x^2\right )^2}-3 i \sqrt {-c d-i \sqrt {a} \sqrt {c} e} \left (4 c d^2-2 i \sqrt {a} \sqrt {c} d e+a e^2\right ) \arctan \left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )+3 i \sqrt {-c d+i \sqrt {a} \sqrt {c} e} \left (4 c d^2+2 i \sqrt {a} \sqrt {c} d e+a e^2\right ) \arctan \left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{32 a^{5/2} c^2} \]
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Time = 2.90 (sec) , antiderivative size = 1025, normalized size of antiderivative = 1.21
method | result | size |
pseudoelliptic | \(\text {Expression too large to display}\) | \(1025\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1749\) |
default | \(\text {Expression too large to display}\) | \(1749\) |
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Time = 0.32 (sec) , antiderivative size = 1037, normalized size of antiderivative = 1.23 \[ \int \frac {(d+e x)^{5/2}}{\left (a+c x^2\right )^3} \, dx=\frac {3 \, {\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt {-\frac {a^{5} c^{3} \sqrt {-\frac {e^{10}}{a^{5} c^{7}}} + 16 \, c^{2} d^{5} + 20 \, a c d^{3} e^{2} + 5 \, a^{2} d e^{4}}{a^{5} c^{3}}} \log \left (27 \, {\left (16 \, c^{2} d^{4} e^{5} + 12 \, a c d^{2} e^{7} + a^{2} e^{9}\right )} \sqrt {e x + d} + 27 \, {\left (2 \, a^{3} c^{2} d e^{6} - {\left (4 \, a^{5} c^{6} d^{2} + a^{6} c^{5} e^{2}\right )} \sqrt {-\frac {e^{10}}{a^{5} c^{7}}}\right )} \sqrt {-\frac {a^{5} c^{3} \sqrt {-\frac {e^{10}}{a^{5} c^{7}}} + 16 \, c^{2} d^{5} + 20 \, a c d^{3} e^{2} + 5 \, a^{2} d e^{4}}{a^{5} c^{3}}}\right ) - 3 \, {\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt {-\frac {a^{5} c^{3} \sqrt {-\frac {e^{10}}{a^{5} c^{7}}} + 16 \, c^{2} d^{5} + 20 \, a c d^{3} e^{2} + 5 \, a^{2} d e^{4}}{a^{5} c^{3}}} \log \left (27 \, {\left (16 \, c^{2} d^{4} e^{5} + 12 \, a c d^{2} e^{7} + a^{2} e^{9}\right )} \sqrt {e x + d} - 27 \, {\left (2 \, a^{3} c^{2} d e^{6} - {\left (4 \, a^{5} c^{6} d^{2} + a^{6} c^{5} e^{2}\right )} \sqrt {-\frac {e^{10}}{a^{5} c^{7}}}\right )} \sqrt {-\frac {a^{5} c^{3} \sqrt {-\frac {e^{10}}{a^{5} c^{7}}} + 16 \, c^{2} d^{5} + 20 \, a c d^{3} e^{2} + 5 \, a^{2} d e^{4}}{a^{5} c^{3}}}\right ) + 3 \, {\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt {\frac {a^{5} c^{3} \sqrt {-\frac {e^{10}}{a^{5} c^{7}}} - 16 \, c^{2} d^{5} - 20 \, a c d^{3} e^{2} - 5 \, a^{2} d e^{4}}{a^{5} c^{3}}} \log \left (27 \, {\left (16 \, c^{2} d^{4} e^{5} + 12 \, a c d^{2} e^{7} + a^{2} e^{9}\right )} \sqrt {e x + d} + 27 \, {\left (2 \, a^{3} c^{2} d e^{6} + {\left (4 \, a^{5} c^{6} d^{2} + a^{6} c^{5} e^{2}\right )} \sqrt {-\frac {e^{10}}{a^{5} c^{7}}}\right )} \sqrt {\frac {a^{5} c^{3} \sqrt {-\frac {e^{10}}{a^{5} c^{7}}} - 16 \, c^{2} d^{5} - 20 \, a c d^{3} e^{2} - 5 \, a^{2} d e^{4}}{a^{5} c^{3}}}\right ) - 3 \, {\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt {\frac {a^{5} c^{3} \sqrt {-\frac {e^{10}}{a^{5} c^{7}}} - 16 \, c^{2} d^{5} - 20 \, a c d^{3} e^{2} - 5 \, a^{2} d e^{4}}{a^{5} c^{3}}} \log \left (27 \, {\left (16 \, c^{2} d^{4} e^{5} + 12 \, a c d^{2} e^{7} + a^{2} e^{9}\right )} \sqrt {e x + d} - 27 \, {\left (2 \, a^{3} c^{2} d e^{6} + {\left (4 \, a^{5} c^{6} d^{2} + a^{6} c^{5} e^{2}\right )} \sqrt {-\frac {e^{10}}{a^{5} c^{7}}}\right )} \sqrt {\frac {a^{5} c^{3} \sqrt {-\frac {e^{10}}{a^{5} c^{7}}} - 16 \, c^{2} d^{5} - 20 \, a c d^{3} e^{2} - 5 \, a^{2} d e^{4}}{a^{5} c^{3}}}\right ) + 4 \, {\left (a c d e x^{2} - 7 \, a^{2} d e + 3 \, {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{3} + {\left (10 \, a c d^{2} - a^{2} e^{2}\right )} x\right )} \sqrt {e x + d}}{64 \, {\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )}} \]
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Timed out. \[ \int \frac {(d+e x)^{5/2}}{\left (a+c x^2\right )^3} \, dx=\text {Timed out} \]
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\[ \int \frac {(d+e x)^{5/2}}{\left (a+c x^2\right )^3} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {5}{2}}}{{\left (c x^{2} + a\right )}^{3}} \,d x } \]
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Time = 0.44 (sec) , antiderivative size = 595, normalized size of antiderivative = 0.70 \[ \int \frac {(d+e x)^{5/2}}{\left (a+c x^2\right )^3} \, dx=\frac {3 \, {\left (4 \, c^{4} d^{4} e + 3 \, a c^{3} d^{2} e^{3} + {\left (2 \, a c d^{2} e + a^{2} e^{3}\right )} c^{2} e^{2} - 2 \, {\left (\sqrt {-a c} c^{2} d^{3} e + \sqrt {-a c} a c d e^{3}\right )} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a^{2} c^{2} d + \sqrt {a^{4} c^{4} d^{2} - {\left (a^{2} c^{2} d^{2} + a^{3} c e^{2}\right )} a^{2} c^{2}}}{a^{2} c^{2}}}}\right )}{32 \, {\left (a^{3} c^{3} e + \sqrt {-a c} a^{2} c^{3} d\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e} {\left | e \right |}} + \frac {3 \, {\left (4 \, c^{4} d^{4} e + 3 \, a c^{3} d^{2} e^{3} + {\left (2 \, a c d^{2} e + a^{2} e^{3}\right )} c^{2} e^{2} + 2 \, {\left (\sqrt {-a c} c^{2} d^{3} e + \sqrt {-a c} a c d e^{3}\right )} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a^{2} c^{2} d - \sqrt {a^{4} c^{4} d^{2} - {\left (a^{2} c^{2} d^{2} + a^{3} c e^{2}\right )} a^{2} c^{2}}}{a^{2} c^{2}}}}\right )}{32 \, {\left (a^{3} c^{3} e - \sqrt {-a c} a^{2} c^{3} d\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e} {\left | e \right |}} + \frac {6 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{2} d^{2} e - 18 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{2} d^{3} e + 18 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{2} d^{4} e - 6 \, \sqrt {e x + d} c^{2} d^{5} e + 3 \, {\left (e x + d\right )}^{\frac {7}{2}} a c e^{3} - 8 \, {\left (e x + d\right )}^{\frac {5}{2}} a c d e^{3} + 17 \, {\left (e x + d\right )}^{\frac {3}{2}} a c d^{2} e^{3} - 12 \, \sqrt {e x + d} a c d^{3} e^{3} - {\left (e x + d\right )}^{\frac {3}{2}} a^{2} e^{5} - 6 \, \sqrt {e x + d} a^{2} d e^{5}}{16 \, {\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} + a e^{2}\right )}^{2} a^{2} c} \]
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Time = 0.53 (sec) , antiderivative size = 1028, normalized size of antiderivative = 1.22 \[ \int \frac {(d+e x)^{5/2}}{\left (a+c x^2\right )^3} \, dx=\frac {\frac {3\,e\,\left (2\,c\,d^2+a\,e^2\right )\,{\left (d+e\,x\right )}^{7/2}}{16\,a^2}+\frac {{\left (d+e\,x\right )}^{3/2}\,\left (-a^2\,e^5+17\,a\,c\,d^2\,e^3+18\,c^2\,d^4\,e\right )}{16\,a^2\,c}-\frac {d\,\left (9\,c\,d^2\,e+4\,a\,e^3\right )\,{\left (d+e\,x\right )}^{5/2}}{8\,a^2}-\frac {3\,\sqrt {d+e\,x}\,\left (a^2\,d\,e^5+2\,a\,c\,d^3\,e^3+c^2\,d^5\,e\right )}{8\,a^2\,c}}{c^2\,{\left (d+e\,x\right )}^4+a^2\,e^4+c^2\,d^4+\left (6\,c^2\,d^2+2\,a\,c\,e^2\right )\,{\left (d+e\,x\right )}^2-\left (4\,c^2\,d^3+4\,a\,c\,d\,e^2\right )\,\left (d+e\,x\right )-4\,c^2\,d\,{\left (d+e\,x\right )}^3+2\,a\,c\,d^2\,e^2}-2\,\mathrm {atanh}\left (\frac {9\,e^8\,\sqrt {d+e\,x}\,\sqrt {-\frac {9\,d^5}{256\,a^5\,c}-\frac {45\,d\,e^4}{4096\,a^3\,c^3}-\frac {45\,d^3\,e^2}{1024\,a^4\,c^2}-\frac {9\,e^5\,\sqrt {-a^{15}\,c^7}}{4096\,a^{10}\,c^7}}}{32\,\left (\frac {27\,e^{11}}{2048\,a\,c^2}+\frac {27\,d^4\,e^7}{512\,a^3}+\frac {135\,d^2\,e^9}{2048\,a^2\,c}-\frac {27\,d\,e^{10}\,\sqrt {-a^{15}\,c^7}}{1024\,a^9\,c^5}-\frac {27\,d^3\,e^8\,\sqrt {-a^{15}\,c^7}}{1024\,a^{10}\,c^4}\right )}+\frac {9\,d\,e^7\,\sqrt {-a^{15}\,c^7}\,\sqrt {d+e\,x}\,\sqrt {-\frac {9\,d^5}{256\,a^5\,c}-\frac {45\,d\,e^4}{4096\,a^3\,c^3}-\frac {45\,d^3\,e^2}{1024\,a^4\,c^2}-\frac {9\,e^5\,\sqrt {-a^{15}\,c^7}}{4096\,a^{10}\,c^7}}}{32\,\left (\frac {27\,a^7\,c\,e^{11}}{2048}+\frac {27\,a^5\,c^3\,d^4\,e^7}{512}+\frac {135\,a^6\,c^2\,d^2\,e^9}{2048}-\frac {27\,d\,e^{10}\,\sqrt {-a^{15}\,c^7}}{1024\,a\,c^2}-\frac {27\,d^3\,e^8\,\sqrt {-a^{15}\,c^7}}{1024\,a^2\,c}\right )}\right )\,\sqrt {-\frac {9\,\left (e^5\,\sqrt {-a^{15}\,c^7}+16\,a^5\,c^6\,d^5+5\,a^7\,c^4\,d\,e^4+20\,a^6\,c^5\,d^3\,e^2\right )}{4096\,a^{10}\,c^7}}-2\,\mathrm {atanh}\left (\frac {9\,e^8\,\sqrt {d+e\,x}\,\sqrt {\frac {9\,e^5\,\sqrt {-a^{15}\,c^7}}{4096\,a^{10}\,c^7}-\frac {45\,d\,e^4}{4096\,a^3\,c^3}-\frac {45\,d^3\,e^2}{1024\,a^4\,c^2}-\frac {9\,d^5}{256\,a^5\,c}}}{32\,\left (\frac {27\,e^{11}}{2048\,a\,c^2}+\frac {27\,d^4\,e^7}{512\,a^3}+\frac {135\,d^2\,e^9}{2048\,a^2\,c}+\frac {27\,d\,e^{10}\,\sqrt {-a^{15}\,c^7}}{1024\,a^9\,c^5}+\frac {27\,d^3\,e^8\,\sqrt {-a^{15}\,c^7}}{1024\,a^{10}\,c^4}\right )}-\frac {9\,d\,e^7\,\sqrt {-a^{15}\,c^7}\,\sqrt {d+e\,x}\,\sqrt {\frac {9\,e^5\,\sqrt {-a^{15}\,c^7}}{4096\,a^{10}\,c^7}-\frac {45\,d\,e^4}{4096\,a^3\,c^3}-\frac {45\,d^3\,e^2}{1024\,a^4\,c^2}-\frac {9\,d^5}{256\,a^5\,c}}}{32\,\left (\frac {27\,a^7\,c\,e^{11}}{2048}+\frac {27\,a^5\,c^3\,d^4\,e^7}{512}+\frac {135\,a^6\,c^2\,d^2\,e^9}{2048}+\frac {27\,d\,e^{10}\,\sqrt {-a^{15}\,c^7}}{1024\,a\,c^2}+\frac {27\,d^3\,e^8\,\sqrt {-a^{15}\,c^7}}{1024\,a^2\,c}\right )}\right )\,\sqrt {-\frac {9\,\left (16\,a^5\,c^6\,d^5-e^5\,\sqrt {-a^{15}\,c^7}+5\,a^7\,c^4\,d\,e^4+20\,a^6\,c^5\,d^3\,e^2\right )}{4096\,a^{10}\,c^7}} \]
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