Optimal. Leaf size=920 \[ \frac {(a e+c d x) \sqrt {d+e x}}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a e \left (c d^2+7 a e^2\right )+6 c d \left (c d^2+2 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac {3 e \left (2 c^2 d^4+5 a c d^2 e^2+7 a^2 e^4+2 \sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{32 \sqrt {2} a^2 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {3 e \left (2 c^2 d^4+5 a c d^2 e^2+7 a^2 e^4+2 \sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{32 \sqrt {2} a^2 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {3 e \left (2 c^2 d^4+5 a c d^2 e^2+7 a^2 e^4-2 \sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{64 \sqrt {2} a^2 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {3 e \left (2 c^2 d^4+5 a c d^2 e^2+7 a^2 e^4-2 \sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{64 \sqrt {2} a^2 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 4.03, antiderivative size = 920, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {755, 837, 841,
1183, 648, 632, 212, 642} \begin {gather*} \frac {\sqrt {d+e x} (a e+c d x)}{4 a \left (c d^2+a e^2\right ) \left (c x^2+a\right )^2}+\frac {3 e \left (2 c^2 d^4+5 a c e^2 d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right ) d+7 a^2 e^4\right ) \tanh ^{-1}\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 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {3 e \left (2 c^2 d^4+5 a c e^2 d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right ) d+7 a^2 e^4\right ) \tanh ^{-1}\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 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {3 e \left (2 c^2 d^4+5 a c e^2 d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right ) d+7 a^2 e^4\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 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {3 e \left (2 c^2 d^4+5 a c e^2 d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right ) d+7 a^2 e^4\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 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\sqrt {d+e x} \left (a e \left (c d^2+7 a e^2\right )+6 c d \left (c d^2+2 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right )^2 \left (c x^2+a\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 632
Rule 642
Rule 648
Rule 755
Rule 837
Rule 841
Rule 1183
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )^3} \, dx &=\frac {(a e+c d x) \sqrt {d+e x}}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}-\frac {\int \frac {\frac {1}{2} \left (-6 c d^2-7 a e^2\right )-\frac {5}{2} c d e x}{\sqrt {d+e x} \left (a+c x^2\right )^2} \, dx}{4 a \left (c d^2+a e^2\right )}\\ &=\frac {(a e+c d x) \sqrt {d+e x}}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a e \left (c d^2+7 a e^2\right )+6 c d \left (c d^2+2 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac {\int \frac {\frac {3}{4} c \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\right )+\frac {3}{2} c^2 d e \left (c d^2+2 a e^2\right ) x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx}{8 a^2 c \left (c d^2+a e^2\right )^2}\\ &=\frac {(a e+c d x) \sqrt {d+e x}}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a e \left (c d^2+7 a e^2\right )+6 c d \left (c d^2+2 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac {\text {Subst}\left (\int \frac {-\frac {3}{2} c^2 d^2 e \left (c d^2+2 a e^2\right )+\frac {3}{4} c e \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\right )+\frac {3}{2} c^2 d e \left (c d^2+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 \left (c d^2+a e^2\right )^2}\\ &=\frac {(a e+c d x) \sqrt {d+e x}}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a e \left (c d^2+7 a e^2\right )+6 c d \left (c d^2+2 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right )^2 \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}{2} c^2 d^2 e \left (c d^2+2 a e^2\right )+\frac {3}{4} c e \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\right )\right )}{\sqrt [4]{c}}-\left (-\frac {3}{2} c^2 d^2 e \left (c d^2+2 a e^2\right )-\frac {3}{2} c^{3/2} d e \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right )+\frac {3}{4} c e \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\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^{5/4} \left (c d^2+a e^2\right )^{5/2} \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}{2} c^2 d^2 e \left (c d^2+2 a e^2\right )+\frac {3}{4} c e \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\right )\right )}{\sqrt [4]{c}}+\left (-\frac {3}{2} c^2 d^2 e \left (c d^2+2 a e^2\right )-\frac {3}{2} c^{3/2} d e \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right )+\frac {3}{4} c e \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\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^{5/4} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ &=\frac {(a e+c d x) \sqrt {d+e x}}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a e \left (c d^2+7 a e^2\right )+6 c d \left (c d^2+2 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac {\left (\frac {3}{2} c^2 d^2 e \left (c d^2+2 a e^2\right )+\frac {3}{2} c^{3/2} d e \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right )-\frac {3}{4} c e \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\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 )}{16 \sqrt {2} a^2 c^{5/4} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (-\frac {3}{2} c^2 d^2 e \left (c d^2+2 a e^2\right )-\frac {3}{2} c^{3/2} d e \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right )+\frac {3}{4} c e \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\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 )}{16 \sqrt {2} a^2 c^{5/4} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (\frac {3}{2} c^2 d^2 e \left (c d^2+2 a e^2\right )+\frac {3}{2} c^{3/2} d e \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right )-\frac {3}{4} c e \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\right )\right )}{\sqrt [4]{c}}+\frac {2 \sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (-\frac {3}{2} c^2 d^2 e \left (c d^2+2 a e^2\right )+\frac {3}{4} c e \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\right )\right )}{\sqrt [4]{c}}\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 )}{16 \sqrt {2} a^2 c^{5/4} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (\frac {2 \sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (-\frac {3}{2} c^2 d^2 e \left (c d^2+2 a e^2\right )+\frac {3}{4} c e \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\right )\right )}{\sqrt [4]{c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (-\frac {3}{2} c^2 d^2 e \left (c d^2+2 a e^2\right )-\frac {3}{2} c^{3/2} d e \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right )+\frac {3}{4} c e \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\right )\right )}{\sqrt [4]{c}}\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 )}{16 \sqrt {2} a^2 c^{5/4} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ &=\frac {(a e+c d x) \sqrt {d+e x}}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a e \left (c d^2+7 a e^2\right )+6 c d \left (c d^2+2 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}-\frac {3 e \left (2 c^2 d^4+5 a c d^2 e^2+7 a^2 e^4-\sqrt {c} d \sqrt {c d^2+a e^2} \left (2 c d^2+4 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 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {3 e \left (2 c^2 d^4+5 a c d^2 e^2+7 a^2 e^4-\sqrt {c} d \sqrt {c d^2+a e^2} \left (2 c d^2+4 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 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\left (\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (\frac {3}{2} c^2 d^2 e \left (c d^2+2 a e^2\right )+\frac {3}{2} c^{3/2} d e \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right )-\frac {3}{4} c e \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\right )\right )}{\sqrt [4]{c}}+\frac {2 \sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (-\frac {3}{2} c^2 d^2 e \left (c d^2+2 a e^2\right )+\frac {3}{4} c e \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\right )\right )}{\sqrt [4]{c}}\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 )}{8 \sqrt {2} a^2 c^{5/4} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\left (\frac {2 \sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (-\frac {3}{2} c^2 d^2 e \left (c d^2+2 a e^2\right )+\frac {3}{4} c e \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\right )\right )}{\sqrt [4]{c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (-\frac {3}{2} c^2 d^2 e \left (c d^2+2 a e^2\right )-\frac {3}{2} c^{3/2} d e \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right )+\frac {3}{4} c e \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\right )\right )}{\sqrt [4]{c}}\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 )}{8 \sqrt {2} a^2 c^{5/4} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ &=\frac {(a e+c d x) \sqrt {d+e x}}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a e \left (c d^2+7 a e^2\right )+6 c d \left (c d^2+2 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac {3 e \left (2 c^2 d^4+5 a c d^2 e^2+7 a^2 e^4+\sqrt {c} d \sqrt {c d^2+a e^2} \left (2 c d^2+4 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 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {3 e \left (2 c^2 d^4+5 a c d^2 e^2+7 a^2 e^4+\sqrt {c} d \sqrt {c d^2+a e^2} \left (2 c d^2+4 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 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {3 e \left (2 c^2 d^4+5 a c d^2 e^2+7 a^2 e^4-\sqrt {c} d \sqrt {c d^2+a e^2} \left (2 c d^2+4 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 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {3 e \left (2 c^2 d^4+5 a c d^2 e^2+7 a^2 e^4-\sqrt {c} d \sqrt {c d^2+a e^2} \left (2 c d^2+4 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 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 1.31, size = 382, normalized size = 0.42 \begin {gather*} \frac {\frac {2 \sqrt {a} \sqrt {d+e x} \left (11 a^3 e^3+6 c^3 d^3 x^3+a^2 c e \left (5 d^2+16 d e x+7 e^2 x^2\right )+a c^2 d x \left (10 d^2+d e x+12 e^2 x^2\right )\right )}{\left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {3 i \left (4 c d^2+10 i \sqrt {a} \sqrt {c} d e-7 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )}{\left (\sqrt {c} d+i \sqrt {a} e\right )^2 \sqrt {-c d-i \sqrt {a} \sqrt {c} e}}-\frac {3 i \left (4 c d^2-10 i \sqrt {a} \sqrt {c} d e-7 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{\left (\sqrt {c} d-i \sqrt {a} e\right )^2 \sqrt {-c d+i \sqrt {a} \sqrt {c} e}}}{32 a^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (c \,x^{2}+a \right )^{3} \sqrt {e x +d}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5408 vs.
\(2 (752) = 1504\).
time = 8.44, size = 5408, normalized size = 5.88 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 3.06, size = 832, normalized size = 0.90 \begin {gather*} -\frac {3 \, {\left (4 \, c d^{2} + 10 \, \sqrt {-a c} d e - 7 \, a e^{2}\right )} {\left | c \right |} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a^{2} c^{3} d^{5} + 2 \, a^{3} c^{2} d^{3} e^{2} + a^{4} c d e^{4} + \sqrt {{\left (a^{2} c^{3} d^{5} + 2 \, a^{3} c^{2} d^{3} e^{2} + a^{4} c d e^{4}\right )}^{2} - {\left (a^{2} c^{3} d^{6} + 3 \, a^{3} c^{2} d^{4} e^{2} + 3 \, a^{4} c d^{2} e^{4} + a^{5} e^{6}\right )} {\left (a^{2} c^{3} d^{4} + 2 \, a^{3} c^{2} d^{2} e^{2} + a^{4} c e^{4}\right )}}}{a^{2} c^{3} d^{4} + 2 \, a^{3} c^{2} d^{2} e^{2} + a^{4} c e^{4}}}}\right )}{32 \, {\left (2 \, a^{3} c d e - \sqrt {-a c} a^{2} c d^{2} + \sqrt {-a c} a^{3} e^{2}\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e}} - \frac {3 \, {\left (4 \, c d^{2} - 10 \, \sqrt {-a c} d e - 7 \, a e^{2}\right )} {\left | c \right |} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a^{2} c^{3} d^{5} + 2 \, a^{3} c^{2} d^{3} e^{2} + a^{4} c d e^{4} - \sqrt {{\left (a^{2} c^{3} d^{5} + 2 \, a^{3} c^{2} d^{3} e^{2} + a^{4} c d e^{4}\right )}^{2} - {\left (a^{2} c^{3} d^{6} + 3 \, a^{3} c^{2} d^{4} e^{2} + 3 \, a^{4} c d^{2} e^{4} + a^{5} e^{6}\right )} {\left (a^{2} c^{3} d^{4} + 2 \, a^{3} c^{2} d^{2} e^{2} + a^{4} c e^{4}\right )}}}{a^{2} c^{3} d^{4} + 2 \, a^{3} c^{2} d^{2} e^{2} + a^{4} c e^{4}}}}\right )}{32 \, {\left (2 \, a^{3} c d e + \sqrt {-a c} a^{2} c d^{2} - \sqrt {-a c} a^{3} e^{2}\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e}} + \frac {6 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{3} d^{3} e - 18 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{3} d^{4} e + 18 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{5} e - 6 \, \sqrt {x e + d} c^{3} d^{6} e + 12 \, {\left (x e + d\right )}^{\frac {7}{2}} a c^{2} d e^{3} - 35 \, {\left (x e + d\right )}^{\frac {5}{2}} a c^{2} d^{2} e^{3} + 44 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{2} d^{3} e^{3} - 21 \, \sqrt {x e + d} a c^{2} d^{4} e^{3} + 7 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} c e^{5} + 2 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} c d e^{5} - 4 \, \sqrt {x e + d} a^{2} c d^{2} e^{5} + 11 \, \sqrt {x e + d} a^{3} e^{7}}{16 \, {\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )} {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + a e^{2}\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 3.60, size = 2500, normalized size = 2.72 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________