Optimal. Leaf size=202 \[ \frac{\left (c d^2-a e^2\right ) \left (3 a e^2+c d^2\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 a^{3/2} d^{5/2} e^{3/2}}-\frac{\left (\frac{c}{a e}-\frac{3 e}{d^2}\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 x}-\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d x^2} \]
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Rubi [A] time = 0.275852, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {849, 834, 806, 724, 206} \[ \frac{\left (c d^2-a e^2\right ) \left (3 a e^2+c d^2\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 a^{3/2} d^{5/2} e^{3/2}}-\frac{\left (\frac{c}{a e}-\frac{3 e}{d^2}\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 x}-\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d x^2} \]
Antiderivative was successfully verified.
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Rule 849
Rule 834
Rule 806
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^3 (d+e x)} \, dx &=\int \frac{a e+c d x}{x^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\\ &=-\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 d x^2}-\frac{\int \frac{-\frac{1}{2} a e \left (c d^2-3 a e^2\right )+a c d e^2 x}{x^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2 a d e}\\ &=-\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 d x^2}-\frac{\left (\frac{c}{a e}-\frac{3 e}{d^2}\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 x}-\frac{\left (\frac{c^2 d^2}{a}+2 c e^2-\frac{3 a e^4}{d^2}\right ) \int \frac{1}{x \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 e}\\ &=-\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 d x^2}-\frac{\left (\frac{c}{a e}-\frac{3 e}{d^2}\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 x}+\frac{\left (\frac{c^2 d^2}{a}+2 c e^2-\frac{3 a e^4}{d^2}\right ) \operatorname{Subst}\left (\int \frac{1}{4 a d e-x^2} \, dx,x,\frac{2 a d e-\left (-c d^2-a e^2\right ) x}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 e}\\ &=-\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 d x^2}-\frac{\left (\frac{c}{a e}-\frac{3 e}{d^2}\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 x}+\frac{\left (c d^2-a e^2\right ) \left (c d^2+3 a e^2\right ) \tanh ^{-1}\left (\frac{2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 a^{3/2} d^{5/2} e^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.17332, size = 162, normalized size = 0.8 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (\frac{\left (-3 a^2 e^4+2 a c d^2 e^2+c^2 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right )}{\sqrt{d+e x} \sqrt{a e+c d x}}+\frac{\sqrt{a} \sqrt{d} \sqrt{e} \left (a e (3 e x-2 d)-c d^2 x\right )}{x^2}\right )}{4 a^{3/2} d^{5/2} e^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.063, size = 882, normalized size = 4.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{{\left (e x + d\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 5.46581, size = 945, normalized size = 4.68 \begin{align*} \left [-\frac{{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4}\right )} \sqrt{a d e} x^{2} \log \left (\frac{8 \, a^{2} d^{2} e^{2} +{\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} - 4 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (2 \, a d e +{\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt{a d e} + 8 \,{\left (a c d^{3} e + a^{2} d e^{3}\right )} x}{x^{2}}\right ) + 4 \,{\left (2 \, a^{2} d^{2} e^{2} +{\left (a c d^{3} e - 3 \, a^{2} d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{16 \, a^{2} d^{3} e^{2} x^{2}}, -\frac{{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4}\right )} \sqrt{-a d e} x^{2} \arctan \left (\frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (2 \, a d e +{\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt{-a d e}}{2 \,{\left (a c d^{2} e^{2} x^{2} + a^{2} d^{2} e^{2} +{\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )}}\right ) + 2 \,{\left (2 \, a^{2} d^{2} e^{2} +{\left (a c d^{3} e - 3 \, a^{2} d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{8 \, a^{2} d^{3} e^{2} x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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