Optimal. Leaf size=158 \[ \frac{5 c^{3/2} d^{3/2} e \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{7/2}}-\frac{5 c d e}{\sqrt{d+e x} \left (c d^2-a e^2\right )^3}-\frac{1}{(d+e x)^{3/2} \left (c d^2-a e^2\right ) (a e+c d x)}-\frac{5 e}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2} \]
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Rubi [A] time = 0.0927356, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108, Rules used = {626, 51, 63, 208} \[ \frac{5 c^{3/2} d^{3/2} e \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{7/2}}-\frac{5 c d e}{\sqrt{d+e x} \left (c d^2-a e^2\right )^3}-\frac{1}{(d+e x)^{3/2} \left (c d^2-a e^2\right ) (a e+c d x)}-\frac{5 e}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 626
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx &=\int \frac{1}{(a e+c d x)^2 (d+e x)^{5/2}} \, dx\\ &=-\frac{1}{\left (c d^2-a e^2\right ) (a e+c d x) (d+e x)^{3/2}}-\frac{(5 e) \int \frac{1}{(a e+c d x) (d+e x)^{5/2}} \, dx}{2 \left (c d^2-a e^2\right )}\\ &=-\frac{5 e}{3 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac{1}{\left (c d^2-a e^2\right ) (a e+c d x) (d+e x)^{3/2}}-\frac{(5 c d e) \int \frac{1}{(a e+c d x) (d+e x)^{3/2}} \, dx}{2 \left (c d^2-a e^2\right )^2}\\ &=-\frac{5 e}{3 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac{1}{\left (c d^2-a e^2\right ) (a e+c d x) (d+e x)^{3/2}}-\frac{5 c d e}{\left (c d^2-a e^2\right )^3 \sqrt{d+e x}}-\frac{\left (5 c^2 d^2 e\right ) \int \frac{1}{(a e+c d x) \sqrt{d+e x}} \, dx}{2 \left (c d^2-a e^2\right )^3}\\ &=-\frac{5 e}{3 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac{1}{\left (c d^2-a e^2\right ) (a e+c d x) (d+e x)^{3/2}}-\frac{5 c d e}{\left (c d^2-a e^2\right )^3 \sqrt{d+e x}}-\frac{\left (5 c^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{c d^2}{e}+a e+\frac{c d x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{\left (c d^2-a e^2\right )^3}\\ &=-\frac{5 e}{3 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac{1}{\left (c d^2-a e^2\right ) (a e+c d x) (d+e x)^{3/2}}-\frac{5 c d e}{\left (c d^2-a e^2\right )^3 \sqrt{d+e x}}+\frac{5 c^{3/2} d^{3/2} e \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0172592, size = 59, normalized size = 0.37 \[ -\frac{2 e \, _2F_1\left (-\frac{3}{2},2;-\frac{1}{2};-\frac{c d (d+e x)}{a e^2-c d^2}\right )}{3 (d+e x)^{3/2} \left (a e^2-c d^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.204, size = 162, normalized size = 1. \begin{align*} -{\frac{2\,e}{3\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}+4\,{\frac{dec}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{3}\sqrt{ex+d}}}+{\frac{{c}^{2}e{d}^{2}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{3} \left ( cdex+a{e}^{2} \right ) }\sqrt{ex+d}}+5\,{\frac{{c}^{2}e{d}^{2}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{3}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{\sqrt{ex+d}cd}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.11454, size = 1751, normalized size = 11.08 \begin{align*} \text{result too large to display} \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(-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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