Optimal. Leaf size=147 \[ -\frac{5 e^2 \sqrt{d+e x}}{8 (a+b x) (b d-a e)^3}+\frac{5 e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 \sqrt{b} (b d-a e)^{7/2}}+\frac{5 e \sqrt{d+e x}}{12 (a+b x)^2 (b d-a e)^2}-\frac{\sqrt{d+e x}}{3 (a+b x)^3 (b d-a e)} \]
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Rubi [A] time = 0.0718597, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {27, 51, 63, 208} \[ -\frac{5 e^2 \sqrt{d+e x}}{8 (a+b x) (b d-a e)^3}+\frac{5 e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 \sqrt{b} (b d-a e)^{7/2}}+\frac{5 e \sqrt{d+e x}}{12 (a+b x)^2 (b d-a e)^2}-\frac{\sqrt{d+e x}}{3 (a+b x)^3 (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{1}{(a+b x)^4 \sqrt{d+e x}} \, dx\\ &=-\frac{\sqrt{d+e x}}{3 (b d-a e) (a+b x)^3}-\frac{(5 e) \int \frac{1}{(a+b x)^3 \sqrt{d+e x}} \, dx}{6 (b d-a e)}\\ &=-\frac{\sqrt{d+e x}}{3 (b d-a e) (a+b x)^3}+\frac{5 e \sqrt{d+e x}}{12 (b d-a e)^2 (a+b x)^2}+\frac{\left (5 e^2\right ) \int \frac{1}{(a+b x)^2 \sqrt{d+e x}} \, dx}{8 (b d-a e)^2}\\ &=-\frac{\sqrt{d+e x}}{3 (b d-a e) (a+b x)^3}+\frac{5 e \sqrt{d+e x}}{12 (b d-a e)^2 (a+b x)^2}-\frac{5 e^2 \sqrt{d+e x}}{8 (b d-a e)^3 (a+b x)}-\frac{\left (5 e^3\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{16 (b d-a e)^3}\\ &=-\frac{\sqrt{d+e x}}{3 (b d-a e) (a+b x)^3}+\frac{5 e \sqrt{d+e x}}{12 (b d-a e)^2 (a+b x)^2}-\frac{5 e^2 \sqrt{d+e x}}{8 (b d-a e)^3 (a+b x)}-\frac{\left (5 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b d}{e}+\frac{b x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{8 (b d-a e)^3}\\ &=-\frac{\sqrt{d+e x}}{3 (b d-a e) (a+b x)^3}+\frac{5 e \sqrt{d+e x}}{12 (b d-a e)^2 (a+b x)^2}-\frac{5 e^2 \sqrt{d+e x}}{8 (b d-a e)^3 (a+b x)}+\frac{5 e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 \sqrt{b} (b d-a e)^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0114543, size = 50, normalized size = 0.34 \[ \frac{2 e^3 \sqrt{d+e x} \, _2F_1\left (\frac{1}{2},4;\frac{3}{2};-\frac{b (d+e x)}{a e-b d}\right )}{(a e-b d)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.196, size = 147, normalized size = 1. \begin{align*}{\frac{{e}^{3}}{ \left ( 3\,ae-3\,bd \right ) \left ( bxe+ae \right ) ^{3}}\sqrt{ex+d}}+{\frac{5\,{e}^{3}}{12\, \left ( ae-bd \right ) ^{2} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{5\,{e}^{3}}{8\, \left ( ae-bd \right ) ^{3} \left ( bxe+ae \right ) }\sqrt{ex+d}}+{\frac{5\,{e}^{3}}{8\, \left ( ae-bd \right ) ^{3}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \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 = 2.11195, size = 1805, normalized size = 12.28 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x\right )^{4} \sqrt{d + e x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13792, size = 315, normalized size = 2.14 \begin{align*} -\frac{5 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{3}}{8 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt{-b^{2} d + a b e}} - \frac{15 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{2} e^{3} - 40 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} d e^{3} + 33 \, \sqrt{x e + d} b^{2} d^{2} e^{3} + 40 \,{\left (x e + d\right )}^{\frac{3}{2}} a b e^{4} - 66 \, \sqrt{x e + d} a b d e^{4} + 33 \, \sqrt{x e + d} a^{2} e^{5}}{24 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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