Optimal. Leaf size=218 \[ -\frac{7 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{3/2} (b d-a e)^{9/2}}+\frac{7 e^4 \sqrt{d+e x}}{128 b (a+b x) (b d-a e)^4}-\frac{7 e^3 \sqrt{d+e x}}{192 b (a+b x)^2 (b d-a e)^3}+\frac{7 e^2 \sqrt{d+e x}}{240 b (a+b x)^3 (b d-a e)^2}-\frac{e \sqrt{d+e x}}{40 b (a+b x)^4 (b d-a e)}-\frac{\sqrt{d+e x}}{5 b (a+b x)^5} \]
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Rubi [A] time = 0.150461, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {27, 47, 51, 63, 208} \[ -\frac{7 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{3/2} (b d-a e)^{9/2}}+\frac{7 e^4 \sqrt{d+e x}}{128 b (a+b x) (b d-a e)^4}-\frac{7 e^3 \sqrt{d+e x}}{192 b (a+b x)^2 (b d-a e)^3}+\frac{7 e^2 \sqrt{d+e x}}{240 b (a+b x)^3 (b d-a e)^2}-\frac{e \sqrt{d+e x}}{40 b (a+b x)^4 (b d-a e)}-\frac{\sqrt{d+e x}}{5 b (a+b x)^5} \]
Antiderivative was successfully verified.
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Rule 27
Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{\sqrt{d+e x}}{(a+b x)^6} \, dx\\ &=-\frac{\sqrt{d+e x}}{5 b (a+b x)^5}+\frac{e \int \frac{1}{(a+b x)^5 \sqrt{d+e x}} \, dx}{10 b}\\ &=-\frac{\sqrt{d+e x}}{5 b (a+b x)^5}-\frac{e \sqrt{d+e x}}{40 b (b d-a e) (a+b x)^4}-\frac{\left (7 e^2\right ) \int \frac{1}{(a+b x)^4 \sqrt{d+e x}} \, dx}{80 b (b d-a e)}\\ &=-\frac{\sqrt{d+e x}}{5 b (a+b x)^5}-\frac{e \sqrt{d+e x}}{40 b (b d-a e) (a+b x)^4}+\frac{7 e^2 \sqrt{d+e x}}{240 b (b d-a e)^2 (a+b x)^3}+\frac{\left (7 e^3\right ) \int \frac{1}{(a+b x)^3 \sqrt{d+e x}} \, dx}{96 b (b d-a e)^2}\\ &=-\frac{\sqrt{d+e x}}{5 b (a+b x)^5}-\frac{e \sqrt{d+e x}}{40 b (b d-a e) (a+b x)^4}+\frac{7 e^2 \sqrt{d+e x}}{240 b (b d-a e)^2 (a+b x)^3}-\frac{7 e^3 \sqrt{d+e x}}{192 b (b d-a e)^3 (a+b x)^2}-\frac{\left (7 e^4\right ) \int \frac{1}{(a+b x)^2 \sqrt{d+e x}} \, dx}{128 b (b d-a e)^3}\\ &=-\frac{\sqrt{d+e x}}{5 b (a+b x)^5}-\frac{e \sqrt{d+e x}}{40 b (b d-a e) (a+b x)^4}+\frac{7 e^2 \sqrt{d+e x}}{240 b (b d-a e)^2 (a+b x)^3}-\frac{7 e^3 \sqrt{d+e x}}{192 b (b d-a e)^3 (a+b x)^2}+\frac{7 e^4 \sqrt{d+e x}}{128 b (b d-a e)^4 (a+b x)}+\frac{\left (7 e^5\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{256 b (b d-a e)^4}\\ &=-\frac{\sqrt{d+e x}}{5 b (a+b x)^5}-\frac{e \sqrt{d+e x}}{40 b (b d-a e) (a+b x)^4}+\frac{7 e^2 \sqrt{d+e x}}{240 b (b d-a e)^2 (a+b x)^3}-\frac{7 e^3 \sqrt{d+e x}}{192 b (b d-a e)^3 (a+b x)^2}+\frac{7 e^4 \sqrt{d+e x}}{128 b (b d-a e)^4 (a+b x)}+\frac{\left (7 e^4\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b d}{e}+\frac{b x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{128 b (b d-a e)^4}\\ &=-\frac{\sqrt{d+e x}}{5 b (a+b x)^5}-\frac{e \sqrt{d+e x}}{40 b (b d-a e) (a+b x)^4}+\frac{7 e^2 \sqrt{d+e x}}{240 b (b d-a e)^2 (a+b x)^3}-\frac{7 e^3 \sqrt{d+e x}}{192 b (b d-a e)^3 (a+b x)^2}+\frac{7 e^4 \sqrt{d+e x}}{128 b (b d-a e)^4 (a+b x)}-\frac{7 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{3/2} (b d-a e)^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.0141032, size = 52, normalized size = 0.24 \[ \frac{2 e^5 (d+e x)^{3/2} \, _2F_1\left (\frac{3}{2},6;\frac{5}{2};-\frac{b (d+e x)}{a e-b d}\right )}{3 (a e-b d)^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.204, size = 337, normalized size = 1.6 \begin{align*}{\frac{7\,{e}^{5}{b}^{3}}{128\, \left ( bxe+ae \right ) ^{5} \left ({a}^{4}{e}^{4}-4\,{a}^{3}bd{e}^{3}+6\,{d}^{2}{e}^{2}{a}^{2}{b}^{2}-4\,a{b}^{3}{d}^{3}e+{b}^{4}{d}^{4} \right ) } \left ( ex+d \right ) ^{{\frac{9}{2}}}}+{\frac{49\,{e}^{5}{b}^{2}}{192\, \left ( bxe+ae \right ) ^{5} \left ({a}^{3}{e}^{3}-3\,d{e}^{2}{a}^{2}b+3\,a{b}^{2}{d}^{2}e-{b}^{3}{d}^{3} \right ) } \left ( ex+d \right ) ^{{\frac{7}{2}}}}+{\frac{7\,{e}^{5}b}{15\, \left ( bxe+ae \right ) ^{5} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) } \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{79\,{e}^{5}}{192\, \left ( bxe+ae \right ) ^{5} \left ( ae-bd \right ) } \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{7\,{e}^{5}}{128\, \left ( bxe+ae \right ) ^{5}b}\sqrt{ex+d}}+{\frac{7\,{e}^{5}}{128\,b \left ({a}^{4}{e}^{4}-4\,{a}^{3}bd{e}^{3}+6\,{d}^{2}{e}^{2}{a}^{2}{b}^{2}-4\,a{b}^{3}{d}^{3}e+{b}^{4}{d}^{4} \right ) }\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 = 1.85345, size = 3452, normalized size = 15.83 \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 [B] time = 1.18188, size = 583, normalized size = 2.67 \begin{align*} \frac{7 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{5}}{128 \,{\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )} \sqrt{-b^{2} d + a b e}} + \frac{105 \,{\left (x e + d\right )}^{\frac{9}{2}} b^{4} e^{5} - 490 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{4} d e^{5} + 896 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} d^{2} e^{5} - 790 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d^{3} e^{5} - 105 \, \sqrt{x e + d} b^{4} d^{4} e^{5} + 490 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{3} e^{6} - 1792 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{3} d e^{6} + 2370 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} d^{2} e^{6} + 420 \, \sqrt{x e + d} a b^{3} d^{3} e^{6} + 896 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{2} e^{7} - 2370 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{2} d e^{7} - 630 \, \sqrt{x e + d} a^{2} b^{2} d^{2} e^{7} + 790 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b e^{8} + 420 \, \sqrt{x e + d} a^{3} b d e^{8} - 105 \, \sqrt{x e + d} a^{4} e^{9}}{1920 \,{\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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