Optimal. Leaf size=213 \[ -\frac{63 e^4 \sqrt{d+e x}}{128 (a+b x) (b d-a e)^5}+\frac{21 e^3 \sqrt{d+e x}}{64 (a+b x)^2 (b d-a e)^4}-\frac{21 e^2 \sqrt{d+e x}}{80 (a+b x)^3 (b d-a e)^3}+\frac{63 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 \sqrt{b} (b d-a e)^{11/2}}+\frac{9 e \sqrt{d+e x}}{40 (a+b x)^4 (b d-a e)^2}-\frac{\sqrt{d+e x}}{5 (a+b x)^5 (b d-a e)} \]
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Rubi [A] time = 0.116132, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {27, 51, 63, 208} \[ -\frac{63 e^4 \sqrt{d+e x}}{128 (a+b x) (b d-a e)^5}+\frac{21 e^3 \sqrt{d+e x}}{64 (a+b x)^2 (b d-a e)^4}-\frac{21 e^2 \sqrt{d+e x}}{80 (a+b x)^3 (b d-a e)^3}+\frac{63 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 \sqrt{b} (b d-a e)^{11/2}}+\frac{9 e \sqrt{d+e x}}{40 (a+b x)^4 (b d-a e)^2}-\frac{\sqrt{d+e x}}{5 (a+b x)^5 (b d-a e)} \]
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 )^3} \, dx &=\int \frac{1}{(a+b x)^6 \sqrt{d+e x}} \, dx\\ &=-\frac{\sqrt{d+e x}}{5 (b d-a e) (a+b x)^5}-\frac{(9 e) \int \frac{1}{(a+b x)^5 \sqrt{d+e x}} \, dx}{10 (b d-a e)}\\ &=-\frac{\sqrt{d+e x}}{5 (b d-a e) (a+b x)^5}+\frac{9 e \sqrt{d+e x}}{40 (b d-a e)^2 (a+b x)^4}+\frac{\left (63 e^2\right ) \int \frac{1}{(a+b x)^4 \sqrt{d+e x}} \, dx}{80 (b d-a e)^2}\\ &=-\frac{\sqrt{d+e x}}{5 (b d-a e) (a+b x)^5}+\frac{9 e \sqrt{d+e x}}{40 (b d-a e)^2 (a+b x)^4}-\frac{21 e^2 \sqrt{d+e x}}{80 (b d-a e)^3 (a+b x)^3}-\frac{\left (21 e^3\right ) \int \frac{1}{(a+b x)^3 \sqrt{d+e x}} \, dx}{32 (b d-a e)^3}\\ &=-\frac{\sqrt{d+e x}}{5 (b d-a e) (a+b x)^5}+\frac{9 e \sqrt{d+e x}}{40 (b d-a e)^2 (a+b x)^4}-\frac{21 e^2 \sqrt{d+e x}}{80 (b d-a e)^3 (a+b x)^3}+\frac{21 e^3 \sqrt{d+e x}}{64 (b d-a e)^4 (a+b x)^2}+\frac{\left (63 e^4\right ) \int \frac{1}{(a+b x)^2 \sqrt{d+e x}} \, dx}{128 (b d-a e)^4}\\ &=-\frac{\sqrt{d+e x}}{5 (b d-a e) (a+b x)^5}+\frac{9 e \sqrt{d+e x}}{40 (b d-a e)^2 (a+b x)^4}-\frac{21 e^2 \sqrt{d+e x}}{80 (b d-a e)^3 (a+b x)^3}+\frac{21 e^3 \sqrt{d+e x}}{64 (b d-a e)^4 (a+b x)^2}-\frac{63 e^4 \sqrt{d+e x}}{128 (b d-a e)^5 (a+b x)}-\frac{\left (63 e^5\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{256 (b d-a e)^5}\\ &=-\frac{\sqrt{d+e x}}{5 (b d-a e) (a+b x)^5}+\frac{9 e \sqrt{d+e x}}{40 (b d-a e)^2 (a+b x)^4}-\frac{21 e^2 \sqrt{d+e x}}{80 (b d-a e)^3 (a+b x)^3}+\frac{21 e^3 \sqrt{d+e x}}{64 (b d-a e)^4 (a+b x)^2}-\frac{63 e^4 \sqrt{d+e x}}{128 (b d-a e)^5 (a+b x)}-\frac{\left (63 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 d-a e)^5}\\ &=-\frac{\sqrt{d+e x}}{5 (b d-a e) (a+b x)^5}+\frac{9 e \sqrt{d+e x}}{40 (b d-a e)^2 (a+b x)^4}-\frac{21 e^2 \sqrt{d+e x}}{80 (b d-a e)^3 (a+b x)^3}+\frac{21 e^3 \sqrt{d+e x}}{64 (b d-a e)^4 (a+b x)^2}-\frac{63 e^4 \sqrt{d+e x}}{128 (b d-a e)^5 (a+b x)}+\frac{63 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 \sqrt{b} (b d-a e)^{11/2}}\\ \end{align*}
Mathematica [C] time = 0.0132862, size = 50, normalized size = 0.23 \[ \frac{2 e^5 \sqrt{d+e x} \, _2F_1\left (\frac{1}{2},6;\frac{3}{2};-\frac{b (d+e x)}{a e-b d}\right )}{(a e-b d)^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.198, size = 211, normalized size = 1. \begin{align*}{\frac{{e}^{5}}{ \left ( 5\,ae-5\,bd \right ) \left ( bxe+ae \right ) ^{5}}\sqrt{ex+d}}+{\frac{9\,{e}^{5}}{40\, \left ( ae-bd \right ) ^{2} \left ( bxe+ae \right ) ^{4}}\sqrt{ex+d}}+{\frac{21\,{e}^{5}}{80\, \left ( ae-bd \right ) ^{3} \left ( bxe+ae \right ) ^{3}}\sqrt{ex+d}}+{\frac{21\,{e}^{5}}{64\, \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{63\,{e}^{5}}{128\, \left ( ae-bd \right ) ^{5} \left ( bxe+ae \right ) }\sqrt{ex+d}}+{\frac{63\,{e}^{5}}{128\, \left ( ae-bd \right ) ^{5}}\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 = 1.91918, size = 3794, normalized size = 17.81 \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 = 1.19648, size = 613, normalized size = 2.88 \begin{align*} -\frac{63 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{5}}{128 \,{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \sqrt{-b^{2} d + a b e}} - \frac{315 \,{\left (x e + d\right )}^{\frac{9}{2}} b^{4} e^{5} - 1470 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{4} d e^{5} + 2688 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} d^{2} e^{5} - 2370 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d^{3} e^{5} + 965 \, \sqrt{x e + d} b^{4} d^{4} e^{5} + 1470 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{3} e^{6} - 5376 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{3} d e^{6} + 7110 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} d^{2} e^{6} - 3860 \, \sqrt{x e + d} a b^{3} d^{3} e^{6} + 2688 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{2} e^{7} - 7110 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{2} d e^{7} + 5790 \, \sqrt{x e + d} a^{2} b^{2} d^{2} e^{7} + 2370 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b e^{8} - 3860 \, \sqrt{x e + d} a^{3} b d e^{8} + 965 \, \sqrt{x e + d} a^{4} e^{9}}{640 \,{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\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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