Optimal. Leaf size=197 \[ -\frac{33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac{231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac{231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac{693 e^5 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{13/2}}-\frac{11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac{693 e^5 \sqrt{d+e x}}{128 b^6} \]
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Rubi [A] time = 0.10289, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {27, 47, 50, 63, 208} \[ -\frac{33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac{231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac{231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac{693 e^5 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{13/2}}-\frac{11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac{693 e^5 \sqrt{d+e x}}{128 b^6} \]
Antiderivative was successfully verified.
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Rule 27
Rule 47
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{(d+e x)^{11/2}}{(a+b x)^6} \, dx\\ &=-\frac{(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac{(11 e) \int \frac{(d+e x)^{9/2}}{(a+b x)^5} \, dx}{10 b}\\ &=-\frac{11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac{\left (99 e^2\right ) \int \frac{(d+e x)^{7/2}}{(a+b x)^4} \, dx}{80 b^2}\\ &=-\frac{33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac{11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac{\left (231 e^3\right ) \int \frac{(d+e x)^{5/2}}{(a+b x)^3} \, dx}{160 b^3}\\ &=-\frac{231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac{33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac{11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac{\left (231 e^4\right ) \int \frac{(d+e x)^{3/2}}{(a+b x)^2} \, dx}{128 b^4}\\ &=-\frac{231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac{231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac{33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac{11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac{\left (693 e^5\right ) \int \frac{\sqrt{d+e x}}{a+b x} \, dx}{256 b^5}\\ &=\frac{693 e^5 \sqrt{d+e x}}{128 b^6}-\frac{231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac{231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac{33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac{11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac{\left (693 e^5 (b d-a e)\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{256 b^6}\\ &=\frac{693 e^5 \sqrt{d+e x}}{128 b^6}-\frac{231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac{231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac{33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac{11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac{\left (693 e^4 (b d-a e)\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^6}\\ &=\frac{693 e^5 \sqrt{d+e x}}{128 b^6}-\frac{231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac{231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac{33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac{11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{11/2}}{5 b (a+b x)^5}-\frac{693 e^5 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{13/2}}\\ \end{align*}
Mathematica [C] time = 0.022983, size = 52, normalized size = 0.26 \[ \frac{2 e^5 (d+e x)^{13/2} \, _2F_1\left (6,\frac{13}{2};\frac{15}{2};-\frac{b (d+e x)}{a e-b d}\right )}{13 (a e-b d)^6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.212, size = 673, normalized size = 3.4 \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(-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 = 2.22922, size = 1976, normalized size = 10.03 \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 = 1.34365, size = 620, normalized size = 3.15 \begin{align*} \frac{693 \,{\left (b d e^{5} - a e^{6}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{128 \, \sqrt{-b^{2} d + a b e} b^{6}} + \frac{2 \, \sqrt{x e + d} e^{5}}{b^{6}} - \frac{4215 \,{\left (x e + d\right )}^{\frac{9}{2}} b^{5} d e^{5} - 13270 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{5} d^{2} e^{5} + 16768 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{5} d^{3} e^{5} - 9770 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{5} d^{4} e^{5} + 2185 \, \sqrt{x e + d} b^{5} d^{5} e^{5} - 4215 \,{\left (x e + d\right )}^{\frac{9}{2}} a b^{4} e^{6} + 26540 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{4} d e^{6} - 50304 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{4} d^{2} e^{6} + 39080 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{4} d^{3} e^{6} - 10925 \, \sqrt{x e + d} a b^{4} d^{4} e^{6} - 13270 \,{\left (x e + d\right )}^{\frac{7}{2}} a^{2} b^{3} e^{7} + 50304 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{3} d e^{7} - 58620 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{3} d^{2} e^{7} + 21850 \, \sqrt{x e + d} a^{2} b^{3} d^{3} e^{7} - 16768 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{3} b^{2} e^{8} + 39080 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b^{2} d e^{8} - 21850 \, \sqrt{x e + d} a^{3} b^{2} d^{2} e^{8} - 9770 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{4} b e^{9} + 10925 \, \sqrt{x e + d} a^{4} b d e^{9} - 2185 \, \sqrt{x e + d} a^{5} e^{10}}{640 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{5} b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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