Optimal. Leaf size=229 \[ -\frac{231 b^2 e^3}{8 \sqrt{d+e x} (b d-a e)^6}+\frac{231 b^{5/2} e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{13/2}}-\frac{77 b e^3}{8 (d+e x)^{3/2} (b d-a e)^5}-\frac{231 e^3}{40 (d+e x)^{5/2} (b d-a e)^4}-\frac{33 e^2}{8 (a+b x) (d+e x)^{5/2} (b d-a e)^3}+\frac{11 e}{12 (a+b x)^2 (d+e x)^{5/2} (b d-a e)^2}-\frac{1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)} \]
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Rubi [A] time = 0.184573, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {27, 51, 63, 208} \[ -\frac{231 b^2 e^3}{8 \sqrt{d+e x} (b d-a e)^6}+\frac{231 b^{5/2} e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{13/2}}-\frac{77 b e^3}{8 (d+e x)^{3/2} (b d-a e)^5}-\frac{231 e^3}{40 (d+e x)^{5/2} (b d-a e)^4}-\frac{33 e^2}{8 (a+b x) (d+e x)^{5/2} (b d-a e)^3}+\frac{11 e}{12 (a+b x)^2 (d+e x)^{5/2} (b d-a e)^2}-\frac{1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{1}{(a+b x)^4 (d+e x)^{7/2}} \, dx\\ &=-\frac{1}{3 (b d-a e) (a+b x)^3 (d+e x)^{5/2}}-\frac{(11 e) \int \frac{1}{(a+b x)^3 (d+e x)^{7/2}} \, dx}{6 (b d-a e)}\\ &=-\frac{1}{3 (b d-a e) (a+b x)^3 (d+e x)^{5/2}}+\frac{11 e}{12 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}+\frac{\left (33 e^2\right ) \int \frac{1}{(a+b x)^2 (d+e x)^{7/2}} \, dx}{8 (b d-a e)^2}\\ &=-\frac{1}{3 (b d-a e) (a+b x)^3 (d+e x)^{5/2}}+\frac{11 e}{12 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}-\frac{33 e^2}{8 (b d-a e)^3 (a+b x) (d+e x)^{5/2}}-\frac{\left (231 e^3\right ) \int \frac{1}{(a+b x) (d+e x)^{7/2}} \, dx}{16 (b d-a e)^3}\\ &=-\frac{231 e^3}{40 (b d-a e)^4 (d+e x)^{5/2}}-\frac{1}{3 (b d-a e) (a+b x)^3 (d+e x)^{5/2}}+\frac{11 e}{12 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}-\frac{33 e^2}{8 (b d-a e)^3 (a+b x) (d+e x)^{5/2}}-\frac{\left (231 b e^3\right ) \int \frac{1}{(a+b x) (d+e x)^{5/2}} \, dx}{16 (b d-a e)^4}\\ &=-\frac{231 e^3}{40 (b d-a e)^4 (d+e x)^{5/2}}-\frac{1}{3 (b d-a e) (a+b x)^3 (d+e x)^{5/2}}+\frac{11 e}{12 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}-\frac{33 e^2}{8 (b d-a e)^3 (a+b x) (d+e x)^{5/2}}-\frac{77 b e^3}{8 (b d-a e)^5 (d+e x)^{3/2}}-\frac{\left (231 b^2 e^3\right ) \int \frac{1}{(a+b x) (d+e x)^{3/2}} \, dx}{16 (b d-a e)^5}\\ &=-\frac{231 e^3}{40 (b d-a e)^4 (d+e x)^{5/2}}-\frac{1}{3 (b d-a e) (a+b x)^3 (d+e x)^{5/2}}+\frac{11 e}{12 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}-\frac{33 e^2}{8 (b d-a e)^3 (a+b x) (d+e x)^{5/2}}-\frac{77 b e^3}{8 (b d-a e)^5 (d+e x)^{3/2}}-\frac{231 b^2 e^3}{8 (b d-a e)^6 \sqrt{d+e x}}-\frac{\left (231 b^3 e^3\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{16 (b d-a e)^6}\\ &=-\frac{231 e^3}{40 (b d-a e)^4 (d+e x)^{5/2}}-\frac{1}{3 (b d-a e) (a+b x)^3 (d+e x)^{5/2}}+\frac{11 e}{12 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}-\frac{33 e^2}{8 (b d-a e)^3 (a+b x) (d+e x)^{5/2}}-\frac{77 b e^3}{8 (b d-a e)^5 (d+e x)^{3/2}}-\frac{231 b^2 e^3}{8 (b d-a e)^6 \sqrt{d+e x}}-\frac{\left (231 b^3 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)^6}\\ &=-\frac{231 e^3}{40 (b d-a e)^4 (d+e x)^{5/2}}-\frac{1}{3 (b d-a e) (a+b x)^3 (d+e x)^{5/2}}+\frac{11 e}{12 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}-\frac{33 e^2}{8 (b d-a e)^3 (a+b x) (d+e x)^{5/2}}-\frac{77 b e^3}{8 (b d-a e)^5 (d+e x)^{3/2}}-\frac{231 b^2 e^3}{8 (b d-a e)^6 \sqrt{d+e x}}+\frac{231 b^{5/2} e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{13/2}}\\ \end{align*}
Mathematica [C] time = 0.021571, size = 52, normalized size = 0.23 \[ -\frac{2 e^3 \, _2F_1\left (-\frac{5}{2},4;-\frac{3}{2};-\frac{b (d+e x)}{a e-b d}\right )}{5 (d+e x)^{5/2} (a e-b d)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.212, size = 344, normalized size = 1.5 \begin{align*} -{\frac{2\,{e}^{3}}{5\, \left ( ae-bd \right ) ^{4}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}}-20\,{\frac{{e}^{3}{b}^{2}}{ \left ( ae-bd \right ) ^{6}\sqrt{ex+d}}}+{\frac{8\,{e}^{3}b}{3\, \left ( ae-bd \right ) ^{5}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}-{\frac{71\,{e}^{3}{b}^{5}}{8\, \left ( ae-bd \right ) ^{6} \left ( bxe+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{59\,{b}^{4}{e}^{4}a}{3\, \left ( ae-bd \right ) ^{6} \left ( bxe+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{59\,{e}^{3}{b}^{5}d}{3\, \left ( ae-bd \right ) ^{6} \left ( bxe+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{89\,{e}^{5}{b}^{3}{a}^{2}}{8\, \left ( ae-bd \right ) ^{6} \left ( bxe+ae \right ) ^{3}}\sqrt{ex+d}}+{\frac{89\,{b}^{4}{e}^{4}ad}{4\, \left ( ae-bd \right ) ^{6} \left ( bxe+ae \right ) ^{3}}\sqrt{ex+d}}-{\frac{89\,{e}^{3}{b}^{5}{d}^{2}}{8\, \left ( ae-bd \right ) ^{6} \left ( bxe+ae \right ) ^{3}}\sqrt{ex+d}}-{\frac{231\,{b}^{3}{e}^{3}}{8\, \left ( ae-bd \right ) ^{6}}\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 = 2.77801, size = 5176, normalized size = 22.6 \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 = 1.20805, size = 635, normalized size = 2.77 \begin{align*} -\frac{231 \, b^{3} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{3}}{8 \,{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \sqrt{-b^{2} d + a b e}} - \frac{2 \,{\left (150 \,{\left (x e + d\right )}^{2} b^{2} e^{3} + 20 \,{\left (x e + d\right )} b^{2} d e^{3} + 3 \, b^{2} d^{2} e^{3} - 20 \,{\left (x e + d\right )} a b e^{4} - 6 \, a b d e^{4} + 3 \, a^{2} e^{5}\right )}}{15 \,{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )}{\left (x e + d\right )}^{\frac{5}{2}}} - \frac{213 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{5} e^{3} - 472 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{5} d e^{3} + 267 \, \sqrt{x e + d} b^{5} d^{2} e^{3} + 472 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{4} e^{4} - 534 \, \sqrt{x e + d} a b^{4} d e^{4} + 267 \, \sqrt{x e + d} a^{2} b^{3} e^{5}}{24 \,{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\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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