Optimal. Leaf size=172 \[ -\frac{21 e^2 (d+e x)^{5/2}}{8 b^3 (a+b x)}+\frac{105 e^3 \sqrt{d+e x} (b d-a e)}{8 b^5}-\frac{105 e^3 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{11/2}}-\frac{3 e (d+e x)^{7/2}}{4 b^2 (a+b x)^2}-\frac{(d+e x)^{9/2}}{3 b (a+b x)^3}+\frac{35 e^3 (d+e x)^{3/2}}{8 b^4} \]
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Rubi [A] time = 0.0973135, antiderivative size = 172, 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, 50, 63, 208} \[ -\frac{21 e^2 (d+e x)^{5/2}}{8 b^3 (a+b x)}+\frac{105 e^3 \sqrt{d+e x} (b d-a e)}{8 b^5}-\frac{105 e^3 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{11/2}}-\frac{3 e (d+e x)^{7/2}}{4 b^2 (a+b x)^2}-\frac{(d+e x)^{9/2}}{3 b (a+b x)^3}+\frac{35 e^3 (d+e x)^{3/2}}{8 b^4} \]
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)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{(d+e x)^{9/2}}{(a+b x)^4} \, dx\\ &=-\frac{(d+e x)^{9/2}}{3 b (a+b x)^3}+\frac{(3 e) \int \frac{(d+e x)^{7/2}}{(a+b x)^3} \, dx}{2 b}\\ &=-\frac{3 e (d+e x)^{7/2}}{4 b^2 (a+b x)^2}-\frac{(d+e x)^{9/2}}{3 b (a+b x)^3}+\frac{\left (21 e^2\right ) \int \frac{(d+e x)^{5/2}}{(a+b x)^2} \, dx}{8 b^2}\\ &=-\frac{21 e^2 (d+e x)^{5/2}}{8 b^3 (a+b x)}-\frac{3 e (d+e x)^{7/2}}{4 b^2 (a+b x)^2}-\frac{(d+e x)^{9/2}}{3 b (a+b x)^3}+\frac{\left (105 e^3\right ) \int \frac{(d+e x)^{3/2}}{a+b x} \, dx}{16 b^3}\\ &=\frac{35 e^3 (d+e x)^{3/2}}{8 b^4}-\frac{21 e^2 (d+e x)^{5/2}}{8 b^3 (a+b x)}-\frac{3 e (d+e x)^{7/2}}{4 b^2 (a+b x)^2}-\frac{(d+e x)^{9/2}}{3 b (a+b x)^3}+\frac{\left (105 e^3 (b d-a e)\right ) \int \frac{\sqrt{d+e x}}{a+b x} \, dx}{16 b^4}\\ &=\frac{105 e^3 (b d-a e) \sqrt{d+e x}}{8 b^5}+\frac{35 e^3 (d+e x)^{3/2}}{8 b^4}-\frac{21 e^2 (d+e x)^{5/2}}{8 b^3 (a+b x)}-\frac{3 e (d+e x)^{7/2}}{4 b^2 (a+b x)^2}-\frac{(d+e x)^{9/2}}{3 b (a+b x)^3}+\frac{\left (105 e^3 (b d-a e)^2\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{16 b^5}\\ &=\frac{105 e^3 (b d-a e) \sqrt{d+e x}}{8 b^5}+\frac{35 e^3 (d+e x)^{3/2}}{8 b^4}-\frac{21 e^2 (d+e x)^{5/2}}{8 b^3 (a+b x)}-\frac{3 e (d+e x)^{7/2}}{4 b^2 (a+b x)^2}-\frac{(d+e x)^{9/2}}{3 b (a+b x)^3}+\frac{\left (105 e^2 (b d-a 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^5}\\ &=\frac{105 e^3 (b d-a e) \sqrt{d+e x}}{8 b^5}+\frac{35 e^3 (d+e x)^{3/2}}{8 b^4}-\frac{21 e^2 (d+e x)^{5/2}}{8 b^3 (a+b x)}-\frac{3 e (d+e x)^{7/2}}{4 b^2 (a+b x)^2}-\frac{(d+e x)^{9/2}}{3 b (a+b x)^3}-\frac{105 e^3 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{11/2}}\\ \end{align*}
Mathematica [C] time = 0.0197256, size = 52, normalized size = 0.3 \[ \frac{2 e^3 (d+e x)^{11/2} \, _2F_1\left (4,\frac{11}{2};\frac{13}{2};-\frac{b (d+e x)}{a e-b d}\right )}{11 (a e-b d)^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.205, size = 525, normalized size = 3.1 \begin{align*}{\frac{2\,{e}^{3}}{3\,{b}^{4}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-8\,{\frac{{e}^{4}a\sqrt{ex+d}}{{b}^{5}}}+8\,{\frac{{e}^{3}\sqrt{ex+d}d}{{b}^{4}}}-{\frac{55\,{a}^{2}{e}^{5}}{8\,{b}^{3} \left ( bxe+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{55\,{e}^{4}ad}{4\,{b}^{2} \left ( bxe+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{55\,{e}^{3}{d}^{2}}{8\,b \left ( bxe+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{35\,{a}^{3}{e}^{6}}{3\,{b}^{4} \left ( bxe+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+35\,{\frac{{e}^{5} \left ( ex+d \right ) ^{3/2}{a}^{2}d}{{b}^{3} \left ( bxe+ae \right ) ^{3}}}-35\,{\frac{{e}^{4} \left ( ex+d \right ) ^{3/2}a{d}^{2}}{{b}^{2} \left ( bxe+ae \right ) ^{3}}}+{\frac{35\,{e}^{3}{d}^{3}}{3\,b \left ( bxe+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{41\,{e}^{7}{a}^{4}}{8\,{b}^{5} \left ( bxe+ae \right ) ^{3}}\sqrt{ex+d}}+{\frac{41\,{a}^{3}{e}^{6}d}{2\,{b}^{4} \left ( bxe+ae \right ) ^{3}}\sqrt{ex+d}}-{\frac{123\,{e}^{5}{d}^{2}{a}^{2}}{4\,{b}^{3} \left ( bxe+ae \right ) ^{3}}\sqrt{ex+d}}+{\frac{41\,{e}^{4}a{d}^{3}}{2\,{b}^{2} \left ( bxe+ae \right ) ^{3}}\sqrt{ex+d}}-{\frac{41\,{e}^{3}{d}^{4}}{8\,b \left ( bxe+ae \right ) ^{3}}\sqrt{ex+d}}+{\frac{105\,{a}^{2}{e}^{5}}{8\,{b}^{5}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}}-{\frac{105\,{e}^{4}ad}{4\,{b}^{4}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}}+{\frac{105\,{e}^{3}{d}^{2}}{8\,{b}^{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 = 1.90122, size = 1528, normalized size = 8.88 \begin{align*} \left [-\frac{315 \,{\left (a^{3} b d e^{3} - a^{4} e^{4} +{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 3 \,{\left (a b^{3} d e^{3} - a^{2} b^{2} e^{4}\right )} x^{2} + 3 \,{\left (a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e + 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) - 2 \,{\left (16 \, b^{4} e^{4} x^{4} - 8 \, b^{4} d^{4} - 18 \, a b^{3} d^{3} e - 63 \, a^{2} b^{2} d^{2} e^{2} + 420 \, a^{3} b d e^{3} - 315 \, a^{4} e^{4} + 16 \,{\left (13 \, b^{4} d e^{3} - 9 \, a b^{3} e^{4}\right )} x^{3} - 3 \,{\left (55 \, b^{4} d^{2} e^{2} - 318 \, a b^{3} d e^{3} + 231 \, a^{2} b^{2} e^{4}\right )} x^{2} - 2 \,{\left (25 \, b^{4} d^{3} e + 90 \, a b^{3} d^{2} e^{2} - 567 \, a^{2} b^{2} d e^{3} + 420 \, a^{3} b e^{4}\right )} x\right )} \sqrt{e x + d}}{48 \,{\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}}, -\frac{315 \,{\left (a^{3} b d e^{3} - a^{4} e^{4} +{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 3 \,{\left (a b^{3} d e^{3} - a^{2} b^{2} e^{4}\right )} x^{2} + 3 \,{\left (a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (-\frac{\sqrt{e x + d} b \sqrt{-\frac{b d - a e}{b}}}{b d - a e}\right ) -{\left (16 \, b^{4} e^{4} x^{4} - 8 \, b^{4} d^{4} - 18 \, a b^{3} d^{3} e - 63 \, a^{2} b^{2} d^{2} e^{2} + 420 \, a^{3} b d e^{3} - 315 \, a^{4} e^{4} + 16 \,{\left (13 \, b^{4} d e^{3} - 9 \, a b^{3} e^{4}\right )} x^{3} - 3 \,{\left (55 \, b^{4} d^{2} e^{2} - 318 \, a b^{3} d e^{3} + 231 \, a^{2} b^{2} e^{4}\right )} x^{2} - 2 \,{\left (25 \, b^{4} d^{3} e + 90 \, a b^{3} d^{2} e^{2} - 567 \, a^{2} b^{2} d e^{3} + 420 \, a^{3} b e^{4}\right )} x\right )} \sqrt{e x + d}}{24 \,{\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}}\right ] \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 = 1.24183, size = 486, normalized size = 2.83 \begin{align*} \frac{105 \,{\left (b^{2} d^{2} e^{3} - 2 \, a b d e^{4} + a^{2} e^{5}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{8 \, \sqrt{-b^{2} d + a b e} b^{5}} - \frac{165 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} d^{2} e^{3} - 280 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d^{3} e^{3} + 123 \, \sqrt{x e + d} b^{4} d^{4} e^{3} - 330 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{3} d e^{4} + 840 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} d^{2} e^{4} - 492 \, \sqrt{x e + d} a b^{3} d^{3} e^{4} + 165 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{2} e^{5} - 840 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{2} d e^{5} + 738 \, \sqrt{x e + d} a^{2} b^{2} d^{2} e^{5} + 280 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b e^{6} - 492 \, \sqrt{x e + d} a^{3} b d e^{6} + 123 \, \sqrt{x e + d} a^{4} e^{7}}{24 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{3} b^{5}} + \frac{2 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} b^{8} e^{3} + 12 \, \sqrt{x e + d} b^{8} d e^{3} - 12 \, \sqrt{x e + d} a b^{7} e^{4}\right )}}{3 \, b^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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