Optimal. Leaf size=126 \[ -\frac{5 e^2 \sqrt{d+e x}}{8 b^3 (a+b x)}-\frac{5 e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{7/2} \sqrt{b d-a e}}-\frac{5 e (d+e x)^{3/2}}{12 b^2 (a+b x)^2}-\frac{(d+e x)^{5/2}}{3 b (a+b x)^3} \]
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Rubi [A] time = 0.0593683, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {27, 47, 63, 208} \[ -\frac{5 e^2 \sqrt{d+e x}}{8 b^3 (a+b x)}-\frac{5 e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{7/2} \sqrt{b d-a e}}-\frac{5 e (d+e x)^{3/2}}{12 b^2 (a+b x)^2}-\frac{(d+e x)^{5/2}}{3 b (a+b x)^3} \]
Antiderivative was successfully verified.
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Rule 27
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{(d+e x)^{5/2}}{(a+b x)^4} \, dx\\ &=-\frac{(d+e x)^{5/2}}{3 b (a+b x)^3}+\frac{(5 e) \int \frac{(d+e x)^{3/2}}{(a+b x)^3} \, dx}{6 b}\\ &=-\frac{5 e (d+e x)^{3/2}}{12 b^2 (a+b x)^2}-\frac{(d+e x)^{5/2}}{3 b (a+b x)^3}+\frac{\left (5 e^2\right ) \int \frac{\sqrt{d+e x}}{(a+b x)^2} \, dx}{8 b^2}\\ &=-\frac{5 e^2 \sqrt{d+e x}}{8 b^3 (a+b x)}-\frac{5 e (d+e x)^{3/2}}{12 b^2 (a+b x)^2}-\frac{(d+e x)^{5/2}}{3 b (a+b x)^3}+\frac{\left (5 e^3\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{16 b^3}\\ &=-\frac{5 e^2 \sqrt{d+e x}}{8 b^3 (a+b x)}-\frac{5 e (d+e x)^{3/2}}{12 b^2 (a+b x)^2}-\frac{(d+e x)^{5/2}}{3 b (a+b x)^3}+\frac{\left (5 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^3}\\ &=-\frac{5 e^2 \sqrt{d+e x}}{8 b^3 (a+b x)}-\frac{5 e (d+e x)^{3/2}}{12 b^2 (a+b x)^2}-\frac{(d+e x)^{5/2}}{3 b (a+b x)^3}-\frac{5 e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{7/2} \sqrt{b d-a e}}\\ \end{align*}
Mathematica [A] time = 0.144229, size = 119, normalized size = 0.94 \[ \frac{5 e^3 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{a e-b d}}\right )}{8 b^{7/2} \sqrt{a e-b d}}-\frac{\sqrt{d+e x} \left (15 a^2 e^2+10 a b e (d+4 e x)+b^2 \left (8 d^2+26 d e x+33 e^2 x^2\right )\right )}{24 b^3 (a+b x)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.2, size = 204, normalized size = 1.6 \begin{align*} -{\frac{11\,{e}^{3}}{8\, \left ( bxe+ae \right ) ^{3}b} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{e}^{4}a}{3\, \left ( bxe+ae \right ) ^{3}{b}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{e}^{3}d}{3\, \left ( bxe+ae \right ) ^{3}b} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{a}^{2}{e}^{5}}{8\, \left ( bxe+ae \right ) ^{3}{b}^{3}}\sqrt{ex+d}}+{\frac{5\,{e}^{4}ad}{4\, \left ( bxe+ae \right ) ^{3}{b}^{2}}\sqrt{ex+d}}-{\frac{5\,{e}^{3}{d}^{2}}{8\, \left ( bxe+ae \right ) ^{3}b}\sqrt{ex+d}}+{\frac{5\,{e}^{3}}{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 = 2.02755, size = 1162, normalized size = 9.22 \begin{align*} \left [\frac{15 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \sqrt{b^{2} d - a b e} \log \left (\frac{b e x + 2 \, b d - a e - 2 \, \sqrt{b^{2} d - a b e} \sqrt{e x + d}}{b x + a}\right ) - 2 \,{\left (8 \, b^{4} d^{3} + 2 \, a b^{3} d^{2} e + 5 \, a^{2} b^{2} d e^{2} - 15 \, a^{3} b e^{3} + 33 \,{\left (b^{4} d e^{2} - a b^{3} e^{3}\right )} x^{2} + 2 \,{\left (13 \, b^{4} d^{2} e + 7 \, a b^{3} d e^{2} - 20 \, a^{2} b^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{48 \,{\left (a^{3} b^{5} d - a^{4} b^{4} e +{\left (b^{8} d - a b^{7} e\right )} x^{3} + 3 \,{\left (a b^{7} d - a^{2} b^{6} e\right )} x^{2} + 3 \,{\left (a^{2} b^{6} d - a^{3} b^{5} e\right )} x\right )}}, \frac{15 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \sqrt{-b^{2} d + a b e} \arctan \left (\frac{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}{b e x + b d}\right ) -{\left (8 \, b^{4} d^{3} + 2 \, a b^{3} d^{2} e + 5 \, a^{2} b^{2} d e^{2} - 15 \, a^{3} b e^{3} + 33 \,{\left (b^{4} d e^{2} - a b^{3} e^{3}\right )} x^{2} + 2 \,{\left (13 \, b^{4} d^{2} e + 7 \, a b^{3} d e^{2} - 20 \, a^{2} b^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{24 \,{\left (a^{3} b^{5} d - a^{4} b^{4} e +{\left (b^{8} d - a b^{7} e\right )} x^{3} + 3 \,{\left (a b^{7} d - a^{2} b^{6} e\right )} x^{2} + 3 \,{\left (a^{2} b^{6} d - a^{3} b^{5} e\right )} x\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 [A] time = 1.25548, size = 223, normalized size = 1.77 \begin{align*} \frac{5 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{3}}{8 \, \sqrt{-b^{2} d + a b e} b^{3}} - \frac{33 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{2} e^{3} - 40 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} d e^{3} + 15 \, \sqrt{x e + d} b^{2} d^{2} e^{3} + 40 \,{\left (x e + d\right )}^{\frac{3}{2}} a b e^{4} - 30 \, \sqrt{x e + d} a b d e^{4} + 15 \, \sqrt{x e + d} a^{2} e^{5}}{24 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{3} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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