Optimal. Leaf size=145 \[ -\frac{35 e^2 (d+e x)^{3/2}}{24 b^3 (a+b x)}-\frac{35 e^3 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{9/2}}-\frac{7 e (d+e x)^{5/2}}{12 b^2 (a+b x)^2}-\frac{(d+e x)^{7/2}}{3 b (a+b x)^3}+\frac{35 e^3 \sqrt{d+e x}}{8 b^4} \]
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Rubi [A] time = 0.0692103, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{35 e^2 (d+e x)^{3/2}}{24 b^3 (a+b x)}-\frac{35 e^3 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{9/2}}-\frac{7 e (d+e x)^{5/2}}{12 b^2 (a+b x)^2}-\frac{(d+e x)^{7/2}}{3 b (a+b x)^3}+\frac{35 e^3 \sqrt{d+e x}}{8 b^4} \]
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)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{(d+e x)^{7/2}}{(a+b x)^4} \, dx\\ &=-\frac{(d+e x)^{7/2}}{3 b (a+b x)^3}+\frac{(7 e) \int \frac{(d+e x)^{5/2}}{(a+b x)^3} \, dx}{6 b}\\ &=-\frac{7 e (d+e x)^{5/2}}{12 b^2 (a+b x)^2}-\frac{(d+e x)^{7/2}}{3 b (a+b x)^3}+\frac{\left (35 e^2\right ) \int \frac{(d+e x)^{3/2}}{(a+b x)^2} \, dx}{24 b^2}\\ &=-\frac{35 e^2 (d+e x)^{3/2}}{24 b^3 (a+b x)}-\frac{7 e (d+e x)^{5/2}}{12 b^2 (a+b x)^2}-\frac{(d+e x)^{7/2}}{3 b (a+b x)^3}+\frac{\left (35 e^3\right ) \int \frac{\sqrt{d+e x}}{a+b x} \, dx}{16 b^3}\\ &=\frac{35 e^3 \sqrt{d+e x}}{8 b^4}-\frac{35 e^2 (d+e x)^{3/2}}{24 b^3 (a+b x)}-\frac{7 e (d+e x)^{5/2}}{12 b^2 (a+b x)^2}-\frac{(d+e x)^{7/2}}{3 b (a+b x)^3}+\frac{\left (35 e^3 (b d-a e)\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{16 b^4}\\ &=\frac{35 e^3 \sqrt{d+e x}}{8 b^4}-\frac{35 e^2 (d+e x)^{3/2}}{24 b^3 (a+b x)}-\frac{7 e (d+e x)^{5/2}}{12 b^2 (a+b x)^2}-\frac{(d+e x)^{7/2}}{3 b (a+b x)^3}+\frac{\left (35 e^2 (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 )}{8 b^4}\\ &=\frac{35 e^3 \sqrt{d+e x}}{8 b^4}-\frac{35 e^2 (d+e x)^{3/2}}{24 b^3 (a+b x)}-\frac{7 e (d+e x)^{5/2}}{12 b^2 (a+b x)^2}-\frac{(d+e x)^{7/2}}{3 b (a+b x)^3}-\frac{35 e^3 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.0186881, size = 52, normalized size = 0.36 \[ \frac{2 e^3 (d+e x)^{9/2} \, _2F_1\left (4,\frac{9}{2};\frac{11}{2};-\frac{b (d+e x)}{a e-b d}\right )}{9 (a e-b d)^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.208, size = 352, normalized size = 2.4 \begin{align*} 2\,{\frac{{e}^{3}\sqrt{ex+d}}{{b}^{4}}}+{\frac{29\,{e}^{4}a}{8\,{b}^{2} \left ( bxe+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{29\,{e}^{3}d}{8\,b \left ( bxe+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{17\,{a}^{2}{e}^{5}}{3\,{b}^{3} \left ( bxe+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{34\,{e}^{4}ad}{3\,{b}^{2} \left ( bxe+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{17\,{e}^{3}{d}^{2}}{3\,b \left ( bxe+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{19\,{a}^{3}{e}^{6}}{8\,{b}^{4} \left ( bxe+ae \right ) ^{3}}\sqrt{ex+d}}-{\frac{57\,{e}^{5}d{a}^{2}}{8\,{b}^{3} \left ( bxe+ae \right ) ^{3}}\sqrt{ex+d}}+{\frac{57\,{e}^{4}a{d}^{2}}{8\,{b}^{2} \left ( bxe+ae \right ) ^{3}}\sqrt{ex+d}}-{\frac{19\,{e}^{3}{d}^{3}}{8\,b \left ( bxe+ae \right ) ^{3}}\sqrt{ex+d}}-{\frac{35\,{e}^{4}a}{8\,{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{35\,{e}^{3}d}{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 = 1.94319, size = 1065, normalized size = 7.34 \begin{align*} \left [\frac{105 \,{\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{\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 (48 \, b^{3} e^{3} x^{3} - 8 \, b^{3} d^{3} - 14 \, a b^{2} d^{2} e - 35 \, a^{2} b d e^{2} + 105 \, a^{3} e^{3} - 3 \,{\left (29 \, b^{3} d e^{2} - 77 \, a b^{2} e^{3}\right )} x^{2} - 2 \,{\left (19 \, b^{3} d^{2} e + 49 \, a b^{2} d e^{2} - 140 \, a^{2} b e^{3}\right )} x\right )} \sqrt{e x + d}}{48 \,{\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}}, -\frac{105 \,{\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{-\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 (48 \, b^{3} e^{3} x^{3} - 8 \, b^{3} d^{3} - 14 \, a b^{2} d^{2} e - 35 \, a^{2} b d e^{2} + 105 \, a^{3} e^{3} - 3 \,{\left (29 \, b^{3} d e^{2} - 77 \, a b^{2} e^{3}\right )} x^{2} - 2 \,{\left (19 \, b^{3} d^{2} e + 49 \, a b^{2} d e^{2} - 140 \, a^{2} b e^{3}\right )} x\right )} \sqrt{e x + d}}{24 \,{\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\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 = 1.24008, size = 335, normalized size = 2.31 \begin{align*} \frac{35 \,{\left (b d e^{3} - a e^{4}\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^{4}} + \frac{2 \, \sqrt{x e + d} e^{3}}{b^{4}} - \frac{87 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{3} d e^{3} - 136 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} d^{2} e^{3} + 57 \, \sqrt{x e + d} b^{3} d^{3} e^{3} - 87 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{2} e^{4} + 272 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{2} d e^{4} - 171 \, \sqrt{x e + d} a b^{2} d^{2} e^{4} - 136 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b e^{5} + 171 \, \sqrt{x e + d} a^{2} b d e^{5} - 57 \, \sqrt{x e + d} a^{3} e^{6}}{24 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{3} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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