Optimal. Leaf size=173 \[ -\frac{35 e^3}{8 \sqrt{d+e x} (b d-a e)^4}-\frac{35 e^2}{24 (a+b x) \sqrt{d+e x} (b d-a e)^3}+\frac{35 \sqrt{b} e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{9/2}}+\frac{7 e}{12 (a+b x)^2 \sqrt{d+e x} (b d-a e)^2}-\frac{1}{3 (a+b x)^3 \sqrt{d+e x} (b d-a e)} \]
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Rubi [A] time = 0.0835521, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {27, 51, 63, 208} \[ -\frac{35 e^3}{8 \sqrt{d+e x} (b d-a e)^4}-\frac{35 e^2}{24 (a+b x) \sqrt{d+e x} (b d-a e)^3}+\frac{35 \sqrt{b} e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{9/2}}+\frac{7 e}{12 (a+b x)^2 \sqrt{d+e x} (b d-a e)^2}-\frac{1}{3 (a+b x)^3 \sqrt{d+e x} (b d-a e)} \]
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)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{1}{(a+b x)^4 (d+e x)^{3/2}} \, dx\\ &=-\frac{1}{3 (b d-a e) (a+b x)^3 \sqrt{d+e x}}-\frac{(7 e) \int \frac{1}{(a+b x)^3 (d+e x)^{3/2}} \, dx}{6 (b d-a e)}\\ &=-\frac{1}{3 (b d-a e) (a+b x)^3 \sqrt{d+e x}}+\frac{7 e}{12 (b d-a e)^2 (a+b x)^2 \sqrt{d+e x}}+\frac{\left (35 e^2\right ) \int \frac{1}{(a+b x)^2 (d+e x)^{3/2}} \, dx}{24 (b d-a e)^2}\\ &=-\frac{1}{3 (b d-a e) (a+b x)^3 \sqrt{d+e x}}+\frac{7 e}{12 (b d-a e)^2 (a+b x)^2 \sqrt{d+e x}}-\frac{35 e^2}{24 (b d-a e)^3 (a+b x) \sqrt{d+e x}}-\frac{\left (35 e^3\right ) \int \frac{1}{(a+b x) (d+e x)^{3/2}} \, dx}{16 (b d-a e)^3}\\ &=-\frac{35 e^3}{8 (b d-a e)^4 \sqrt{d+e x}}-\frac{1}{3 (b d-a e) (a+b x)^3 \sqrt{d+e x}}+\frac{7 e}{12 (b d-a e)^2 (a+b x)^2 \sqrt{d+e x}}-\frac{35 e^2}{24 (b d-a e)^3 (a+b x) \sqrt{d+e x}}-\frac{\left (35 b e^3\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{16 (b d-a e)^4}\\ &=-\frac{35 e^3}{8 (b d-a e)^4 \sqrt{d+e x}}-\frac{1}{3 (b d-a e) (a+b x)^3 \sqrt{d+e x}}+\frac{7 e}{12 (b d-a e)^2 (a+b x)^2 \sqrt{d+e x}}-\frac{35 e^2}{24 (b d-a e)^3 (a+b x) \sqrt{d+e x}}-\frac{\left (35 b 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)^4}\\ &=-\frac{35 e^3}{8 (b d-a e)^4 \sqrt{d+e x}}-\frac{1}{3 (b d-a e) (a+b x)^3 \sqrt{d+e x}}+\frac{7 e}{12 (b d-a e)^2 (a+b x)^2 \sqrt{d+e x}}-\frac{35 e^2}{24 (b d-a e)^3 (a+b x) \sqrt{d+e x}}+\frac{35 \sqrt{b} e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.0147443, size = 50, normalized size = 0.29 \[ -\frac{2 e^3 \, _2F_1\left (-\frac{1}{2},4;\frac{1}{2};-\frac{b (d+e x)}{a e-b d}\right )}{\sqrt{d+e x} (a e-b d)^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.21, size = 292, normalized size = 1.7 \begin{align*} -2\,{\frac{{e}^{3}}{ \left ( ae-bd \right ) ^{4}\sqrt{ex+d}}}-{\frac{19\,{b}^{3}{e}^{3}}{8\, \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{17\,{e}^{4}{b}^{2}a}{3\, \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{17\,{b}^{3}{e}^{3}d}{3\, \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{29\,{e}^{5}b{a}^{2}}{8\, \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{3}}\sqrt{ex+d}}+{\frac{29\,{e}^{4}{b}^{2}ad}{4\, \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{3}}\sqrt{ex+d}}-{\frac{29\,{b}^{3}{e}^{3}{d}^{2}}{8\, \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{3}}\sqrt{ex+d}}-{\frac{35\,{e}^{3}b}{8\, \left ( ae-bd \right ) ^{4}}\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 = 2.46893, size = 2441, normalized size = 14.11 \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 = 1.22019, size = 437, normalized size = 2.53 \begin{align*} -\frac{35 \, b \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{3}}{8 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \sqrt{-b^{2} d + a b e}} - \frac{2 \, e^{3}}{{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \sqrt{x e + d}} - \frac{57 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{3} e^{3} - 136 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} d e^{3} + 87 \, \sqrt{x e + d} b^{3} d^{2} e^{3} + 136 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{2} e^{4} - 174 \, \sqrt{x e + d} a b^{2} d e^{4} + 87 \, \sqrt{x e + d} a^{2} b e^{5}}{24 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\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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