Optimal. Leaf size=99 \[ -\frac{3 e}{\sqrt{d+e x} (b d-a e)^2}-\frac{1}{(a+b x) \sqrt{d+e x} (b d-a e)}+\frac{3 \sqrt{b} e \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{5/2}} \]
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Rubi [A] time = 0.0539409, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {27, 51, 63, 208} \[ -\frac{3 e}{\sqrt{d+e x} (b d-a e)^2}-\frac{1}{(a+b x) \sqrt{d+e x} (b d-a e)}+\frac{3 \sqrt{b} e \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{5/2}} \]
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 )} \, dx &=\int \frac{1}{(a+b x)^2 (d+e x)^{3/2}} \, dx\\ &=-\frac{1}{(b d-a e) (a+b x) \sqrt{d+e x}}-\frac{(3 e) \int \frac{1}{(a+b x) (d+e x)^{3/2}} \, dx}{2 (b d-a e)}\\ &=-\frac{3 e}{(b d-a e)^2 \sqrt{d+e x}}-\frac{1}{(b d-a e) (a+b x) \sqrt{d+e x}}-\frac{(3 b e) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{2 (b d-a e)^2}\\ &=-\frac{3 e}{(b d-a e)^2 \sqrt{d+e x}}-\frac{1}{(b d-a e) (a+b x) \sqrt{d+e x}}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b d}{e}+\frac{b x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{(b d-a e)^2}\\ &=-\frac{3 e}{(b d-a e)^2 \sqrt{d+e x}}-\frac{1}{(b d-a e) (a+b x) \sqrt{d+e x}}+\frac{3 \sqrt{b} e \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0131378, size = 48, normalized size = 0.48 \[ -\frac{2 e \, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};-\frac{b (d+e x)}{a e-b d}\right )}{\sqrt{d+e x} (a e-b d)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.207, size = 101, normalized size = 1. \begin{align*} -2\,{\frac{e}{ \left ( ae-bd \right ) ^{2}\sqrt{ex+d}}}-{\frac{be}{ \left ( ae-bd \right ) ^{2} \left ( bxe+ae \right ) }\sqrt{ex+d}}-3\,{\frac{be}{ \left ( ae-bd \right ) ^{2}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) } \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 = 2.01781, size = 883, normalized size = 8.92 \begin{align*} \left [\frac{3 \,{\left (b e^{2} x^{2} + a d e +{\left (b d e + a e^{2}\right )} x\right )} \sqrt{\frac{b}{b d - a e}} \log \left (\frac{b e x + 2 \, b d - a e + 2 \,{\left (b d - a e\right )} \sqrt{e x + d} \sqrt{\frac{b}{b d - a e}}}{b x + a}\right ) - 2 \,{\left (3 \, b e x + b d + 2 \, a e\right )} \sqrt{e x + d}}{2 \,{\left (a b^{2} d^{3} - 2 \, a^{2} b d^{2} e + a^{3} d e^{2} +{\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x^{2} +{\left (b^{3} d^{3} - a b^{2} d^{2} e - a^{2} b d e^{2} + a^{3} e^{3}\right )} x\right )}}, \frac{3 \,{\left (b e^{2} x^{2} + a d e +{\left (b d e + a e^{2}\right )} x\right )} \sqrt{-\frac{b}{b d - a e}} \arctan \left (-\frac{{\left (b d - a e\right )} \sqrt{e x + d} \sqrt{-\frac{b}{b d - a e}}}{b e x + b d}\right ) -{\left (3 \, b e x + b d + 2 \, a e\right )} \sqrt{e x + d}}{a b^{2} d^{3} - 2 \, a^{2} b d^{2} e + a^{3} d e^{2} +{\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x^{2} +{\left (b^{3} d^{3} - a b^{2} d^{2} e - a^{2} b d e^{2} + a^{3} e^{3}\right )} x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x\right )^{2} \left (d + e x\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18709, size = 207, normalized size = 2.09 \begin{align*} -\frac{3 \, b \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e}{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \sqrt{-b^{2} d + a b e}} - \frac{3 \,{\left (x e + d\right )} b e - 2 \, b d e + 2 \, a e^{2}}{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )}{\left ({\left (x e + d\right )}^{\frac{3}{2}} b - \sqrt{x e + d} b d + \sqrt{x e + d} a e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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