Optimal. Leaf size=119 \[ \frac{2 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}-\frac{2 \sqrt{b} (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{3/2}} \]
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Rubi [A] time = 0.0666665, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {646, 51, 63, 208} \[ \frac{2 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}-\frac{2 \sqrt{b} (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 646
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}} \, dx &=\frac{\left (a b+b^2 x\right ) \int \frac{1}{\left (a b+b^2 x\right ) (d+e x)^{3/2}} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 (a+b x)}{(b d-a e) \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (b \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right ) \sqrt{d+e x}} \, dx}{(b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 (a+b x)}{(b d-a e) \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (2 b \left (a b+b^2 x\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a b-\frac{b^2 d}{e}+\frac{b^2 x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{e (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 (a+b x)}{(b d-a e) \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 \sqrt{b} (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [C] time = 0.0165321, size = 62, normalized size = 0.52 \[ -\frac{2 (a+b x) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{b (d+e x)}{b d-a e}\right )}{\sqrt{(a+b x)^2} \sqrt{d+e x} (a e-b d)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.233, size = 90, normalized size = 0.8 \begin{align*} -2\,{\frac{bx+a}{\sqrt{ \left ( bx+a \right ) ^{2}} \left ( ae-bd \right ) \sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}} \left ( b\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{{\left (b x + a\right )}^{2}}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65611, size = 456, normalized size = 3.83 \begin{align*} \left [-\frac{{\left (e x + d\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 \, \sqrt{e x + d}}{b d^{2} - a d e +{\left (b d e - a e^{2}\right )} x}, -\frac{2 \,{\left ({\left (e x + d\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 ) - \sqrt{e x + d}\right )}}{b d^{2} - a d e +{\left (b d e - a e^{2}\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 (d + e x\right )^{\frac{3}{2}} \sqrt{\left (a + b x\right )^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17445, size = 109, normalized size = 0.92 \begin{align*} 2 \,{\left (\frac{b \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e}{\left (b d - a e\right )}} + \frac{1}{{\left (b d - a e\right )} \sqrt{x e + d}}\right )} \mathrm{sgn}\left (b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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