Optimal. Leaf size=161 \[ \frac{2 (a+b x) \sqrt{d+e x} (b d-a e)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (d+e x)^{3/2}}{3 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 (a+b x) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.0780594, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {646, 50, 63, 208} \[ \frac{2 (a+b x) \sqrt{d+e x} (b d-a e)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (d+e x)^{3/2}}{3 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 (a+b x) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 646
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(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{(d+e x)^{3/2}}{a b+b^2 x} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 (a+b x) (d+e x)^{3/2}}{3 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (\left (b^2 d-a b e\right ) \left (a b+b^2 x\right )\right ) \int \frac{\sqrt{d+e x}}{a b+b^2 x} \, dx}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 (b d-a e) (a+b x) \sqrt{d+e x}}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (d+e x)^{3/2}}{3 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (\left (b^2 d-a b e\right )^2 \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right ) \sqrt{d+e x}} \, dx}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 (b d-a e) (a+b x) \sqrt{d+e x}}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (d+e x)^{3/2}}{3 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (2 \left (b^2 d-a b e\right )^2 \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 )}{b^4 e \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 (b d-a e) (a+b x) \sqrt{d+e x}}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (d+e x)^{3/2}}{3 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 (b d-a e)^{3/2} (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0552392, size = 96, normalized size = 0.6 \[ \frac{2 (a+b x) \left (\sqrt{b} \sqrt{d+e x} (-3 a e+4 b d+b e x)-3 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )\right )}{3 b^{5/2} \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.233, size = 188, normalized size = 1.2 \begin{align*}{\frac{2\,bx+2\,a}{3\,{b}^{2}} \left ( \sqrt{ \left ( ae-bd \right ) b} \left ( ex+d \right ) ^{{\frac{3}{2}}}b+3\,\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ){a}^{2}{e}^{2}-6\,\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) abde+3\,\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ){b}^{2}{d}^{2}-3\,\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}ae+3\,\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}bd \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{\frac{3}{2}}}{\sqrt{{\left (b x + a\right )}^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68801, size = 424, normalized size = 2.63 \begin{align*} \left [-\frac{3 \,{\left (b d - a e\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 (b e x + 4 \, b d - 3 \, a e\right )} \sqrt{e x + d}}{3 \, b^{2}}, -\frac{2 \,{\left (3 \,{\left (b d - a e\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 (b e x + 4 \, b d - 3 \, a e\right )} \sqrt{e x + d}\right )}}{3 \, b^{2}}\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 = 1.19885, size = 200, normalized size = 1.24 \begin{align*} \frac{2 \,{\left (b^{2} d^{2} \mathrm{sgn}\left (b x + a\right ) - 2 \, a b d e \mathrm{sgn}\left (b x + a\right ) + a^{2} e^{2} \mathrm{sgn}\left (b x + a\right )\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b^{2}} + \frac{2 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} b^{2} \mathrm{sgn}\left (b x + a\right ) + 3 \, \sqrt{x e + d} b^{2} d \mathrm{sgn}\left (b x + a\right ) - 3 \, \sqrt{x e + d} a b e \mathrm{sgn}\left (b x + a\right )\right )}}{3 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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