Optimal. Leaf size=168 \[ \frac{2 b (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}+\frac{2 (a+b x)}{3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}-\frac{2 b^{3/2} (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{5/2}} \]
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Rubi [A] time = 0.0708907, antiderivative size = 168, 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, 51, 63, 208} \[ \frac{2 b (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}+\frac{2 (a+b x)}{3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}-\frac{2 b^{3/2} (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{5/2}} \]
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)^{5/2} \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)^{5/2}} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 (a+b x)}{3 (b d-a e) (d+e x)^{3/2} \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 ) (d+e x)^{3/2}} \, dx}{(b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 (a+b x)}{3 (b d-a e) (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 b (a+b x)}{(b d-a e)^2 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (b^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 d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 (a+b x)}{3 (b d-a e) (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 b (a+b x)}{(b d-a e)^2 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (2 b^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 )}{e (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 (a+b x)}{3 (b d-a e) (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 b (a+b x)}{(b d-a e)^2 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 b^{3/2} (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [C] time = 0.0173199, size = 64, normalized size = 0.38 \[ \frac{2 (a+b x) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{b (d+e x)}{b d-a e}\right )}{3 \sqrt{(a+b x)^2} (d+e x)^{3/2} (b d-a e)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.249, size = 130, normalized size = 0.8 \begin{align*}{\frac{2\,bx+2\,a}{3\, \left ( ae-bd \right ) ^{2}} \left ( 3\,{b}^{2}\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) \left ( ex+d \right ) ^{3/2}+3\,\sqrt{ \left ( ae-bd \right ) b}xbe-\sqrt{ \left ( ae-bd \right ) b}ae+4\,\sqrt{ \left ( ae-bd \right ) b}bd \right ) \left ( ex+d \right ) ^{-{\frac{3}{2}}}{\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{1}{\sqrt{{\left (b x + a\right )}^{2}}{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59893, size = 841, normalized size = 5.01 \begin{align*} \left [\frac{3 \,{\left (b e^{2} x^{2} + 2 \, b d e x + b d^{2}\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 + 4 \, b d - a e\right )} \sqrt{e x + d}}{3 \,{\left (b^{2} d^{4} - 2 \, a b d^{3} e + a^{2} d^{2} e^{2} +{\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} x^{2} + 2 \,{\left (b^{2} d^{3} e - 2 \, a b d^{2} e^{2} + a^{2} d e^{3}\right )} x\right )}}, -\frac{2 \,{\left (3 \,{\left (b e^{2} x^{2} + 2 \, b d e x + b d^{2}\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 + 4 \, b d - a e\right )} \sqrt{e x + d}\right )}}{3 \,{\left (b^{2} d^{4} - 2 \, a b d^{3} e + a^{2} d^{2} e^{2} +{\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} x^{2} + 2 \,{\left (b^{2} d^{3} e - 2 \, a b d^{2} e^{2} + a^{2} d e^{3}\right )} x\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 [A] time = 1.22782, size = 170, normalized size = 1.01 \begin{align*} \frac{2}{3} \,{\left (\frac{3 \, b^{2} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{{\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 + b d - a e}{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )}{\left (x e + d\right )}^{\frac{3}{2}}}\right )} \mathrm{sgn}\left (b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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