Optimal. Leaf size=217 \[ \frac{e^2 \sqrt{d+e x}}{8 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{e^3 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{5/2}}-\frac{e \sqrt{d+e x}}{12 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{\sqrt{d+e x}}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.14836, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {768, 646, 51, 63, 208} \[ \frac{e^2 \sqrt{d+e x}}{8 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{e^3 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{5/2}}-\frac{e \sqrt{d+e x}}{12 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{\sqrt{d+e x}}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 768
Rule 646
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x) \sqrt{d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=-\frac{\sqrt{d+e x}}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}+\frac{e \int \frac{1}{\sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx}{6 b}\\ &=-\frac{\sqrt{d+e x}}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}+\frac{\left (b e \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right )^3 \sqrt{d+e x}} \, dx}{6 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{\sqrt{d+e x}}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}-\frac{e \sqrt{d+e x}}{12 b (b d-a e) (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (e^2 \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right )^2 \sqrt{d+e x}} \, dx}{8 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{\sqrt{d+e x}}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}+\frac{e^2 \sqrt{d+e x}}{8 b (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e \sqrt{d+e x}}{12 b (b d-a e) (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (e^3 \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right ) \sqrt{d+e x}} \, dx}{16 b (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{\sqrt{d+e x}}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}+\frac{e^2 \sqrt{d+e x}}{8 b (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e \sqrt{d+e x}}{12 b (b d-a e) (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (e^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 )}{8 b (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{\sqrt{d+e x}}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}+\frac{e^2 \sqrt{d+e x}}{8 b (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e \sqrt{d+e x}}{12 b (b d-a e) (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e^3 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{3/2} (b d-a e)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [C] time = 0.0243075, size = 68, normalized size = 0.31 \[ \frac{2 e^3 (a+b x) (d+e x)^{3/2} \, _2F_1\left (\frac{3}{2},4;\frac{5}{2};-\frac{b (d+e x)}{a e-b d}\right )}{3 \sqrt{(a+b x)^2} (a e-b d)^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.018, size = 326, normalized size = 1.5 \begin{align*}{\frac{ \left ( bx+a \right ) ^{2}}{24\, \left ( ae-bd \right ) ^{2}b} \left ( 3\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ){x}^{3}{b}^{3}{e}^{3}+9\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ){x}^{2}a{b}^{2}{e}^{3}+3\,\sqrt{ \left ( ae-bd \right ) b} \left ( ex+d \right ) ^{5/2}{b}^{2}+9\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) x{a}^{2}b{e}^{3}+8\,\sqrt{ \left ( ae-bd \right ) b} \left ( ex+d \right ) ^{3/2}abe-8\,\sqrt{ \left ( ae-bd \right ) b} \left ( ex+d \right ) ^{3/2}{b}^{2}d+3\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ){a}^{3}{e}^{3}-3\,\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}{a}^{2}{e}^{2}+6\,\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}abde-3\,\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}{b}^{2}{d}^{2} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \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 (b x + a\right )} \sqrt{e x + d}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.09628, size = 1581, normalized size = 7.29 \begin{align*} \left [\frac{3 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \sqrt{b^{2} d - a b e} \log \left (\frac{b e x + 2 \, b d - a e - 2 \, \sqrt{b^{2} d - a b e} \sqrt{e x + d}}{b x + a}\right ) - 2 \,{\left (8 \, b^{4} d^{3} - 22 \, a b^{3} d^{2} e + 17 \, a^{2} b^{2} d e^{2} - 3 \, a^{3} b e^{3} - 3 \,{\left (b^{4} d e^{2} - a b^{3} e^{3}\right )} x^{2} + 2 \,{\left (b^{4} d^{2} e - 5 \, a b^{3} d e^{2} + 4 \, a^{2} b^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{48 \,{\left (a^{3} b^{5} d^{3} - 3 \, a^{4} b^{4} d^{2} e + 3 \, a^{5} b^{3} d e^{2} - a^{6} b^{2} e^{3} +{\left (b^{8} d^{3} - 3 \, a b^{7} d^{2} e + 3 \, a^{2} b^{6} d e^{2} - a^{3} b^{5} e^{3}\right )} x^{3} + 3 \,{\left (a b^{7} d^{3} - 3 \, a^{2} b^{6} d^{2} e + 3 \, a^{3} b^{5} d e^{2} - a^{4} b^{4} e^{3}\right )} x^{2} + 3 \,{\left (a^{2} b^{6} d^{3} - 3 \, a^{3} b^{5} d^{2} e + 3 \, a^{4} b^{4} d e^{2} - a^{5} b^{3} e^{3}\right )} x\right )}}, \frac{3 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \sqrt{-b^{2} d + a b e} \arctan \left (\frac{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}{b e x + b d}\right ) -{\left (8 \, b^{4} d^{3} - 22 \, a b^{3} d^{2} e + 17 \, a^{2} b^{2} d e^{2} - 3 \, a^{3} b e^{3} - 3 \,{\left (b^{4} d e^{2} - a b^{3} e^{3}\right )} x^{2} + 2 \,{\left (b^{4} d^{2} e - 5 \, a b^{3} d e^{2} + 4 \, a^{2} b^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{24 \,{\left (a^{3} b^{5} d^{3} - 3 \, a^{4} b^{4} d^{2} e + 3 \, a^{5} b^{3} d e^{2} - a^{6} b^{2} e^{3} +{\left (b^{8} d^{3} - 3 \, a b^{7} d^{2} e + 3 \, a^{2} b^{6} d e^{2} - a^{3} b^{5} e^{3}\right )} x^{3} + 3 \,{\left (a b^{7} d^{3} - 3 \, a^{2} b^{6} d^{2} e + 3 \, a^{3} b^{5} d e^{2} - a^{4} b^{4} e^{3}\right )} x^{2} + 3 \,{\left (a^{2} b^{6} d^{3} - 3 \, a^{3} b^{5} d^{2} e + 3 \, a^{4} b^{4} d e^{2} - a^{5} b^{3} 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 [B] time = 1.2416, size = 463, normalized size = 2.13 \begin{align*} \frac{\arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{3}}{8 \,{\left (b^{3} d^{2} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 2 \, a b^{2} d e \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + a^{2} b e^{2} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )\right )} \sqrt{-b^{2} d + a b e}} + \frac{3 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{2} e^{3} - 8 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} d e^{3} - 3 \, \sqrt{x e + d} b^{2} d^{2} e^{3} + 8 \,{\left (x e + d\right )}^{\frac{3}{2}} a b e^{4} + 6 \, \sqrt{x e + d} a b d e^{4} - 3 \, \sqrt{x e + d} a^{2} e^{5}}{24 \,{\left (b^{3} d^{2} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 2 \, a b^{2} d e \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + a^{2} b e^{2} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )\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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