Optimal. Leaf size=263 \[ \frac{2 (a+b x) \sqrt{d+e x} (b d-a e)^3}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (d+e x)^{3/2} (b d-a e)^2}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (d+e x)^{5/2} (b d-a e)}{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (d+e x)^{7/2}}{7 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 (a+b x) (b d-a e)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.16931, antiderivative size = 263, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {646, 50, 63, 208} \[ \frac{2 (a+b x) \sqrt{d+e x} (b d-a e)^3}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (d+e x)^{3/2} (b d-a e)^2}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (d+e x)^{5/2} (b d-a e)}{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (d+e x)^{7/2}}{7 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 (a+b x) (b d-a e)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]
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)^{7/2}}{\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)^{7/2}}{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)^{7/2}}{7 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{(d+e x)^{5/2}}{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) (d+e x)^{5/2}}{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (d+e x)^{7/2}}{7 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{(d+e x)^{3/2}}{a b+b^2 x} \, dx}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 (b d-a e)^2 (a+b x) (d+e x)^{3/2}}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (b d-a e) (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (d+e x)^{7/2}}{7 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (\left (b^2 d-a b e\right )^3 \left (a b+b^2 x\right )\right ) \int \frac{\sqrt{d+e x}}{a b+b^2 x} \, dx}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 (b d-a e)^3 (a+b x) \sqrt{d+e x}}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (b d-a e)^2 (a+b x) (d+e x)^{3/2}}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (b d-a e) (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (d+e x)^{7/2}}{7 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (\left (b^2 d-a b e\right )^4 \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right ) \sqrt{d+e x}} \, dx}{b^8 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 (b d-a e)^3 (a+b x) \sqrt{d+e x}}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (b d-a e)^2 (a+b x) (d+e x)^{3/2}}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (b d-a e) (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (d+e x)^{7/2}}{7 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (2 \left (b^2 d-a b e\right )^4 \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^8 e \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 (b d-a e)^3 (a+b x) \sqrt{d+e x}}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (b d-a e)^2 (a+b x) (d+e x)^{3/2}}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (b d-a e) (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (d+e x)^{7/2}}{7 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 (b d-a e)^{7/2} (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.255444, size = 150, normalized size = 0.57 \[ \frac{(a+b x) \left (\frac{14 (b d-a e) \left (5 (b d-a e) \left (\sqrt{b} \sqrt{d+e x} (-3 a e+4 b d+b e x)-3 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )\right )+3 b^{5/2} (d+e x)^{5/2}\right )}{15 b^{7/2}}+2 (d+e x)^{7/2}\right )}{7 b \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.232, size = 462, normalized size = 1.8 \begin{align*}{\frac{2\,bx+2\,a}{105\,{b}^{4}} \left ( 15\,\sqrt{ \left ( ae-bd \right ) b} \left ( ex+d \right ) ^{7/2}{b}^{3}-21\,\sqrt{ \left ( ae-bd \right ) b} \left ( ex+d \right ) ^{5/2}a{b}^{2}e+21\,\sqrt{ \left ( ae-bd \right ) b} \left ( ex+d \right ) ^{5/2}{b}^{3}d+105\,\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ){a}^{4}{e}^{4}-420\,\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ){a}^{3}bd{e}^{3}+630\,\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ){a}^{2}{b}^{2}{d}^{2}{e}^{2}-420\,\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) a{b}^{3}{d}^{3}e+105\,\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ){b}^{4}{d}^{4}+35\,\sqrt{ \left ( ae-bd \right ) b} \left ( ex+d \right ) ^{3/2}{a}^{2}b{e}^{2}-70\,\sqrt{ \left ( ae-bd \right ) b} \left ( ex+d \right ) ^{3/2}a{b}^{2}de+35\,\sqrt{ \left ( ae-bd \right ) b} \left ( ex+d \right ) ^{3/2}{b}^{3}{d}^{2}-105\,\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}{a}^{3}{e}^{3}+315\,\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}{a}^{2}bd{e}^{2}-315\,\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}a{b}^{2}{d}^{2}e+105\,\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}{b}^{3}{d}^{3} \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{7}{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 = 1.70972, size = 940, normalized size = 3.57 \begin{align*} \left [-\frac{105 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\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 (15 \, b^{3} e^{3} x^{3} + 176 \, b^{3} d^{3} - 406 \, a b^{2} d^{2} e + 350 \, a^{2} b d e^{2} - 105 \, a^{3} e^{3} + 3 \,{\left (22 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} +{\left (122 \, b^{3} d^{2} e - 112 \, a b^{2} d e^{2} + 35 \, a^{2} b e^{3}\right )} x\right )} \sqrt{e x + d}}{105 \, b^{4}}, -\frac{2 \,{\left (105 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\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 (15 \, b^{3} e^{3} x^{3} + 176 \, b^{3} d^{3} - 406 \, a b^{2} d^{2} e + 350 \, a^{2} b d e^{2} - 105 \, a^{3} e^{3} + 3 \,{\left (22 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} +{\left (122 \, b^{3} d^{2} e - 112 \, a b^{2} d e^{2} + 35 \, a^{2} b e^{3}\right )} x\right )} \sqrt{e x + d}\right )}}{105 \, b^{4}}\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 = 1.17778, size = 478, normalized size = 1.82 \begin{align*} \frac{2 \,{\left (b^{4} d^{4} \mathrm{sgn}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e \mathrm{sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) - 4 \, a^{3} b d e^{3} \mathrm{sgn}\left (b x + a\right ) + a^{4} e^{4} \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^{4}} + \frac{2 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{6} \mathrm{sgn}\left (b x + a\right ) + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{6} d \mathrm{sgn}\left (b x + a\right ) + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{6} d^{2} \mathrm{sgn}\left (b x + a\right ) + 105 \, \sqrt{x e + d} b^{6} d^{3} \mathrm{sgn}\left (b x + a\right ) - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{5} e \mathrm{sgn}\left (b x + a\right ) - 70 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{5} d e \mathrm{sgn}\left (b x + a\right ) - 315 \, \sqrt{x e + d} a b^{5} d^{2} e \mathrm{sgn}\left (b x + a\right ) + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + 315 \, \sqrt{x e + d} a^{2} b^{4} d e^{2} \mathrm{sgn}\left (b x + a\right ) - 105 \, \sqrt{x e + d} a^{3} b^{3} e^{3} \mathrm{sgn}\left (b x + a\right )\right )}}{105 \, b^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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