Optimal. Leaf size=278 \[ \frac{35 e^3 \sqrt{d+e x}}{64 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{35 e^2 \sqrt{d+e x}}{96 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{35 e^4 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}}+\frac{7 e \sqrt{d+e x}}{24 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{\sqrt{d+e x}}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]
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Rubi [A] time = 0.142837, antiderivative size = 278, 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, 51, 63, 208} \[ \frac{35 e^3 \sqrt{d+e x}}{64 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{35 e^2 \sqrt{d+e x}}{96 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{35 e^4 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}}+\frac{7 e \sqrt{d+e x}}{24 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{\sqrt{d+e x}}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 646
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac{\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right )^5 \sqrt{d+e x}} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{\sqrt{d+e x}}{4 (b d-a e) (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (7 b^3 e \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right )^4 \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}}{4 (b d-a e) (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e \sqrt{d+e x}}{24 (b d-a e)^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (35 b^2 e^2 \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right )^3 \sqrt{d+e x}} \, dx}{48 (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{\sqrt{d+e x}}{4 (b d-a e) (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e \sqrt{d+e x}}{24 (b d-a e)^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^2 \sqrt{d+e x}}{96 (b d-a e)^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (35 b e^3 \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right )^2 \sqrt{d+e x}} \, dx}{64 (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{35 e^3 \sqrt{d+e x}}{64 (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\sqrt{d+e x}}{4 (b d-a e) (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e \sqrt{d+e x}}{24 (b d-a e)^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^2 \sqrt{d+e x}}{96 (b d-a e)^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (35 e^4 \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right ) \sqrt{d+e x}} \, dx}{128 (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{35 e^3 \sqrt{d+e x}}{64 (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\sqrt{d+e x}}{4 (b d-a e) (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e \sqrt{d+e x}}{24 (b d-a e)^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^2 \sqrt{d+e x}}{96 (b d-a e)^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (35 e^3 \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 )}{64 (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{35 e^3 \sqrt{d+e x}}{64 (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\sqrt{d+e x}}{4 (b d-a e) (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e \sqrt{d+e x}}{24 (b d-a e)^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^2 \sqrt{d+e x}}{96 (b d-a e)^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^4 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 \sqrt{b} (b d-a e)^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [C] time = 0.0240033, size = 65, normalized size = 0.23 \[ -\frac{2 e^4 (a+b x) \sqrt{d+e x} \, _2F_1\left (\frac{1}{2},5;\frac{3}{2};\frac{b (d+e x)}{b d-a e}\right )}{\sqrt{(a+b x)^2} (b d-a e)^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.268, size = 497, normalized size = 1.8 \begin{align*}{\frac{bx+a}{192\, \left ( ae-bd \right ) ^{4}} \left ( 105\,\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ){x}^{4}{b}^{4}{e}^{4}+420\,\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ){x}^{3}a{b}^{3}{e}^{4}+105\,\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}{x}^{3}{b}^{3}{e}^{3}+630\,\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ){x}^{2}{a}^{2}{b}^{2}{e}^{4}+385\,\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}{x}^{2}a{b}^{2}{e}^{3}-70\,\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}{x}^{2}{b}^{3}d{e}^{2}+420\,\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) x{a}^{3}b{e}^{4}+511\,\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}x{a}^{2}b{e}^{3}-252\,\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}xa{b}^{2}d{e}^{2}+56\,\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}x{b}^{3}{d}^{2}e+105\,\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ){a}^{4}{e}^{4}+279\,\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}{a}^{3}{e}^{3}-326\,\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}{a}^{2}bd{e}^{2}+200\,\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}a{b}^{2}{d}^{2}e-48\,\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}{b}^{3}{d}^{3} \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{1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{5}{2}} \sqrt{e x + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.73517, size = 2709, normalized size = 9.74 \begin{align*} \text{result too large to display} \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 = 1.26337, size = 744, normalized size = 2.68 \begin{align*} \frac{35 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{4}}{64 \,{\left (b^{4} d^{4} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a b^{3} d^{3} e \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a^{3} b d e^{3} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + a^{4} e^{4} \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{105 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{3} e^{4} - 385 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{3} d e^{4} + 511 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} d^{2} e^{4} - 279 \, \sqrt{x e + d} b^{3} d^{3} e^{4} + 385 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{2} e^{5} - 1022 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{2} d e^{5} + 837 \, \sqrt{x e + d} a b^{2} d^{2} e^{5} + 511 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b e^{6} - 837 \, \sqrt{x e + d} a^{2} b d e^{6} + 279 \, \sqrt{x e + d} a^{3} e^{7}}{192 \,{\left (b^{4} d^{4} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a b^{3} d^{3} e \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a^{3} b d e^{3} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + a^{4} e^{4} \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 )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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