Optimal. Leaf size=292 \[ \frac{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^6 (a+b x) (d+e x)^2}-\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{3 e^6 (a+b x) (d+e x)^3}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{4 e^6 (a+b x) (d+e x)^4}+\frac{b^5 x \sqrt{a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)}-\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^6 (a+b x)}-\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x) (d+e x)} \]
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Rubi [A] time = 0.42146, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^6 (a+b x) (d+e x)^2}-\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{3 e^6 (a+b x) (d+e x)^3}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{4 e^6 (a+b x) (d+e x)^4}+\frac{b^5 x \sqrt{a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)}-\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^6 (a+b x)}-\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x) (d+e x)} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^5,x]
[Out]
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Rubi in Sympy [A] time = 33.8206, size = 230, normalized size = 0.79 \[ \frac{5 b^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{5}} + \frac{5 b^{4} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{e^{6} \left (a + b x\right )} - \frac{5 b^{3} \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{6 e^{4} \left (d + e x\right )} - \frac{5 b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{6 e^{3} \left (d + e x\right )^{2}} - \frac{b \left (5 a + 5 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{12 e^{2} \left (d + e x\right )^{3}} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{4 e \left (d + e x\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**5,x)
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Mathematica [A] time = 0.298375, size = 243, normalized size = 0.83 \[ -\frac{\sqrt{(a+b x)^2} \left (3 a^5 e^5+5 a^4 b e^4 (d+4 e x)+10 a^3 b^2 e^3 \left (d^2+4 d e x+6 e^2 x^2\right )+30 a^2 b^3 e^2 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )-5 a b^4 d e \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )+60 b^4 (d+e x)^4 (b d-a e) \log (d+e x)+b^5 \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )\right )}{12 e^6 (a+b x) (d+e x)^4} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^5,x]
[Out]
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Maple [B] time = 0.023, size = 458, normalized size = 1.6 \[{\frac{60\,\ln \left ( ex+d \right ){x}^{4}a{b}^{4}{e}^{5}-60\,\ln \left ( ex+d \right ){x}^{4}{b}^{5}d{e}^{4}-240\,\ln \left ( ex+d \right ){x}^{3}{b}^{5}{d}^{2}{e}^{3}-240\,\ln \left ( ex+d \right ) x{b}^{5}{d}^{4}e-30\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}-3\,{a}^{5}{e}^{5}-77\,{b}^{5}{d}^{5}+360\,\ln \left ( ex+d \right ){x}^{2}a{b}^{4}{d}^{2}{e}^{3}+240\,\ln \left ( ex+d \right ){x}^{3}a{b}^{4}d{e}^{4}-360\,\ln \left ( ex+d \right ){x}^{2}{b}^{5}{d}^{3}{e}^{2}+240\,\ln \left ( ex+d \right ) xa{b}^{4}{d}^{3}{e}^{2}-60\,\ln \left ( ex+d \right ){b}^{5}{d}^{5}+48\,{x}^{4}{b}^{5}d{e}^{4}-120\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}-48\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}-60\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}-252\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}-20\,x{a}^{4}b{e}^{5}-248\,x{b}^{5}{d}^{4}e-5\,{a}^{4}bd{e}^{4}-10\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}+125\,a{b}^{4}{d}^{4}e+240\,{x}^{3}a{b}^{4}d{e}^{4}-180\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+540\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}+60\,\ln \left ( ex+d \right ) a{b}^{4}{d}^{4}e-40\,x{a}^{3}{b}^{2}d{e}^{4}-120\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}+12\,{x}^{5}{b}^{5}{e}^{5}+440\,xa{b}^{4}{d}^{3}{e}^{2}}{12\, \left ( bx+a \right ) ^{5}{e}^{6} \left ( ex+d \right ) ^{4}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^5,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)/(e*x + d)^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.212377, size = 556, normalized size = 1.9 \[ \frac{12 \, b^{5} e^{5} x^{5} + 48 \, b^{5} d e^{4} x^{4} - 77 \, b^{5} d^{5} + 125 \, a b^{4} d^{4} e - 30 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} - 5 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} - 24 \,{\left (2 \, b^{5} d^{2} e^{3} - 10 \, a b^{4} d e^{4} + 5 \, a^{2} b^{3} e^{5}\right )} x^{3} - 12 \,{\left (21 \, b^{5} d^{3} e^{2} - 45 \, a b^{4} d^{2} e^{3} + 15 \, a^{2} b^{3} d e^{4} + 5 \, a^{3} b^{2} e^{5}\right )} x^{2} - 4 \,{\left (62 \, b^{5} d^{4} e - 110 \, a b^{4} d^{3} e^{2} + 30 \, a^{2} b^{3} d^{2} e^{3} + 10 \, a^{3} b^{2} d e^{4} + 5 \, a^{4} b e^{5}\right )} x - 60 \,{\left (b^{5} d^{5} - a b^{4} d^{4} e +{\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 4 \,{\left (b^{5} d^{2} e^{3} - a b^{4} d e^{4}\right )} x^{3} + 6 \,{\left (b^{5} d^{3} e^{2} - a b^{4} d^{2} e^{3}\right )} x^{2} + 4 \,{\left (b^{5} d^{4} e - a b^{4} d^{3} e^{2}\right )} x\right )} \log \left (e x + d\right )}{12 \,{\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)/(e*x + d)^5,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**5,x)
[Out]
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GIAC/XCAS [A] time = 0.222076, size = 500, normalized size = 1.71 \[ b^{5} x e^{\left (-5\right )}{\rm sign}\left (b x + a\right ) - 5 \,{\left (b^{5} d{\rm sign}\left (b x + a\right ) - a b^{4} e{\rm sign}\left (b x + a\right )\right )} e^{\left (-6\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) - \frac{{\left (77 \, b^{5} d^{5}{\rm sign}\left (b x + a\right ) - 125 \, a b^{4} d^{4} e{\rm sign}\left (b x + a\right ) + 30 \, a^{2} b^{3} d^{3} e^{2}{\rm sign}\left (b x + a\right ) + 10 \, a^{3} b^{2} d^{2} e^{3}{\rm sign}\left (b x + a\right ) + 5 \, a^{4} b d e^{4}{\rm sign}\left (b x + a\right ) + 3 \, a^{5} e^{5}{\rm sign}\left (b x + a\right ) + 120 \,{\left (b^{5} d^{2} e^{3}{\rm sign}\left (b x + a\right ) - 2 \, a b^{4} d e^{4}{\rm sign}\left (b x + a\right ) + a^{2} b^{3} e^{5}{\rm sign}\left (b x + a\right )\right )} x^{3} + 60 \,{\left (5 \, b^{5} d^{3} e^{2}{\rm sign}\left (b x + a\right ) - 9 \, a b^{4} d^{2} e^{3}{\rm sign}\left (b x + a\right ) + 3 \, a^{2} b^{3} d e^{4}{\rm sign}\left (b x + a\right ) + a^{3} b^{2} e^{5}{\rm sign}\left (b x + a\right )\right )} x^{2} + 20 \,{\left (13 \, b^{5} d^{4} e{\rm sign}\left (b x + a\right ) - 22 \, a b^{4} d^{3} e^{2}{\rm sign}\left (b x + a\right ) + 6 \, a^{2} b^{3} d^{2} e^{3}{\rm sign}\left (b x + a\right ) + 2 \, a^{3} b^{2} d e^{4}{\rm sign}\left (b x + a\right ) + a^{4} b e^{5}{\rm sign}\left (b x + a\right )\right )} x\right )} e^{\left (-6\right )}}{12 \,{\left (x e + d\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)/(e*x + d)^5,x, algorithm="giac")
[Out]