Optimal. Leaf size=292 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^6 (a+b x) (d+e x)}+\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 \log (d+e x)}{e^6 (a+b x)}-\frac{10 b^2 x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^5 (a+b x)}+\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4}{4 e^6 (a+b x)}-\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)}{3 e^6 (a+b x)}+\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^2}{e^6 (a+b x)} \]
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Rubi [A] time = 0.463625, 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 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^6 (a+b x) (d+e x)}+\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 \log (d+e x)}{e^6 (a+b x)}-\frac{10 b^2 x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^5 (a+b x)}+\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4}{4 e^6 (a+b x)}-\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)}{3 e^6 (a+b x)}+\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^2}{e^6 (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^2,x]
[Out]
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Rubi in Sympy [A] time = 38.3728, size = 224, normalized size = 0.77 \[ \frac{5 b \left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{4 e^{2}} + \frac{5 b \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3 e^{3}} + \frac{5 b \left (3 a + 3 b x\right ) \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{6 e^{4}} + \frac{5 b \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{5}} + \frac{5 b \left (a e - b d\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{e^{6} \left (a + b x\right )} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{e \left (d + e x\right )} \]
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)**2,x)
[Out]
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Mathematica [A] time = 0.328236, size = 246, normalized size = 0.84 \[ \frac{\sqrt{(a+b x)^2} \left (-12 a^5 e^5+60 a^4 b d e^4+120 a^3 b^2 e^3 \left (-d^2+d e x+e^2 x^2\right )+60 a^2 b^3 e^2 \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+20 a b^4 e \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+60 b (d+e x) (b d-a e)^4 \log (d+e x)+b^5 \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )\right )}{12 e^6 (a+b x) (d+e x)} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^2,x]
[Out]
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Maple [B] time = 0.024, size = 456, normalized size = 1.6 \[{\frac{60\,\ln \left ( ex+d \right ) x{a}^{4}b{e}^{5}+60\,\ln \left ( ex+d \right ) x{b}^{5}{d}^{4}e+120\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}-12\,{a}^{5}{e}^{5}+12\,{b}^{5}{d}^{5}-240\,\ln \left ( ex+d \right ) x{a}^{3}{b}^{2}d{e}^{4}+360\,\ln \left ( ex+d \right ) x{a}^{2}{b}^{3}{d}^{2}{e}^{3}-240\,\ln \left ( ex+d \right ) xa{b}^{4}{d}^{3}{e}^{2}+60\,\ln \left ( ex+d \right ){b}^{5}{d}^{5}+20\,{x}^{4}a{b}^{4}{e}^{5}-5\,{x}^{4}{b}^{5}d{e}^{4}+60\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+10\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+120\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}-30\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}-48\,x{b}^{5}{d}^{4}e+60\,{a}^{4}bd{e}^{4}-120\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-60\,a{b}^{4}{d}^{4}e-40\,{x}^{3}a{b}^{4}d{e}^{4}-180\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+120\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}+60\,\ln \left ( ex+d \right ){a}^{4}bd{e}^{4}-240\,\ln \left ( ex+d \right ){a}^{3}{b}^{2}{d}^{2}{e}^{3}+360\,\ln \left ( ex+d \right ){a}^{2}{b}^{3}{d}^{3}{e}^{2}-240\,\ln \left ( ex+d \right ) a{b}^{4}{d}^{4}e+120\,x{a}^{3}{b}^{2}d{e}^{4}-240\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}+3\,{x}^{5}{b}^{5}{e}^{5}+180\,xa{b}^{4}{d}^{3}{e}^{2}}{12\, \left ( bx+a \right ) ^{5}{e}^{6} \left ( ex+d \right ) } \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]
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)^2,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)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.211425, size = 504, normalized size = 1.73 \[ \frac{3 \, b^{5} e^{5} x^{5} + 12 \, b^{5} d^{5} - 60 \, a b^{4} d^{4} e + 120 \, a^{2} b^{3} d^{3} e^{2} - 120 \, a^{3} b^{2} d^{2} e^{3} + 60 \, a^{4} b d e^{4} - 12 \, a^{5} e^{5} - 5 \,{\left (b^{5} d e^{4} - 4 \, a b^{4} e^{5}\right )} x^{4} + 10 \,{\left (b^{5} d^{2} e^{3} - 4 \, a b^{4} d e^{4} + 6 \, a^{2} b^{3} e^{5}\right )} x^{3} - 30 \,{\left (b^{5} d^{3} e^{2} - 4 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} - 4 \, a^{3} b^{2} e^{5}\right )} x^{2} - 12 \,{\left (4 \, b^{5} d^{4} e - 15 \, a b^{4} d^{3} e^{2} + 20 \, a^{2} b^{3} d^{2} e^{3} - 10 \, a^{3} b^{2} d e^{4}\right )} x + 60 \,{\left (b^{5} d^{5} - 4 \, a b^{4} d^{4} e + 6 \, a^{2} b^{3} d^{3} e^{2} - 4 \, a^{3} b^{2} d^{2} e^{3} + a^{4} b d e^{4} +{\left (b^{5} d^{4} e - 4 \, a b^{4} d^{3} e^{2} + 6 \, a^{2} b^{3} d^{2} e^{3} - 4 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x\right )} \log \left (e x + d\right )}{12 \,{\left (e^{7} x + d e^{6}\right )}} \]
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)^2,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)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.221998, size = 516, normalized size = 1.77 \[ 5 \,{\left (b^{5} d^{4}{\rm sign}\left (b x + a\right ) - 4 \, a b^{4} d^{3} e{\rm sign}\left (b x + a\right ) + 6 \, a^{2} b^{3} d^{2} e^{2}{\rm sign}\left (b x + a\right ) - 4 \, a^{3} b^{2} d e^{3}{\rm sign}\left (b x + a\right ) + a^{4} b e^{4}{\rm sign}\left (b x + a\right )\right )} e^{\left (-6\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{12} \,{\left (3 \, b^{5} x^{4} e^{6}{\rm sign}\left (b x + a\right ) - 8 \, b^{5} d x^{3} e^{5}{\rm sign}\left (b x + a\right ) + 18 \, b^{5} d^{2} x^{2} e^{4}{\rm sign}\left (b x + a\right ) - 48 \, b^{5} d^{3} x e^{3}{\rm sign}\left (b x + a\right ) + 20 \, a b^{4} x^{3} e^{6}{\rm sign}\left (b x + a\right ) - 60 \, a b^{4} d x^{2} e^{5}{\rm sign}\left (b x + a\right ) + 180 \, a b^{4} d^{2} x e^{4}{\rm sign}\left (b x + a\right ) + 60 \, a^{2} b^{3} x^{2} e^{6}{\rm sign}\left (b x + a\right ) - 240 \, a^{2} b^{3} d x e^{5}{\rm sign}\left (b x + a\right ) + 120 \, a^{3} b^{2} x e^{6}{\rm sign}\left (b x + a\right )\right )} e^{\left (-8\right )} + \frac{{\left (b^{5} d^{5}{\rm sign}\left (b x + a\right ) - 5 \, a b^{4} d^{4} e{\rm sign}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{2}{\rm sign}\left (b x + a\right ) - 10 \, a^{3} b^{2} d^{2} e^{3}{\rm sign}\left (b x + a\right ) + 5 \, a^{4} b d e^{4}{\rm sign}\left (b x + a\right ) - a^{5} e^{5}{\rm sign}\left (b x + a\right )\right )} e^{\left (-6\right )}}{x e + d} \]
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)^2,x, algorithm="giac")
[Out]