Optimal. Leaf size=344 \[ -\frac{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^7 (a+b x) (d+e x)^3}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{2 e^7 (a+b x) (d+e x)^4}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{5 e^7 (a+b x) (d+e x)^5}+\frac{b^6 x \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}-\frac{6 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^7 (a+b x)}-\frac{15 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^7 (a+b x) (d+e x)}+\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^7 (a+b x) (d+e x)^2} \]
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Rubi [A] time = 0.595363, antiderivative size = 344, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^7 (a+b x) (d+e x)^3}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{2 e^7 (a+b x) (d+e x)^4}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{5 e^7 (a+b x) (d+e x)^5}+\frac{b^6 x \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}-\frac{6 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^7 (a+b x)}-\frac{15 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^7 (a+b x) (d+e x)}+\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^7 (a+b x) (d+e x)^2} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^6,x]
[Out]
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Rubi in Sympy [A] time = 51.037, size = 265, normalized size = 0.77 \[ \frac{6 b^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{6}} + \frac{6 b^{5} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{e^{7} \left (a + b x\right )} - \frac{b^{4} \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{5} \left (d + e x\right )} - \frac{b^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{e^{4} \left (d + e x\right )^{2}} - \frac{b^{2} \left (5 a + 5 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{10 e^{3} \left (d + e x\right )^{3}} - \frac{3 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{10 e^{2} \left (d + e x\right )^{4}} - \frac{\left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{5 e \left (d + e x\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**6,x)
[Out]
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Mathematica [A] time = 0.344156, size = 315, normalized size = 0.92 \[ -\frac{\sqrt{(a+b x)^2} \left (2 a^6 e^6+3 a^5 b e^5 (d+5 e x)+5 a^4 b^2 e^4 \left (d^2+5 d e x+10 e^2 x^2\right )+10 a^3 b^3 e^3 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+30 a^2 b^4 e^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )-a b^5 d e \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )+60 b^5 (d+e x)^5 (b d-a e) \log (d+e x)+b^6 \left (87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right )\right )}{10 e^7 (a+b x) (d+e x)^5} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^6,x]
[Out]
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Maple [B] time = 0.027, size = 603, normalized size = 1.8 \[{\frac{-300\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+60\,\ln \left ( ex+d \right ) a{b}^{5}{d}^{5}e-300\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+1100\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}-50\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+300\,\ln \left ( ex+d \right ){x}^{4}a{b}^{5}d{e}^{5}-375\,x{b}^{6}{d}^{5}e-50\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}-2\,{a}^{6}{e}^{6}-87\,{b}^{6}{d}^{6}+137\,{d}^{5}a{b}^{5}e-10\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}-60\,\ln \left ( ex+d \right ){b}^{6}{d}^{6}+10\,{x}^{6}{b}^{6}{e}^{6}-600\,\ln \left ( ex+d \right ){x}^{3}{b}^{6}{d}^{3}{e}^{3}-100\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+300\,{x}^{4}a{b}^{5}d{e}^{5}+600\,\ln \left ( ex+d \right ){x}^{2}a{b}^{5}{d}^{3}{e}^{3}+300\,\ln \left ( ex+d \right ) xa{b}^{5}{d}^{4}{e}^{2}+600\,\ln \left ( ex+d \right ){x}^{3}a{b}^{5}{d}^{2}{e}^{4}-30\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}-150\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+625\,xa{b}^{5}{d}^{4}{e}^{2}+900\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}-25\,x{a}^{4}{b}^{2}d{e}^{5}-300\,\ln \left ( ex+d \right ) x{b}^{6}{d}^{5}e-600\,\ln \left ( ex+d \right ){x}^{2}{b}^{6}{d}^{4}{e}^{2}+60\,\ln \left ( ex+d \right ){x}^{5}a{b}^{5}{e}^{6}-60\,\ln \left ( ex+d \right ){x}^{5}{b}^{6}d{e}^{5}-5\,{b}^{2}{a}^{4}{d}^{2}{e}^{4}-300\,\ln \left ( ex+d \right ){x}^{4}{b}^{6}{d}^{2}{e}^{4}-3\,{a}^{5}bd{e}^{5}+50\,{x}^{5}{b}^{6}d{e}^{5}-400\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}-100\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}-150\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}-50\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}-600\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}-15\,x{a}^{5}b{e}^{6}}{10\, \left ( bx+a \right ) ^{5}{e}^{7} \left ( ex+d \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^6,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)*(b*x + a)/(e*x + d)^6,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.303409, size = 732, normalized size = 2.13 \[ \frac{10 \, b^{6} e^{6} x^{6} + 50 \, b^{6} d e^{5} x^{5} - 87 \, b^{6} d^{6} + 137 \, a b^{5} d^{5} e - 30 \, a^{2} b^{4} d^{4} e^{2} - 10 \, a^{3} b^{3} d^{3} e^{3} - 5 \, a^{4} b^{2} d^{2} e^{4} - 3 \, a^{5} b d e^{5} - 2 \, a^{6} e^{6} - 50 \,{\left (b^{6} d^{2} e^{4} - 6 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} - 100 \,{\left (4 \, b^{6} d^{3} e^{3} - 9 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} - 50 \,{\left (12 \, b^{6} d^{4} e^{2} - 22 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} + 2 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} - 5 \,{\left (75 \, b^{6} d^{5} e - 125 \, a b^{5} d^{4} e^{2} + 30 \, a^{2} b^{4} d^{3} e^{3} + 10 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + 3 \, a^{5} b e^{6}\right )} x - 60 \,{\left (b^{6} d^{6} - a b^{5} d^{5} e +{\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 5 \,{\left (b^{6} d^{2} e^{4} - a b^{5} d e^{5}\right )} x^{4} + 10 \,{\left (b^{6} d^{3} e^{3} - a b^{5} d^{2} e^{4}\right )} x^{3} + 10 \,{\left (b^{6} d^{4} e^{2} - a b^{5} d^{3} e^{3}\right )} x^{2} + 5 \,{\left (b^{6} d^{5} e - a b^{5} d^{4} e^{2}\right )} x\right )} \log \left (e x + d\right )}{10 \,{\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)/(e*x + d)^6,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*x+a)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**6,x)
[Out]
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GIAC/XCAS [A] time = 0.293729, size = 674, normalized size = 1.96 \[ b^{6} x e^{\left (-6\right )}{\rm sign}\left (b x + a\right ) - 6 \,{\left (b^{6} d{\rm sign}\left (b x + a\right ) - a b^{5} e{\rm sign}\left (b x + a\right )\right )} e^{\left (-7\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) - \frac{{\left (87 \, b^{6} d^{6}{\rm sign}\left (b x + a\right ) - 137 \, a b^{5} d^{5} e{\rm sign}\left (b x + a\right ) + 30 \, a^{2} b^{4} d^{4} e^{2}{\rm sign}\left (b x + a\right ) + 10 \, a^{3} b^{3} d^{3} e^{3}{\rm sign}\left (b x + a\right ) + 5 \, a^{4} b^{2} d^{2} e^{4}{\rm sign}\left (b x + a\right ) + 3 \, a^{5} b d e^{5}{\rm sign}\left (b x + a\right ) + 2 \, a^{6} e^{6}{\rm sign}\left (b x + a\right ) + 150 \,{\left (b^{6} d^{2} e^{4}{\rm sign}\left (b x + a\right ) - 2 \, a b^{5} d e^{5}{\rm sign}\left (b x + a\right ) + a^{2} b^{4} e^{6}{\rm sign}\left (b x + a\right )\right )} x^{4} + 100 \,{\left (5 \, b^{6} d^{3} e^{3}{\rm sign}\left (b x + a\right ) - 9 \, a b^{5} d^{2} e^{4}{\rm sign}\left (b x + a\right ) + 3 \, a^{2} b^{4} d e^{5}{\rm sign}\left (b x + a\right ) + a^{3} b^{3} e^{6}{\rm sign}\left (b x + a\right )\right )} x^{3} + 50 \,{\left (13 \, b^{6} d^{4} e^{2}{\rm sign}\left (b x + a\right ) - 22 \, a b^{5} d^{3} e^{3}{\rm sign}\left (b x + a\right ) + 6 \, a^{2} b^{4} d^{2} e^{4}{\rm sign}\left (b x + a\right ) + 2 \, a^{3} b^{3} d e^{5}{\rm sign}\left (b x + a\right ) + a^{4} b^{2} e^{6}{\rm sign}\left (b x + a\right )\right )} x^{2} + 5 \,{\left (77 \, b^{6} d^{5} e{\rm sign}\left (b x + a\right ) - 125 \, a b^{5} d^{4} e^{2}{\rm sign}\left (b x + a\right ) + 30 \, a^{2} b^{4} d^{3} e^{3}{\rm sign}\left (b x + a\right ) + 10 \, a^{3} b^{3} d^{2} e^{4}{\rm sign}\left (b x + a\right ) + 5 \, a^{4} b^{2} d e^{5}{\rm sign}\left (b x + a\right ) + 3 \, a^{5} b e^{6}{\rm sign}\left (b x + a\right )\right )} x\right )} e^{\left (-7\right )}}{10 \,{\left (x e + d\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)/(e*x + d)^6,x, algorithm="giac")
[Out]