Optimal. Leaf size=345 \[ -\frac{15 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^7 (a+b x) (d+e x)}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^7 (a+b x) (d+e x)^2}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{3 e^7 (a+b x) (d+e x)^3}+\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3}{3 e^7 (a+b x)}-\frac{3 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)}{e^7 (a+b x)}+\frac{15 b^4 x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x)}-\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 \log (d+e x)}{e^7 (a+b x)} \]
[Out]
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Rubi [A] time = 0.61602, antiderivative size = 345, 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 \[ -\frac{15 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^7 (a+b x) (d+e x)}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^7 (a+b x) (d+e x)^2}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{3 e^7 (a+b x) (d+e x)^3}+\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3}{3 e^7 (a+b x)}-\frac{3 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)}{e^7 (a+b x)}+\frac{15 b^4 x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x)}-\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 \log (d+e x)}{e^7 (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^4,x]
[Out]
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Rubi in Sympy [A] time = 60.8191, size = 269, normalized size = 0.78 \[ \frac{20 b^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3 e^{4}} + \frac{10 b^{3} \left (3 a + 3 b x\right ) \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3 e^{5}} + \frac{20 b^{3} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{6}} + \frac{20 b^{3} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{e^{7} \left (a + b x\right )} - \frac{5 b^{2} \left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{e^{3} \left (d + e x\right )} - \frac{b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{e^{2} \left (d + e x\right )^{2}} - \frac{\left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{3 e \left (d + e x\right )^{3}} \]
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)**4,x)
[Out]
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Mathematica [A] time = 0.304703, size = 320, normalized size = 0.93 \[ -\frac{\sqrt{(a+b x)^2} \left (a^6 e^6+3 a^5 b e^5 (d+3 e x)+15 a^4 b^2 e^4 \left (d^2+3 d e x+3 e^2 x^2\right )-10 a^3 b^3 d e^3 \left (11 d^2+27 d e x+18 e^2 x^2\right )+15 a^2 b^4 e^2 \left (13 d^4+27 d^3 e x+9 d^2 e^2 x^2-9 d e^3 x^3-3 e^4 x^4\right )-3 a b^5 e \left (47 d^5+81 d^4 e x-9 d^3 e^2 x^2-63 d^2 e^3 x^3-15 d e^4 x^4+3 e^5 x^5\right )+60 b^3 (d+e x)^3 (b d-a e)^3 \log (d+e x)+b^6 \left (37 d^6+51 d^5 e x-39 d^4 e^2 x^2-73 d^3 e^3 x^3-15 d^2 e^4 x^4+3 d e^5 x^5-e^6 x^6\right )\right )}{3 e^7 (a+b x) (d+e x)^3} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^4,x]
[Out]
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Maple [B] time = 0.027, size = 692, normalized size = 2. \[{\frac{135\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+60\,\ln \left ( ex+d \right ){a}^{3}{b}^{3}{d}^{3}{e}^{3}-180\,\ln \left ( ex+d \right ){a}^{2}{b}^{4}{d}^{4}{e}^{2}+180\,\ln \left ( ex+d \right ) a{b}^{5}{d}^{5}e-135\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-27\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+270\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-51\,x{b}^{6}{d}^{5}e+15\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}-{a}^{6}{e}^{6}-37\,{b}^{6}{d}^{6}+141\,{d}^{5}a{b}^{5}e+110\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}-60\,\ln \left ( ex+d \right ){b}^{6}{d}^{6}+{x}^{6}{b}^{6}{e}^{6}-60\,\ln \left ( ex+d \right ){x}^{3}{b}^{6}{d}^{3}{e}^{3}+60\,\ln \left ( ex+d \right ){x}^{3}{a}^{3}{b}^{3}{e}^{6}+180\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}-45\,{x}^{4}a{b}^{5}d{e}^{5}+180\,\ln \left ( ex+d \right ){x}^{2}{a}^{3}{b}^{3}d{e}^{5}-540\,\ln \left ( ex+d \right ){x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+540\,\ln \left ( ex+d \right ){x}^{2}a{b}^{5}{d}^{3}{e}^{3}+180\,\ln \left ( ex+d \right ) x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-540\,\ln \left ( ex+d \right ) x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+540\,\ln \left ( ex+d \right ) xa{b}^{5}{d}^{4}{e}^{2}+180\,\ln \left ( ex+d \right ){x}^{3}a{b}^{5}{d}^{2}{e}^{4}-180\,\ln \left ( ex+d \right ){x}^{3}{a}^{2}{b}^{4}d{e}^{5}-195\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}-405\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+243\,xa{b}^{5}{d}^{4}{e}^{2}-189\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}-45\,x{a}^{4}{b}^{2}d{e}^{5}-180\,\ln \left ( ex+d \right ) x{b}^{6}{d}^{5}e-180\,\ln \left ( ex+d \right ){x}^{2}{b}^{6}{d}^{4}{e}^{2}-15\,{b}^{2}{a}^{4}{d}^{2}{e}^{4}-3\,{a}^{5}bd{e}^{5}+9\,{x}^{5}a{b}^{5}{e}^{6}-3\,{x}^{5}{b}^{6}d{e}^{5}+73\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+45\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}-45\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+39\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}-9\,x{a}^{5}b{e}^{6}}{3\, \left ( bx+a \right ) ^{5}{e}^{7} \left ( ex+d \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]
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)^4,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)^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.298268, size = 778, normalized size = 2.26 \[ \frac{b^{6} e^{6} x^{6} - 37 \, b^{6} d^{6} + 141 \, a b^{5} d^{5} e - 195 \, a^{2} b^{4} d^{4} e^{2} + 110 \, a^{3} b^{3} d^{3} e^{3} - 15 \, a^{4} b^{2} d^{2} e^{4} - 3 \, a^{5} b d e^{5} - a^{6} e^{6} - 3 \,{\left (b^{6} d e^{5} - 3 \, a b^{5} e^{6}\right )} x^{5} + 15 \,{\left (b^{6} d^{2} e^{4} - 3 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} +{\left (73 \, b^{6} d^{3} e^{3} - 189 \, a b^{5} d^{2} e^{4} + 135 \, a^{2} b^{4} d e^{5}\right )} x^{3} + 3 \,{\left (13 \, b^{6} d^{4} e^{2} - 9 \, a b^{5} d^{3} e^{3} - 45 \, a^{2} b^{4} d^{2} e^{4} + 60 \, a^{3} b^{3} d e^{5} - 15 \, a^{4} b^{2} e^{6}\right )} x^{2} - 3 \,{\left (17 \, b^{6} d^{5} e - 81 \, a b^{5} d^{4} e^{2} + 135 \, a^{2} b^{4} d^{3} e^{3} - 90 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} + 3 \, a^{5} b e^{6}\right )} x - 60 \,{\left (b^{6} d^{6} - 3 \, a b^{5} d^{5} e + 3 \, a^{2} b^{4} d^{4} e^{2} - a^{3} b^{3} d^{3} e^{3} +{\left (b^{6} d^{3} e^{3} - 3 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} - a^{3} b^{3} e^{6}\right )} x^{3} + 3 \,{\left (b^{6} d^{4} e^{2} - 3 \, a b^{5} d^{3} e^{3} + 3 \, a^{2} b^{4} d^{2} e^{4} - a^{3} b^{3} d e^{5}\right )} x^{2} + 3 \,{\left (b^{6} d^{5} e - 3 \, a b^{5} d^{4} e^{2} + 3 \, a^{2} b^{4} d^{3} e^{3} - a^{3} b^{3} d^{2} e^{4}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} \]
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)^4,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)**4,x)
[Out]
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GIAC/XCAS [A] time = 0.29678, size = 679, normalized size = 1.97 \[ -20 \,{\left (b^{6} d^{3}{\rm sign}\left (b x + a\right ) - 3 \, a b^{5} d^{2} e{\rm sign}\left (b x + a\right ) + 3 \, a^{2} b^{4} d e^{2}{\rm sign}\left (b x + a\right ) - a^{3} b^{3} e^{3}{\rm sign}\left (b x + a\right )\right )} e^{\left (-7\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{3} \,{\left (b^{6} x^{3} e^{8}{\rm sign}\left (b x + a\right ) - 6 \, b^{6} d x^{2} e^{7}{\rm sign}\left (b x + a\right ) + 30 \, b^{6} d^{2} x e^{6}{\rm sign}\left (b x + a\right ) + 9 \, a b^{5} x^{2} e^{8}{\rm sign}\left (b x + a\right ) - 72 \, a b^{5} d x e^{7}{\rm sign}\left (b x + a\right ) + 45 \, a^{2} b^{4} x e^{8}{\rm sign}\left (b x + a\right )\right )} e^{\left (-12\right )} - \frac{{\left (37 \, b^{6} d^{6}{\rm sign}\left (b x + a\right ) - 141 \, a b^{5} d^{5} e{\rm sign}\left (b x + a\right ) + 195 \, a^{2} b^{4} d^{4} e^{2}{\rm sign}\left (b x + a\right ) - 110 \, a^{3} b^{3} d^{3} e^{3}{\rm sign}\left (b x + a\right ) + 15 \, a^{4} b^{2} d^{2} e^{4}{\rm sign}\left (b x + a\right ) + 3 \, a^{5} b d e^{5}{\rm sign}\left (b x + a\right ) + a^{6} e^{6}{\rm sign}\left (b x + a\right ) + 45 \,{\left (b^{6} d^{4} e^{2}{\rm sign}\left (b x + a\right ) - 4 \, a b^{5} d^{3} e^{3}{\rm sign}\left (b x + a\right ) + 6 \, a^{2} b^{4} d^{2} e^{4}{\rm sign}\left (b x + a\right ) - 4 \, a^{3} b^{3} d e^{5}{\rm sign}\left (b x + a\right ) + a^{4} b^{2} e^{6}{\rm sign}\left (b x + a\right )\right )} x^{2} + 9 \,{\left (9 \, b^{6} d^{5} e{\rm sign}\left (b x + a\right ) - 35 \, a b^{5} d^{4} e^{2}{\rm sign}\left (b x + a\right ) + 50 \, a^{2} b^{4} d^{3} e^{3}{\rm sign}\left (b x + a\right ) - 30 \, a^{3} b^{3} d^{2} e^{4}{\rm sign}\left (b x + a\right ) + 5 \, a^{4} b^{2} d e^{5}{\rm sign}\left (b x + a\right ) + a^{5} b e^{6}{\rm sign}\left (b x + a\right )\right )} x\right )} e^{\left (-7\right )}}{3 \,{\left (x e + d\right )}^{3}} \]
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)^4,x, algorithm="giac")
[Out]