Optimal. Leaf size=227 \[ \frac{e^4 (a+b x)^4 (-6 a B e+A b e+5 b B d)}{4 b^7}+\frac{5 e^3 (a+b x)^3 (b d-a e) (-3 a B e+A b e+2 b B d)}{3 b^7}+\frac{5 e^2 (a+b x)^2 (b d-a e)^2 (-2 a B e+A b e+b B d)}{b^7}-\frac{(A b-a B) (b d-a e)^5}{b^7 (a+b x)}+\frac{(b d-a e)^4 \log (a+b x) (-6 a B e+5 A b e+b B d)}{b^7}+\frac{5 e x (b d-a e)^3 (-3 a B e+2 A b e+b B d)}{b^6}+\frac{B e^5 (a+b x)^5}{5 b^7} \]
[Out]
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Rubi [A] time = 0.803545, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{e^4 (a+b x)^4 (-6 a B e+A b e+5 b B d)}{4 b^7}+\frac{5 e^3 (a+b x)^3 (b d-a e) (-3 a B e+A b e+2 b B d)}{3 b^7}+\frac{5 e^2 (a+b x)^2 (b d-a e)^2 (-2 a B e+A b e+b B d)}{b^7}-\frac{(A b-a B) (b d-a e)^5}{b^7 (a+b x)}+\frac{(b d-a e)^4 \log (a+b x) (-6 a B e+5 A b e+b B d)}{b^7}+\frac{5 e x (b d-a e)^3 (-3 a B e+2 A b e+b B d)}{b^6}+\frac{B e^5 (a+b x)^5}{5 b^7} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(d + e*x)^5)/(a + b*x)^2,x]
[Out]
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Rubi in Sympy [A] time = 105.298, size = 230, normalized size = 1.01 \[ \frac{B e^{5} \left (a + b x\right )^{5}}{5 b^{7}} - \frac{5 e x \left (a e - b d\right )^{3} \left (2 A b e - 3 B a e + B b d\right )}{b^{6}} + \frac{e^{4} \left (a + b x\right )^{4} \left (A b e - 6 B a e + 5 B b d\right )}{4 b^{7}} - \frac{5 e^{3} \left (a + b x\right )^{3} \left (a e - b d\right ) \left (A b e - 3 B a e + 2 B b d\right )}{3 b^{7}} + \frac{5 e^{2} \left (a + b x\right )^{2} \left (a e - b d\right )^{2} \left (A b e - 2 B a e + B b d\right )}{b^{7}} + \frac{\left (a e - b d\right )^{4} \left (5 A b e - 6 B a e + B b d\right ) \log{\left (a + b x \right )}}{b^{7}} + \frac{\left (A b - B a\right ) \left (a e - b d\right )^{5}}{b^{7} \left (a + b x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**5/(b*x+a)**2,x)
[Out]
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Mathematica [B] time = 0.46199, size = 500, normalized size = 2.2 \[ \frac{-5 A b \left (-12 a^5 e^5+12 a^4 b e^4 (5 d+4 e x)+30 a^3 b^2 e^3 \left (-4 d^2-6 d e x+e^2 x^2\right )-10 a^2 b^3 e^2 \left (-12 d^3-24 d^2 e x+12 d e^2 x^2+e^3 x^3\right )+5 a b^4 e \left (-12 d^4-24 d^3 e x+36 d^2 e^2 x^2+8 d e^3 x^3+e^4 x^4\right )+b^5 \left (12 d^5-120 d^3 e^2 x^2-60 d^2 e^3 x^3-20 d e^4 x^4-3 e^5 x^5\right )\right )+B \left (-60 a^6 e^5+300 a^5 b e^4 (d+e x)+60 a^4 b^2 e^3 \left (-10 d^2-20 d e x+3 e^2 x^2\right )+30 a^3 b^3 e^2 \left (20 d^3+60 d^2 e x-25 d e^2 x^2-2 e^3 x^3\right )+10 a^2 b^4 e \left (-30 d^4-120 d^3 e x+120 d^2 e^2 x^2+25 d e^3 x^3+3 e^4 x^4\right )+a b^5 \left (60 d^5+300 d^4 e x-900 d^3 e^2 x^2-400 d^2 e^3 x^3-125 d e^4 x^4-18 e^5 x^5\right )+b^6 e x^2 \left (300 d^4+300 d^3 e x+200 d^2 e^2 x^2+75 d e^3 x^3+12 e^4 x^4\right )\right )+60 (a+b x) (b d-a e)^4 \log (a+b x) (-6 a B e+5 A b e+b B d)}{60 b^7 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(d + e*x)^5)/(a + b*x)^2,x]
[Out]
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Maple [B] time = 0.019, size = 787, normalized size = 3.5 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^5/(b*x+a)^2,x)
[Out]
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Maxima [A] time = 1.3716, size = 782, normalized size = 3.44 \[ \frac{{\left (B a b^{5} - A b^{6}\right )} d^{5} - 5 \,{\left (B a^{2} b^{4} - A a b^{5}\right )} d^{4} e + 10 \,{\left (B a^{3} b^{3} - A a^{2} b^{4}\right )} d^{3} e^{2} - 10 \,{\left (B a^{4} b^{2} - A a^{3} b^{3}\right )} d^{2} e^{3} + 5 \,{\left (B a^{5} b - A a^{4} b^{2}\right )} d e^{4} -{\left (B a^{6} - A a^{5} b\right )} e^{5}}{b^{8} x + a b^{7}} + \frac{12 \, B b^{4} e^{5} x^{5} + 15 \,{\left (5 \, B b^{4} d e^{4} -{\left (2 \, B a b^{3} - A b^{4}\right )} e^{5}\right )} x^{4} + 20 \,{\left (10 \, B b^{4} d^{2} e^{3} - 5 \,{\left (2 \, B a b^{3} - A b^{4}\right )} d e^{4} +{\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 30 \,{\left (10 \, B b^{4} d^{3} e^{2} - 10 \,{\left (2 \, B a b^{3} - A b^{4}\right )} d^{2} e^{3} + 5 \,{\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d e^{4} -{\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 60 \,{\left (5 \, B b^{4} d^{4} e - 10 \,{\left (2 \, B a b^{3} - A b^{4}\right )} d^{3} e^{2} + 10 \,{\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d^{2} e^{3} - 5 \,{\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} d e^{4} +{\left (5 \, B a^{4} - 4 \, A a^{3} b\right )} e^{5}\right )} x}{60 \, b^{6}} + \frac{{\left (B b^{5} d^{5} - 5 \,{\left (2 \, B a b^{4} - A b^{5}\right )} d^{4} e + 10 \,{\left (3 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} d^{3} e^{2} - 10 \,{\left (4 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} d^{2} e^{3} + 5 \,{\left (5 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} d e^{4} -{\left (6 \, B a^{5} - 5 \, A a^{4} b\right )} e^{5}\right )} \log \left (b x + a\right )}{b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^5/(b*x + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.212266, size = 1123, normalized size = 4.95 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^5/(b*x + a)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 13.6166, size = 552, normalized size = 2.43 \[ \frac{B e^{5} x^{5}}{5 b^{2}} - \frac{- A a^{5} b e^{5} + 5 A a^{4} b^{2} d e^{4} - 10 A a^{3} b^{3} d^{2} e^{3} + 10 A a^{2} b^{4} d^{3} e^{2} - 5 A a b^{5} d^{4} e + A b^{6} d^{5} + B a^{6} e^{5} - 5 B a^{5} b d e^{4} + 10 B a^{4} b^{2} d^{2} e^{3} - 10 B a^{3} b^{3} d^{3} e^{2} + 5 B a^{2} b^{4} d^{4} e - B a b^{5} d^{5}}{a b^{7} + b^{8} x} - \frac{x^{4} \left (- A b e^{5} + 2 B a e^{5} - 5 B b d e^{4}\right )}{4 b^{3}} + \frac{x^{3} \left (- 2 A a b e^{5} + 5 A b^{2} d e^{4} + 3 B a^{2} e^{5} - 10 B a b d e^{4} + 10 B b^{2} d^{2} e^{3}\right )}{3 b^{4}} - \frac{x^{2} \left (- 3 A a^{2} b e^{5} + 10 A a b^{2} d e^{4} - 10 A b^{3} d^{2} e^{3} + 4 B a^{3} e^{5} - 15 B a^{2} b d e^{4} + 20 B a b^{2} d^{2} e^{3} - 10 B b^{3} d^{3} e^{2}\right )}{2 b^{5}} + \frac{x \left (- 4 A a^{3} b e^{5} + 15 A a^{2} b^{2} d e^{4} - 20 A a b^{3} d^{2} e^{3} + 10 A b^{4} d^{3} e^{2} + 5 B a^{4} e^{5} - 20 B a^{3} b d e^{4} + 30 B a^{2} b^{2} d^{2} e^{3} - 20 B a b^{3} d^{3} e^{2} + 5 B b^{4} d^{4} e\right )}{b^{6}} - \frac{\left (a e - b d\right )^{4} \left (- 5 A b e + 6 B a e - B b d\right ) \log{\left (a + b x \right )}}{b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)**5/(b*x+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.234027, size = 953, normalized size = 4.2 \[ \frac{{\left (b x + a\right )}^{5}{\left (12 \, B e^{5} + \frac{15 \,{\left (5 \, B b^{2} d e^{4} - 6 \, B a b e^{5} + A b^{2} e^{5}\right )}}{{\left (b x + a\right )} b} + \frac{100 \,{\left (2 \, B b^{4} d^{2} e^{3} - 5 \, B a b^{3} d e^{4} + A b^{4} d e^{4} + 3 \, B a^{2} b^{2} e^{5} - A a b^{3} e^{5}\right )}}{{\left (b x + a\right )}^{2} b^{2}} + \frac{300 \,{\left (B b^{6} d^{3} e^{2} - 4 \, B a b^{5} d^{2} e^{3} + A b^{6} d^{2} e^{3} + 5 \, B a^{2} b^{4} d e^{4} - 2 \, A a b^{5} d e^{4} - 2 \, B a^{3} b^{3} e^{5} + A a^{2} b^{4} e^{5}\right )}}{{\left (b x + a\right )}^{3} b^{3}} + \frac{300 \,{\left (B b^{8} d^{4} e - 6 \, B a b^{7} d^{3} e^{2} + 2 \, A b^{8} d^{3} e^{2} + 12 \, B a^{2} b^{6} d^{2} e^{3} - 6 \, A a b^{7} d^{2} e^{3} - 10 \, B a^{3} b^{5} d e^{4} + 6 \, A a^{2} b^{6} d e^{4} + 3 \, B a^{4} b^{4} e^{5} - 2 \, A a^{3} b^{5} e^{5}\right )}}{{\left (b x + a\right )}^{4} b^{4}}\right )}}{60 \, b^{7}} - \frac{{\left (B b^{5} d^{5} - 10 \, B a b^{4} d^{4} e + 5 \, A b^{5} d^{4} e + 30 \, B a^{2} b^{3} d^{3} e^{2} - 20 \, A a b^{4} d^{3} e^{2} - 40 \, B a^{3} b^{2} d^{2} e^{3} + 30 \, A a^{2} b^{3} d^{2} e^{3} + 25 \, B a^{4} b d e^{4} - 20 \, A a^{3} b^{2} d e^{4} - 6 \, B a^{5} e^{5} + 5 \, A a^{4} b e^{5}\right )}{\rm ln}\left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{7}} + \frac{\frac{B a b^{10} d^{5}}{b x + a} - \frac{A b^{11} d^{5}}{b x + a} - \frac{5 \, B a^{2} b^{9} d^{4} e}{b x + a} + \frac{5 \, A a b^{10} d^{4} e}{b x + a} + \frac{10 \, B a^{3} b^{8} d^{3} e^{2}}{b x + a} - \frac{10 \, A a^{2} b^{9} d^{3} e^{2}}{b x + a} - \frac{10 \, B a^{4} b^{7} d^{2} e^{3}}{b x + a} + \frac{10 \, A a^{3} b^{8} d^{2} e^{3}}{b x + a} + \frac{5 \, B a^{5} b^{6} d e^{4}}{b x + a} - \frac{5 \, A a^{4} b^{7} d e^{4}}{b x + a} - \frac{B a^{6} b^{5} e^{5}}{b x + a} + \frac{A a^{5} b^{6} e^{5}}{b x + a}}{b^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^5/(b*x + a)^2,x, algorithm="giac")
[Out]