Optimal. Leaf size=86 \[ \frac{(a+b x)^3 (-4 a B e+A b e+3 b B d)}{12 e (d+e x)^3 (b d-a e)^2}-\frac{(a+b x)^3 (B d-A e)}{4 e (d+e x)^4 (b d-a e)} \]
[Out]
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Rubi [A] time = 0.106339, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{(a+b x)^3 (-4 a B e+A b e+3 b B d)}{12 e (d+e x)^3 (b d-a e)^2}-\frac{(a+b x)^3 (B d-A e)}{4 e (d+e x)^4 (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)^2*(A + B*x))/(d + e*x)^5,x]
[Out]
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Rubi in Sympy [A] time = 11.8358, size = 73, normalized size = 0.85 \[ - \frac{\left (a + b x\right )^{3} \left (- A b e + B \left (4 a e - 3 b d\right )\right )}{12 e \left (d + e x\right )^{3} \left (a e - b d\right )^{2}} - \frac{\left (a + b x\right )^{3} \left (A e - B d\right )}{4 e \left (d + e x\right )^{4} \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**2*(B*x+A)/(e*x+d)**5,x)
[Out]
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Mathematica [A] time = 0.111387, size = 125, normalized size = 1.45 \[ -\frac{a^2 e^2 (3 A e+B (d+4 e x))+2 a b e \left (A e (d+4 e x)+B \left (d^2+4 d e x+6 e^2 x^2\right )\right )+b^2 \left (A e \left (d^2+4 d e x+6 e^2 x^2\right )+3 B \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )}{12 e^4 (d+e x)^4} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)^2*(A + B*x))/(d + e*x)^5,x]
[Out]
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Maple [B] time = 0.007, size = 166, normalized size = 1.9 \[ -{\frac{2\,Aab{e}^{2}-2\,Ad{b}^{2}e+B{a}^{2}{e}^{2}-4\,Bdabe+3\,{b}^{2}B{d}^{2}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{B{b}^{2}}{{e}^{4} \left ( ex+d \right ) }}-{\frac{b \left ( Abe+2\,Bae-3\,Bbd \right ) }{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{{a}^{2}A{e}^{3}-2\,Aabd{e}^{2}+A{d}^{2}{b}^{2}e-B{a}^{2}d{e}^{2}+2\,B{d}^{2}abe-{b}^{2}B{d}^{3}}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^2*(B*x+A)/(e*x+d)^5,x)
[Out]
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Maxima [A] time = 1.3678, size = 252, normalized size = 2.93 \[ -\frac{12 \, B b^{2} e^{3} x^{3} + 3 \, B b^{2} d^{3} + 3 \, A a^{2} e^{3} +{\left (2 \, B a b + A b^{2}\right )} d^{2} e +{\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 6 \,{\left (3 \, B b^{2} d e^{2} +{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 4 \,{\left (3 \, B b^{2} d^{2} e +{\left (2 \, B a b + A b^{2}\right )} d e^{2} +{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{12 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^2/(e*x + d)^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.201244, size = 252, normalized size = 2.93 \[ -\frac{12 \, B b^{2} e^{3} x^{3} + 3 \, B b^{2} d^{3} + 3 \, A a^{2} e^{3} +{\left (2 \, B a b + A b^{2}\right )} d^{2} e +{\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 6 \,{\left (3 \, B b^{2} d e^{2} +{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 4 \,{\left (3 \, B b^{2} d^{2} e +{\left (2 \, B a b + A b^{2}\right )} d e^{2} +{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{12 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^2/(e*x + d)^5,x, algorithm="fricas")
[Out]
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Sympy [A] time = 51.9108, size = 221, normalized size = 2.57 \[ - \frac{3 A a^{2} e^{3} + 2 A a b d e^{2} + A b^{2} d^{2} e + B a^{2} d e^{2} + 2 B a b d^{2} e + 3 B b^{2} d^{3} + 12 B b^{2} e^{3} x^{3} + x^{2} \left (6 A b^{2} e^{3} + 12 B a b e^{3} + 18 B b^{2} d e^{2}\right ) + x \left (8 A a b e^{3} + 4 A b^{2} d e^{2} + 4 B a^{2} e^{3} + 8 B a b d e^{2} + 12 B b^{2} d^{2} e\right )}{12 d^{4} e^{4} + 48 d^{3} e^{5} x + 72 d^{2} e^{6} x^{2} + 48 d e^{7} x^{3} + 12 e^{8} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**2*(B*x+A)/(e*x+d)**5,x)
[Out]
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GIAC/XCAS [A] time = 0.216092, size = 347, normalized size = 4.03 \[ -\frac{1}{12} \,{\left (\frac{12 \, B b^{2} e^{8}}{x e + d} - \frac{18 \, B b^{2} d e^{8}}{{\left (x e + d\right )}^{2}} + \frac{12 \, B b^{2} d^{2} e^{8}}{{\left (x e + d\right )}^{3}} - \frac{3 \, B b^{2} d^{3} e^{8}}{{\left (x e + d\right )}^{4}} + \frac{12 \, B a b e^{9}}{{\left (x e + d\right )}^{2}} + \frac{6 \, A b^{2} e^{9}}{{\left (x e + d\right )}^{2}} - \frac{16 \, B a b d e^{9}}{{\left (x e + d\right )}^{3}} - \frac{8 \, A b^{2} d e^{9}}{{\left (x e + d\right )}^{3}} + \frac{6 \, B a b d^{2} e^{9}}{{\left (x e + d\right )}^{4}} + \frac{3 \, A b^{2} d^{2} e^{9}}{{\left (x e + d\right )}^{4}} + \frac{4 \, B a^{2} e^{10}}{{\left (x e + d\right )}^{3}} + \frac{8 \, A a b e^{10}}{{\left (x e + d\right )}^{3}} - \frac{3 \, B a^{2} d e^{10}}{{\left (x e + d\right )}^{4}} - \frac{6 \, A a b d e^{10}}{{\left (x e + d\right )}^{4}} + \frac{3 \, A a^{2} e^{11}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-12\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^2/(e*x + d)^5,x, algorithm="giac")
[Out]