Optimal. Leaf size=253 \[ \frac{2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )}{3 e^6 (d+e x)^3}-\frac{A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{4 e^6 (d+e x)^4}+\frac{d^2 (B d-A e) (c d-b e)^2}{6 e^6 (d+e x)^6}+\frac{c (-A c e-2 b B e+5 B c d)}{2 e^6 (d+e x)^2}-\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{5 e^6 (d+e x)^5}-\frac{B c^2}{e^6 (d+e x)} \]
[Out]
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Rubi [A] time = 0.719744, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )}{3 e^6 (d+e x)^3}-\frac{A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{4 e^6 (d+e x)^4}+\frac{d^2 (B d-A e) (c d-b e)^2}{6 e^6 (d+e x)^6}+\frac{c (-A c e-2 b B e+5 B c d)}{2 e^6 (d+e x)^2}-\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{5 e^6 (d+e x)^5}-\frac{B c^2}{e^6 (d+e x)} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(b*x + c*x^2)^2)/(d + e*x)^7,x]
[Out]
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Rubi in Sympy [A] time = 118.875, size = 269, normalized size = 1.06 \[ - \frac{B c^{2}}{e^{6} \left (d + e x\right )} - \frac{c \left (A c e + 2 B b e - 5 B c d\right )}{2 e^{6} \left (d + e x\right )^{2}} - \frac{d^{2} \left (A e - B d\right ) \left (b e - c d\right )^{2}}{6 e^{6} \left (d + e x\right )^{6}} + \frac{d \left (b e - c d\right ) \left (2 A b e^{2} - 4 A c d e - 3 B b d e + 5 B c d^{2}\right )}{5 e^{6} \left (d + e x\right )^{5}} - \frac{2 A b c e^{2} - 4 A c^{2} d e + B b^{2} e^{2} - 8 B b c d e + 10 B c^{2} d^{2}}{3 e^{6} \left (d + e x\right )^{3}} - \frac{A b^{2} e^{3} - 6 A b c d e^{2} + 6 A c^{2} d^{2} e - 3 B b^{2} d e^{2} + 12 B b c d^{2} e - 10 B c^{2} d^{3}}{4 e^{6} \left (d + e x\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+b*x)**2/(e*x+d)**7,x)
[Out]
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Mathematica [A] time = 0.234187, size = 257, normalized size = 1.02 \[ -\frac{A e \left (b^2 e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+2 b c e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+2 c^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )\right )+B \left (b^2 e^2 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+4 b c e \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )+10 c^2 \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )\right )}{60 e^6 (d+e x)^6} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(b*x + c*x^2)^2)/(d + e*x)^7,x]
[Out]
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Maple [A] time = 0.011, size = 307, normalized size = 1.2 \[ -{\frac{{d}^{2} \left ( A{b}^{2}{e}^{3}-2\,Abcd{e}^{2}+A{c}^{2}{d}^{2}e-{b}^{2}Bd{e}^{2}+2\,B{d}^{2}bce-B{c}^{2}{d}^{3} \right ) }{6\,{e}^{6} \left ( ex+d \right ) ^{6}}}+{\frac{d \left ( 2\,A{b}^{2}{e}^{3}-6\,Abcd{e}^{2}+4\,A{c}^{2}{d}^{2}e-3\,{b}^{2}Bd{e}^{2}+8\,B{d}^{2}bce-5\,B{c}^{2}{d}^{3} \right ) }{5\,{e}^{6} \left ( ex+d \right ) ^{5}}}-{\frac{A{b}^{2}{e}^{3}-6\,Abcd{e}^{2}+6\,A{c}^{2}{d}^{2}e-3\,{b}^{2}Bd{e}^{2}+12\,B{d}^{2}bce-10\,B{c}^{2}{d}^{3}}{4\,{e}^{6} \left ( ex+d \right ) ^{4}}}-{\frac{2\,Abc{e}^{2}-4\,A{c}^{2}de+B{e}^{2}{b}^{2}-8\,Bdbce+10\,B{c}^{2}{d}^{2}}{3\,{e}^{6} \left ( ex+d \right ) ^{3}}}-{\frac{c \left ( Ace+2\,bBe-5\,Bcd \right ) }{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}-{\frac{B{c}^{2}}{{e}^{6} \left ( ex+d \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^7,x)
[Out]
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Maxima [A] time = 0.706111, size = 459, normalized size = 1.81 \[ -\frac{60 \, B c^{2} e^{5} x^{5} + 10 \, B c^{2} d^{5} + A b^{2} d^{2} e^{3} + 2 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e +{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} + 30 \,{\left (5 \, B c^{2} d e^{4} +{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 20 \,{\left (10 \, B c^{2} d^{2} e^{3} + 2 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} +{\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} + 15 \,{\left (10 \, B c^{2} d^{3} e^{2} + A b^{2} e^{5} + 2 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} +{\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 6 \,{\left (10 \, B c^{2} d^{4} e + A b^{2} d e^{4} + 2 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} +{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x}{60 \,{\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2*(B*x + A)/(e*x + d)^7,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.300742, size = 459, normalized size = 1.81 \[ -\frac{60 \, B c^{2} e^{5} x^{5} + 10 \, B c^{2} d^{5} + A b^{2} d^{2} e^{3} + 2 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e +{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} + 30 \,{\left (5 \, B c^{2} d e^{4} +{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 20 \,{\left (10 \, B c^{2} d^{2} e^{3} + 2 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} +{\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} + 15 \,{\left (10 \, B c^{2} d^{3} e^{2} + A b^{2} e^{5} + 2 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} +{\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 6 \,{\left (10 \, B c^{2} d^{4} e + A b^{2} d e^{4} + 2 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} +{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x}{60 \,{\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2*(B*x + A)/(e*x + d)^7,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)*(c*x**2+b*x)**2/(e*x+d)**7,x)
[Out]
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GIAC/XCAS [A] time = 0.279285, size = 429, normalized size = 1.7 \[ -\frac{{\left (60 \, B c^{2} x^{5} e^{5} + 150 \, B c^{2} d x^{4} e^{4} + 200 \, B c^{2} d^{2} x^{3} e^{3} + 150 \, B c^{2} d^{3} x^{2} e^{2} + 60 \, B c^{2} d^{4} x e + 10 \, B c^{2} d^{5} + 60 \, B b c x^{4} e^{5} + 30 \, A c^{2} x^{4} e^{5} + 80 \, B b c d x^{3} e^{4} + 40 \, A c^{2} d x^{3} e^{4} + 60 \, B b c d^{2} x^{2} e^{3} + 30 \, A c^{2} d^{2} x^{2} e^{3} + 24 \, B b c d^{3} x e^{2} + 12 \, A c^{2} d^{3} x e^{2} + 4 \, B b c d^{4} e + 2 \, A c^{2} d^{4} e + 20 \, B b^{2} x^{3} e^{5} + 40 \, A b c x^{3} e^{5} + 15 \, B b^{2} d x^{2} e^{4} + 30 \, A b c d x^{2} e^{4} + 6 \, B b^{2} d^{2} x e^{3} + 12 \, A b c d^{2} x e^{3} + B b^{2} d^{3} e^{2} + 2 \, A b c d^{3} e^{2} + 15 \, A b^{2} x^{2} e^{5} + 6 \, A b^{2} d x e^{4} + A b^{2} d^{2} e^{3}\right )} e^{\left (-6\right )}}{60 \,{\left (x e + d\right )}^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2*(B*x + A)/(e*x + d)^7,x, algorithm="giac")
[Out]