3.242 \(\int \frac{\left (b x+c x^2\right )^2}{(d+e x)^7} \, dx\)

Optimal. Leaf size=137 \[ -\frac{b^2 e^2-6 b c d e+6 c^2 d^2}{4 e^5 (d+e x)^4}-\frac{d^2 (c d-b e)^2}{6 e^5 (d+e x)^6}+\frac{2 c (2 c d-b e)}{3 e^5 (d+e x)^3}+\frac{2 d (c d-b e) (2 c d-b e)}{5 e^5 (d+e x)^5}-\frac{c^2}{2 e^5 (d+e x)^2} \]

[Out]

-(d^2*(c*d - b*e)^2)/(6*e^5*(d + e*x)^6) + (2*d*(c*d - b*e)*(2*c*d - b*e))/(5*e^
5*(d + e*x)^5) - (6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)/(4*e^5*(d + e*x)^4) + (2*c*(2
*c*d - b*e))/(3*e^5*(d + e*x)^3) - c^2/(2*e^5*(d + e*x)^2)

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Rubi [A]  time = 0.255919, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{b^2 e^2-6 b c d e+6 c^2 d^2}{4 e^5 (d+e x)^4}-\frac{d^2 (c d-b e)^2}{6 e^5 (d+e x)^6}+\frac{2 c (2 c d-b e)}{3 e^5 (d+e x)^3}+\frac{2 d (c d-b e) (2 c d-b e)}{5 e^5 (d+e x)^5}-\frac{c^2}{2 e^5 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^2/(d + e*x)^7,x]

[Out]

-(d^2*(c*d - b*e)^2)/(6*e^5*(d + e*x)^6) + (2*d*(c*d - b*e)*(2*c*d - b*e))/(5*e^
5*(d + e*x)^5) - (6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)/(4*e^5*(d + e*x)^4) + (2*c*(2
*c*d - b*e))/(3*e^5*(d + e*x)^3) - c^2/(2*e^5*(d + e*x)^2)

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Rubi in Sympy [A]  time = 40.3159, size = 128, normalized size = 0.93 \[ - \frac{c^{2}}{2 e^{5} \left (d + e x\right )^{2}} - \frac{2 c \left (b e - 2 c d\right )}{3 e^{5} \left (d + e x\right )^{3}} - \frac{d^{2} \left (b e - c d\right )^{2}}{6 e^{5} \left (d + e x\right )^{6}} + \frac{2 d \left (b e - 2 c d\right ) \left (b e - c d\right )}{5 e^{5} \left (d + e x\right )^{5}} - \frac{b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}}{4 e^{5} \left (d + e x\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**2/(e*x+d)**7,x)

[Out]

-c**2/(2*e**5*(d + e*x)**2) - 2*c*(b*e - 2*c*d)/(3*e**5*(d + e*x)**3) - d**2*(b*
e - c*d)**2/(6*e**5*(d + e*x)**6) + 2*d*(b*e - 2*c*d)*(b*e - c*d)/(5*e**5*(d + e
*x)**5) - (b**2*e**2 - 6*b*c*d*e + 6*c**2*d**2)/(4*e**5*(d + e*x)**4)

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Mathematica [A]  time = 0.0718704, size = 116, normalized size = 0.85 \[ -\frac{b^2 e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+2 b c e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+2 c^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )}{60 e^5 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^2/(d + e*x)^7,x]

[Out]

-(b^2*e^2*(d^2 + 6*d*e*x + 15*e^2*x^2) + 2*b*c*e*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2
+ 20*e^3*x^3) + 2*c^2*(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))/(60*e^5*(d + e*x)^6)

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Maple [A]  time = 0.008, size = 143, normalized size = 1. \[{\frac{2\,d \left ({b}^{2}{e}^{2}-3\,bcde+2\,{c}^{2}{d}^{2} \right ) }{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}-{\frac{{c}^{2}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{2\,c \left ( be-2\,cd \right ) }{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{{d}^{2} \left ({b}^{2}{e}^{2}-2\,bcde+{c}^{2}{d}^{2} \right ) }{6\,{e}^{5} \left ( ex+d \right ) ^{6}}}-{\frac{{b}^{2}{e}^{2}-6\,bcde+6\,{c}^{2}{d}^{2}}{4\,{e}^{5} \left ( ex+d \right ) ^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^2/(e*x+d)^7,x)

[Out]

2/5*d*(b^2*e^2-3*b*c*d*e+2*c^2*d^2)/e^5/(e*x+d)^5-1/2*c^2/e^5/(e*x+d)^2-2/3*c*(b
*e-2*c*d)/e^5/(e*x+d)^3-1/6*d^2*(b^2*e^2-2*b*c*d*e+c^2*d^2)/e^5/(e*x+d)^6-1/4*(b
^2*e^2-6*b*c*d*e+6*c^2*d^2)/e^5/(e*x+d)^4

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Maxima [A]  time = 0.717577, size = 258, normalized size = 1.88 \[ -\frac{30 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 40 \,{\left (c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 15 \,{\left (2 \, c^{2} d^{2} e^{2} + 2 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} + 6 \,{\left (2 \, c^{2} d^{3} e + 2 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x}{60 \,{\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2/(e*x + d)^7,x, algorithm="maxima")

[Out]

-1/60*(30*c^2*e^4*x^4 + 2*c^2*d^4 + 2*b*c*d^3*e + b^2*d^2*e^2 + 40*(c^2*d*e^3 +
b*c*e^4)*x^3 + 15*(2*c^2*d^2*e^2 + 2*b*c*d*e^3 + b^2*e^4)*x^2 + 6*(2*c^2*d^3*e +
2*b*c*d^2*e^2 + b^2*d*e^3)*x)/(e^11*x^6 + 6*d*e^10*x^5 + 15*d^2*e^9*x^4 + 20*d^
3*e^8*x^3 + 15*d^4*e^7*x^2 + 6*d^5*e^6*x + d^6*e^5)

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Fricas [A]  time = 0.211875, size = 258, normalized size = 1.88 \[ -\frac{30 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 40 \,{\left (c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 15 \,{\left (2 \, c^{2} d^{2} e^{2} + 2 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} + 6 \,{\left (2 \, c^{2} d^{3} e + 2 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x}{60 \,{\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2/(e*x + d)^7,x, algorithm="fricas")

[Out]

-1/60*(30*c^2*e^4*x^4 + 2*c^2*d^4 + 2*b*c*d^3*e + b^2*d^2*e^2 + 40*(c^2*d*e^3 +
b*c*e^4)*x^3 + 15*(2*c^2*d^2*e^2 + 2*b*c*d*e^3 + b^2*e^4)*x^2 + 6*(2*c^2*d^3*e +
2*b*c*d^2*e^2 + b^2*d*e^3)*x)/(e^11*x^6 + 6*d*e^10*x^5 + 15*d^2*e^9*x^4 + 20*d^
3*e^8*x^3 + 15*d^4*e^7*x^2 + 6*d^5*e^6*x + d^6*e^5)

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Sympy [A]  time = 15.4872, size = 204, normalized size = 1.49 \[ - \frac{b^{2} d^{2} e^{2} + 2 b c d^{3} e + 2 c^{2} d^{4} + 30 c^{2} e^{4} x^{4} + x^{3} \left (40 b c e^{4} + 40 c^{2} d e^{3}\right ) + x^{2} \left (15 b^{2} e^{4} + 30 b c d e^{3} + 30 c^{2} d^{2} e^{2}\right ) + x \left (6 b^{2} d e^{3} + 12 b c d^{2} e^{2} + 12 c^{2} d^{3} e\right )}{60 d^{6} e^{5} + 360 d^{5} e^{6} x + 900 d^{4} e^{7} x^{2} + 1200 d^{3} e^{8} x^{3} + 900 d^{2} e^{9} x^{4} + 360 d e^{10} x^{5} + 60 e^{11} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**2/(e*x+d)**7,x)

[Out]

-(b**2*d**2*e**2 + 2*b*c*d**3*e + 2*c**2*d**4 + 30*c**2*e**4*x**4 + x**3*(40*b*c
*e**4 + 40*c**2*d*e**3) + x**2*(15*b**2*e**4 + 30*b*c*d*e**3 + 30*c**2*d**2*e**2
) + x*(6*b**2*d*e**3 + 12*b*c*d**2*e**2 + 12*c**2*d**3*e))/(60*d**6*e**5 + 360*d
**5*e**6*x + 900*d**4*e**7*x**2 + 1200*d**3*e**8*x**3 + 900*d**2*e**9*x**4 + 360
*d*e**10*x**5 + 60*e**11*x**6)

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GIAC/XCAS [A]  time = 0.210511, size = 178, normalized size = 1.3 \[ -\frac{{\left (30 \, c^{2} x^{4} e^{4} + 40 \, c^{2} d x^{3} e^{3} + 30 \, c^{2} d^{2} x^{2} e^{2} + 12 \, c^{2} d^{3} x e + 2 \, c^{2} d^{4} + 40 \, b c x^{3} e^{4} + 30 \, b c d x^{2} e^{3} + 12 \, b c d^{2} x e^{2} + 2 \, b c d^{3} e + 15 \, b^{2} x^{2} e^{4} + 6 \, b^{2} d x e^{3} + b^{2} d^{2} e^{2}\right )} e^{\left (-5\right )}}{60 \,{\left (x e + d\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2/(e*x + d)^7,x, algorithm="giac")

[Out]

-1/60*(30*c^2*x^4*e^4 + 40*c^2*d*x^3*e^3 + 30*c^2*d^2*x^2*e^2 + 12*c^2*d^3*x*e +
2*c^2*d^4 + 40*b*c*x^3*e^4 + 30*b*c*d*x^2*e^3 + 12*b*c*d^2*x*e^2 + 2*b*c*d^3*e
+ 15*b^2*x^2*e^4 + 6*b^2*d*x*e^3 + b^2*d^2*e^2)*e^(-5)/(x*e + d)^6