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

Optimal. Leaf size=107 \[ -\frac{d^2 (c d-b e)^2}{e^5 (d+e x)}-\frac{2 d (c d-b e) (2 c d-b e) \log (d+e x)}{e^5}+\frac{x (c d-b e) (3 c d-b e)}{e^4}-\frac{c x^2 (c d-b e)}{e^3}+\frac{c^2 x^3}{3 e^2} \]

[Out]

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

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Rubi [A]  time = 0.264408, antiderivative size = 107, 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 \[ -\frac{d^2 (c d-b e)^2}{e^5 (d+e x)}-\frac{2 d (c d-b e) (2 c d-b e) \log (d+e x)}{e^5}+\frac{x (c d-b e) (3 c d-b e)}{e^4}-\frac{c x^2 (c d-b e)}{e^3}+\frac{c^2 x^3}{3 e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{c^{2} x^{3}}{3 e^{2}} + \frac{2 c \left (b e - c d\right ) \int x\, dx}{e^{3}} - \frac{d^{2} \left (b e - c d\right )^{2}}{e^{5} \left (d + e x\right )} - \frac{2 d \left (b e - 2 c d\right ) \left (b e - c d\right ) \log{\left (d + e x \right )}}{e^{5}} + \left (b e - 3 c d\right ) \left (b e - c d\right ) \int \frac{1}{e^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c**2*x**3/(3*e**2) + 2*c*(b*e - c*d)*Integral(x, x)/e**3 - d**2*(b*e - c*d)**2/(
e**5*(d + e*x)) - 2*d*(b*e - 2*c*d)*(b*e - c*d)*log(d + e*x)/e**5 + (b*e - 3*c*d
)*(b*e - c*d)*Integral(e**(-4), x)

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Mathematica [A]  time = 0.166236, size = 114, normalized size = 1.07 \[ \frac{3 e x \left (b^2 e^2-4 b c d e+3 c^2 d^2\right )-6 d \left (b^2 e^2-3 b c d e+2 c^2 d^2\right ) \log (d+e x)-\frac{3 d^2 (c d-b e)^2}{d+e x}-3 c e^2 x^2 (c d-b e)+c^2 e^3 x^3}{3 e^5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 164, normalized size = 1.5 \[{\frac{{c}^{2}{x}^{3}}{3\,{e}^{2}}}+{\frac{{x}^{2}bc}{{e}^{2}}}-{\frac{d{c}^{2}{x}^{2}}{{e}^{3}}}+{\frac{{b}^{2}x}{{e}^{2}}}-4\,{\frac{bcdx}{{e}^{3}}}+3\,{\frac{{c}^{2}{d}^{2}x}{{e}^{4}}}-2\,{\frac{d\ln \left ( ex+d \right ){b}^{2}}{{e}^{3}}}+6\,{\frac{{d}^{2}\ln \left ( ex+d \right ) bc}{{e}^{4}}}-4\,{\frac{{d}^{3}\ln \left ( ex+d \right ){c}^{2}}{{e}^{5}}}-{\frac{{b}^{2}{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}+2\,{\frac{{d}^{3}bc}{{e}^{4} \left ( ex+d \right ) }}-{\frac{{c}^{2}{d}^{4}}{{e}^{5} \left ( ex+d \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^2/(e*x+d)^2,x)

[Out]

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

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Maxima [A]  time = 0.690122, size = 186, normalized size = 1.74 \[ -\frac{c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}}{e^{6} x + d e^{5}} + \frac{c^{2} e^{2} x^{3} - 3 \,{\left (c^{2} d e - b c e^{2}\right )} x^{2} + 3 \,{\left (3 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2}\right )} x}{3 \, e^{4}} - \frac{2 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} \log \left (e x + d\right )}{e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.242456, size = 274, normalized size = 2.56 \[ \frac{c^{2} e^{4} x^{4} - 3 \, c^{2} d^{4} + 6 \, b c d^{3} e - 3 \, b^{2} d^{2} e^{2} -{\left (2 \, c^{2} d e^{3} - 3 \, b c e^{4}\right )} x^{3} + 3 \,{\left (2 \, c^{2} d^{2} e^{2} - 3 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} + 3 \,{\left (3 \, c^{2} d^{3} e - 4 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x - 6 \,{\left (2 \, c^{2} d^{4} - 3 \, b c d^{3} e + b^{2} d^{2} e^{2} +{\left (2 \, c^{2} d^{3} e - 3 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (e^{6} x + d e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.884, size = 122, normalized size = 1.14 \[ \frac{c^{2} x^{3}}{3 e^{2}} - \frac{2 d \left (b e - 2 c d\right ) \left (b e - c d\right ) \log{\left (d + e x \right )}}{e^{5}} - \frac{b^{2} d^{2} e^{2} - 2 b c d^{3} e + c^{2} d^{4}}{d e^{5} + e^{6} x} + \frac{x^{2} \left (b c e - c^{2} d\right )}{e^{3}} + \frac{x \left (b^{2} e^{2} - 4 b c d e + 3 c^{2} d^{2}\right )}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.21053, size = 248, normalized size = 2.32 \[ \frac{1}{3} \,{\left (c^{2} - \frac{3 \,{\left (2 \, c^{2} d e - b c e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac{3 \,{\left (6 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} + b^{2} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}}\right )}{\left (x e + d\right )}^{3} e^{\left (-5\right )} + 2 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} e^{\left (-5\right )}{\rm ln}\left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) -{\left (\frac{c^{2} d^{4} e^{3}}{x e + d} - \frac{2 \, b c d^{3} e^{4}}{x e + d} + \frac{b^{2} d^{2} e^{5}}{x e + d}\right )} e^{\left (-8\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(c^2 - 3*(2*c^2*d*e - b*c*e^2)*e^(-1)/(x*e + d) + 3*(6*c^2*d^2*e^2 - 6*b*c*d
*e^3 + b^2*e^4)*e^(-2)/(x*e + d)^2)*(x*e + d)^3*e^(-5) + 2*(2*c^2*d^3 - 3*b*c*d^
2*e + b^2*d*e^2)*e^(-5)*ln(abs(x*e + d)*e^(-1)/(x*e + d)^2) - (c^2*d^4*e^3/(x*e
+ d) - 2*b*c*d^3*e^4/(x*e + d) + b^2*d^2*e^5/(x*e + d))*e^(-8)