3.238 \(\int \frac{\left (b x+c x^2\right )^2}{(d+e x)^3} \, dx\)

Optimal. Leaf size=119 \[ \frac{\left (b^2 e^2-6 b c d e+6 c^2 d^2\right ) \log (d+e x)}{e^5}-\frac{d^2 (c d-b e)^2}{2 e^5 (d+e x)^2}+\frac{2 d (2 c d-b e) (c d-b e)}{e^5 (d+e x)}-\frac{c x (3 c d-2 b e)}{e^4}+\frac{c^2 x^2}{2 e^3} \]

[Out]

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

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Rubi [A]  time = 0.275539, antiderivative size = 119, 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 \[ \frac{\left (b^2 e^2-6 b c d e+6 c^2 d^2\right ) \log (d+e x)}{e^5}-\frac{d^2 (c d-b e)^2}{2 e^5 (d+e x)^2}+\frac{2 d (2 c d-b e) (c d-b e)}{e^5 (d+e x)}-\frac{c x (3 c d-2 b e)}{e^4}+\frac{c^2 x^2}{2 e^3} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^2/(d + e*x)^3,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{c^{2} \int x\, dx}{e^{3}} - \frac{d^{2} \left (b e - c d\right )^{2}}{2 e^{5} \left (d + e x\right )^{2}} + \frac{2 d \left (b e - 2 c d\right ) \left (b e - c d\right )}{e^{5} \left (d + e x\right )} + \frac{\left (2 b e - 3 c d\right ) \int c\, dx}{e^{4}} + \frac{\left (b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right ) \log{\left (d + e x \right )}}{e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**2/(e*x+d)**3,x)

[Out]

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

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Mathematica [A]  time = 0.122005, size = 116, normalized size = 0.97 \[ \frac{\frac{4 d \left (b^2 e^2-3 b c d e+2 c^2 d^2\right )}{d+e x}+2 \left (b^2 e^2-6 b c d e+6 c^2 d^2\right ) \log (d+e x)-\frac{d^2 (c d-b e)^2}{(d+e x)^2}-2 c e x (3 c d-2 b e)+c^2 e^2 x^2}{2 e^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^2/(d + e*x)^3,x]

[Out]

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

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Maple [A]  time = 0.013, size = 178, normalized size = 1.5 \[{\frac{{c}^{2}{x}^{2}}{2\,{e}^{3}}}+2\,{\frac{bcx}{{e}^{3}}}-3\,{\frac{d{c}^{2}x}{{e}^{4}}}-{\frac{{b}^{2}{d}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{{d}^{3}bc}{{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{{c}^{2}{d}^{4}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}+{\frac{{b}^{2}\ln \left ( ex+d \right ) }{{e}^{3}}}-6\,{\frac{\ln \left ( ex+d \right ) bcd}{{e}^{4}}}+6\,{\frac{\ln \left ( ex+d \right ){c}^{2}{d}^{2}}{{e}^{5}}}+2\,{\frac{{b}^{2}d}{{e}^{3} \left ( ex+d \right ) }}-6\,{\frac{{d}^{2}bc}{{e}^{4} \left ( ex+d \right ) }}+4\,{\frac{{c}^{2}{d}^{3}}{{e}^{5} \left ( ex+d \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^2/(e*x+d)^3,x)

[Out]

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

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Maxima [A]  time = 0.711896, size = 198, normalized size = 1.66 \[ \frac{7 \, c^{2} d^{4} - 10 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2} + 4 \,{\left (2 \, c^{2} d^{3} e - 3 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x}{2 \,{\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} + \frac{c^{2} e x^{2} - 2 \,{\left (3 \, c^{2} d - 2 \, b c e\right )} x}{2 \, e^{4}} + \frac{{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )} \log \left (e x + d\right )}{e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2/(e*x + d)^3,x, algorithm="maxima")

[Out]

1/2*(7*c^2*d^4 - 10*b*c*d^3*e + 3*b^2*d^2*e^2 + 4*(2*c^2*d^3*e - 3*b*c*d^2*e^2 +
b^2*d*e^3)*x)/(e^7*x^2 + 2*d*e^6*x + d^2*e^5) + 1/2*(c^2*e*x^2 - 2*(3*c^2*d - 2
*b*c*e)*x)/e^4 + (6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*log(e*x + d)/e^5

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Fricas [A]  time = 0.224926, size = 321, normalized size = 2.7 \[ \frac{c^{2} e^{4} x^{4} + 7 \, c^{2} d^{4} - 10 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2} - 4 \,{\left (c^{2} d e^{3} - b c e^{4}\right )} x^{3} -{\left (11 \, c^{2} d^{2} e^{2} - 8 \, b c d e^{3}\right )} x^{2} + 2 \,{\left (c^{2} d^{3} e - 4 \, b c d^{2} e^{2} + 2 \, b^{2} d e^{3}\right )} x + 2 \,{\left (6 \, c^{2} d^{4} - 6 \, b c d^{3} e + b^{2} d^{2} e^{2} +{\left (6 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} + 2 \,{\left (6 \, c^{2} d^{3} e - 6 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2/(e*x + d)^3,x, algorithm="fricas")

[Out]

1/2*(c^2*e^4*x^4 + 7*c^2*d^4 - 10*b*c*d^3*e + 3*b^2*d^2*e^2 - 4*(c^2*d*e^3 - b*c
*e^4)*x^3 - (11*c^2*d^2*e^2 - 8*b*c*d*e^3)*x^2 + 2*(c^2*d^3*e - 4*b*c*d^2*e^2 +
2*b^2*d*e^3)*x + 2*(6*c^2*d^4 - 6*b*c*d^3*e + b^2*d^2*e^2 + (6*c^2*d^2*e^2 - 6*b
*c*d*e^3 + b^2*e^4)*x^2 + 2*(6*c^2*d^3*e - 6*b*c*d^2*e^2 + b^2*d*e^3)*x)*log(e*x
+ d))/(e^7*x^2 + 2*d*e^6*x + d^2*e^5)

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Sympy [A]  time = 4.24406, size = 153, normalized size = 1.29 \[ \frac{c^{2} x^{2}}{2 e^{3}} + \frac{3 b^{2} d^{2} e^{2} - 10 b c d^{3} e + 7 c^{2} d^{4} + x \left (4 b^{2} d e^{3} - 12 b c d^{2} e^{2} + 8 c^{2} d^{3} e\right )}{2 d^{2} e^{5} + 4 d e^{6} x + 2 e^{7} x^{2}} + \frac{x \left (2 b c e - 3 c^{2} d\right )}{e^{4}} + \frac{\left (b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right ) \log{\left (d + e x \right )}}{e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**2/(e*x+d)**3,x)

[Out]

c**2*x**2/(2*e**3) + (3*b**2*d**2*e**2 - 10*b*c*d**3*e + 7*c**2*d**4 + x*(4*b**2
*d*e**3 - 12*b*c*d**2*e**2 + 8*c**2*d**3*e))/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*
x**2) + x*(2*b*c*e - 3*c**2*d)/e**4 + (b**2*e**2 - 6*b*c*d*e + 6*c**2*d**2)*log(
d + e*x)/e**5

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GIAC/XCAS [A]  time = 0.20985, size = 181, normalized size = 1.52 \[{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )} e^{\left (-5\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (c^{2} x^{2} e^{3} - 6 \, c^{2} d x e^{2} + 4 \, b c x e^{3}\right )} e^{\left (-6\right )} + \frac{{\left (7 \, c^{2} d^{4} - 10 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2} + 4 \,{\left (2 \, c^{2} d^{3} e - 3 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x\right )} e^{\left (-5\right )}}{2 \,{\left (x e + d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2/(e*x + d)^3,x, algorithm="giac")

[Out]

(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*e^(-5)*ln(abs(x*e + d)) + 1/2*(c^2*x^2*e^3 - 6
*c^2*d*x*e^2 + 4*b*c*x*e^3)*e^(-6) + 1/2*(7*c^2*d^4 - 10*b*c*d^3*e + 3*b^2*d^2*e
^2 + 4*(2*c^2*d^3*e - 3*b*c*d^2*e^2 + b^2*d*e^3)*x)*e^(-5)/(x*e + d)^2