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

Optimal. Leaf size=93 \[ \frac{d^2 (c d-b e)^2 \log (d+e x)}{e^5}-\frac{d x (c d-b e)^2}{e^4}+\frac{x^2 (c d-b e)^2}{2 e^3}-\frac{c x^3 (c d-2 b e)}{3 e^2}+\frac{c^2 x^4}{4 e} \]

[Out]

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

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Rubi [A]  time = 0.209353, antiderivative size = 93, 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 \[ \frac{d^2 (c d-b e)^2 \log (d+e x)}{e^5}-\frac{d x (c d-b e)^2}{e^4}+\frac{x^2 (c d-b e)^2}{2 e^3}-\frac{c x^3 (c d-2 b e)}{3 e^2}+\frac{c^2 x^4}{4 e} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.073342, size = 106, normalized size = 1.14 \[ \frac{\left (b^2 d^2 e^2-2 b c d^3 e+c^2 d^4\right ) \log (d+e x)}{e^5}-\frac{d x (c d-b e)^2}{e^4}+\frac{x^2 (b e-c d)^2}{2 e^3}-\frac{c x^3 (c d-2 b e)}{3 e^2}+\frac{c^2 x^4}{4 e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 152, normalized size = 1.6 \[{\frac{{c}^{2}{x}^{4}}{4\,e}}+{\frac{2\,b{x}^{3}c}{3\,e}}-{\frac{d{c}^{2}{x}^{3}}{3\,{e}^{2}}}+{\frac{{b}^{2}{x}^{2}}{2\,e}}-{\frac{{x}^{2}bcd}{{e}^{2}}}+{\frac{{c}^{2}{x}^{2}{d}^{2}}{2\,{e}^{3}}}-{\frac{{b}^{2}dx}{{e}^{2}}}+2\,{\frac{{d}^{2}bcx}{{e}^{3}}}-{\frac{{c}^{2}{d}^{3}x}{{e}^{4}}}+{\frac{{d}^{2}\ln \left ( ex+d \right ){b}^{2}}{{e}^{3}}}-2\,{\frac{{d}^{3}\ln \left ( ex+d \right ) bc}{{e}^{4}}}+{\frac{{d}^{4}\ln \left ( ex+d \right ){c}^{2}}{{e}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^2/(e*x+d),x)

[Out]

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

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Maxima [A]  time = 0.69262, size = 177, normalized size = 1.9 \[ \frac{3 \, c^{2} e^{3} x^{4} - 4 \,{\left (c^{2} d e^{2} - 2 \, b c e^{3}\right )} x^{3} + 6 \,{\left (c^{2} d^{2} e - 2 \, b c d e^{2} + b^{2} e^{3}\right )} x^{2} - 12 \,{\left (c^{2} d^{3} - 2 \, b c d^{2} e + b^{2} d e^{2}\right )} x}{12 \, e^{4}} + \frac{{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \log \left (e x + d\right )}{e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(3*c^2*e^3*x^4 - 4*(c^2*d*e^2 - 2*b*c*e^3)*x^3 + 6*(c^2*d^2*e - 2*b*c*d*e^2
+ b^2*e^3)*x^2 - 12*(c^2*d^3 - 2*b*c*d^2*e + b^2*d*e^2)*x)/e^4 + (c^2*d^4 - 2*b
*c*d^3*e + b^2*d^2*e^2)*log(e*x + d)/e^5

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Fricas [A]  time = 0.215725, size = 180, normalized size = 1.94 \[ \frac{3 \, c^{2} e^{4} x^{4} - 4 \,{\left (c^{2} d e^{3} - 2 \, b c e^{4}\right )} x^{3} + 6 \,{\left (c^{2} d^{2} e^{2} - 2 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} - 12 \,{\left (c^{2} d^{3} e - 2 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x + 12 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \log \left (e x + d\right )}{12 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(3*c^2*e^4*x^4 - 4*(c^2*d*e^3 - 2*b*c*e^4)*x^3 + 6*(c^2*d^2*e^2 - 2*b*c*d*e
^3 + b^2*e^4)*x^2 - 12*(c^2*d^3*e - 2*b*c*d^2*e^2 + b^2*d*e^3)*x + 12*(c^2*d^4 -
2*b*c*d^3*e + b^2*d^2*e^2)*log(e*x + d))/e^5

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Sympy [A]  time = 1.78719, size = 112, normalized size = 1.2 \[ \frac{c^{2} x^{4}}{4 e} + \frac{d^{2} \left (b e - c d\right )^{2} \log{\left (d + e x \right )}}{e^{5}} + \frac{x^{3} \left (2 b c e - c^{2} d\right )}{3 e^{2}} + \frac{x^{2} \left (b^{2} e^{2} - 2 b c d e + c^{2} d^{2}\right )}{2 e^{3}} - \frac{x \left (b^{2} d e^{2} - 2 b c d^{2} e + c^{2} d^{3}\right )}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.206033, size = 181, normalized size = 1.95 \[{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} e^{\left (-5\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{12} \,{\left (3 \, c^{2} x^{4} e^{3} - 4 \, c^{2} d x^{3} e^{2} + 6 \, c^{2} d^{2} x^{2} e - 12 \, c^{2} d^{3} x + 8 \, b c x^{3} e^{3} - 12 \, b c d x^{2} e^{2} + 24 \, b c d^{2} x e + 6 \, b^{2} x^{2} e^{3} - 12 \, b^{2} d x e^{2}\right )} e^{\left (-4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2)*e^(-5)*ln(abs(x*e + d)) + 1/12*(3*c^2*x^4*
e^3 - 4*c^2*d*x^3*e^2 + 6*c^2*d^2*x^2*e - 12*c^2*d^3*x + 8*b*c*x^3*e^3 - 12*b*c*
d*x^2*e^2 + 24*b*c*d^2*x*e + 6*b^2*x^2*e^3 - 12*b^2*d*x*e^2)*e^(-4)