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

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

[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 = 0.0815552, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.053, Rules used = {698} $\frac{d^2 (c d-b e)^2 \log (d+e x)}{e^5}-\frac{c x^3 (c d-2 b e)}{3 e^2}+\frac{x^2 (c d-b e)^2}{2 e^3}-\frac{d x (c d-b e)^2}{e^4}+\frac{c^2 x^4}{4 e}$

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

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{\left (b x+c x^2\right )^2}{d+e x} \, dx &=\int \left (-\frac{d (c d-b e)^2}{e^4}+\frac{(-c d+b e)^2 x}{e^3}-\frac{c (c d-2 b e) x^2}{e^2}+\frac{c^2 x^3}{e}+\frac{d^2 (c d-b e)^2}{e^4 (d+e x)}\right ) \, dx\\ &=-\frac{d (c d-b e)^2 x}{e^4}+\frac{(c d-b e)^2 x^2}{2 e^3}-\frac{c (c d-2 b e) x^3}{3 e^2}+\frac{c^2 x^4}{4 e}+\frac{d^2 (c d-b e)^2 \log (d+e x)}{e^5}\\ \end{align*}

Mathematica [A]  time = 0.0420754, size = 106, normalized size = 1.14 $\frac{\left (b^2 d^2 e^2-2 b c d^3 e+c^2 d^4\right ) \log (d+e x)}{e^5}-\frac{c x^3 (c d-2 b e)}{3 e^2}+\frac{x^2 (b e-c d)^2}{2 e^3}-\frac{d x (c d-b e)^2}{e^4}+\frac{c^2 x^4}{4 e}$

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 = 0.045, size = 152, normalized size = 1.6 \begin{align*}{\frac{{c}^{2}{x}^{4}}{4\,e}}+{\frac{2\,bc{x}^{3}}{3\,e}}-{\frac{{c}^{2}d{x}^{3}}{3\,{e}^{2}}}+{\frac{{b}^{2}{x}^{2}}{2\,e}}-{\frac{bc{x}^{2}d}{{e}^{2}}}+{\frac{{c}^{2}{x}^{2}{d}^{2}}{2\,{e}^{3}}}-{\frac{{b}^{2}dx}{{e}^{2}}}+2\,{\frac{bc{d}^{2}x}{{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}}} \end{align*}

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*b^2*d*x+
2/e^3*b*c*d^2*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 = 1.15742, size = 177, normalized size = 1.9 \begin{align*} \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}} \end{align*}

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 = 1.53304, size = 279, normalized size = 3. \begin{align*} \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}} \end{align*}

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 = 0.958715, size = 112, normalized size = 1.2 \begin{align*} \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}} \end{align*}

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 [A]  time = 1.28223, size = 181, normalized size = 1.95 \begin{align*}{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} e^{\left (-5\right )} \log \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 )} \end{align*}

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)*log(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)