### 3.249 $$\int \frac{(b x+c x^2)^3}{d+e x} \, dx$$

Optimal. Leaf size=151 $\frac{c x^4 \left (3 b^2 e^2-3 b c d e+c^2 d^2\right )}{4 e^3}-\frac{c^2 x^5 (c d-3 b e)}{5 e^2}-\frac{d^2 x (c d-b e)^3}{e^6}+\frac{d^3 (c d-b e)^3 \log (d+e x)}{e^7}-\frac{x^3 (c d-b e)^3}{3 e^4}+\frac{d x^2 (c d-b e)^3}{2 e^5}+\frac{c^3 x^6}{6 e}$

[Out]

-((d^2*(c*d - b*e)^3*x)/e^6) + (d*(c*d - b*e)^3*x^2)/(2*e^5) - ((c*d - b*e)^3*x^3)/(3*e^4) + (c*(c^2*d^2 - 3*b
*c*d*e + 3*b^2*e^2)*x^4)/(4*e^3) - (c^2*(c*d - 3*b*e)*x^5)/(5*e^2) + (c^3*x^6)/(6*e) + (d^3*(c*d - b*e)^3*Log[
d + e*x])/e^7

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Rubi [A]  time = 0.151383, antiderivative size = 151, 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} $\frac{c x^4 \left (3 b^2 e^2-3 b c d e+c^2 d^2\right )}{4 e^3}-\frac{c^2 x^5 (c d-3 b e)}{5 e^2}-\frac{d^2 x (c d-b e)^3}{e^6}+\frac{d^3 (c d-b e)^3 \log (d+e x)}{e^7}-\frac{x^3 (c d-b e)^3}{3 e^4}+\frac{d x^2 (c d-b e)^3}{2 e^5}+\frac{c^3 x^6}{6 e}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*x + c*x^2)^3/(d + e*x),x]

[Out]

-((d^2*(c*d - b*e)^3*x)/e^6) + (d*(c*d - b*e)^3*x^2)/(2*e^5) - ((c*d - b*e)^3*x^3)/(3*e^4) + (c*(c^2*d^2 - 3*b
*c*d*e + 3*b^2*e^2)*x^4)/(4*e^3) - (c^2*(c*d - 3*b*e)*x^5)/(5*e^2) + (c^3*x^6)/(6*e) + (d^3*(c*d - b*e)^3*Log[
d + e*x])/e^7

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 )^3}{d+e x} \, dx &=\int \left (-\frac{d^2 (c d-b e)^3}{e^6}+\frac{d (c d-b e)^3 x}{e^5}+\frac{(-c d+b e)^3 x^2}{e^4}+\frac{c \left (c^2 d^2-3 b c d e+3 b^2 e^2\right ) x^3}{e^3}-\frac{c^2 (c d-3 b e) x^4}{e^2}+\frac{c^3 x^5}{e}+\frac{d^3 (c d-b e)^3}{e^6 (d+e x)}\right ) \, dx\\ &=-\frac{d^2 (c d-b e)^3 x}{e^6}+\frac{d (c d-b e)^3 x^2}{2 e^5}-\frac{(c d-b e)^3 x^3}{3 e^4}+\frac{c \left (c^2 d^2-3 b c d e+3 b^2 e^2\right ) x^4}{4 e^3}-\frac{c^2 (c d-3 b e) x^5}{5 e^2}+\frac{c^3 x^6}{6 e}+\frac{d^3 (c d-b e)^3 \log (d+e x)}{e^7}\\ \end{align*}

Mathematica [A]  time = 0.0830686, size = 144, normalized size = 0.95 $\frac{15 c e^4 x^4 \left (3 b^2 e^2-3 b c d e+c^2 d^2\right )-12 c^2 e^5 x^5 (c d-3 b e)-60 d^2 e x (c d-b e)^3+60 d^3 (c d-b e)^3 \log (d+e x)+20 e^3 x^3 (b e-c d)^3+30 d e^2 x^2 (c d-b e)^3+10 c^3 e^6 x^6}{60 e^7}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-60*d^2*e*(c*d - b*e)^3*x + 30*d*e^2*(c*d - b*e)^3*x^2 + 20*e^3*(-(c*d) + b*e)^3*x^3 + 15*c*e^4*(c^2*d^2 - 3*
b*c*d*e + 3*b^2*e^2)*x^4 - 12*c^2*e^5*(c*d - 3*b*e)*x^5 + 10*c^3*e^6*x^6 + 60*d^3*(c*d - b*e)^3*Log[d + e*x])/
(60*e^7)

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Maple [B]  time = 0.046, size = 302, normalized size = 2. \begin{align*}{\frac{{c}^{3}{x}^{6}}{6\,e}}+{\frac{3\,b{x}^{5}{c}^{2}}{5\,e}}-{\frac{{c}^{3}d{x}^{5}}{5\,{e}^{2}}}+{\frac{3\,{x}^{4}{b}^{2}c}{4\,e}}-{\frac{3\,b{x}^{4}{c}^{2}d}{4\,{e}^{2}}}+{\frac{{x}^{4}{c}^{3}{d}^{2}}{4\,{e}^{3}}}+{\frac{{x}^{3}{b}^{3}}{3\,e}}-{\frac{{b}^{2}c{x}^{3}d}{{e}^{2}}}+{\frac{b{x}^{3}{c}^{2}{d}^{2}}{{e}^{3}}}-{\frac{{x}^{3}{c}^{3}{d}^{3}}{3\,{e}^{4}}}-{\frac{{x}^{2}{b}^{3}d}{2\,{e}^{2}}}+{\frac{3\,{b}^{2}{x}^{2}c{d}^{2}}{2\,{e}^{3}}}-{\frac{3\,b{x}^{2}{c}^{2}{d}^{3}}{2\,{e}^{4}}}+{\frac{{x}^{2}{c}^{3}{d}^{4}}{2\,{e}^{5}}}+{\frac{{b}^{3}{d}^{2}x}{{e}^{3}}}-3\,{\frac{{b}^{2}c{d}^{3}x}{{e}^{4}}}+3\,{\frac{b{c}^{2}{d}^{4}x}{{e}^{5}}}-{\frac{{c}^{3}{d}^{5}x}{{e}^{6}}}-{\frac{{d}^{3}\ln \left ( ex+d \right ){b}^{3}}{{e}^{4}}}+3\,{\frac{{d}^{4}\ln \left ( ex+d \right ){b}^{2}c}{{e}^{5}}}-3\,{\frac{{d}^{5}\ln \left ( ex+d \right ) b{c}^{2}}{{e}^{6}}}+{\frac{{d}^{6}\ln \left ( ex+d \right ){c}^{3}}{{e}^{7}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^3/(e*x+d),x)

[Out]

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

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Maxima [A]  time = 1.12637, size = 356, normalized size = 2.36 \begin{align*} \frac{10 \, c^{3} e^{5} x^{6} - 12 \,{\left (c^{3} d e^{4} - 3 \, b c^{2} e^{5}\right )} x^{5} + 15 \,{\left (c^{3} d^{2} e^{3} - 3 \, b c^{2} d e^{4} + 3 \, b^{2} c e^{5}\right )} x^{4} - 20 \,{\left (c^{3} d^{3} e^{2} - 3 \, b c^{2} d^{2} e^{3} + 3 \, b^{2} c d e^{4} - b^{3} e^{5}\right )} x^{3} + 30 \,{\left (c^{3} d^{4} e - 3 \, b c^{2} d^{3} e^{2} + 3 \, b^{2} c d^{2} e^{3} - b^{3} d e^{4}\right )} x^{2} - 60 \,{\left (c^{3} d^{5} - 3 \, b c^{2} d^{4} e + 3 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )} x}{60 \, e^{6}} + \frac{{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} \log \left (e x + d\right )}{e^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(10*c^3*e^5*x^6 - 12*(c^3*d*e^4 - 3*b*c^2*e^5)*x^5 + 15*(c^3*d^2*e^3 - 3*b*c^2*d*e^4 + 3*b^2*c*e^5)*x^4 -
20*(c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*b^2*c*d*e^4 - b^3*e^5)*x^3 + 30*(c^3*d^4*e - 3*b*c^2*d^3*e^2 + 3*b^2*c*
d^2*e^3 - b^3*d*e^4)*x^2 - 60*(c^3*d^5 - 3*b*c^2*d^4*e + 3*b^2*c*d^3*e^2 - b^3*d^2*e^3)*x)/e^6 + (c^3*d^6 - 3*
b*c^2*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3)*log(e*x + d)/e^7

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Fricas [A]  time = 1.66888, size = 537, normalized size = 3.56 \begin{align*} \frac{10 \, c^{3} e^{6} x^{6} - 12 \,{\left (c^{3} d e^{5} - 3 \, b c^{2} e^{6}\right )} x^{5} + 15 \,{\left (c^{3} d^{2} e^{4} - 3 \, b c^{2} d e^{5} + 3 \, b^{2} c e^{6}\right )} x^{4} - 20 \,{\left (c^{3} d^{3} e^{3} - 3 \, b c^{2} d^{2} e^{4} + 3 \, b^{2} c d e^{5} - b^{3} e^{6}\right )} x^{3} + 30 \,{\left (c^{3} d^{4} e^{2} - 3 \, b c^{2} d^{3} e^{3} + 3 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} - 60 \,{\left (c^{3} d^{5} e - 3 \, b c^{2} d^{4} e^{2} + 3 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x + 60 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} \log \left (e x + d\right )}{60 \, e^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(10*c^3*e^6*x^6 - 12*(c^3*d*e^5 - 3*b*c^2*e^6)*x^5 + 15*(c^3*d^2*e^4 - 3*b*c^2*d*e^5 + 3*b^2*c*e^6)*x^4 -
20*(c^3*d^3*e^3 - 3*b*c^2*d^2*e^4 + 3*b^2*c*d*e^5 - b^3*e^6)*x^3 + 30*(c^3*d^4*e^2 - 3*b*c^2*d^3*e^3 + 3*b^2*
c*d^2*e^4 - b^3*d*e^5)*x^2 - 60*(c^3*d^5*e - 3*b*c^2*d^4*e^2 + 3*b^2*c*d^3*e^3 - b^3*d^2*e^4)*x + 60*(c^3*d^6
- 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3)*log(e*x + d))/e^7

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Sympy [A]  time = 1.73305, size = 231, normalized size = 1.53 \begin{align*} \frac{c^{3} x^{6}}{6 e} - \frac{d^{3} \left (b e - c d\right )^{3} \log{\left (d + e x \right )}}{e^{7}} + \frac{x^{5} \left (3 b c^{2} e - c^{3} d\right )}{5 e^{2}} + \frac{x^{4} \left (3 b^{2} c e^{2} - 3 b c^{2} d e + c^{3} d^{2}\right )}{4 e^{3}} + \frac{x^{3} \left (b^{3} e^{3} - 3 b^{2} c d e^{2} + 3 b c^{2} d^{2} e - c^{3} d^{3}\right )}{3 e^{4}} - \frac{x^{2} \left (b^{3} d e^{3} - 3 b^{2} c d^{2} e^{2} + 3 b c^{2} d^{3} e - c^{3} d^{4}\right )}{2 e^{5}} + \frac{x \left (b^{3} d^{2} e^{3} - 3 b^{2} c d^{3} e^{2} + 3 b c^{2} d^{4} e - c^{3} d^{5}\right )}{e^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.30434, size = 365, normalized size = 2.42 \begin{align*}{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{60} \,{\left (10 \, c^{3} x^{6} e^{5} - 12 \, c^{3} d x^{5} e^{4} + 15 \, c^{3} d^{2} x^{4} e^{3} - 20 \, c^{3} d^{3} x^{3} e^{2} + 30 \, c^{3} d^{4} x^{2} e - 60 \, c^{3} d^{5} x + 36 \, b c^{2} x^{5} e^{5} - 45 \, b c^{2} d x^{4} e^{4} + 60 \, b c^{2} d^{2} x^{3} e^{3} - 90 \, b c^{2} d^{3} x^{2} e^{2} + 180 \, b c^{2} d^{4} x e + 45 \, b^{2} c x^{4} e^{5} - 60 \, b^{2} c d x^{3} e^{4} + 90 \, b^{2} c d^{2} x^{2} e^{3} - 180 \, b^{2} c d^{3} x e^{2} + 20 \, b^{3} x^{3} e^{5} - 30 \, b^{3} d x^{2} e^{4} + 60 \, b^{3} d^{2} x e^{3}\right )} e^{\left (-6\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3)*e^(-7)*log(abs(x*e + d)) + 1/60*(10*c^3*x^6*e^5 - 12
*c^3*d*x^5*e^4 + 15*c^3*d^2*x^4*e^3 - 20*c^3*d^3*x^3*e^2 + 30*c^3*d^4*x^2*e - 60*c^3*d^5*x + 36*b*c^2*x^5*e^5
- 45*b*c^2*d*x^4*e^4 + 60*b*c^2*d^2*x^3*e^3 - 90*b*c^2*d^3*x^2*e^2 + 180*b*c^2*d^4*x*e + 45*b^2*c*x^4*e^5 - 60
*b^2*c*d*x^3*e^4 + 90*b^2*c*d^2*x^2*e^3 - 180*b^2*c*d^3*x*e^2 + 20*b^3*x^3*e^5 - 30*b^3*d*x^2*e^4 + 60*b^3*d^2
*x*e^3)*e^(-6)