### 3.1142 $$\int \frac{(a+b x+c x^2)^3}{b d+2 c d x} \, dx$$

Optimal. Leaf size=100 $-\frac{3 \left (b^2-4 a c\right ) (b+2 c x)^4}{512 c^4 d}+\frac{3 \left (b^2-4 a c\right )^2 (b+2 c x)^2}{256 c^4 d}-\frac{\left (b^2-4 a c\right )^3 \log (b+2 c x)}{128 c^4 d}+\frac{(b+2 c x)^6}{768 c^4 d}$

[Out]

(3*(b^2 - 4*a*c)^2*(b + 2*c*x)^2)/(256*c^4*d) - (3*(b^2 - 4*a*c)*(b + 2*c*x)^4)/(512*c^4*d) + (b + 2*c*x)^6/(7
68*c^4*d) - ((b^2 - 4*a*c)^3*Log[b + 2*c*x])/(128*c^4*d)

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Rubi [A]  time = 0.117778, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.042, Rules used = {683} $-\frac{3 \left (b^2-4 a c\right ) (b+2 c x)^4}{512 c^4 d}+\frac{3 \left (b^2-4 a c\right )^2 (b+2 c x)^2}{256 c^4 d}-\frac{\left (b^2-4 a c\right )^3 \log (b+2 c x)}{128 c^4 d}+\frac{(b+2 c x)^6}{768 c^4 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(3*(b^2 - 4*a*c)^2*(b + 2*c*x)^2)/(256*c^4*d) - (3*(b^2 - 4*a*c)*(b + 2*c*x)^4)/(512*c^4*d) + (b + 2*c*x)^6/(7
68*c^4*d) - ((b^2 - 4*a*c)^3*Log[b + 2*c*x])/(128*c^4*d)

Rule 683

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] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps

\begin{align*} \int \frac{\left (a+b x+c x^2\right )^3}{b d+2 c d x} \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^3}{64 c^3 (b d+2 c d x)}+\frac{3 \left (-b^2+4 a c\right )^2 (b d+2 c d x)}{64 c^3 d^2}+\frac{3 \left (-b^2+4 a c\right ) (b d+2 c d x)^3}{64 c^3 d^4}+\frac{(b d+2 c d x)^5}{64 c^3 d^6}\right ) \, dx\\ &=\frac{3 \left (b^2-4 a c\right )^2 (b+2 c x)^2}{256 c^4 d}-\frac{3 \left (b^2-4 a c\right ) (b+2 c x)^4}{512 c^4 d}+\frac{(b+2 c x)^6}{768 c^4 d}-\frac{\left (b^2-4 a c\right )^3 \log (b+2 c x)}{128 c^4 d}\\ \end{align*}

Mathematica [A]  time = 0.0451798, size = 111, normalized size = 1.11 $\frac{2 c x (b+c x) \left (8 c^2 \left (18 a^2+9 a c x^2+2 c^2 x^4\right )+2 b^2 c \left (5 c x^2-18 a\right )+8 b c^2 x \left (9 a+4 c x^2\right )-6 b^3 c x+3 b^4\right )-3 \left (b^2-4 a c\right )^3 \log (b+2 c x)}{384 c^4 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*c*x*(b + c*x)*(3*b^4 - 6*b^3*c*x + 8*b*c^2*x*(9*a + 4*c*x^2) + 2*b^2*c*(-18*a + 5*c*x^2) + 8*c^2*(18*a^2 +
9*a*c*x^2 + 2*c^2*x^4)) - 3*(b^2 - 4*a*c)^3*Log[b + 2*c*x])/(384*c^4*d)

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Maple [B]  time = 0.041, size = 222, normalized size = 2.2 \begin{align*}{\frac{{c}^{2}{x}^{6}}{12\,d}}+{\frac{bc{x}^{5}}{4\,d}}+{\frac{3\,a{x}^{4}c}{8\,d}}+{\frac{7\,{x}^{4}{b}^{2}}{32\,d}}+{\frac{3\,a{x}^{3}b}{4\,d}}+{\frac{{x}^{3}{b}^{3}}{48\,cd}}+{\frac{3\,{a}^{2}{x}^{2}}{4\,d}}+{\frac{3\,a{b}^{2}{x}^{2}}{16\,cd}}-{\frac{{x}^{2}{b}^{4}}{64\,{c}^{2}d}}+{\frac{3\,b{a}^{2}x}{4\,cd}}-{\frac{3\,a{b}^{3}x}{16\,{c}^{2}d}}+{\frac{{b}^{5}x}{64\,d{c}^{3}}}+{\frac{\ln \left ( 2\,cx+b \right ){a}^{3}}{2\,cd}}-{\frac{3\,\ln \left ( 2\,cx+b \right ){a}^{2}{b}^{2}}{8\,{c}^{2}d}}+{\frac{3\,\ln \left ( 2\,cx+b \right ) a{b}^{4}}{32\,d{c}^{3}}}-{\frac{\ln \left ( 2\,cx+b \right ){b}^{6}}{128\,d{c}^{4}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12/d*c^2*x^6+1/4/d*c*b*x^5+3/8/d*x^4*a*c+7/32/d*x^4*b^2+3/4/d*x^3*a*b+1/48/d/c*x^3*b^3+3/4/d*a^2*x^2+3/16/d/
c*x^2*a*b^2-1/64/d/c^2*x^2*b^4+3/4/d/c*a^2*b*x-3/16/d/c^2*a*b^3*x+1/64/d/c^3*b^5*x+1/2/d/c*ln(2*c*x+b)*a^3-3/8
/d/c^2*ln(2*c*x+b)*a^2*b^2+3/32/d/c^3*ln(2*c*x+b)*a*b^4-1/128/d/c^4*ln(2*c*x+b)*b^6

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Maxima [A]  time = 1.14785, size = 220, normalized size = 2.2 \begin{align*} \frac{16 \, c^{5} x^{6} + 48 \, b c^{4} x^{5} + 6 \,{\left (7 \, b^{2} c^{3} + 12 \, a c^{4}\right )} x^{4} + 4 \,{\left (b^{3} c^{2} + 36 \, a b c^{3}\right )} x^{3} - 3 \,{\left (b^{4} c - 12 \, a b^{2} c^{2} - 48 \, a^{2} c^{3}\right )} x^{2} + 3 \,{\left (b^{5} - 12 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} x}{192 \, c^{3} d} - \frac{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \log \left (2 \, c x + b\right )}{128 \, c^{4} d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*(16*c^5*x^6 + 48*b*c^4*x^5 + 6*(7*b^2*c^3 + 12*a*c^4)*x^4 + 4*(b^3*c^2 + 36*a*b*c^3)*x^3 - 3*(b^4*c - 12
*a*b^2*c^2 - 48*a^2*c^3)*x^2 + 3*(b^5 - 12*a*b^3*c + 48*a^2*b*c^2)*x)/(c^3*d) - 1/128*(b^6 - 12*a*b^4*c + 48*a
^2*b^2*c^2 - 64*a^3*c^3)*log(2*c*x + b)/(c^4*d)

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Fricas [A]  time = 1.97344, size = 356, normalized size = 3.56 \begin{align*} \frac{32 \, c^{6} x^{6} + 96 \, b c^{5} x^{5} + 12 \,{\left (7 \, b^{2} c^{4} + 12 \, a c^{5}\right )} x^{4} + 8 \,{\left (b^{3} c^{3} + 36 \, a b c^{4}\right )} x^{3} - 6 \,{\left (b^{4} c^{2} - 12 \, a b^{2} c^{3} - 48 \, a^{2} c^{4}\right )} x^{2} + 6 \,{\left (b^{5} c - 12 \, a b^{3} c^{2} + 48 \, a^{2} b c^{3}\right )} x - 3 \,{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \log \left (2 \, c x + b\right )}{384 \, c^{4} d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(32*c^6*x^6 + 96*b*c^5*x^5 + 12*(7*b^2*c^4 + 12*a*c^5)*x^4 + 8*(b^3*c^3 + 36*a*b*c^4)*x^3 - 6*(b^4*c^2 -
12*a*b^2*c^3 - 48*a^2*c^4)*x^2 + 6*(b^5*c - 12*a*b^3*c^2 + 48*a^2*b*c^3)*x - 3*(b^6 - 12*a*b^4*c + 48*a^2*b^2
*c^2 - 64*a^3*c^3)*log(2*c*x + b))/(c^4*d)

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Sympy [A]  time = 1.08956, size = 141, normalized size = 1.41 \begin{align*} \frac{b c x^{5}}{4 d} + \frac{c^{2} x^{6}}{12 d} + \frac{x^{4} \left (12 a c + 7 b^{2}\right )}{32 d} + \frac{x^{3} \left (36 a b c + b^{3}\right )}{48 c d} + \frac{x^{2} \left (48 a^{2} c^{2} + 12 a b^{2} c - b^{4}\right )}{64 c^{2} d} + \frac{x \left (48 a^{2} b c^{2} - 12 a b^{3} c + b^{5}\right )}{64 c^{3} d} + \frac{\left (4 a c - b^{2}\right )^{3} \log{\left (b + 2 c x \right )}}{128 c^{4} d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*c*x**5/(4*d) + c**2*x**6/(12*d) + x**4*(12*a*c + 7*b**2)/(32*d) + x**3*(36*a*b*c + b**3)/(48*c*d) + x**2*(48
*a**2*c**2 + 12*a*b**2*c - b**4)/(64*c**2*d) + x*(48*a**2*b*c**2 - 12*a*b**3*c + b**5)/(64*c**3*d) + (4*a*c -
b**2)**3*log(b + 2*c*x)/(128*c**4*d)

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Giac [B]  time = 1.16337, size = 288, normalized size = 2.88 \begin{align*} -\frac{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \log \left ({\left | 2 \, c x + b \right |}\right )}{128 \, c^{4} d} + \frac{16 \, c^{8} d^{5} x^{6} + 48 \, b c^{7} d^{5} x^{5} + 42 \, b^{2} c^{6} d^{5} x^{4} + 72 \, a c^{7} d^{5} x^{4} + 4 \, b^{3} c^{5} d^{5} x^{3} + 144 \, a b c^{6} d^{5} x^{3} - 3 \, b^{4} c^{4} d^{5} x^{2} + 36 \, a b^{2} c^{5} d^{5} x^{2} + 144 \, a^{2} c^{6} d^{5} x^{2} + 3 \, b^{5} c^{3} d^{5} x - 36 \, a b^{3} c^{4} d^{5} x + 144 \, a^{2} b c^{5} d^{5} x}{192 \, c^{6} d^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/128*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*log(abs(2*c*x + b))/(c^4*d) + 1/192*(16*c^8*d^5*x^6 +
48*b*c^7*d^5*x^5 + 42*b^2*c^6*d^5*x^4 + 72*a*c^7*d^5*x^4 + 4*b^3*c^5*d^5*x^3 + 144*a*b*c^6*d^5*x^3 - 3*b^4*c^4
*d^5*x^2 + 36*a*b^2*c^5*d^5*x^2 + 144*a^2*c^6*d^5*x^2 + 3*b^5*c^3*d^5*x - 36*a*b^3*c^4*d^5*x + 144*a^2*b*c^5*d
^5*x)/(c^6*d^6)