### 3.1802 $$\int \frac{(a+b x)^3}{a c+(b c+a d) x+b d x^2} \, dx$$

Optimal. Leaf size=50 $-\frac{b x (b c-a d)}{d^2}+\frac{(b c-a d)^2 \log (c+d x)}{d^3}+\frac{(a+b x)^2}{2 d}$

[Out]

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

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Rubi [A]  time = 0.0254921, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.069, Rules used = {626, 43} $-\frac{b x (b c-a d)}{d^2}+\frac{(b c-a d)^2 \log (c+d x)}{d^3}+\frac{(a+b x)^2}{2 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 626

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0158795, size = 43, normalized size = 0.86 $\frac{b d x (4 a d-2 b c+b d x)+2 (b c-a d)^2 \log (c+d x)}{2 d^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.041, size = 74, normalized size = 1.5 \begin{align*}{\frac{{b}^{2}{x}^{2}}{2\,d}}+2\,{\frac{abx}{d}}-{\frac{{b}^{2}xc}{{d}^{2}}}+{\frac{\ln \left ( dx+c \right ){a}^{2}}{d}}-2\,{\frac{\ln \left ( dx+c \right ) cab}{{d}^{2}}}+{\frac{\ln \left ( dx+c \right ){b}^{2}{c}^{2}}{{d}^{3}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^2/d*x^2+2*b/d*a*x-b^2/d^2*x*c+1/d*ln(d*x+c)*a^2-2/d^2*ln(d*x+c)*c*a*b+1/d^3*ln(d*x+c)*b^2*c^2

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Maxima [A]  time = 1.04225, size = 81, normalized size = 1.62 \begin{align*} \frac{b^{2} d x^{2} - 2 \,{\left (b^{2} c - 2 \, a b d\right )} x}{2 \, d^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (d x + c\right )}{d^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^2*d*x^2 - 2*(b^2*c - 2*a*b*d)*x)/d^2 + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*log(d*x + c)/d^3

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Fricas [A]  time = 1.5661, size = 135, normalized size = 2.7 \begin{align*} \frac{b^{2} d^{2} x^{2} - 2 \,{\left (b^{2} c d - 2 \, a b d^{2}\right )} x + 2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (d x + c\right )}{2 \, d^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^2*d^2*x^2 - 2*(b^2*c*d - 2*a*b*d^2)*x + 2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*log(d*x + c))/d^3

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Sympy [A]  time = 0.602858, size = 44, normalized size = 0.88 \begin{align*} \frac{b^{2} x^{2}}{2 d} + \frac{x \left (2 a b d - b^{2} c\right )}{d^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (c + d x \right )}}{d^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b**2*x**2/(2*d) + x*(2*a*b*d - b**2*c)/d**2 + (a*d - b*c)**2*log(c + d*x)/d**3

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Giac [A]  time = 1.19037, size = 81, normalized size = 1.62 \begin{align*} \frac{b^{2} d x^{2} - 2 \, b^{2} c x + 4 \, a b d x}{2 \, d^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^2*d*x^2 - 2*b^2*c*x + 4*a*b*d*x)/d^2 + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*log(abs(d*x + c))/d^3