### 3.1790 $$\int \frac{(a c+(b c+a d) x+b d x^2)^3}{(a+b x)^4} \, dx$$

Optimal. Leaf size=73 $\frac{d x (b c-a d)^2}{b^3}+\frac{(c+d x)^2 (b c-a d)}{2 b^2}+\frac{(b c-a d)^3 \log (a+b x)}{b^4}+\frac{(c+d x)^3}{3 b}$

[Out]

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

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Rubi [A]  time = 0.0371784, antiderivative size = 73, 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} $\frac{d x (b c-a d)^2}{b^3}+\frac{(c+d x)^2 (b c-a d)}{2 b^2}+\frac{(b c-a d)^3 \log (a+b x)}{b^4}+\frac{(c+d x)^3}{3 b}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0287097, size = 74, normalized size = 1.01 $\frac{b d x \left (6 a^2 d^2-3 a b d (6 c+d x)+b^2 \left (18 c^2+9 c d x+2 d^2 x^2\right )\right )+6 (b c-a d)^3 \log (a+b x)}{6 b^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.042, size = 133, normalized size = 1.8 \begin{align*}{\frac{{d}^{3}{x}^{3}}{3\,b}}-{\frac{{d}^{3}{x}^{2}a}{2\,{b}^{2}}}+{\frac{3\,{d}^{2}{x}^{2}c}{2\,b}}+{\frac{{d}^{3}{a}^{2}x}{{b}^{3}}}-3\,{\frac{ac{d}^{2}x}{{b}^{2}}}+3\,{\frac{{c}^{2}dx}{b}}-{\frac{\ln \left ( bx+a \right ){a}^{3}{d}^{3}}{{b}^{4}}}+3\,{\frac{\ln \left ( bx+a \right ) c{a}^{2}{d}^{2}}{{b}^{3}}}-3\,{\frac{\ln \left ( bx+a \right ) a{c}^{2}d}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ){c}^{3}}{b}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((a*c+(a*d+b*c)*x+b*d*x^2)^3/(b*x+a)^4,x)

[Out]

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

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Maxima [A]  time = 1.03305, size = 154, normalized size = 2.11 \begin{align*} \frac{2 \, b^{2} d^{3} x^{3} + 3 \,{\left (3 \, b^{2} c d^{2} - a b d^{3}\right )} x^{2} + 6 \,{\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} x}{6 \, b^{3}} + \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{b^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x^2)^3/(b*x+a)^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.52819, size = 238, normalized size = 3.26 \begin{align*} \frac{2 \, b^{3} d^{3} x^{3} + 3 \,{\left (3 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 6 \,{\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x + 6 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{6 \, b^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x^2)^3/(b*x+a)^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.799061, size = 82, normalized size = 1.12 \begin{align*} \frac{d^{3} x^{3}}{3 b} - \frac{x^{2} \left (a d^{3} - 3 b c d^{2}\right )}{2 b^{2}} + \frac{x \left (a^{2} d^{3} - 3 a b c d^{2} + 3 b^{2} c^{2} d\right )}{b^{3}} - \frac{\left (a d - b c\right )^{3} \log{\left (a + b x \right )}}{b^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x**2)**3/(b*x+a)**4,x)

[Out]

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

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Giac [A]  time = 1.26836, size = 155, normalized size = 2.12 \begin{align*} \frac{2 \, b^{2} d^{3} x^{3} + 9 \, b^{2} c d^{2} x^{2} - 3 \, a b d^{3} x^{2} + 18 \, b^{2} c^{2} d x - 18 \, a b c d^{2} x + 6 \, a^{2} d^{3} x}{6 \, b^{3}} + \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x^2)^3/(b*x+a)^4,x, algorithm="giac")

[Out]

1/6*(2*b^2*d^3*x^3 + 9*b^2*c*d^2*x^2 - 3*a*b*d^3*x^2 + 18*b^2*c^2*d*x - 18*a*b*c*d^2*x + 6*a^2*d^3*x)/b^3 + (b
^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*log(abs(b*x + a))/b^4