### 3.1801 $$\int \frac{(a+b x)^4}{a c+(b c+a d) x+b d x^2} \, dx$$

Optimal. Leaf size=74 $\frac{b x (b c-a d)^2}{d^3}-\frac{(a+b x)^2 (b c-a d)}{2 d^2}-\frac{(b c-a d)^3 \log (c+d x)}{d^4}+\frac{(a+b x)^3}{3 d}$

[Out]

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

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Rubi [A]  time = 0.0361576, antiderivative size = 74, 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} $\frac{b x (b c-a d)^2}{d^3}-\frac{(a+b x)^2 (b c-a d)}{2 d^2}-\frac{(b c-a d)^3 \log (c+d x)}{d^4}+\frac{(a+b x)^3}{3 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [A]  time = 0.026245, size = 74, normalized size = 1. $\frac{b d x \left (18 a^2 d^2+9 a b d (d x-2 c)+b^2 \left (6 c^2-3 c d x+2 d^2 x^2\right )\right )-6 (b c-a d)^3 \log (c+d x)}{6 d^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.04, size = 133, normalized size = 1.8 \begin{align*}{\frac{{b}^{3}{x}^{3}}{3\,d}}+{\frac{3\,a{b}^{2}{x}^{2}}{2\,d}}-{\frac{{b}^{3}{x}^{2}c}{2\,{d}^{2}}}+3\,{\frac{b{a}^{2}x}{d}}-3\,{\frac{ac{b}^{2}x}{{d}^{2}}}+{\frac{{b}^{3}{c}^{2}x}{{d}^{3}}}+{\frac{\ln \left ( dx+c \right ){a}^{3}}{d}}-3\,{\frac{\ln \left ( dx+c \right ) cb{a}^{2}}{{d}^{2}}}+3\,{\frac{\ln \left ( dx+c \right ) a{c}^{2}{b}^{2}}{{d}^{3}}}-{\frac{\ln \left ( dx+c \right ){b}^{3}{c}^{3}}{{d}^{4}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.06319, size = 154, normalized size = 2.08 \begin{align*} \frac{2 \, b^{3} d^{2} x^{3} - 3 \,{\left (b^{3} c d - 3 \, a b^{2} d^{2}\right )} x^{2} + 6 \,{\left (b^{3} c^{2} - 3 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x}{6 \, d^{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 (d x + c\right )}{d^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.57452, size = 238, normalized size = 3.22 \begin{align*} \frac{2 \, b^{3} d^{3} x^{3} - 3 \,{\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} x^{2} + 6 \,{\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 3 \, 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 (d x + c\right )}{6 \, d^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.536191, size = 82, normalized size = 1.11 \begin{align*} \frac{b^{3} x^{3}}{3 d} + \frac{x^{2} \left (3 a b^{2} d - b^{3} c\right )}{2 d^{2}} + \frac{x \left (3 a^{2} b d^{2} - 3 a b^{2} c d + b^{3} c^{2}\right )}{d^{3}} + \frac{\left (a d - b c\right )^{3} \log{\left (c + d x \right )}}{d^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15981, size = 157, normalized size = 2.12 \begin{align*} \frac{2 \, b^{3} d^{2} x^{3} - 3 \, b^{3} c d x^{2} + 9 \, a b^{2} d^{2} x^{2} + 6 \, b^{3} c^{2} x - 18 \, a b^{2} c d x + 18 \, a^{2} b d^{2} x}{6 \, d^{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 | d x + c \right |}\right )}{d^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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