### 3.1811 $$\int \frac{(a+b x)^5}{(a c+(b c+a d) x+b d x^2)^2} \, dx$$

Optimal. Leaf size=75 $-\frac{b^2 x (2 b c-3 a d)}{d^3}+\frac{(b c-a d)^3}{d^4 (c+d x)}+\frac{3 b (b c-a d)^2 \log (c+d x)}{d^4}+\frac{b^3 x^2}{2 d^2}$

[Out]

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

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Rubi [A]  time = 0.0664855, antiderivative size = 75, 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} $-\frac{b^2 x (2 b c-3 a d)}{d^3}+\frac{(b c-a d)^3}{d^4 (c+d x)}+\frac{3 b (b c-a d)^2 \log (c+d x)}{d^4}+\frac{b^3 x^2}{2 d^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x)^5/(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]

[Out]

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

Mathematica [A]  time = 0.0341325, size = 114, normalized size = 1.52 $\frac{3 a^2 b c d^2-a^3 d^3-3 a b^2 c^2 d+b^3 c^3}{d^4 (c+d x)}+\frac{3 \left (a^2 b d^2-2 a b^2 c d+b^3 c^2\right ) \log (c+d x)}{d^4}-\frac{b^2 x (2 b c-3 a d)}{d^3}+\frac{b^3 x^2}{2 d^2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x)^5/(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]

[Out]

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

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Maple [B]  time = 0.046, size = 149, normalized size = 2. \begin{align*}{\frac{{b}^{3}{x}^{2}}{2\,{d}^{2}}}+3\,{\frac{{b}^{2}ax}{{d}^{2}}}-2\,{\frac{{b}^{3}cx}{{d}^{3}}}+3\,{\frac{b\ln \left ( dx+c \right ){a}^{2}}{{d}^{2}}}-6\,{\frac{{b}^{2}\ln \left ( dx+c \right ) ca}{{d}^{3}}}+3\,{\frac{{b}^{3}\ln \left ( dx+c \right ){c}^{2}}{{d}^{4}}}-{\frac{{a}^{3}}{d \left ( dx+c \right ) }}+3\,{\frac{bc{a}^{2}}{{d}^{2} \left ( dx+c \right ) }}-3\,{\frac{a{b}^{2}{c}^{2}}{{d}^{3} \left ( dx+c \right ) }}+{\frac{{b}^{3}{c}^{3}}{{d}^{4} \left ( dx+c \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x)

[Out]

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

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Maxima [A]  time = 1.05338, size = 158, normalized size = 2.11 \begin{align*} \frac{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}}{d^{5} x + c d^{4}} + \frac{b^{3} d x^{2} - 2 \,{\left (2 \, b^{3} c - 3 \, a b^{2} d\right )} x}{2 \, d^{3}} + \frac{3 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\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)^5/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x, algorithm="maxima")

[Out]

(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)/(d^5*x + c*d^4) + 1/2*(b^3*d*x^2 - 2*(2*b^3*c - 3*a*b^2*d)
*x)/d^3 + 3*(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*log(d*x + c)/d^4

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Fricas [B]  time = 1.52327, size = 354, normalized size = 4.72 \begin{align*} \frac{b^{3} d^{3} x^{3} + 2 \, b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3} - 3 \,{\left (b^{3} c d^{2} - 2 \, a b^{2} d^{3}\right )} x^{2} - 2 \,{\left (2 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2}\right )} x + 6 \,{\left (b^{3} c^{3} - 2 \, a b^{2} c^{2} d + a^{2} b c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \,{\left (d^{5} x + c d^{4}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x, algorithm="fricas")

[Out]

1/2*(b^3*d^3*x^3 + 2*b^3*c^3 - 6*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 2*a^3*d^3 - 3*(b^3*c*d^2 - 2*a*b^2*d^3)*x^2 - 2
*(2*b^3*c^2*d - 3*a*b^2*c*d^2)*x + 6*(b^3*c^3 - 2*a*b^2*c^2*d + a^2*b*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2
*b*d^3)*x)*log(d*x + c))/(d^5*x + c*d^4)

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Sympy [A]  time = 1.13319, size = 100, normalized size = 1.33 \begin{align*} \frac{b^{3} x^{2}}{2 d^{2}} + \frac{3 b \left (a d - b c\right )^{2} \log{\left (c + d x \right )}}{d^{4}} - \frac{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}}{c d^{4} + d^{5} x} + \frac{x \left (3 a b^{2} d - 2 b^{3} c\right )}{d^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5/(a*c+(a*d+b*c)*x+b*d*x**2)**2,x)

[Out]

b**3*x**2/(2*d**2) + 3*b*(a*d - b*c)**2*log(c + d*x)/d**4 - (a**3*d**3 - 3*a**2*b*c*d**2 + 3*a*b**2*c**2*d - b
**3*c**3)/(c*d**4 + d**5*x) + x*(3*a*b**2*d - 2*b**3*c)/d**3

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Giac [A]  time = 1.17548, size = 159, normalized size = 2.12 \begin{align*} \frac{3 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{4}} + \frac{b^{3} d^{2} x^{2} - 4 \, b^{3} c d x + 6 \, a b^{2} d^{2} x}{2 \, d^{4}} + \frac{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}}{{\left (d x + c\right )} d^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x, algorithm="giac")

[Out]

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