### 3.1799 $$\int \frac{(a+b x)^6}{a c+(b c+a d) x+b d x^2} \, dx$$

Optimal. Leaf size=122 $\frac{b x (b c-a d)^4}{d^5}-\frac{(a+b x)^2 (b c-a d)^3}{2 d^4}+\frac{(a+b x)^3 (b c-a d)^2}{3 d^3}-\frac{(a+b x)^4 (b c-a d)}{4 d^2}-\frac{(b c-a d)^5 \log (c+d x)}{d^6}+\frac{(a+b x)^5}{5 d}$

[Out]

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

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Rubi [A]  time = 0.0626873, antiderivative size = 122, 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} $\frac{b x (b c-a d)^4}{d^5}-\frac{(a+b x)^2 (b c-a d)^3}{2 d^4}+\frac{(a+b x)^3 (b c-a d)^2}{3 d^3}-\frac{(a+b x)^4 (b c-a d)}{4 d^2}-\frac{(b c-a d)^5 \log (c+d x)}{d^6}+\frac{(a+b x)^5}{5 d}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x)^6/(a*c + (b*c + a*d)*x + b*d*x^2),x]

[Out]

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

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

Mathematica [A]  time = 0.0641482, size = 167, normalized size = 1.37 $\frac{b d x \left (100 a^2 b^2 d^2 \left (6 c^2-3 c d x+2 d^2 x^2\right )+300 a^3 b d^3 (d x-2 c)+300 a^4 d^4+25 a b^3 d \left (6 c^2 d x-12 c^3-4 c d^2 x^2+3 d^3 x^3\right )+b^4 \left (20 c^2 d^2 x^2-30 c^3 d x+60 c^4-15 c d^3 x^3+12 d^4 x^4\right )\right )-60 (b c-a d)^5 \log (c+d x)}{60 d^6}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x)^6/(a*c + (b*c + a*d)*x + b*d*x^2),x]

[Out]

(b*d*x*(300*a^4*d^4 + 300*a^3*b*d^3*(-2*c + d*x) + 100*a^2*b^2*d^2*(6*c^2 - 3*c*d*x + 2*d^2*x^2) + 25*a*b^3*d*
(-12*c^3 + 6*c^2*d*x - 4*c*d^2*x^2 + 3*d^3*x^3) + b^4*(60*c^4 - 30*c^3*d*x + 20*c^2*d^2*x^2 - 15*c*d^3*x^3 + 1
2*d^4*x^4)) - 60*(b*c - a*d)^5*Log[c + d*x])/(60*d^6)

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Maple [B]  time = 0.043, size = 302, normalized size = 2.5 \begin{align*}{\frac{{b}^{5}{x}^{5}}{5\,d}}+{\frac{5\,a{b}^{4}{x}^{4}}{4\,d}}-{\frac{{b}^{5}{x}^{4}c}{4\,{d}^{2}}}+{\frac{10\,{a}^{2}{b}^{3}{x}^{3}}{3\,d}}-{\frac{5\,{b}^{4}{x}^{3}ac}{3\,{d}^{2}}}+{\frac{{b}^{5}{x}^{3}{c}^{2}}{3\,{d}^{3}}}+5\,{\frac{{a}^{3}{b}^{2}{x}^{2}}{d}}-5\,{\frac{{b}^{3}{x}^{2}{a}^{2}c}{{d}^{2}}}+{\frac{5\,{b}^{4}{x}^{2}a{c}^{2}}{2\,{d}^{3}}}-{\frac{{b}^{5}{x}^{2}{c}^{3}}{2\,{d}^{4}}}+5\,{\frac{b{a}^{4}x}{d}}-10\,{\frac{{a}^{3}{b}^{2}cx}{{d}^{2}}}+10\,{\frac{{a}^{2}{b}^{3}{c}^{2}x}{{d}^{3}}}-5\,{\frac{a{b}^{4}{c}^{3}x}{{d}^{4}}}+{\frac{{b}^{5}{c}^{4}x}{{d}^{5}}}+{\frac{\ln \left ( dx+c \right ){a}^{5}}{d}}-5\,{\frac{\ln \left ( dx+c \right ){a}^{4}bc}{{d}^{2}}}+10\,{\frac{\ln \left ( dx+c \right ){a}^{3}{b}^{2}{c}^{2}}{{d}^{3}}}-10\,{\frac{\ln \left ( dx+c \right ){a}^{2}{b}^{3}{c}^{3}}{{d}^{4}}}+5\,{\frac{\ln \left ( dx+c \right ) a{b}^{4}{c}^{4}}{{d}^{5}}}-{\frac{\ln \left ( dx+c \right ){b}^{5}{c}^{5}}{{d}^{6}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^6/(a*c+(a*d+b*c)*x+b*d*x^2),x)

[Out]

1/5*b^5/d*x^5+5/4*b^4/d*x^4*a-1/4*b^5/d^2*x^4*c+10/3*b^3/d*x^3*a^2-5/3*b^4/d^2*x^3*a*c+1/3*b^5/d^3*x^3*c^2+5*b
^2/d*x^2*a^3-5*b^3/d^2*x^2*a^2*c+5/2*b^4/d^3*x^2*a*c^2-1/2*b^5/d^4*x^2*c^3+5*b/d*a^4*x-10*b^2/d^2*a^3*c*x+10*b
^3/d^3*a^2*c^2*x-5*b^4/d^4*a*c^3*x+b^5/d^5*c^4*x+1/d*ln(d*x+c)*a^5-5/d^2*ln(d*x+c)*a^4*b*c+10/d^3*ln(d*x+c)*a^
3*b^2*c^2-10/d^4*ln(d*x+c)*a^2*b^3*c^3+5/d^5*ln(d*x+c)*a*b^4*c^4-1/d^6*ln(d*x+c)*b^5*c^5

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Maxima [B]  time = 1.04921, size = 348, normalized size = 2.85 \begin{align*} \frac{12 \, b^{5} d^{4} x^{5} - 15 \,{\left (b^{5} c d^{3} - 5 \, a b^{4} d^{4}\right )} x^{4} + 20 \,{\left (b^{5} c^{2} d^{2} - 5 \, a b^{4} c d^{3} + 10 \, a^{2} b^{3} d^{4}\right )} x^{3} - 30 \,{\left (b^{5} c^{3} d - 5 \, a b^{4} c^{2} d^{2} + 10 \, a^{2} b^{3} c d^{3} - 10 \, a^{3} b^{2} d^{4}\right )} x^{2} + 60 \,{\left (b^{5} c^{4} - 5 \, a b^{4} c^{3} d + 10 \, a^{2} b^{3} c^{2} d^{2} - 10 \, a^{3} b^{2} c d^{3} + 5 \, a^{4} b d^{4}\right )} x}{60 \, d^{5}} - \frac{{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} \log \left (d x + c\right )}{d^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6/(a*c+(a*d+b*c)*x+b*d*x^2),x, algorithm="maxima")

[Out]

1/60*(12*b^5*d^4*x^5 - 15*(b^5*c*d^3 - 5*a*b^4*d^4)*x^4 + 20*(b^5*c^2*d^2 - 5*a*b^4*c*d^3 + 10*a^2*b^3*d^4)*x^
3 - 30*(b^5*c^3*d - 5*a*b^4*c^2*d^2 + 10*a^2*b^3*c*d^3 - 10*a^3*b^2*d^4)*x^2 + 60*(b^5*c^4 - 5*a*b^4*c^3*d + 1
0*a^2*b^3*c^2*d^2 - 10*a^3*b^2*c*d^3 + 5*a^4*b*d^4)*x)/d^5 - (b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 1
0*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*log(d*x + c)/d^6

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Fricas [B]  time = 1.61941, size = 537, normalized size = 4.4 \begin{align*} \frac{12 \, b^{5} d^{5} x^{5} - 15 \,{\left (b^{5} c d^{4} - 5 \, a b^{4} d^{5}\right )} x^{4} + 20 \,{\left (b^{5} c^{2} d^{3} - 5 \, a b^{4} c d^{4} + 10 \, a^{2} b^{3} d^{5}\right )} x^{3} - 30 \,{\left (b^{5} c^{3} d^{2} - 5 \, a b^{4} c^{2} d^{3} + 10 \, a^{2} b^{3} c d^{4} - 10 \, a^{3} b^{2} d^{5}\right )} x^{2} + 60 \,{\left (b^{5} c^{4} d - 5 \, a b^{4} c^{3} d^{2} + 10 \, a^{2} b^{3} c^{2} d^{3} - 10 \, a^{3} b^{2} c d^{4} + 5 \, a^{4} b d^{5}\right )} x - 60 \,{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} \log \left (d x + c\right )}{60 \, d^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6/(a*c+(a*d+b*c)*x+b*d*x^2),x, algorithm="fricas")

[Out]

1/60*(12*b^5*d^5*x^5 - 15*(b^5*c*d^4 - 5*a*b^4*d^5)*x^4 + 20*(b^5*c^2*d^3 - 5*a*b^4*c*d^4 + 10*a^2*b^3*d^5)*x^
3 - 30*(b^5*c^3*d^2 - 5*a*b^4*c^2*d^3 + 10*a^2*b^3*c*d^4 - 10*a^3*b^2*d^5)*x^2 + 60*(b^5*c^4*d - 5*a*b^4*c^3*d
^2 + 10*a^2*b^3*c^2*d^3 - 10*a^3*b^2*c*d^4 + 5*a^4*b*d^5)*x - 60*(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2
- 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*log(d*x + c))/d^6

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Sympy [A]  time = 1.05147, size = 202, normalized size = 1.66 \begin{align*} \frac{b^{5} x^{5}}{5 d} + \frac{x^{4} \left (5 a b^{4} d - b^{5} c\right )}{4 d^{2}} + \frac{x^{3} \left (10 a^{2} b^{3} d^{2} - 5 a b^{4} c d + b^{5} c^{2}\right )}{3 d^{3}} + \frac{x^{2} \left (10 a^{3} b^{2} d^{3} - 10 a^{2} b^{3} c d^{2} + 5 a b^{4} c^{2} d - b^{5} c^{3}\right )}{2 d^{4}} + \frac{x \left (5 a^{4} b d^{4} - 10 a^{3} b^{2} c d^{3} + 10 a^{2} b^{3} c^{2} d^{2} - 5 a b^{4} c^{3} d + b^{5} c^{4}\right )}{d^{5}} + \frac{\left (a d - b c\right )^{5} \log{\left (c + d x \right )}}{d^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**6/(a*c+(a*d+b*c)*x+b*d*x**2),x)

[Out]

b**5*x**5/(5*d) + x**4*(5*a*b**4*d - b**5*c)/(4*d**2) + x**3*(10*a**2*b**3*d**2 - 5*a*b**4*c*d + b**5*c**2)/(3
*d**3) + x**2*(10*a**3*b**2*d**3 - 10*a**2*b**3*c*d**2 + 5*a*b**4*c**2*d - b**5*c**3)/(2*d**4) + x*(5*a**4*b*d
**4 - 10*a**3*b**2*c*d**3 + 10*a**2*b**3*c**2*d**2 - 5*a*b**4*c**3*d + b**5*c**4)/d**5 + (a*d - b*c)**5*log(c
+ d*x)/d**6

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Giac [B]  time = 1.25377, size = 369, normalized size = 3.02 \begin{align*} \frac{12 \, b^{5} d^{4} x^{5} - 15 \, b^{5} c d^{3} x^{4} + 75 \, a b^{4} d^{4} x^{4} + 20 \, b^{5} c^{2} d^{2} x^{3} - 100 \, a b^{4} c d^{3} x^{3} + 200 \, a^{2} b^{3} d^{4} x^{3} - 30 \, b^{5} c^{3} d x^{2} + 150 \, a b^{4} c^{2} d^{2} x^{2} - 300 \, a^{2} b^{3} c d^{3} x^{2} + 300 \, a^{3} b^{2} d^{4} x^{2} + 60 \, b^{5} c^{4} x - 300 \, a b^{4} c^{3} d x + 600 \, a^{2} b^{3} c^{2} d^{2} x - 600 \, a^{3} b^{2} c d^{3} x + 300 \, a^{4} b d^{4} x}{60 \, d^{5}} - \frac{{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6/(a*c+(a*d+b*c)*x+b*d*x^2),x, algorithm="giac")

[Out]

1/60*(12*b^5*d^4*x^5 - 15*b^5*c*d^3*x^4 + 75*a*b^4*d^4*x^4 + 20*b^5*c^2*d^2*x^3 - 100*a*b^4*c*d^3*x^3 + 200*a^
2*b^3*d^4*x^3 - 30*b^5*c^3*d*x^2 + 150*a*b^4*c^2*d^2*x^2 - 300*a^2*b^3*c*d^3*x^2 + 300*a^3*b^2*d^4*x^2 + 60*b^
5*c^4*x - 300*a*b^4*c^3*d*x + 600*a^2*b^3*c^2*d^2*x - 600*a^3*b^2*c*d^3*x + 300*a^4*b*d^4*x)/d^5 - (b^5*c^5 -
5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*log(abs(d*x + c))/d^6