\(\int \frac {(a+b x)^6}{(a c+(b c+a d) x+b d x^2)^2} \, dx\) [1810]

Optimal result

Integrand size = 29, antiderivative size = 104 \[ \int \frac {(a+b x)^6}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\frac {6 b^2 (b c-a d)^2 x}{d^4}-\frac {(b c-a d)^4}{d^5 (c+d x)}-\frac {2 b^3 (b c-a d) (c+d x)^2}{d^5}+\frac {b^4 (c+d x)^3}{3 d^5}-\frac {4 b (b c-a d)^3 \log (c+d x)}{d^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {640, 45} \[ \int \frac {(a+b x)^6}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=-\frac {2 b^3 (c+d x)^2 (b c-a d)}{d^5}+\frac {6 b^2 x (b c-a d)^2}{d^4}-\frac {(b c-a d)^4}{d^5 (c+d x)}-\frac {4 b (b c-a d)^3 \log (c+d x)}{d^5}+\frac {b^4 (c+d x)^3}{3 d^5} \]

[In]

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

[Out]

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

Rule 45

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])

Rule 640

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c/e)*x)^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]

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.59 \[ \int \frac {(a+b x)^6}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\frac {12 a^3 b c d^3-3 a^4 d^4+18 a^2 b^2 d^2 \left (-c^2+c d x+d^2 x^2\right )+6 a b^3 d \left (2 c^3-4 c^2 d x-3 c d^2 x^2+d^3 x^3\right )+b^4 \left (-3 c^4+9 c^3 d x+6 c^2 d^2 x^2-2 c d^3 x^3+d^4 x^4\right )-12 b (b c-a d)^3 (c+d x) \log (c+d x)}{3 d^5 (c+d x)} \]

[In]

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

[Out]

(12*a^3*b*c*d^3 - 3*a^4*d^4 + 18*a^2*b^2*d^2*(-c^2 + c*d*x + d^2*x^2) + 6*a*b^3*d*(2*c^3 - 4*c^2*d*x - 3*c*d^2
*x^2 + d^3*x^3) + b^4*(-3*c^4 + 9*c^3*d*x + 6*c^2*d^2*x^2 - 2*c*d^3*x^3 + d^4*x^4) - 12*b*(b*c - a*d)^3*(c + d
*x)*Log[c + d*x])/(3*d^5*(c + d*x))

Maple [A] (verified)

Time = 2.58 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.68

method	result	size
default	\(\frac {b^{2} \left (\frac {1}{3} d^{2} x^{3} b^{2}+2 x^{2} a b \,d^{2}-x^{2} b^{2} c d +6 a^{2} d^{2} x -8 a b c d x +3 b^{2} c^{2} x \right )}{d^{4}}+\frac {4 b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (d x +c \right )}{d^{5}}-\frac {a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}{d^{5} \left (d x +c \right )}\)	\(175\)
risch	\(\frac {b^{4} x^{3}}{3 d^{2}}+\frac {2 b^{3} x^{2} a}{d^{2}}-\frac {b^{4} x^{2} c}{d^{3}}+\frac {6 b^{2} a^{2} x}{d^{2}}-\frac {8 b^{3} a c x}{d^{3}}+\frac {3 b^{4} c^{2} x}{d^{4}}+\frac {4 b \ln \left (d x +c \right ) a^{3}}{d^{2}}-\frac {12 b^{2} \ln \left (d x +c \right ) a^{2} c}{d^{3}}+\frac {12 b^{3} \ln \left (d x +c \right ) a \,c^{2}}{d^{4}}-\frac {4 b^{4} \ln \left (d x +c \right ) c^{3}}{d^{5}}-\frac {a^{4}}{d \left (d x +c \right )}+\frac {4 a^{3} b c}{d^{2} \left (d x +c \right )}-\frac {6 a^{2} b^{2} c^{2}}{d^{3} \left (d x +c \right )}+\frac {4 a \,b^{3} c^{3}}{d^{4} \left (d x +c \right )}-\frac {b^{4} c^{4}}{d^{5} \left (d x +c \right )}\)	\(230\)
norman	\(\frac {\frac {b^{5} x^{5}}{3 d}+\frac {2 b^{3} \left (12 a^{2} d^{2}-10 a b c d +3 b^{2} c^{2}\right ) x^{3}}{3 d^{3}}+\frac {b^{4} \left (7 a d -2 b c \right ) x^{4}}{3 d^{2}}-\frac {a \left (a^{4} b \,d^{4}+2 a^{3} b^{2} c \,d^{3}+6 a^{2} b^{3} c^{2} d^{2}-10 a \,b^{4} c^{3} d +4 b^{5} c^{4}\right )}{d^{5} b}-\frac {\left (7 a^{4} b^{2} d^{4}-4 a^{3} c \,d^{3} b^{3}+8 a^{2} c^{2} d^{2} b^{4}-10 a \,c^{3} d \,b^{5}+4 c^{4} b^{6}\right ) x}{d^{5} b}}{\left (b x +a \right ) \left (d x +c \right )}+\frac {4 b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (d x +c \right )}{d^{5}}\)	\(258\)
parallelrisch	\(\frac {b^{4} x^{4} d^{4}+6 x^{3} a \,b^{3} d^{4}-2 x^{3} b^{4} c \,d^{3}+12 \ln \left (d x +c \right ) x \,a^{3} b \,d^{4}-36 \ln \left (d x +c \right ) x \,a^{2} b^{2} c \,d^{3}+36 \ln \left (d x +c \right ) x a \,b^{3} c^{2} d^{2}-12 \ln \left (d x +c \right ) x \,b^{4} c^{3} d +18 x^{2} a^{2} b^{2} d^{4}-18 x^{2} a \,b^{3} c \,d^{3}+6 x^{2} b^{4} c^{2} d^{2}+12 \ln \left (d x +c \right ) a^{3} b c \,d^{3}-36 \ln \left (d x +c \right ) a^{2} b^{2} c^{2} d^{2}+36 \ln \left (d x +c \right ) a \,b^{3} c^{3} d -12 \ln \left (d x +c \right ) b^{4} c^{4}-3 a^{4} d^{4}+12 a^{3} b c \,d^{3}-36 a^{2} b^{2} c^{2} d^{2}+36 a \,b^{3} c^{3} d -12 b^{4} c^{4}}{3 d^{5} \left (d x +c \right )}\)	\(275\)

[In]

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

[Out]

b^2/d^4*(1/3*d^2*x^3*b^2+2*x^2*a*b*d^2-x^2*b^2*c*d+6*a^2*d^2*x-8*a*b*c*d*x+3*b^2*c^2*x)+4*b/d^5*(a^3*d^3-3*a^2
*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*ln(d*x+c)-(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)/d^5/
(d*x+c)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (102) = 204\).

Time = 0.27 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.57 \[ \int \frac {(a+b x)^6}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\frac {b^{4} d^{4} x^{4} - 3 \, b^{4} c^{4} + 12 \, a b^{3} c^{3} d - 18 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} - 3 \, a^{4} d^{4} - 2 \, {\left (b^{4} c d^{3} - 3 \, a b^{3} d^{4}\right )} x^{3} + 6 \, {\left (b^{4} c^{2} d^{2} - 3 \, a b^{3} c d^{3} + 3 \, a^{2} b^{2} d^{4}\right )} x^{2} + 3 \, {\left (3 \, b^{4} c^{3} d - 8 \, a b^{3} c^{2} d^{2} + 6 \, a^{2} b^{2} c d^{3}\right )} x - 12 \, {\left (b^{4} c^{4} - 3 \, a b^{3} c^{3} d + 3 \, a^{2} b^{2} c^{2} d^{2} - a^{3} b c d^{3} + {\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x\right )} \log \left (d x + c\right )}{3 \, {\left (d^{6} x + c d^{5}\right )}} \]

[In]

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

[Out]

1/3*(b^4*d^4*x^4 - 3*b^4*c^4 + 12*a*b^3*c^3*d - 18*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 - 3*a^4*d^4 - 2*(b^4*c*d^3
- 3*a*b^3*d^4)*x^3 + 6*(b^4*c^2*d^2 - 3*a*b^3*c*d^3 + 3*a^2*b^2*d^4)*x^2 + 3*(3*b^4*c^3*d - 8*a*b^3*c^2*d^2 +
6*a^2*b^2*c*d^3)*x - 12*(b^4*c^4 - 3*a*b^3*c^3*d + 3*a^2*b^2*c^2*d^2 - a^3*b*c*d^3 + (b^4*c^3*d - 3*a*b^3*c^2
*d^2 + 3*a^2*b^2*c*d^3 - a^3*b*d^4)*x)*log(d*x + c))/(d^6*x + c*d^5)

Sympy [A] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.49 \[ \int \frac {(a+b x)^6}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\frac {b^{4} x^{3}}{3 d^{2}} + \frac {4 b \left (a d - b c\right )^{3} \log {\left (c + d x \right )}}{d^{5}} + x^{2} \cdot \left (\frac {2 a b^{3}}{d^{2}} - \frac {b^{4} c}{d^{3}}\right ) + x \left (\frac {6 a^{2} b^{2}}{d^{2}} - \frac {8 a b^{3} c}{d^{3}} + \frac {3 b^{4} c^{2}}{d^{4}}\right ) + \frac {- a^{4} d^{4} + 4 a^{3} b c d^{3} - 6 a^{2} b^{2} c^{2} d^{2} + 4 a b^{3} c^{3} d - b^{4} c^{4}}{c d^{5} + d^{6} x} \]

[In]

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

[Out]

b**4*x**3/(3*d**2) + 4*b*(a*d - b*c)**3*log(c + d*x)/d**5 + x**2*(2*a*b**3/d**2 - b**4*c/d**3) + x*(6*a**2*b**
2/d**2 - 8*a*b**3*c/d**3 + 3*b**4*c**2/d**4) + (-a**4*d**4 + 4*a**3*b*c*d**3 - 6*a**2*b**2*c**2*d**2 + 4*a*b**
3*c**3*d - b**4*c**4)/(c*d**5 + d**6*x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.76 \[ \int \frac {(a+b x)^6}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=-\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{d^{6} x + c d^{5}} + \frac {b^{4} d^{2} x^{3} - 3 \, {\left (b^{4} c d - 2 \, a b^{3} d^{2}\right )} x^{2} + 3 \, {\left (3 \, b^{4} c^{2} - 8 \, a b^{3} c d + 6 \, a^{2} b^{2} d^{2}\right )} x}{3 \, d^{4}} - \frac {4 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \log \left (d x + c\right )}{d^{5}} \]

[In]

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

[Out]

-(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)/(d^6*x + c*d^5) + 1/3*(b^4*d^2*x^3 -
3*(b^4*c*d - 2*a*b^3*d^2)*x^2 + 3*(3*b^4*c^2 - 8*a*b^3*c*d + 6*a^2*b^2*d^2)*x)/d^4 - 4*(b^4*c^3 - 3*a*b^3*c^2*
d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*log(d*x + c)/d^5

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.81 \[ \int \frac {(a+b x)^6}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=-\frac {4 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{5}} + \frac {b^{4} d^{4} x^{3} - 3 \, b^{4} c d^{3} x^{2} + 6 \, a b^{3} d^{4} x^{2} + 9 \, b^{4} c^{2} d^{2} x - 24 \, a b^{3} c d^{3} x + 18 \, a^{2} b^{2} d^{4} x}{3 \, d^{6}} - \frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{{\left (d x + c\right )} d^{5}} \]

[In]

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

[Out]

-4*(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*log(abs(d*x + c))/d^5 + 1/3*(b^4*d^4*x^3 - 3*b^4*c*
d^3*x^2 + 6*a*b^3*d^4*x^2 + 9*b^4*c^2*d^2*x - 24*a*b^3*c*d^3*x + 18*a^2*b^2*d^4*x)/d^6 - (b^4*c^4 - 4*a*b^3*c^
3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)/((d*x + c)*d^5)

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.95 \[ \int \frac {(a+b x)^6}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=x^2\,\left (\frac {2\,a\,b^3}{d^2}-\frac {b^4\,c}{d^3}\right )-x\,\left (\frac {2\,c\,\left (\frac {4\,a\,b^3}{d^2}-\frac {2\,b^4\,c}{d^3}\right )}{d}-\frac {6\,a^2\,b^2}{d^2}+\frac {b^4\,c^2}{d^4}\right )+\frac {b^4\,x^3}{3\,d^2}-\frac {\ln \left (c+d\,x\right )\,\left (-4\,a^3\,b\,d^3+12\,a^2\,b^2\,c\,d^2-12\,a\,b^3\,c^2\,d+4\,b^4\,c^3\right )}{d^5}-\frac {a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4}{d\,\left (x\,d^5+c\,d^4\right )} \]

[In]

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

[Out]

x^2*((2*a*b^3)/d^2 - (b^4*c)/d^3) - x*((2*c*((4*a*b^3)/d^2 - (2*b^4*c)/d^3))/d - (6*a^2*b^2)/d^2 + (b^4*c^2)/d
^4) + (b^4*x^3)/(3*d^2) - (log(c + d*x)*(4*b^4*c^3 - 4*a^3*b*d^3 + 12*a^2*b^2*c*d^2 - 12*a*b^3*c^2*d))/d^5 - (
a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)/(d*(c*d^4 + d^5*x))