\(\int \frac {(a+b x)^7}{(a c+(b c+a d) x+b d x^2)^3} \, dx\) [1819]

Optimal result

Integrand size = 29, antiderivative size = 103 \[ \int \frac {(a+b x)^7}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=-\frac {b^3 (3 b c-4 a d) x}{d^4}+\frac {b^4 x^2}{2 d^3}-\frac {(b c-a d)^4}{2 d^5 (c+d x)^2}+\frac {4 b (b c-a d)^3}{d^5 (c+d x)}+\frac {6 b^2 (b c-a d)^2 \log (c+d x)}{d^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 103, 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} \[ \int \frac {(a+b x)^7}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=-\frac {b^3 x (3 b c-4 a d)}{d^4}+\frac {6 b^2 (b c-a d)^2 \log (c+d x)}{d^5}+\frac {4 b (b c-a d)^3}{d^5 (c+d x)}-\frac {(b c-a d)^4}{2 d^5 (c+d x)^2}+\frac {b^4 x^2}{2 d^3} \]

[In]

Int[(a + b*x)^7/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.62 \[ \int \frac {(a+b x)^7}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\frac {-a^4 d^4-4 a^3 b d^3 (c+2 d x)+6 a^2 b^2 c d^2 (3 c+4 d x)+4 a b^3 d \left (-5 c^3-4 c^2 d x+4 c d^2 x^2+2 d^3 x^3\right )+b^4 \left (7 c^4+2 c^3 d x-11 c^2 d^2 x^2-4 c d^3 x^3+d^4 x^4\right )+12 b^2 (b c-a d)^2 (c+d x)^2 \log (c+d x)}{2 d^5 (c+d x)^2} \]

[In]

Integrate[(a + b*x)^7/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]

[Out]

(-(a^4*d^4) - 4*a^3*b*d^3*(c + 2*d*x) + 6*a^2*b^2*c*d^2*(3*c + 4*d*x) + 4*a*b^3*d*(-5*c^3 - 4*c^2*d*x + 4*c*d^
2*x^2 + 2*d^3*x^3) + b^4*(7*c^4 + 2*c^3*d*x - 11*c^2*d^2*x^2 - 4*c*d^3*x^3 + d^4*x^4) + 12*b^2*(b*c - a*d)^2*(
c + d*x)^2*Log[c + d*x])/(2*d^5*(c + d*x)^2)

Maple [A] (verified)

Time = 2.49 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.67

method	result	size
default	\(\frac {b^{3} \left (\frac {1}{2} b d \,x^{2}+4 a d x -3 b c x \right )}{d^{4}}-\frac {a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}{2 d^{5} \left (d x +c \right )^{2}}+\frac {6 b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (d x +c \right )}{d^{5}}-\frac {4 b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{d^{5} \left (d x +c \right )}\)	\(172\)
risch	\(\frac {b^{4} x^{2}}{2 d^{3}}+\frac {4 b^{3} a x}{d^{3}}-\frac {3 b^{4} c x}{d^{4}}+\frac {\left (-4 a^{3} b \,d^{3}+12 a^{2} b^{2} c \,d^{2}-12 a \,b^{3} c^{2} d +4 b^{4} c^{3}\right ) x -\frac {a^{4} d^{4}+4 a^{3} b c \,d^{3}-18 a^{2} b^{2} c^{2} d^{2}+20 a \,b^{3} c^{3} d -7 b^{4} c^{4}}{2 d}}{d^{4} \left (d x +c \right )^{2}}+\frac {6 b^{2} \ln \left (d x +c \right ) a^{2}}{d^{3}}-\frac {12 b^{3} \ln \left (d x +c \right ) a c}{d^{4}}+\frac {6 b^{4} \ln \left (d x +c \right ) c^{2}}{d^{5}}\)	\(192\)
parallelrisch	\(\frac {b^{4} x^{4} d^{4}+12 \ln \left (d x +c \right ) x^{2} a^{2} b^{2} d^{4}-24 \ln \left (d x +c \right ) x^{2} a \,b^{3} c \,d^{3}+12 \ln \left (d x +c \right ) x^{2} b^{4} c^{2} d^{2}+8 x^{3} a \,b^{3} d^{4}-4 x^{3} b^{4} c \,d^{3}+24 \ln \left (d x +c \right ) x \,a^{2} b^{2} c \,d^{3}-48 \ln \left (d x +c \right ) x a \,b^{3} c^{2} d^{2}+24 \ln \left (d x +c \right ) x \,b^{4} c^{3} d +12 \ln \left (d x +c \right ) a^{2} b^{2} c^{2} d^{2}-24 \ln \left (d x +c \right ) a \,b^{3} c^{3} d +12 \ln \left (d x +c \right ) b^{4} c^{4}-8 a^{3} b \,d^{4} x +24 a^{2} b^{2} c \,d^{3} x -48 a \,b^{3} c^{2} d^{2} x +24 b^{4} c^{3} d x -a^{4} d^{4}-4 a^{3} b c \,d^{3}+18 a^{2} b^{2} c^{2} d^{2}-36 a \,b^{3} c^{3} d +18 b^{4} c^{4}}{2 d^{5} \left (d x +c \right )^{2}}\)	\(307\)
norman	\(\frac {\frac {b^{5} \left (5 a d -2 b c \right ) x^{5}}{d^{2}}+\frac {b^{6} x^{6}}{2 d}-\frac {a^{2} \left (a^{4} b^{2} d^{4}+4 a^{3} c \,d^{3} b^{3}-a^{2} c^{2} d^{2} b^{4}+28 a \,c^{3} d \,b^{5}-18 c^{4} b^{6}\right )}{2 b^{2} d^{5}}-\frac {\left (34 a^{4} b^{4} d^{4}+16 a^{3} b^{5} c \,d^{3}+63 a^{2} b^{6} c^{2} d^{2}-20 a \,b^{7} c^{3} d -18 b^{8} c^{4}\right ) x^{2}}{2 d^{5} b^{2}}-\frac {\left (17 a^{3} b^{4} d^{3}-a^{2} b^{5} c \,d^{2}+16 a \,c^{2} d \,b^{6}-12 c^{3} b^{7}\right ) x^{3}}{d^{4} b}-\frac {a \left (5 a^{4} b^{3} d^{4}+9 a^{3} b^{4} c \,d^{3}+15 a^{2} b^{5} c^{2} d^{2}+16 a \,b^{6} c^{3} d -18 b^{7} c^{4}\right ) x}{d^{5} b^{2}}}{\left (b x +a \right )^{2} \left (d x +c \right )^{2}}+\frac {6 b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (d x +c \right )}{d^{5}}\)	\(335\)

[In]

int((b*x+a)^7/(b*d*x^2+(a*d+b*c)*x+a*c)^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (99) = 198\).

Time = 0.30 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.83 \[ \int \frac {(a+b x)^7}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\frac {b^{4} d^{4} x^{4} + 7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4} - 4 \, {\left (b^{4} c d^{3} - 2 \, a b^{3} d^{4}\right )} x^{3} - {\left (11 \, b^{4} c^{2} d^{2} - 16 \, a b^{3} c d^{3}\right )} x^{2} + 2 \, {\left (b^{4} c^{3} d - 8 \, a b^{3} c^{2} d^{2} + 12 \, a^{2} b^{2} c d^{3} - 4 \, a^{3} b d^{4}\right )} x + 12 \, {\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + a^{2} b^{2} c^{2} d^{2} + {\left (b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{2} + 2 \, {\left (b^{4} c^{3} d - 2 \, a b^{3} c^{2} d^{2} + a^{2} b^{2} c d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (d^{7} x^{2} + 2 \, c d^{6} x + c^{2} d^{5}\right )}} \]

[In]

integrate((b*x+a)^7/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="fricas")

[Out]

1/2*(b^4*d^4*x^4 + 7*b^4*c^4 - 20*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 - a^4*d^4 - 4*(b^4*c*d^3 -
2*a*b^3*d^4)*x^3 - (11*b^4*c^2*d^2 - 16*a*b^3*c*d^3)*x^2 + 2*(b^4*c^3*d - 8*a*b^3*c^2*d^2 + 12*a^2*b^2*c*d^3 -
4*a^3*b*d^4)*x + 12*(b^4*c^4 - 2*a*b^3*c^3*d + a^2*b^2*c^2*d^2 + (b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*
x^2 + 2*(b^4*c^3*d - 2*a*b^3*c^2*d^2 + a^2*b^2*c*d^3)*x)*log(d*x + c))/(d^7*x^2 + 2*c*d^6*x + c^2*d^5)

Sympy [A] (verification not implemented)

Time = 0.87 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.80 \[ \int \frac {(a+b x)^7}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\frac {b^{4} x^{2}}{2 d^{3}} + \frac {6 b^{2} \left (a d - b c\right )^{2} \log {\left (c + d x \right )}}{d^{5}} + x \left (\frac {4 a b^{3}}{d^{3}} - \frac {3 b^{4} c}{d^{4}}\right ) + \frac {- a^{4} d^{4} - 4 a^{3} b c d^{3} + 18 a^{2} b^{2} c^{2} d^{2} - 20 a b^{3} c^{3} d + 7 b^{4} c^{4} + x \left (- 8 a^{3} b d^{4} + 24 a^{2} b^{2} c d^{3} - 24 a b^{3} c^{2} d^{2} + 8 b^{4} c^{3} d\right )}{2 c^{2} d^{5} + 4 c d^{6} x + 2 d^{7} x^{2}} \]

[In]

integrate((b*x+a)**7/(a*c+(a*d+b*c)*x+b*d*x**2)**3,x)

[Out]

b**4*x**2/(2*d**3) + 6*b**2*(a*d - b*c)**2*log(c + d*x)/d**5 + x*(4*a*b**3/d**3 - 3*b**4*c/d**4) + (-a**4*d**4
- 4*a**3*b*c*d**3 + 18*a**2*b**2*c**2*d**2 - 20*a*b**3*c**3*d + 7*b**4*c**4 + x*(-8*a**3*b*d**4 + 24*a**2*b**
2*c*d**3 - 24*a*b**3*c**2*d**2 + 8*b**4*c**3*d))/(2*c**2*d**5 + 4*c*d**6*x + 2*d**7*x**2)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.85 \[ \int \frac {(a+b x)^7}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\frac {7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4} + 8 \, {\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x}{2 \, {\left (d^{7} x^{2} + 2 \, c d^{6} x + c^{2} d^{5}\right )}} + \frac {b^{4} d x^{2} - 2 \, {\left (3 \, b^{4} c - 4 \, a b^{3} d\right )} x}{2 \, d^{4}} + \frac {6 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \log \left (d x + c\right )}{d^{5}} \]

[In]

integrate((b*x+a)^7/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="maxima")

[Out]

1/2*(7*b^4*c^4 - 20*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 - a^4*d^4 + 8*(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)/(d^7*x^2 + 2*c*d^6*x + c^2*d^5) + 1/2*(b^4*d*x^2 - 2*(3*b^4*c - 4*a*b^3*d)
*x)/d^4 + 6*(b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*log(d*x + c)/d^5

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.78 \[ \int \frac {(a+b x)^7}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\frac {6 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{5}} + \frac {b^{4} d^{3} x^{2} - 6 \, b^{4} c d^{2} x + 8 \, a b^{3} d^{3} x}{2 \, d^{6}} + \frac {7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4} + 8 \, {\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x}{2 \, {\left (d x + c\right )}^{2} d^{5}} \]

[In]

integrate((b*x+a)^7/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="giac")

[Out]

6*(b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*log(abs(d*x + c))/d^5 + 1/2*(b^4*d^3*x^2 - 6*b^4*c*d^2*x + 8*a*b^3*d^3
*x)/d^6 + 1/2*(7*b^4*c^4 - 20*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 - a^4*d^4 + 8*(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)/((d*x + c)^2*d^5)

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.90 \[ \int \frac {(a+b x)^7}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=x\,\left (\frac {4\,a\,b^3}{d^3}-\frac {3\,b^4\,c}{d^4}\right )-\frac {\frac {a^4\,d^4+4\,a^3\,b\,c\,d^3-18\,a^2\,b^2\,c^2\,d^2+20\,a\,b^3\,c^3\,d-7\,b^4\,c^4}{2\,d}-x\,\left (-4\,a^3\,b\,d^3+12\,a^2\,b^2\,c\,d^2-12\,a\,b^3\,c^2\,d+4\,b^4\,c^3\right )}{c^2\,d^4+2\,c\,d^5\,x+d^6\,x^2}+\frac {b^4\,x^2}{2\,d^3}+\frac {\ln \left (c+d\,x\right )\,\left (6\,a^2\,b^2\,d^2-12\,a\,b^3\,c\,d+6\,b^4\,c^2\right )}{d^5} \]

[In]

int((a + b*x)^7/(a*c + x*(a*d + b*c) + b*d*x^2)^3,x)

[Out]

x*((4*a*b^3)/d^3 - (3*b^4*c)/d^4) - ((a^4*d^4 - 7*b^4*c^4 - 18*a^2*b^2*c^2*d^2 + 20*a*b^3*c^3*d + 4*a^3*b*c*d^
3)/(2*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))/(c^2*d^4 + d^6*x^2 + 2*c*d^5*x) +
(b^4*x^2)/(2*d^3) + (log(c + d*x)*(6*b^4*c^2 + 6*a^2*b^2*d^2 - 12*a*b^3*c*d))/d^5