\(\int \frac {(a c+(b c+a d) x+b d x^2)^3}{(a+b x)^7} \, dx\) [1793]

Optimal result

Integrand size = 29, antiderivative size = 86 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^7} \, dx=-\frac {(b c-a d)^3}{3 b^4 (a+b x)^3}-\frac {3 d (b c-a d)^2}{2 b^4 (a+b x)^2}-\frac {3 d^2 (b c-a d)}{b^4 (a+b x)}+\frac {d^3 \log (a+b x)}{b^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 86, 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} \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^7} \, dx=-\frac {3 d^2 (b c-a d)}{b^4 (a+b x)}-\frac {3 d (b c-a d)^2}{2 b^4 (a+b x)^2}-\frac {(b c-a d)^3}{3 b^4 (a+b x)^3}+\frac {d^3 \log (a+b x)}{b^4} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^7} \, dx=\frac {-\frac {(b c-a d) \left (11 a^2 d^2+a b d (5 c+27 d x)+b^2 \left (2 c^2+9 c d x+18 d^2 x^2\right )\right )}{(a+b x)^3}+6 d^3 \log (a+b x)}{6 b^4} \]

[In]

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

[Out]

(-(((b*c - a*d)*(11*a^2*d^2 + a*b*d*(5*c + 27*d*x) + b^2*(2*c^2 + 9*c*d*x + 18*d^2*x^2)))/(a + b*x)^3) + 6*d^3
*Log[a + b*x])/(6*b^4)

Maple [A] (verified)

Time = 2.52 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.34

method	result	size
risch	\(\frac {\frac {3 d^{2} \left (a d -b c \right ) x^{2}}{b^{2}}+\frac {3 d \left (3 a^{2} d^{2}-2 a b c d -b^{2} c^{2}\right ) x}{2 b^{3}}+\frac {11 a^{3} d^{3}-6 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d -2 b^{3} c^{3}}{6 b^{4}}}{\left (b x +a \right )^{3}}+\frac {d^{3} \ln \left (b x +a \right )}{b^{4}}\)	\(115\)
default	\(-\frac {-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}}{3 b^{4} \left (b x +a \right )^{3}}+\frac {d^{3} \ln \left (b x +a \right )}{b^{4}}-\frac {3 d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 b^{4} \left (b x +a \right )^{2}}+\frac {3 d^{2} \left (a d -b c \right )}{b^{4} \left (b x +a \right )}\)	\(120\)
parallelrisch	\(\frac {6 \ln \left (b x +a \right ) x^{3} b^{3} d^{3}+18 \ln \left (b x +a \right ) x^{2} a \,b^{2} d^{3}+18 \ln \left (b x +a \right ) x \,a^{2} b \,d^{3}+18 x^{2} a \,b^{2} d^{3}-18 x^{2} b^{3} c \,d^{2}+6 \ln \left (b x +a \right ) a^{3} d^{3}+27 x \,a^{2} b \,d^{3}-18 x a \,b^{2} c \,d^{2}-9 x \,b^{3} c^{2} d +11 a^{3} d^{3}-6 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d -2 b^{3} c^{3}}{6 b^{4} \left (b x +a \right )^{3}}\)	\(170\)
norman	\(\frac {\frac {a \left (22 a^{3} b^{2} d^{3}-15 a^{2} b^{3} c \,d^{2}-6 a \,b^{4} c^{2} d -b^{5} c^{3}\right ) x^{2}}{b^{4}}+\frac {a^{2} \left (10 a^{3} b^{2} d^{3}-6 a^{2} b^{3} c \,d^{2}-3 a \,b^{4} c^{2} d -b^{5} c^{3}\right ) x}{b^{5}}+\frac {a^{3} \left (11 a^{3} b^{2} d^{3}-6 a^{2} b^{3} c \,d^{2}-3 a \,b^{4} c^{2} d -2 b^{5} c^{3}\right )}{6 b^{6}}+\frac {3 \left (a \,b^{2} d^{3}-b^{3} c \,d^{2}\right ) x^{5}}{b}+\frac {3 \left (9 a^{2} b^{2} d^{3}-8 a \,b^{3} c \,d^{2}-b^{4} c^{2} d \right ) x^{4}}{2 b^{2}}+\frac {\left (73 a^{3} b^{2} d^{3}-57 a^{2} b^{3} c \,d^{2}-15 a \,b^{4} c^{2} d -b^{5} c^{3}\right ) x^{3}}{3 b^{3}}}{\left (b x +a \right )^{6}}+\frac {d^{3} \ln \left (b x +a \right )}{b^{4}}\)	\(289\)

[In]

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

[Out]

(3*d^2*(a*d-b*c)/b^2*x^2+3/2*d*(3*a^2*d^2-2*a*b*c*d-b^2*c^2)/b^3*x+1/6*(11*a^3*d^3-6*a^2*b*c*d^2-3*a*b^2*c^2*d
-2*b^3*c^3)/b^4)/(b*x+a)^3+d^3*ln(b*x+a)/b^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 176 vs. \(2 (82) = 164\).

Time = 0.28 (sec) , antiderivative size = 176, normalized size of antiderivative = 2.05 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^7} \, dx=-\frac {2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} - 11 \, a^{3} d^{3} + 18 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 9 \, {\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3}\right )} x - 6 \, {\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \log \left (b x + a\right )}{6 \, {\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} \]

[In]

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

[Out]

-1/6*(2*b^3*c^3 + 3*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 11*a^3*d^3 + 18*(b^3*c*d^2 - a*b^2*d^3)*x^2 + 9*(b^3*c^2*d +
2*a*b^2*c*d^2 - 3*a^2*b*d^3)*x - 6*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a))/(b
^7*x^3 + 3*a*b^6*x^2 + 3*a^2*b^5*x + a^3*b^4)

Sympy [A] (verification not implemented)

Time = 0.80 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.72 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^7} \, dx=\frac {11 a^{3} d^{3} - 6 a^{2} b c d^{2} - 3 a b^{2} c^{2} d - 2 b^{3} c^{3} + x^{2} \cdot \left (18 a b^{2} d^{3} - 18 b^{3} c d^{2}\right ) + x \left (27 a^{2} b d^{3} - 18 a b^{2} c d^{2} - 9 b^{3} c^{2} d\right )}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} + \frac {d^{3} \log {\left (a + b x \right )}}{b^{4}} \]

[In]

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

[Out]

(11*a**3*d**3 - 6*a**2*b*c*d**2 - 3*a*b**2*c**2*d - 2*b**3*c**3 + x**2*(18*a*b**2*d**3 - 18*b**3*c*d**2) + x*(
27*a**2*b*d**3 - 18*a*b**2*c*d**2 - 9*b**3*c**2*d))/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x*
*3) + d**3*log(a + b*x)/b**4

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.65 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^7} \, dx=-\frac {2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} - 11 \, a^{3} d^{3} + 18 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 9 \, {\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3}\right )} x}{6 \, {\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} + \frac {d^{3} \log \left (b x + a\right )}{b^{4}} \]

[In]

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

[Out]

-1/6*(2*b^3*c^3 + 3*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 11*a^3*d^3 + 18*(b^3*c*d^2 - a*b^2*d^3)*x^2 + 9*(b^3*c^2*d +
2*a*b^2*c*d^2 - 3*a^2*b*d^3)*x)/(b^7*x^3 + 3*a*b^6*x^2 + 3*a^2*b^5*x + a^3*b^4) + d^3*log(b*x + a)/b^4

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.37 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^7} \, dx=\frac {d^{3} \log \left ({\left | b x + a \right |}\right )}{b^{4}} - \frac {18 \, {\left (b^{2} c d^{2} - a b d^{3}\right )} x^{2} + 9 \, {\left (b^{2} c^{2} d + 2 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x + \frac {2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} - 11 \, a^{3} d^{3}}{b}}{6 \, {\left (b x + a\right )}^{3} b^{3}} \]

[In]

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

[Out]

d^3*log(abs(b*x + a))/b^4 - 1/6*(18*(b^2*c*d^2 - a*b*d^3)*x^2 + 9*(b^2*c^2*d + 2*a*b*c*d^2 - 3*a^2*d^3)*x + (2
*b^3*c^3 + 3*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 11*a^3*d^3)/b)/((b*x + a)^3*b^3)

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.60 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^7} \, dx=\frac {d^3\,\ln \left (a+b\,x\right )}{b^4}-\frac {\frac {-11\,a^3\,d^3+6\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d+2\,b^3\,c^3}{6\,b^4}+\frac {3\,x\,\left (-3\,a^2\,d^3+2\,a\,b\,c\,d^2+b^2\,c^2\,d\right )}{2\,b^3}-\frac {3\,d^2\,x^2\,\left (a\,d-b\,c\right )}{b^2}}{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3} \]

[In]

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

[Out]

(d^3*log(a + b*x))/b^4 - ((2*b^3*c^3 - 11*a^3*d^3 + 3*a*b^2*c^2*d + 6*a^2*b*c*d^2)/(6*b^4) + (3*x*(b^2*c^2*d -
3*a^2*d^3 + 2*a*b*c*d^2))/(2*b^3) - (3*d^2*x^2*(a*d - b*c))/b^2)/(a^3 + b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x)