\(\int \frac {(a c+(b c+a d) x+b d x^2)^3}{(a+b x)^4} \, dx\) [1790]

Optimal result

Integrand size = 29, antiderivative size = 73 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^4} \, dx=\frac {d (b c-a d)^2 x}{b^3}+\frac {(b c-a d) (c+d x)^2}{2 b^2}+\frac {(c+d x)^3}{3 b}+\frac {(b c-a d)^3 \log (a+b x)}{b^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 73, 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} \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^4} \, dx=\frac {(b c-a d)^3 \log (a+b x)}{b^4}+\frac {d x (b c-a d)^2}{b^3}+\frac {(c+d x)^2 (b c-a d)}{2 b^2}+\frac {(c+d x)^3}{3 b} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^4} \, dx=\frac {b d x \left (6 a^2 d^2-3 a b d (6 c+d x)+b^2 \left (18 c^2+9 c d x+2 d^2 x^2\right )\right )+6 (b c-a d)^3 \log (a+b x)}{6 b^4} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.36 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.49

method	result	size
default	\(\frac {d \left (\frac {1}{3} d^{2} x^{3} b^{2}-\frac {1}{2} x^{2} a b \,d^{2}+\frac {3}{2} x^{2} b^{2} c d +a^{2} d^{2} x -3 a b c d x +3 b^{2} c^{2} x \right )}{b^{3}}+\frac {\left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{4}}\)	\(109\)
risch	\(\frac {d^{3} x^{3}}{3 b}-\frac {d^{3} x^{2} a}{2 b^{2}}+\frac {3 d^{2} x^{2} c}{2 b}+\frac {d^{3} a^{2} x}{b^{3}}-\frac {3 d^{2} a c x}{b^{2}}+\frac {3 d \,c^{2} x}{b}-\frac {\ln \left (b x +a \right ) a^{3} d^{3}}{b^{4}}+\frac {3 \ln \left (b x +a \right ) a^{2} c \,d^{2}}{b^{3}}-\frac {3 \ln \left (b x +a \right ) a \,c^{2} d}{b^{2}}+\frac {\ln \left (b x +a \right ) c^{3}}{b}\)	\(133\)
parallelrisch	\(-\frac {-2 d^{3} x^{3} b^{3}+3 x^{2} a \,b^{2} d^{3}-9 x^{2} b^{3} c \,d^{2}+6 \ln \left (b x +a \right ) a^{3} d^{3}-18 \ln \left (b x +a \right ) a^{2} b c \,d^{2}+18 \ln \left (b x +a \right ) a \,b^{2} c^{2} d -6 \ln \left (b x +a \right ) b^{3} c^{3}-6 x \,a^{2} b \,d^{3}+18 x a \,b^{2} c \,d^{2}-18 x \,b^{3} c^{2} d}{6 b^{4}}\)	\(133\)
norman	\(\frac {\left (\frac {1}{2} a b \,d^{3}+\frac {3}{2} b^{2} c \,d^{2}\right ) x^{5}+\left (\frac {1}{2} a^{2} d^{3}+\frac {3}{2} a b c \,d^{2}+3 b^{2} c^{2} d \right ) x^{4}-\frac {a^{3} \left (11 a^{3} d^{3}-27 a^{2} b c \,d^{2}+54 a \,b^{2} c^{2} d \right )}{6 b^{4}}+\frac {b^{2} d^{3} x^{6}}{3}-\frac {3 a \left (a^{3} d^{3}-2 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d \right ) x^{2}}{b^{2}}-\frac {3 a^{2} \left (3 a^{3} d^{3}-7 a^{2} b c \,d^{2}+16 a \,b^{2} c^{2} d \right ) x}{2 b^{3}}}{\left (b x +a \right )^{3}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{4}}\)	\(232\)

[In]

int((b*d*x^2+(a*d+b*c)*x+a*c)^3/(b*x+a)^4,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.59 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^4} \, dx=\frac {2 \, b^{3} d^{3} x^{3} + 3 \, {\left (3 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 6 \, {\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x + 6 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{6 \, b^{4}} \]

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x^2)^3/(b*x+a)^4,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^4} \, dx=x^{2} \left (- \frac {a d^{3}}{2 b^{2}} + \frac {3 c d^{2}}{2 b}\right ) + x \left (\frac {a^{2} d^{3}}{b^{3}} - \frac {3 a c d^{2}}{b^{2}} + \frac {3 c^{2} d}{b}\right ) + \frac {d^{3} x^{3}}{3 b} - \frac {\left (a d - b c\right )^{3} \log {\left (a + b x \right )}}{b^{4}} \]

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x**2)**3/(b*x+a)**4,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.56 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^4} \, dx=\frac {2 \, b^{2} d^{3} x^{3} + 3 \, {\left (3 \, b^{2} c d^{2} - a b d^{3}\right )} x^{2} + 6 \, {\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} x}{6 \, b^{3}} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{b^{4}} \]

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x^2)^3/(b*x+a)^4,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.58 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^4} \, dx=\frac {2 \, b^{2} d^{3} x^{3} + 9 \, b^{2} c d^{2} x^{2} - 3 \, a b d^{3} x^{2} + 18 \, b^{2} c^{2} d x - 18 \, a b c d^{2} x + 6 \, a^{2} d^{3} x}{6 \, b^{3}} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{4}} \]

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x^2)^3/(b*x+a)^4,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.62 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^4} \, dx=x\,\left (\frac {3\,c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{b}\right )-x^2\,\left (\frac {a\,d^3}{2\,b^2}-\frac {3\,c\,d^2}{2\,b}\right )+\frac {d^3\,x^3}{3\,b}-\frac {\ln \left (a+b\,x\right )\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{b^4} \]

[In]

int((a*c + x*(a*d + b*c) + b*d*x^2)^3/(a + b*x)^4,x)

[Out]

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