\(\int \frac {(a c+(b c+a d) x+b d x^2)^3}{(a+b x)^8} \, dx\) [1794]

Optimal result

Integrand size = 29, antiderivative size = 28 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^8} \, dx=-\frac {(c+d x)^4}{4 (b c-a d) (a+b x)^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {640, 37} \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^8} \, dx=-\frac {(c+d x)^4}{4 (a+b x)^4 (b c-a d)} \]

[In]

Int[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^8,x]

[Out]

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

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)^5} \, dx \\ & = -\frac {(c+d x)^4}{4 (b c-a d) (a+b x)^4} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(91\) vs. \(2(28)=56\).

Time = 0.02 (sec) , antiderivative size = 91, normalized size of antiderivative = 3.25 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^8} \, dx=-\frac {a^3 d^3+a^2 b d^2 (c+4 d x)+a b^2 d \left (c^2+4 c d x+6 d^2 x^2\right )+b^3 \left (c^3+4 c^2 d x+6 c d^2 x^2+4 d^3 x^3\right )}{4 b^4 (a+b x)^4} \]

[In]

Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^8,x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(103\) vs. \(2(26)=52\).

Time = 2.33 (sec) , antiderivative size = 104, normalized size of antiderivative = 3.71

method	result	size
risch	\(\frac {-\frac {d^{3} x^{3}}{b}-\frac {3 d^{2} \left (a d +b c \right ) x^{2}}{2 b^{2}}-\frac {d \left (a^{2} d^{2}+a b c d +b^{2} c^{2}\right ) x}{b^{3}}-\frac {a^{3} d^{3}+a^{2} b c \,d^{2}+a \,b^{2} c^{2} d +b^{3} c^{3}}{4 b^{4}}}{\left (b x +a \right )^{4}}\)	\(104\)
gosper	\(-\frac {4 d^{3} x^{3} b^{3}+6 x^{2} a \,b^{2} d^{3}+6 x^{2} b^{3} c \,d^{2}+4 x \,a^{2} b \,d^{3}+4 x a \,b^{2} c \,d^{2}+4 x \,b^{3} c^{2} d +a^{3} d^{3}+a^{2} b c \,d^{2}+a \,b^{2} c^{2} d +b^{3} c^{3}}{4 b^{4} \left (b x +a \right )^{4}}\)	\(112\)
parallelrisch	\(\frac {-4 d^{3} x^{3} b^{3}-6 x^{2} a \,b^{2} d^{3}-6 x^{2} b^{3} c \,d^{2}-4 x \,a^{2} b \,d^{3}-4 x a \,b^{2} c \,d^{2}-4 x \,b^{3} c^{2} d -a^{3} d^{3}-a^{2} b c \,d^{2}-a \,b^{2} c^{2} d -b^{3} c^{3}}{4 b^{4} \left (b x +a \right )^{4}}\)	\(116\)
default	\(-\frac {d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{b^{4} \left (b x +a \right )^{3}}-\frac {-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}}{4 b^{4} \left (b x +a \right )^{4}}+\frac {3 d^{2} \left (a d -b c \right )}{2 b^{4} \left (b x +a \right )^{2}}-\frac {d^{3}}{b^{4} \left (b x +a \right )}\)	\(122\)
norman	\(\frac {-b^{2} d^{3} x^{6}+\frac {a^{3} \left (-a^{3} b^{3} d^{3}-a^{2} b^{4} c \,d^{2}-c^{2} d a \,b^{5}-c^{3} b^{6}\right )}{4 b^{7}}+\frac {3 \left (-3 a \,b^{3} d^{3}-b^{4} c \,d^{2}\right ) x^{5}}{2 b^{2}}+\frac {\left (-17 a^{2} b^{3} d^{3}-11 a \,b^{4} c \,d^{2}-2 b^{5} c^{2} d \right ) x^{4}}{2 b^{3}}+\frac {\left (-35 a^{3} b^{3} d^{3}-31 a^{2} b^{4} c \,d^{2}-13 c^{2} d a \,b^{5}-c^{3} b^{6}\right ) x^{3}}{4 b^{4}}+\frac {3 a \left (-7 a^{3} b^{3} d^{3}-7 a^{2} b^{4} c \,d^{2}-5 c^{2} d a \,b^{5}-c^{3} b^{6}\right ) x^{2}}{4 b^{5}}+\frac {a^{2} \left (-7 a^{3} b^{3} d^{3}-7 a^{2} b^{4} c \,d^{2}-7 c^{2} d a \,b^{5}-3 c^{3} b^{6}\right ) x}{4 b^{6}}}{\left (b x +a \right )^{7}}\)	\(289\)

[In]

int((b*d*x^2+(a*d+b*c)*x+a*c)^3/(b*x+a)^8,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (26) = 52\).

Time = 0.28 (sec) , antiderivative size = 143, normalized size of antiderivative = 5.11 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^8} \, dx=-\frac {4 \, b^{3} d^{3} x^{3} + b^{3} c^{3} + a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3} + 6 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 4 \, {\left (b^{3} c^{2} d + a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{4 \, {\left (b^{8} x^{4} + 4 \, a b^{7} x^{3} + 6 \, a^{2} b^{6} x^{2} + 4 \, a^{3} b^{5} x + a^{4} b^{4}\right )}} \]

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x^2)^3/(b*x+a)^8,x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (22) = 44\).

Time = 2.60 (sec) , antiderivative size = 155, normalized size of antiderivative = 5.54 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^8} \, dx=\frac {- a^{3} d^{3} - a^{2} b c d^{2} - a b^{2} c^{2} d - b^{3} c^{3} - 4 b^{3} d^{3} x^{3} + x^{2} \left (- 6 a b^{2} d^{3} - 6 b^{3} c d^{2}\right ) + x \left (- 4 a^{2} b d^{3} - 4 a b^{2} c d^{2} - 4 b^{3} c^{2} d\right )}{4 a^{4} b^{4} + 16 a^{3} b^{5} x + 24 a^{2} b^{6} x^{2} + 16 a b^{7} x^{3} + 4 b^{8} x^{4}} \]

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x**2)**3/(b*x+a)**8,x)

[Out]

(-a**3*d**3 - a**2*b*c*d**2 - a*b**2*c**2*d - b**3*c**3 - 4*b**3*d**3*x**3 + x**2*(-6*a*b**2*d**3 - 6*b**3*c*d
**2) + x*(-4*a**2*b*d**3 - 4*a*b**2*c*d**2 - 4*b**3*c**2*d))/(4*a**4*b**4 + 16*a**3*b**5*x + 24*a**2*b**6*x**2
+ 16*a*b**7*x**3 + 4*b**8*x**4)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (26) = 52\).

Time = 0.21 (sec) , antiderivative size = 143, normalized size of antiderivative = 5.11 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^8} \, dx=-\frac {4 \, b^{3} d^{3} x^{3} + b^{3} c^{3} + a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3} + 6 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 4 \, {\left (b^{3} c^{2} d + a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{4 \, {\left (b^{8} x^{4} + 4 \, a b^{7} x^{3} + 6 \, a^{2} b^{6} x^{2} + 4 \, a^{3} b^{5} x + a^{4} b^{4}\right )}} \]

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x^2)^3/(b*x+a)^8,x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (26) = 52\).

Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 3.96 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^8} \, dx=-\frac {4 \, b^{3} d^{3} x^{3} + 6 \, b^{3} c d^{2} x^{2} + 6 \, a b^{2} d^{3} x^{2} + 4 \, b^{3} c^{2} d x + 4 \, a b^{2} c d^{2} x + 4 \, a^{2} b d^{3} x + b^{3} c^{3} + a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3}}{4 \, {\left (b x + a\right )}^{4} b^{4}} \]

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x^2)^3/(b*x+a)^8,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.74 (sec) , antiderivative size = 135, normalized size of antiderivative = 4.82 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^8} \, dx=-\frac {\frac {a^3\,d^3+a^2\,b\,c\,d^2+a\,b^2\,c^2\,d+b^3\,c^3}{4\,b^4}+\frac {d^3\,x^3}{b}+\frac {d\,x\,\left (a^2\,d^2+a\,b\,c\,d+b^2\,c^2\right )}{b^3}+\frac {3\,d^2\,x^2\,\left (a\,d+b\,c\right )}{2\,b^2}}{a^4+4\,a^3\,b\,x+6\,a^2\,b^2\,x^2+4\,a\,b^3\,x^3+b^4\,x^4} \]

[In]

int((a*c + x*(a*d + b*c) + b*d*x^2)^3/(a + b*x)^8,x)

[Out]

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