\(\int \frac {(a c+(b c+a d) x+b d x^2)^2}{(a+b x)^6} \, dx\) [1778]

Optimal result

Integrand size = 29, antiderivative size = 28 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^6} \, dx=-\frac {(c+d x)^3}{3 (b c-a d) (a+b x)^3} \]

[Out]

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

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} \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^6} \, dx=-\frac {(c+d x)^3}{3 (a+b x)^3 (b c-a d)} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.89 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^6} \, dx=-\frac {a^2 d^2+a b d (c+3 d x)+b^2 \left (c^2+3 c d x+3 d^2 x^2\right )}{3 b^3 (a+b x)^3} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(59\) vs. \(2(26)=52\).

Time = 2.36 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.14

method	result	size
gosper	\(-\frac {3 d^{2} x^{2} b^{2}+3 x a b \,d^{2}+3 x \,b^{2} c d +a^{2} d^{2}+a b c d +b^{2} c^{2}}{3 b^{3} \left (b x +a \right )^{3}}\)	\(60\)
risch	\(\frac {-\frac {d^{2} x^{2}}{b}-\frac {d \left (a d +b c \right ) x}{b^{2}}-\frac {a^{2} d^{2}+a b c d +b^{2} c^{2}}{3 b^{3}}}{\left (b x +a \right )^{3}}\)	\(60\)
parallelrisch	\(\frac {-3 d^{2} x^{2} b^{2}-3 x a b \,d^{2}-3 x \,b^{2} c d -a^{2} d^{2}-a b c d -b^{2} c^{2}}{3 b^{3} \left (b x +a \right )^{3}}\)	\(63\)
default	\(-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{3 b^{3} \left (b x +a \right )^{3}}+\frac {\left (a d -b c \right ) d}{b^{3} \left (b x +a \right )^{2}}-\frac {d^{2}}{b^{3} \left (b x +a \right )}\)	\(70\)
norman	\(\frac {-b \,d^{2} x^{4}+\frac {\left (-3 a \,b^{2} d^{2}-b^{3} d c \right ) x^{3}}{b^{2}}+\frac {a^{2} \left (-b^{2} d^{2} a^{2}-a \,b^{3} c d -b^{4} c^{2}\right )}{3 b^{5}}+\frac {\left (-10 b^{2} d^{2} a^{2}-7 a \,b^{3} c d -b^{4} c^{2}\right ) x^{2}}{3 b^{3}}+\frac {a \left (-5 b^{2} d^{2} a^{2}-5 a \,b^{3} c d -2 b^{4} c^{2}\right ) x}{3 b^{4}}}{\left (b x +a \right )^{5}}\)	\(150\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (26) = 52\).

Time = 0.27 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.00 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^6} \, dx=-\frac {3 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + a b c d + a^{2} d^{2} + 3 \, {\left (b^{2} c d + a b d^{2}\right )} x}{3 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (22) = 44\).

Time = 0.35 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.14 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^6} \, dx=\frac {- a^{2} d^{2} - a b c d - b^{2} c^{2} - 3 b^{2} d^{2} x^{2} + x \left (- 3 a b d^{2} - 3 b^{2} c d\right )}{3 a^{3} b^{3} + 9 a^{2} b^{4} x + 9 a b^{5} x^{2} + 3 b^{6} x^{3}} \]

[In]

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

[Out]

(-a**2*d**2 - a*b*c*d - b**2*c**2 - 3*b**2*d**2*x**2 + x*(-3*a*b*d**2 - 3*b**2*c*d))/(3*a**3*b**3 + 9*a**2*b**
4*x + 9*a*b**5*x**2 + 3*b**6*x**3)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (26) = 52\).

Time = 0.22 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.00 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^6} \, dx=-\frac {3 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + a b c d + a^{2} d^{2} + 3 \, {\left (b^{2} c d + a b d^{2}\right )} x}{3 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (26) = 52\).

Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.11 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^6} \, dx=-\frac {3 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c d x + 3 \, a b d^{2} x + b^{2} c^{2} + a b c d + a^{2} d^{2}}{3 \, {\left (b x + a\right )}^{3} b^{3}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.86 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^6} \, dx=-\frac {\frac {a^2\,d^2+a\,b\,c\,d+b^2\,c^2}{3\,b^3}+\frac {d^2\,x^2}{b}+\frac {d\,x\,\left (a\,d+b\,c\right )}{b^2}}{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3} \]

[In]

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

[Out]

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