\(\int \frac {(c+d x)^2}{(a+b x)^6} \, dx\) [1256]

Optimal result

Integrand size = 15, antiderivative size = 65 \[ \int \frac {(c+d x)^2}{(a+b x)^6} \, dx=-\frac {(b c-a d)^2}{5 b^3 (a+b x)^5}-\frac {d (b c-a d)}{2 b^3 (a+b x)^4}-\frac {d^2}{3 b^3 (a+b x)^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \[ \int \frac {(c+d x)^2}{(a+b x)^6} \, dx=-\frac {d (b c-a d)}{2 b^3 (a+b x)^4}-\frac {(b c-a d)^2}{5 b^3 (a+b x)^5}-\frac {d^2}{3 b^3 (a+b x)^3} \]

[In]

Int[(c + d*x)^2/(a + b*x)^6,x]

[Out]

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

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])

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.88 \[ \int \frac {(c+d x)^2}{(a+b x)^6} \, dx=-\frac {a^2 d^2+a b d (3 c+5 d x)+b^2 \left (6 c^2+15 c d x+10 d^2 x^2\right )}{30 b^3 (a+b x)^5} \]

[In]

Integrate[(c + d*x)^2/(a + b*x)^6,x]

[Out]

-1/30*(a^2*d^2 + a*b*d*(3*c + 5*d*x) + b^2*(6*c^2 + 15*c*d*x + 10*d^2*x^2))/(b^3*(a + b*x)^5)

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.95

method	result	size
gosper	\(-\frac {10 d^{2} x^{2} b^{2}+5 x a b \,d^{2}+15 x \,b^{2} c d +a^{2} d^{2}+3 a b c d +6 b^{2} c^{2}}{30 b^{3} \left (b x +a \right )^{5}}\)	\(62\)
risch	\(\frac {-\frac {d^{2} x^{2}}{3 b}-\frac {d \left (a d +3 b c \right ) x}{6 b^{2}}-\frac {a^{2} d^{2}+3 a b c d +6 b^{2} c^{2}}{30 b^{3}}}{\left (b x +a \right )^{5}}\)	\(63\)
parallelrisch	\(\frac {-10 d^{2} x^{2} b^{4}-5 a \,b^{3} d^{2} x -15 b^{4} c d x -a^{2} b^{2} d^{2}-3 a \,b^{3} c d -6 b^{4} c^{2}}{30 b^{5} \left (b x +a \right )^{5}}\)	\(70\)
default	\(-\frac {d^{2}}{3 b^{3} \left (b x +a \right )^{3}}+\frac {d \left (a d -b c \right )}{2 b^{3} \left (b x +a \right )^{4}}-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{5 b^{3} \left (b x +a \right )^{5}}\)	\(71\)
norman	\(\frac {-\frac {d^{2} x^{2}}{3 b}+\frac {\left (-a \,b^{2} d^{2}-3 b^{3} c d \right ) x}{6 b^{4}}+\frac {-a^{2} b^{2} d^{2}-3 a \,b^{3} c d -6 b^{4} c^{2}}{30 b^{5}}}{\left (b x +a \right )^{5}}\)	\(77\)

[In]

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

[Out]

-1/30/b^3*(10*b^2*d^2*x^2+5*a*b*d^2*x+15*b^2*c*d*x+a^2*d^2+3*a*b*c*d+6*b^2*c^2)/(b*x+a)^5

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.68 \[ \int \frac {(c+d x)^2}{(a+b x)^6} \, dx=-\frac {10 \, b^{2} d^{2} x^{2} + 6 \, b^{2} c^{2} + 3 \, a b c d + a^{2} d^{2} + 5 \, {\left (3 \, b^{2} c d + a b d^{2}\right )} x}{30 \, {\left (b^{8} x^{5} + 5 \, a b^{7} x^{4} + 10 \, a^{2} b^{6} x^{3} + 10 \, a^{3} b^{5} x^{2} + 5 \, a^{4} b^{4} x + a^{5} b^{3}\right )}} \]

[In]

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

[Out]

-1/30*(10*b^2*d^2*x^2 + 6*b^2*c^2 + 3*a*b*c*d + a^2*d^2 + 5*(3*b^2*c*d + a*b*d^2)*x)/(b^8*x^5 + 5*a*b^7*x^4 +
10*a^2*b^6*x^3 + 10*a^3*b^5*x^2 + 5*a^4*b^4*x + a^5*b^3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (56) = 112\).

Time = 0.56 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.78 \[ \int \frac {(c+d x)^2}{(a+b x)^6} \, dx=\frac {- a^{2} d^{2} - 3 a b c d - 6 b^{2} c^{2} - 10 b^{2} d^{2} x^{2} + x \left (- 5 a b d^{2} - 15 b^{2} c d\right )}{30 a^{5} b^{3} + 150 a^{4} b^{4} x + 300 a^{3} b^{5} x^{2} + 300 a^{2} b^{6} x^{3} + 150 a b^{7} x^{4} + 30 b^{8} x^{5}} \]

[In]

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

[Out]

(-a**2*d**2 - 3*a*b*c*d - 6*b**2*c**2 - 10*b**2*d**2*x**2 + x*(-5*a*b*d**2 - 15*b**2*c*d))/(30*a**5*b**3 + 150
*a**4*b**4*x + 300*a**3*b**5*x**2 + 300*a**2*b**6*x**3 + 150*a*b**7*x**4 + 30*b**8*x**5)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.68 \[ \int \frac {(c+d x)^2}{(a+b x)^6} \, dx=-\frac {10 \, b^{2} d^{2} x^{2} + 6 \, b^{2} c^{2} + 3 \, a b c d + a^{2} d^{2} + 5 \, {\left (3 \, b^{2} c d + a b d^{2}\right )} x}{30 \, {\left (b^{8} x^{5} + 5 \, a b^{7} x^{4} + 10 \, a^{2} b^{6} x^{3} + 10 \, a^{3} b^{5} x^{2} + 5 \, a^{4} b^{4} x + a^{5} b^{3}\right )}} \]

[In]

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

[Out]

-1/30*(10*b^2*d^2*x^2 + 6*b^2*c^2 + 3*a*b*c*d + a^2*d^2 + 5*(3*b^2*c*d + a*b*d^2)*x)/(b^8*x^5 + 5*a*b^7*x^4 +
10*a^2*b^6*x^3 + 10*a^3*b^5*x^2 + 5*a^4*b^4*x + a^5*b^3)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.94 \[ \int \frac {(c+d x)^2}{(a+b x)^6} \, dx=-\frac {10 \, b^{2} d^{2} x^{2} + 15 \, b^{2} c d x + 5 \, a b d^{2} x + 6 \, b^{2} c^{2} + 3 \, a b c d + a^{2} d^{2}}{30 \, {\left (b x + a\right )}^{5} b^{3}} \]

[In]

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

[Out]

-1/30*(10*b^2*d^2*x^2 + 15*b^2*c*d*x + 5*a*b*d^2*x + 6*b^2*c^2 + 3*a*b*c*d + a^2*d^2)/((b*x + a)^5*b^3)

Mupad [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.65 \[ \int \frac {(c+d x)^2}{(a+b x)^6} \, dx=-\frac {\frac {a^2\,d^2+3\,a\,b\,c\,d+6\,b^2\,c^2}{30\,b^3}+\frac {d^2\,x^2}{3\,b}+\frac {d\,x\,\left (a\,d+3\,b\,c\right )}{6\,b^2}}{a^5+5\,a^4\,b\,x+10\,a^3\,b^2\,x^2+10\,a^2\,b^3\,x^3+5\,a\,b^4\,x^4+b^5\,x^5} \]

[In]

int((c + d*x)^2/(a + b*x)^6,x)

[Out]

-((a^2*d^2 + 6*b^2*c^2 + 3*a*b*c*d)/(30*b^3) + (d^2*x^2)/(3*b) + (d*x*(a*d + 3*b*c))/(6*b^2))/(a^5 + b^5*x^5 +
5*a*b^4*x^4 + 10*a^3*b^2*x^2 + 10*a^2*b^3*x^3 + 5*a^4*b*x)