\(\int \frac {(c+d x)^2}{(a+b x)^5} \, dx\) [1255]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.03 (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} \[ \int \frac {(c+d x)^2}{(a+b x)^5} \, dx=-\frac {2 d (b c-a d)}{3 b^3 (a+b x)^3}-\frac {(b c-a d)^2}{4 b^3 (a+b x)^4}-\frac {d^2}{2 b^3 (a+b x)^2} \]

[In]

Int[(c + d*x)^2/(a + b*x)^5,x]

[Out]

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

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)^5}+\frac {2 d (b c-a d)}{b^2 (a+b x)^4}+\frac {d^2}{b^2 (a+b x)^3}\right ) \, dx \\ & = -\frac {(b c-a d)^2}{4 b^3 (a+b x)^4}-\frac {2 d (b c-a d)}{3 b^3 (a+b x)^3}-\frac {d^2}{2 b^3 (a+b x)^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.86 \[ \int \frac {(c+d x)^2}{(a+b x)^5} \, dx=-\frac {a^2 d^2+2 a b d (c+2 d x)+b^2 \left (3 c^2+8 c d x+6 d^2 x^2\right )}{12 b^3 (a+b x)^4} \]

[In]

Integrate[(c + d*x)^2/(a + b*x)^5,x]

[Out]

-1/12*(a^2*d^2 + 2*a*b*d*(c + 2*d*x) + b^2*(3*c^2 + 8*c*d*x + 6*d^2*x^2))/(b^3*(a + b*x)^4)

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.95

method	result	size
gosper	\(-\frac {6 d^{2} x^{2} b^{2}+4 x a b \,d^{2}+8 x \,b^{2} c d +a^{2} d^{2}+2 a b c d +3 b^{2} c^{2}}{12 b^{3} \left (b x +a \right )^{4}}\)	\(62\)
risch	\(\frac {-\frac {d^{2} x^{2}}{2 b}-\frac {d \left (a d +2 b c \right ) x}{3 b^{2}}-\frac {a^{2} d^{2}+2 a b c d +3 b^{2} c^{2}}{12 b^{3}}}{\left (b x +a \right )^{4}}\)	\(63\)
parallelrisch	\(\frac {-6 d^{2} x^{2} b^{3}-4 a \,b^{2} d^{2} x -8 b^{3} c d x -a^{2} b \,d^{2}-2 a \,b^{2} c d -3 b^{3} c^{2}}{12 b^{4} \left (b x +a \right )^{4}}\)	\(68\)
default	\(\frac {2 d \left (a d -b c \right )}{3 b^{3} \left (b x +a \right )^{3}}-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{4 b^{3} \left (b x +a \right )^{4}}-\frac {d^{2}}{2 b^{3} \left (b x +a \right )^{2}}\)	\(71\)
norman	\(\frac {-\frac {d^{2} x^{2}}{2 b}+\frac {\left (-a b \,d^{2}-2 b^{2} c d \right ) x}{3 b^{3}}+\frac {-a^{2} b \,d^{2}-2 a \,b^{2} c d -3 b^{3} c^{2}}{12 b^{4}}}{\left (b x +a \right )^{4}}\)	\(73\)

[In]

int((d*x+c)^2/(b*x+a)^5,x,method=_RETURNVERBOSE)

[Out]

-1/12/b^3*(6*b^2*d^2*x^2+4*a*b*d^2*x+8*b^2*c*d*x+a^2*d^2+2*a*b*c*d+3*b^2*c^2)/(b*x+a)^4

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.51 \[ \int \frac {(c+d x)^2}{(a+b x)^5} \, dx=-\frac {6 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b^{2} c d + a b d^{2}\right )} x}{12 \, {\left (b^{7} x^{4} + 4 \, a b^{6} x^{3} + 6 \, a^{2} b^{5} x^{2} + 4 \, a^{3} b^{4} x + a^{4} b^{3}\right )}} \]

[In]

integrate((d*x+c)^2/(b*x+a)^5,x, algorithm="fricas")

[Out]

-1/12*(6*b^2*d^2*x^2 + 3*b^2*c^2 + 2*a*b*c*d + a^2*d^2 + 4*(2*b^2*c*d + a*b*d^2)*x)/(b^7*x^4 + 4*a*b^6*x^3 + 6
*a^2*b^5*x^2 + 4*a^3*b^4*x + a^4*b^3)

Sympy [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.60 \[ \int \frac {(c+d x)^2}{(a+b x)^5} \, dx=\frac {- a^{2} d^{2} - 2 a b c d - 3 b^{2} c^{2} - 6 b^{2} d^{2} x^{2} + x \left (- 4 a b d^{2} - 8 b^{2} c d\right )}{12 a^{4} b^{3} + 48 a^{3} b^{4} x + 72 a^{2} b^{5} x^{2} + 48 a b^{6} x^{3} + 12 b^{7} x^{4}} \]

[In]

integrate((d*x+c)**2/(b*x+a)**5,x)

[Out]

(-a**2*d**2 - 2*a*b*c*d - 3*b**2*c**2 - 6*b**2*d**2*x**2 + x*(-4*a*b*d**2 - 8*b**2*c*d))/(12*a**4*b**3 + 48*a*
*3*b**4*x + 72*a**2*b**5*x**2 + 48*a*b**6*x**3 + 12*b**7*x**4)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.51 \[ \int \frac {(c+d x)^2}{(a+b x)^5} \, dx=-\frac {6 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b^{2} c d + a b d^{2}\right )} x}{12 \, {\left (b^{7} x^{4} + 4 \, a b^{6} x^{3} + 6 \, a^{2} b^{5} x^{2} + 4 \, a^{3} b^{4} x + a^{4} b^{3}\right )}} \]

[In]

integrate((d*x+c)^2/(b*x+a)^5,x, algorithm="maxima")

[Out]

-1/12*(6*b^2*d^2*x^2 + 3*b^2*c^2 + 2*a*b*c*d + a^2*d^2 + 4*(2*b^2*c*d + a*b*d^2)*x)/(b^7*x^4 + 4*a*b^6*x^3 + 6
*a^2*b^5*x^2 + 4*a^3*b^4*x + a^4*b^3)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.48 \[ \int \frac {(c+d x)^2}{(a+b x)^5} \, dx=-\frac {\frac {3 \, c^{2}}{{\left (b x + a\right )}^{4}} + \frac {8 \, c d}{{\left (b x + a\right )}^{3} b} - \frac {6 \, a c d}{{\left (b x + a\right )}^{4} b} + \frac {6 \, d^{2}}{{\left (b x + a\right )}^{2} b^{2}} - \frac {8 \, a d^{2}}{{\left (b x + a\right )}^{3} b^{2}} + \frac {3 \, a^{2} d^{2}}{{\left (b x + a\right )}^{4} b^{2}}}{12 \, b} \]

[In]

integrate((d*x+c)^2/(b*x+a)^5,x, algorithm="giac")

[Out]

-1/12*(3*c^2/(b*x + a)^4 + 8*c*d/((b*x + a)^3*b) - 6*a*c*d/((b*x + a)^4*b) + 6*d^2/((b*x + a)^2*b^2) - 8*a*d^2
/((b*x + a)^3*b^2) + 3*a^2*d^2/((b*x + a)^4*b^2))/b

Mupad [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.60 \[ \int \frac {(c+d x)^2}{(a+b x)^5} \, dx=\frac {{\left (c+d\,x\right )}^3\,\left (4\,a\,d-3\,b\,c+b\,d\,x\right )}{12\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^4} \]

[In]

int((c + d*x)^2/(a + b*x)^5,x)

[Out]

((c + d*x)^3*(4*a*d - 3*b*c + b*d*x))/(12*(a*d - b*c)^2*(a + b*x)^4)