\(\int \frac {(a c+(b c+a d) x+b d x^2)^2}{(a+b x)^{10}} \, dx\) [1782]

Optimal result

Integrand size = 29, antiderivative size = 65 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^{10}} \, dx=-\frac {(b c-a d)^2}{7 b^3 (a+b x)^7}-\frac {d (b c-a d)}{3 b^3 (a+b x)^6}-\frac {d^2}{5 b^3 (a+b x)^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 65, 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} \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^{10}} \, dx=-\frac {d (b c-a d)}{3 b^3 (a+b x)^6}-\frac {(b c-a d)^2}{7 b^3 (a+b x)^7}-\frac {d^2}{5 b^3 (a+b x)^5} \]

[In]

Int[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x)^10,x]

[Out]

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

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)^2}{(a+b x)^8} \, dx \\ & = \int \left (\frac {(b c-a d)^2}{b^2 (a+b x)^8}+\frac {2 d (b c-a d)}{b^2 (a+b x)^7}+\frac {d^2}{b^2 (a+b x)^6}\right ) \, dx \\ & = -\frac {(b c-a d)^2}{7 b^3 (a+b x)^7}-\frac {d (b c-a d)}{3 b^3 (a+b x)^6}-\frac {d^2}{5 b^3 (a+b x)^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^{10}} \, dx=-\frac {a^2 d^2+a b d (5 c+7 d x)+b^2 \left (15 c^2+35 c d x+21 d^2 x^2\right )}{105 b^3 (a+b x)^7} \]

[In]

Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x)^10,x]

[Out]

-1/105*(a^2*d^2 + a*b*d*(5*c + 7*d*x) + b^2*(15*c^2 + 35*c*d*x + 21*d^2*x^2))/(b^3*(a + b*x)^7)

Maple [A] (verified)

Time = 2.80 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.95

method	result	size
gosper	\(-\frac {21 d^{2} x^{2} b^{2}+7 x a b \,d^{2}+35 x \,b^{2} c d +a^{2} d^{2}+5 a b c d +15 b^{2} c^{2}}{105 b^{3} \left (b x +a \right )^{7}}\)	\(62\)
risch	\(\frac {-\frac {d^{2} x^{2}}{5 b}-\frac {d \left (a d +5 b c \right ) x}{15 b^{2}}-\frac {a^{2} d^{2}+5 a b c d +15 b^{2} c^{2}}{105 b^{3}}}{\left (b x +a \right )^{7}}\)	\(63\)
parallelrisch	\(\frac {-21 d^{2} x^{2} b^{6}-7 a \,b^{5} d^{2} x -35 b^{6} c d x -a^{2} b^{4} d^{2}-5 a c d \,b^{5}-15 c^{2} b^{6}}{105 b^{7} \left (b x +a \right )^{7}}\)	\(70\)
default	\(-\frac {d^{2}}{5 b^{3} \left (b x +a \right )^{5}}+\frac {\left (a d -b c \right ) d}{3 b^{3} \left (b x +a \right )^{6}}-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{7 b^{3} \left (b x +a \right )^{7}}\)	\(71\)
norman	\(\frac {\frac {a^{2} \left (-a^{2} b^{6} d^{2}-5 a \,b^{7} c d -15 c^{2} b^{8}\right )}{105 b^{9}}-\frac {b \,d^{2} x^{4}}{5}+\frac {\left (-7 a \,b^{6} d^{2}-5 c d \,b^{7}\right ) x^{3}}{15 b^{6}}+\frac {\left (-12 a^{2} b^{6} d^{2}-25 a \,b^{7} c d -5 c^{2} b^{8}\right ) x^{2}}{35 b^{7}}+\frac {a \left (-3 a^{2} b^{6} d^{2}-15 a \,b^{7} c d -10 c^{2} b^{8}\right ) x}{35 b^{8}}}{\left (b x +a \right )^{9}}\)	\(151\)

[In]

int((b*d*x^2+(a*d+b*c)*x+a*c)^2/(b*x+a)^10,x,method=_RETURNVERBOSE)

[Out]

-1/105/b^3*(21*b^2*d^2*x^2+7*a*b*d^2*x+35*b^2*c*d*x+a^2*d^2+5*a*b*c*d+15*b^2*c^2)/(b*x+a)^7

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (59) = 118\).

Time = 0.27 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.02 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^{10}} \, dx=-\frac {21 \, b^{2} d^{2} x^{2} + 15 \, b^{2} c^{2} + 5 \, a b c d + a^{2} d^{2} + 7 \, {\left (5 \, b^{2} c d + a b d^{2}\right )} x}{105 \, {\left (b^{10} x^{7} + 7 \, a b^{9} x^{6} + 21 \, a^{2} b^{8} x^{5} + 35 \, a^{3} b^{7} x^{4} + 35 \, a^{4} b^{6} x^{3} + 21 \, a^{5} b^{5} x^{2} + 7 \, a^{6} b^{4} x + a^{7} b^{3}\right )}} \]

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x^2)^2/(b*x+a)^10,x, algorithm="fricas")

[Out]

-1/105*(21*b^2*d^2*x^2 + 15*b^2*c^2 + 5*a*b*c*d + a^2*d^2 + 7*(5*b^2*c*d + a*b*d^2)*x)/(b^10*x^7 + 7*a*b^9*x^6
+ 21*a^2*b^8*x^5 + 35*a^3*b^7*x^4 + 35*a^4*b^6*x^3 + 21*a^5*b^5*x^2 + 7*a^6*b^4*x + a^7*b^3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (56) = 112\).

Time = 2.87 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.14 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^{10}} \, dx=\frac {- a^{2} d^{2} - 5 a b c d - 15 b^{2} c^{2} - 21 b^{2} d^{2} x^{2} + x \left (- 7 a b d^{2} - 35 b^{2} c d\right )}{105 a^{7} b^{3} + 735 a^{6} b^{4} x + 2205 a^{5} b^{5} x^{2} + 3675 a^{4} b^{6} x^{3} + 3675 a^{3} b^{7} x^{4} + 2205 a^{2} b^{8} x^{5} + 735 a b^{9} x^{6} + 105 b^{10} x^{7}} \]

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x**2)**2/(b*x+a)**10,x)

[Out]

(-a**2*d**2 - 5*a*b*c*d - 15*b**2*c**2 - 21*b**2*d**2*x**2 + x*(-7*a*b*d**2 - 35*b**2*c*d))/(105*a**7*b**3 + 7
35*a**6*b**4*x + 2205*a**5*b**5*x**2 + 3675*a**4*b**6*x**3 + 3675*a**3*b**7*x**4 + 2205*a**2*b**8*x**5 + 735*a
*b**9*x**6 + 105*b**10*x**7)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (59) = 118\).

Time = 0.19 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.02 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^{10}} \, dx=-\frac {21 \, b^{2} d^{2} x^{2} + 15 \, b^{2} c^{2} + 5 \, a b c d + a^{2} d^{2} + 7 \, {\left (5 \, b^{2} c d + a b d^{2}\right )} x}{105 \, {\left (b^{10} x^{7} + 7 \, a b^{9} x^{6} + 21 \, a^{2} b^{8} x^{5} + 35 \, a^{3} b^{7} x^{4} + 35 \, a^{4} b^{6} x^{3} + 21 \, a^{5} b^{5} x^{2} + 7 \, a^{6} b^{4} x + a^{7} b^{3}\right )}} \]

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x^2)^2/(b*x+a)^10,x, algorithm="maxima")

[Out]

-1/105*(21*b^2*d^2*x^2 + 15*b^2*c^2 + 5*a*b*c*d + a^2*d^2 + 7*(5*b^2*c*d + a*b*d^2)*x)/(b^10*x^7 + 7*a*b^9*x^6
+ 21*a^2*b^8*x^5 + 35*a^3*b^7*x^4 + 35*a^4*b^6*x^3 + 21*a^5*b^5*x^2 + 7*a^6*b^4*x + a^7*b^3)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^{10}} \, dx=-\frac {21 \, b^{2} d^{2} x^{2} + 35 \, b^{2} c d x + 7 \, a b d^{2} x + 15 \, b^{2} c^{2} + 5 \, a b c d + a^{2} d^{2}}{105 \, {\left (b x + a\right )}^{7} b^{3}} \]

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x^2)^2/(b*x+a)^10,x, algorithm="giac")

[Out]

-1/105*(21*b^2*d^2*x^2 + 35*b^2*c*d*x + 7*a*b*d^2*x + 15*b^2*c^2 + 5*a*b*c*d + a^2*d^2)/((b*x + a)^7*b^3)

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.98 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^{10}} \, dx=-\frac {\frac {a^2\,d^2+5\,a\,b\,c\,d+15\,b^2\,c^2}{105\,b^3}+\frac {d^2\,x^2}{5\,b}+\frac {d\,x\,\left (a\,d+5\,b\,c\right )}{15\,b^2}}{a^7+7\,a^6\,b\,x+21\,a^5\,b^2\,x^2+35\,a^4\,b^3\,x^3+35\,a^3\,b^4\,x^4+21\,a^2\,b^5\,x^5+7\,a\,b^6\,x^6+b^7\,x^7} \]

[In]

int((a*c + x*(a*d + b*c) + b*d*x^2)^2/(a + b*x)^10,x)

[Out]

-((a^2*d^2 + 15*b^2*c^2 + 5*a*b*c*d)/(105*b^3) + (d^2*x^2)/(5*b) + (d*x*(a*d + 5*b*c))/(15*b^2))/(a^7 + b^7*x^
7 + 7*a*b^6*x^6 + 21*a^5*b^2*x^2 + 35*a^4*b^3*x^3 + 35*a^3*b^4*x^4 + 21*a^2*b^5*x^5 + 7*a^6*b*x)