\(\int \frac {(a+b x)^2 (A+B x)}{(d+e x)^8} \, dx\) [1034]

Optimal result

Integrand size = 20, antiderivative size = 120 \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^8} \, dx=\frac {(b d-a e)^2 (B d-A e)}{7 e^4 (d+e x)^7}-\frac {(b d-a e) (3 b B d-2 A b e-a B e)}{6 e^4 (d+e x)^6}+\frac {b (3 b B d-A b e-2 a B e)}{5 e^4 (d+e x)^5}-\frac {b^2 B}{4 e^4 (d+e x)^4} \]

[Out]

1/7*(-a*e+b*d)^2*(-A*e+B*d)/e^4/(e*x+d)^7-1/6*(-a*e+b*d)*(-2*A*b*e-B*a*e+3*B*b*d)/e^4/(e*x+d)^6+1/5*b*(-A*b*e-
2*B*a*e+3*B*b*d)/e^4/(e*x+d)^5-1/4*b^2*B/e^4/(e*x+d)^4

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^8} \, dx=\frac {b (-2 a B e-A b e+3 b B d)}{5 e^4 (d+e x)^5}-\frac {(b d-a e) (-a B e-2 A b e+3 b B d)}{6 e^4 (d+e x)^6}+\frac {(b d-a e)^2 (B d-A e)}{7 e^4 (d+e x)^7}-\frac {b^2 B}{4 e^4 (d+e x)^4} \]

[In]

Int[((a + b*x)^2*(A + B*x))/(d + e*x)^8,x]

[Out]

((b*d - a*e)^2*(B*d - A*e))/(7*e^4*(d + e*x)^7) - ((b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e))/(6*e^4*(d + e*x)^6
) + (b*(3*b*B*d - A*b*e - 2*a*B*e))/(5*e^4*(d + e*x)^5) - (b^2*B)/(4*e^4*(d + e*x)^4)

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b d+a e)^2 (-B d+A e)}{e^3 (d+e x)^8}+\frac {(-b d+a e) (-3 b B d+2 A b e+a B e)}{e^3 (d+e x)^7}+\frac {b (-3 b B d+A b e+2 a B e)}{e^3 (d+e x)^6}+\frac {b^2 B}{e^3 (d+e x)^5}\right ) \, dx \\ & = \frac {(b d-a e)^2 (B d-A e)}{7 e^4 (d+e x)^7}-\frac {(b d-a e) (3 b B d-2 A b e-a B e)}{6 e^4 (d+e x)^6}+\frac {b (3 b B d-A b e-2 a B e)}{5 e^4 (d+e x)^5}-\frac {b^2 B}{4 e^4 (d+e x)^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^8} \, dx=-\frac {10 a^2 e^2 (6 A e+B (d+7 e x))+4 a b e \left (5 A e (d+7 e x)+2 B \left (d^2+7 d e x+21 e^2 x^2\right )\right )+b^2 \left (4 A e \left (d^2+7 d e x+21 e^2 x^2\right )+3 B \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )\right )}{420 e^4 (d+e x)^7} \]

[In]

Integrate[((a + b*x)^2*(A + B*x))/(d + e*x)^8,x]

[Out]

-1/420*(10*a^2*e^2*(6*A*e + B*(d + 7*e*x)) + 4*a*b*e*(5*A*e*(d + 7*e*x) + 2*B*(d^2 + 7*d*e*x + 21*e^2*x^2)) +
b^2*(4*A*e*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3*B*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3)))/(e^4*(d + e*x)^7
)

Maple [A] (verified)

Time = 0.68 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.31

method	result	size
risch	\(\frac {-\frac {b^{2} B \,x^{3}}{4 e}-\frac {b \left (4 A b e +8 B a e +3 B b d \right ) x^{2}}{20 e^{2}}-\frac {\left (20 A a b \,e^{2}+4 A \,b^{2} d e +10 B \,a^{2} e^{2}+8 B a b d e +3 b^{2} B \,d^{2}\right ) x}{60 e^{3}}-\frac {60 a^{2} A \,e^{3}+20 A a b d \,e^{2}+4 A \,b^{2} d^{2} e +10 B \,a^{2} d \,e^{2}+8 B a b \,d^{2} e +3 b^{2} B \,d^{3}}{420 e^{4}}}{\left (e x +d \right )^{7}}\)	\(157\)
default	\(-\frac {a^{2} A \,e^{3}-2 A a b d \,e^{2}+A \,b^{2} d^{2} e -B \,a^{2} d \,e^{2}+2 B a b \,d^{2} e -b^{2} B \,d^{3}}{7 e^{4} \left (e x +d \right )^{7}}-\frac {b \left (A b e +2 B a e -3 B b d \right )}{5 e^{4} \left (e x +d \right )^{5}}-\frac {2 A a b \,e^{2}-2 A \,b^{2} d e +B \,a^{2} e^{2}-4 B a b d e +3 b^{2} B \,d^{2}}{6 e^{4} \left (e x +d \right )^{6}}-\frac {b^{2} B}{4 e^{4} \left (e x +d \right )^{4}}\)	\(166\)
gosper	\(-\frac {105 b^{2} B \,x^{3} e^{3}+84 A \,x^{2} b^{2} e^{3}+168 B \,x^{2} a b \,e^{3}+63 B \,x^{2} b^{2} d \,e^{2}+140 A x a b \,e^{3}+28 A x \,b^{2} d \,e^{2}+70 B x \,a^{2} e^{3}+56 B x a b d \,e^{2}+21 B x \,b^{2} d^{2} e +60 a^{2} A \,e^{3}+20 A a b d \,e^{2}+4 A \,b^{2} d^{2} e +10 B \,a^{2} d \,e^{2}+8 B a b \,d^{2} e +3 b^{2} B \,d^{3}}{420 e^{4} \left (e x +d \right )^{7}}\)	\(169\)
parallelrisch	\(-\frac {105 b^{2} B \,x^{3} e^{6}+84 A \,b^{2} e^{6} x^{2}+168 B a b \,e^{6} x^{2}+63 B \,b^{2} d \,e^{5} x^{2}+140 A a b \,e^{6} x +28 A \,b^{2} d \,e^{5} x +70 B \,a^{2} e^{6} x +56 B a b d \,e^{5} x +21 B \,b^{2} d^{2} e^{4} x +60 a^{2} A \,e^{6}+20 A a b d \,e^{5}+4 A \,b^{2} d^{2} e^{4}+10 B \,a^{2} d \,e^{5}+8 B a b \,d^{2} e^{4}+3 B \,b^{2} d^{3} e^{3}}{420 e^{7} \left (e x +d \right )^{7}}\)	\(178\)
norman	\(\frac {-\frac {b^{2} B \,x^{3}}{4 e}-\frac {\left (4 A \,b^{2} e^{4}+8 B a b \,e^{4}+3 b^{2} B d \,e^{3}\right ) x^{2}}{20 e^{5}}-\frac {\left (20 a b A \,e^{5}+4 A \,b^{2} d \,e^{4}+10 a^{2} B \,e^{5}+8 B a b d \,e^{4}+3 B \,b^{2} d^{2} e^{3}\right ) x}{60 e^{6}}-\frac {60 a^{2} A \,e^{6}+20 A a b d \,e^{5}+4 A \,b^{2} d^{2} e^{4}+10 B \,a^{2} d \,e^{5}+8 B a b \,d^{2} e^{4}+3 B \,b^{2} d^{3} e^{3}}{420 e^{7}}}{\left (e x +d \right )^{7}}\)	\(182\)

[In]

int((b*x+a)^2*(B*x+A)/(e*x+d)^8,x,method=_RETURNVERBOSE)

[Out]

(-1/4*b^2*B/e*x^3-1/20*b/e^2*(4*A*b*e+8*B*a*e+3*B*b*d)*x^2-1/60/e^3*(20*A*a*b*e^2+4*A*b^2*d*e+10*B*a^2*e^2+8*B
*a*b*d*e+3*B*b^2*d^2)*x-1/420/e^4*(60*A*a^2*e^3+20*A*a*b*d*e^2+4*A*b^2*d^2*e+10*B*a^2*d*e^2+8*B*a*b*d^2*e+3*B*
b^2*d^3))/(e*x+d)^7

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (112) = 224\).

Time = 0.22 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.88 \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^8} \, dx=-\frac {105 \, B b^{2} e^{3} x^{3} + 3 \, B b^{2} d^{3} + 60 \, A a^{2} e^{3} + 4 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e + 10 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 21 \, {\left (3 \, B b^{2} d e^{2} + 4 \, {\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 7 \, {\left (3 \, B b^{2} d^{2} e + 4 \, {\left (2 \, B a b + A b^{2}\right )} d e^{2} + 10 \, {\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{420 \, {\left (e^{11} x^{7} + 7 \, d e^{10} x^{6} + 21 \, d^{2} e^{9} x^{5} + 35 \, d^{3} e^{8} x^{4} + 35 \, d^{4} e^{7} x^{3} + 21 \, d^{5} e^{6} x^{2} + 7 \, d^{6} e^{5} x + d^{7} e^{4}\right )}} \]

[In]

integrate((b*x+a)^2*(B*x+A)/(e*x+d)^8,x, algorithm="fricas")

[Out]

-1/420*(105*B*b^2*e^3*x^3 + 3*B*b^2*d^3 + 60*A*a^2*e^3 + 4*(2*B*a*b + A*b^2)*d^2*e + 10*(B*a^2 + 2*A*a*b)*d*e^
2 + 21*(3*B*b^2*d*e^2 + 4*(2*B*a*b + A*b^2)*e^3)*x^2 + 7*(3*B*b^2*d^2*e + 4*(2*B*a*b + A*b^2)*d*e^2 + 10*(B*a^
2 + 2*A*a*b)*e^3)*x)/(e^11*x^7 + 7*d*e^10*x^6 + 21*d^2*e^9*x^5 + 35*d^3*e^8*x^4 + 35*d^4*e^7*x^3 + 21*d^5*e^6*
x^2 + 7*d^6*e^5*x + d^7*e^4)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^8} \, dx=\text {Timed out} \]

[In]

integrate((b*x+a)**2*(B*x+A)/(e*x+d)**8,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (112) = 224\).

Time = 0.22 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.88 \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^8} \, dx=-\frac {105 \, B b^{2} e^{3} x^{3} + 3 \, B b^{2} d^{3} + 60 \, A a^{2} e^{3} + 4 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e + 10 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 21 \, {\left (3 \, B b^{2} d e^{2} + 4 \, {\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 7 \, {\left (3 \, B b^{2} d^{2} e + 4 \, {\left (2 \, B a b + A b^{2}\right )} d e^{2} + 10 \, {\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{420 \, {\left (e^{11} x^{7} + 7 \, d e^{10} x^{6} + 21 \, d^{2} e^{9} x^{5} + 35 \, d^{3} e^{8} x^{4} + 35 \, d^{4} e^{7} x^{3} + 21 \, d^{5} e^{6} x^{2} + 7 \, d^{6} e^{5} x + d^{7} e^{4}\right )}} \]

[In]

integrate((b*x+a)^2*(B*x+A)/(e*x+d)^8,x, algorithm="maxima")

[Out]

-1/420*(105*B*b^2*e^3*x^3 + 3*B*b^2*d^3 + 60*A*a^2*e^3 + 4*(2*B*a*b + A*b^2)*d^2*e + 10*(B*a^2 + 2*A*a*b)*d*e^
2 + 21*(3*B*b^2*d*e^2 + 4*(2*B*a*b + A*b^2)*e^3)*x^2 + 7*(3*B*b^2*d^2*e + 4*(2*B*a*b + A*b^2)*d*e^2 + 10*(B*a^
2 + 2*A*a*b)*e^3)*x)/(e^11*x^7 + 7*d*e^10*x^6 + 21*d^2*e^9*x^5 + 35*d^3*e^8*x^4 + 35*d^4*e^7*x^3 + 21*d^5*e^6*
x^2 + 7*d^6*e^5*x + d^7*e^4)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.40 \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^8} \, dx=-\frac {105 \, B b^{2} e^{3} x^{3} + 63 \, B b^{2} d e^{2} x^{2} + 168 \, B a b e^{3} x^{2} + 84 \, A b^{2} e^{3} x^{2} + 21 \, B b^{2} d^{2} e x + 56 \, B a b d e^{2} x + 28 \, A b^{2} d e^{2} x + 70 \, B a^{2} e^{3} x + 140 \, A a b e^{3} x + 3 \, B b^{2} d^{3} + 8 \, B a b d^{2} e + 4 \, A b^{2} d^{2} e + 10 \, B a^{2} d e^{2} + 20 \, A a b d e^{2} + 60 \, A a^{2} e^{3}}{420 \, {\left (e x + d\right )}^{7} e^{4}} \]

[In]

integrate((b*x+a)^2*(B*x+A)/(e*x+d)^8,x, algorithm="giac")

[Out]

-1/420*(105*B*b^2*e^3*x^3 + 63*B*b^2*d*e^2*x^2 + 168*B*a*b*e^3*x^2 + 84*A*b^2*e^3*x^2 + 21*B*b^2*d^2*e*x + 56*
B*a*b*d*e^2*x + 28*A*b^2*d*e^2*x + 70*B*a^2*e^3*x + 140*A*a*b*e^3*x + 3*B*b^2*d^3 + 8*B*a*b*d^2*e + 4*A*b^2*d^
2*e + 10*B*a^2*d*e^2 + 20*A*a*b*d*e^2 + 60*A*a^2*e^3)/((e*x + d)^7*e^4)

Mupad [B] (verification not implemented)

Time = 1.29 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.86 \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^8} \, dx=-\frac {\frac {10\,B\,a^2\,d\,e^2+60\,A\,a^2\,e^3+8\,B\,a\,b\,d^2\,e+20\,A\,a\,b\,d\,e^2+3\,B\,b^2\,d^3+4\,A\,b^2\,d^2\,e}{420\,e^4}+\frac {x\,\left (10\,B\,a^2\,e^2+8\,B\,a\,b\,d\,e+20\,A\,a\,b\,e^2+3\,B\,b^2\,d^2+4\,A\,b^2\,d\,e\right )}{60\,e^3}+\frac {b\,x^2\,\left (4\,A\,b\,e+8\,B\,a\,e+3\,B\,b\,d\right )}{20\,e^2}+\frac {B\,b^2\,x^3}{4\,e}}{d^7+7\,d^6\,e\,x+21\,d^5\,e^2\,x^2+35\,d^4\,e^3\,x^3+35\,d^3\,e^4\,x^4+21\,d^2\,e^5\,x^5+7\,d\,e^6\,x^6+e^7\,x^7} \]

[In]

int(((A + B*x)*(a + b*x)^2)/(d + e*x)^8,x)

[Out]

-((60*A*a^2*e^3 + 3*B*b^2*d^3 + 4*A*b^2*d^2*e + 10*B*a^2*d*e^2 + 20*A*a*b*d*e^2 + 8*B*a*b*d^2*e)/(420*e^4) + (
x*(10*B*a^2*e^2 + 3*B*b^2*d^2 + 20*A*a*b*e^2 + 4*A*b^2*d*e + 8*B*a*b*d*e))/(60*e^3) + (b*x^2*(4*A*b*e + 8*B*a*
e + 3*B*b*d))/(20*e^2) + (B*b^2*x^3)/(4*e))/(d^7 + e^7*x^7 + 7*d*e^6*x^6 + 21*d^5*e^2*x^2 + 35*d^4*e^3*x^3 + 3
5*d^3*e^4*x^4 + 21*d^2*e^5*x^5 + 7*d^6*e*x)