3.1034 \(\int \frac {(a+b x)^2 (A+B x)}{(d+e x)^8} \, dx\)

Optimal. Leaf size=120 \[ \frac {b (-2 a B e-A b e+3 b B d)}{5 e^4 (d+e x)^5}-\frac {(b d-a e) (-a B e-2 A b e+3 b B d)}{6 e^4 (d+e x)^6}+\frac {(b d-a e)^2 (B d-A e)}{7 e^4 (d+e x)^7}-\frac {b^2 B}{4 e^4 (d+e x)^4} \]

[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

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Rubi [A]  time = 0.08, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \[ \frac {b (-2 a B e-A b e+3 b B d)}{5 e^4 (d+e x)^5}-\frac {(b d-a e) (-a B e-2 A b e+3 b B d)}{6 e^4 (d+e x)^6}+\frac {(b d-a e)^2 (B d-A e)}{7 e^4 (d+e x)^7}-\frac {b^2 B}{4 e^4 (d+e x)^4} \]

Antiderivative was successfully verified.

[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 77

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*} \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^8} \, dx &=\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*}

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Mathematica [A]  time = 0.06, size = 129, normalized size = 1.08 \[ -\frac {10 a^2 e^2 (6 A e+B (d+7 e x))+4 a b e \left (5 A e (d+7 e x)+2 B \left (d^2+7 d e x+21 e^2 x^2\right )\right )+b^2 \left (4 A e \left (d^2+7 d e x+21 e^2 x^2\right )+3 B \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )\right )}{420 e^4 (d+e x)^7} \]

Antiderivative was successfully verified.

[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
)

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fricas [B]  time = 1.06, size = 225, normalized size = 1.88 \[ -\frac {105 \, B b^{2} e^{3} x^{3} + 3 \, B b^{2} d^{3} + 60 \, A a^{2} e^{3} + 4 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e + 10 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 21 \, {\left (3 \, B b^{2} d e^{2} + 4 \, {\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 7 \, {\left (3 \, B b^{2} d^{2} e + 4 \, {\left (2 \, B a b + A b^{2}\right )} d e^{2} + 10 \, {\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{420 \, {\left (e^{11} x^{7} + 7 \, d e^{10} x^{6} + 21 \, d^{2} e^{9} x^{5} + 35 \, d^{3} e^{8} x^{4} + 35 \, d^{4} e^{7} x^{3} + 21 \, d^{5} e^{6} x^{2} + 7 \, d^{6} e^{5} x + d^{7} e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 1.17, size = 160, normalized size = 1.33 \[ -\frac {{\left (105 \, B b^{2} x^{3} e^{3} + 63 \, B b^{2} d x^{2} e^{2} + 21 \, B b^{2} d^{2} x e + 3 \, B b^{2} d^{3} + 168 \, B a b x^{2} e^{3} + 84 \, A b^{2} x^{2} e^{3} + 56 \, B a b d x e^{2} + 28 \, A b^{2} d x e^{2} + 8 \, B a b d^{2} e + 4 \, A b^{2} d^{2} e + 70 \, B a^{2} x e^{3} + 140 \, A a b x e^{3} + 10 \, B a^{2} d e^{2} + 20 \, A a b d e^{2} + 60 \, A a^{2} e^{3}\right )} e^{\left (-4\right )}}{420 \, {\left (x e + d\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 166, normalized size = 1.38 \[ -\frac {B \,b^{2}}{4 \left (e x +d \right )^{4} e^{4}}-\frac {\left (A b e +2 B a e -3 B b d \right ) b}{5 \left (e x +d \right )^{5} e^{4}}-\frac {2 A a b \,e^{2}-2 A d \,b^{2} e +B \,a^{2} e^{2}-4 B d a b e +3 B \,b^{2} d^{2}}{6 \left (e x +d \right )^{6} e^{4}}-\frac {A \,a^{2} e^{3}-2 A d a b \,e^{2}+A \,d^{2} b^{2} e -B d \,a^{2} e^{2}+2 B \,d^{2} a b e -B \,b^{2} d^{3}}{7 \left (e x +d \right )^{7} e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.55, size = 225, normalized size = 1.88 \[ -\frac {105 \, B b^{2} e^{3} x^{3} + 3 \, B b^{2} d^{3} + 60 \, A a^{2} e^{3} + 4 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e + 10 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 21 \, {\left (3 \, B b^{2} d e^{2} + 4 \, {\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 7 \, {\left (3 \, B b^{2} d^{2} e + 4 \, {\left (2 \, B a b + A b^{2}\right )} d e^{2} + 10 \, {\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{420 \, {\left (e^{11} x^{7} + 7 \, d e^{10} x^{6} + 21 \, d^{2} e^{9} x^{5} + 35 \, d^{3} e^{8} x^{4} + 35 \, d^{4} e^{7} x^{3} + 21 \, d^{5} e^{6} x^{2} + 7 \, d^{6} e^{5} x + d^{7} e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 1.10, size = 223, normalized size = 1.86 \[ -\frac {\frac {10\,B\,a^2\,d\,e^2+60\,A\,a^2\,e^3+8\,B\,a\,b\,d^2\,e+20\,A\,a\,b\,d\,e^2+3\,B\,b^2\,d^3+4\,A\,b^2\,d^2\,e}{420\,e^4}+\frac {x\,\left (10\,B\,a^2\,e^2+8\,B\,a\,b\,d\,e+20\,A\,a\,b\,e^2+3\,B\,b^2\,d^2+4\,A\,b^2\,d\,e\right )}{60\,e^3}+\frac {b\,x^2\,\left (4\,A\,b\,e+8\,B\,a\,e+3\,B\,b\,d\right )}{20\,e^2}+\frac {B\,b^2\,x^3}{4\,e}}{d^7+7\,d^6\,e\,x+21\,d^5\,e^2\,x^2+35\,d^4\,e^3\,x^3+35\,d^3\,e^4\,x^4+21\,d^2\,e^5\,x^5+7\,d\,e^6\,x^6+e^7\,x^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [B]  time = 67.99, size = 262, normalized size = 2.18 \[ \frac {- 60 A a^{2} e^{3} - 20 A a b d e^{2} - 4 A b^{2} d^{2} e - 10 B a^{2} d e^{2} - 8 B a b d^{2} e - 3 B b^{2} d^{3} - 105 B b^{2} e^{3} x^{3} + x^{2} \left (- 84 A b^{2} e^{3} - 168 B a b e^{3} - 63 B b^{2} d e^{2}\right ) + x \left (- 140 A a b e^{3} - 28 A b^{2} d e^{2} - 70 B a^{2} e^{3} - 56 B a b d e^{2} - 21 B b^{2} d^{2} e\right )}{420 d^{7} e^{4} + 2940 d^{6} e^{5} x + 8820 d^{5} e^{6} x^{2} + 14700 d^{4} e^{7} x^{3} + 14700 d^{3} e^{8} x^{4} + 8820 d^{2} e^{9} x^{5} + 2940 d e^{10} x^{6} + 420 e^{11} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-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 - 105
*B*b**2*e**3*x**3 + x**2*(-84*A*b**2*e**3 - 168*B*a*b*e**3 - 63*B*b**2*d*e**2) + x*(-140*A*a*b*e**3 - 28*A*b**
2*d*e**2 - 70*B*a**2*e**3 - 56*B*a*b*d*e**2 - 21*B*b**2*d**2*e))/(420*d**7*e**4 + 2940*d**6*e**5*x + 8820*d**5
*e**6*x**2 + 14700*d**4*e**7*x**3 + 14700*d**3*e**8*x**4 + 8820*d**2*e**9*x**5 + 2940*d*e**10*x**6 + 420*e**11
*x**7)

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