3.1021 \(\int \frac {(a+b x) (A+B x)}{(d+e x)^6} \, dx\)

Optimal. Leaf size=77 \[ \frac {-a B e-A b e+2 b B d}{4 e^3 (d+e x)^4}-\frac {(b d-a e) (B d-A e)}{5 e^3 (d+e x)^5}-\frac {b B}{3 e^3 (d+e x)^3} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {77} \[ \frac {-a B e-A b e+2 b B d}{4 e^3 (d+e x)^4}-\frac {(b d-a e) (B d-A e)}{5 e^3 (d+e x)^5}-\frac {b B}{3 e^3 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(A + B*x))/(d + e*x)^6,x]

[Out]

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

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

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Mathematica [A]  time = 0.03, size = 65, normalized size = 0.84 \[ -\frac {3 a e (4 A e+B (d+5 e x))+b \left (3 A e (d+5 e x)+2 B \left (d^2+5 d e x+10 e^2 x^2\right )\right )}{60 e^3 (d+e x)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(A + B*x))/(d + e*x)^6,x]

[Out]

-1/60*(3*a*e*(4*A*e + B*(d + 5*e*x)) + b*(3*A*e*(d + 5*e*x) + 2*B*(d^2 + 5*d*e*x + 10*e^2*x^2)))/(e^3*(d + e*x
)^5)

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fricas [A]  time = 0.73, size = 117, normalized size = 1.52 \[ -\frac {20 \, B b e^{2} x^{2} + 2 \, B b d^{2} + 12 \, A a e^{2} + 3 \, {\left (B a + A b\right )} d e + 5 \, {\left (2 \, B b d e + 3 \, {\left (B a + A b\right )} e^{2}\right )} x}{60 \, {\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)/(e*x+d)^6,x, algorithm="fricas")

[Out]

-1/60*(20*B*b*e^2*x^2 + 2*B*b*d^2 + 12*A*a*e^2 + 3*(B*a + A*b)*d*e + 5*(2*B*b*d*e + 3*(B*a + A*b)*e^2)*x)/(e^8
*x^5 + 5*d*e^7*x^4 + 10*d^2*e^6*x^3 + 10*d^3*e^5*x^2 + 5*d^4*e^4*x + d^5*e^3)

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giac [A]  time = 1.21, size = 71, normalized size = 0.92 \[ -\frac {{\left (20 \, B b x^{2} e^{2} + 10 \, B b d x e + 2 \, B b d^{2} + 15 \, B a x e^{2} + 15 \, A b x e^{2} + 3 \, B a d e + 3 \, A b d e + 12 \, A a e^{2}\right )} e^{\left (-3\right )}}{60 \, {\left (x e + d\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)/(e*x+d)^6,x, algorithm="giac")

[Out]

-1/60*(20*B*b*x^2*e^2 + 10*B*b*d*x*e + 2*B*b*d^2 + 15*B*a*x*e^2 + 15*A*b*x*e^2 + 3*B*a*d*e + 3*A*b*d*e + 12*A*
a*e^2)*e^(-3)/(x*e + d)^5

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maple [A]  time = 0.00, size = 79, normalized size = 1.03 \[ -\frac {B b}{3 \left (e x +d \right )^{3} e^{3}}-\frac {A b e +B a e -2 B b d}{4 \left (e x +d \right )^{4} e^{3}}-\frac {A a \,e^{2}-A d b e -B d a e +B b \,d^{2}}{5 \left (e x +d \right )^{5} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(B*x+A)/(e*x+d)^6,x)

[Out]

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

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maxima [A]  time = 0.55, size = 117, normalized size = 1.52 \[ -\frac {20 \, B b e^{2} x^{2} + 2 \, B b d^{2} + 12 \, A a e^{2} + 3 \, {\left (B a + A b\right )} d e + 5 \, {\left (2 \, B b d e + 3 \, {\left (B a + A b\right )} e^{2}\right )} x}{60 \, {\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)/(e*x+d)^6,x, algorithm="maxima")

[Out]

-1/60*(20*B*b*e^2*x^2 + 2*B*b*d^2 + 12*A*a*e^2 + 3*(B*a + A*b)*d*e + 5*(2*B*b*d*e + 3*(B*a + A*b)*e^2)*x)/(e^8
*x^5 + 5*d*e^7*x^4 + 10*d^2*e^6*x^3 + 10*d^3*e^5*x^2 + 5*d^4*e^4*x + d^5*e^3)

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mupad [B]  time = 1.07, size = 118, normalized size = 1.53 \[ -\frac {\frac {12\,A\,a\,e^2+2\,B\,b\,d^2+3\,A\,b\,d\,e+3\,B\,a\,d\,e}{60\,e^3}+\frac {x\,\left (3\,A\,b\,e+3\,B\,a\,e+2\,B\,b\,d\right )}{12\,e^2}+\frac {B\,b\,x^2}{3\,e}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x))/(d + e*x)^6,x)

[Out]

-((12*A*a*e^2 + 2*B*b*d^2 + 3*A*b*d*e + 3*B*a*d*e)/(60*e^3) + (x*(3*A*b*e + 3*B*a*e + 2*B*b*d))/(12*e^2) + (B*
b*x^2)/(3*e))/(d^5 + e^5*x^5 + 5*d*e^4*x^4 + 10*d^3*e^2*x^2 + 10*d^2*e^3*x^3 + 5*d^4*e*x)

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sympy [A]  time = 4.74, size = 134, normalized size = 1.74 \[ \frac {- 12 A a e^{2} - 3 A b d e - 3 B a d e - 2 B b d^{2} - 20 B b e^{2} x^{2} + x \left (- 15 A b e^{2} - 15 B a e^{2} - 10 B b d e\right )}{60 d^{5} e^{3} + 300 d^{4} e^{4} x + 600 d^{3} e^{5} x^{2} + 600 d^{2} e^{6} x^{3} + 300 d e^{7} x^{4} + 60 e^{8} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)/(e*x+d)**6,x)

[Out]

(-12*A*a*e**2 - 3*A*b*d*e - 3*B*a*d*e - 2*B*b*d**2 - 20*B*b*e**2*x**2 + x*(-15*A*b*e**2 - 15*B*a*e**2 - 10*B*b
*d*e))/(60*d**5*e**3 + 300*d**4*e**4*x + 600*d**3*e**5*x**2 + 600*d**2*e**6*x**3 + 300*d*e**7*x**4 + 60*e**8*x
**5)

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