3.1032 \(\int \frac{(a+b x)^2 (A+B x)}{(d+e x)^6} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0935494, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ \frac{b (-2 a B e-A b e+3 b B d)}{3 e^4 (d+e x)^3}-\frac{(b d-a e) (-a B e-2 A b e+3 b B d)}{4 e^4 (d+e x)^4}+\frac{(b d-a e)^2 (B d-A e)}{5 e^4 (d+e x)^5}-\frac{b^2 B}{2 e^4 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^2*(A + B*x))/(d + e*x)^6,x]

[Out]

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

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

Mathematica [A]  time = 0.0584187, size = 129, normalized size = 1.08 \[ -\frac{3 a^2 e^2 (4 A e+B (d+5 e x))+2 a b e \left (3 A e (d+5 e x)+2 B \left (d^2+5 d e x+10 e^2 x^2\right )\right )+b^2 \left (2 A e \left (d^2+5 d e x+10 e^2 x^2\right )+3 B \left (5 d^2 e x+d^3+10 d e^2 x^2+10 e^3 x^3\right )\right )}{60 e^4 (d+e x)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^2*(A + B*x))/(d + e*x)^6,x]

[Out]

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

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Maple [A]  time = 0.006, size = 166, normalized size = 1.4 \begin{align*} -{\frac{b \left ( Abe+2\,Bae-3\,Bbd \right ) }{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{B{b}^{2}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{{a}^{2}A{e}^{3}-2\,Adab{e}^{2}+A{d}^{2}{b}^{2}e-Bd{a}^{2}{e}^{2}+2\,B{d}^{2}abe-{b}^{2}B{d}^{3}}{5\,{e}^{4} \left ( ex+d \right ) ^{5}}}-{\frac{2\,Aba{e}^{2}-2\,Ad{b}^{2}e+B{a}^{2}{e}^{2}-4\,Bdabe+3\,{b}^{2}B{d}^{2}}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2*(B*x+A)/(e*x+d)^6,x)

[Out]

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

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Maxima [A]  time = 1.24212, size = 274, normalized size = 2.28 \begin{align*} -\frac{30 \, B b^{2} e^{3} x^{3} + 3 \, B b^{2} d^{3} + 12 \, A a^{2} e^{3} + 2 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e + 3 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 10 \,{\left (3 \, B b^{2} d e^{2} + 2 \,{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 5 \,{\left (3 \, B b^{2} d^{2} e + 2 \,{\left (2 \, B a b + A b^{2}\right )} d e^{2} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{60 \,{\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)/(e*x+d)^6,x, algorithm="maxima")

[Out]

-1/60*(30*B*b^2*e^3*x^3 + 3*B*b^2*d^3 + 12*A*a^2*e^3 + 2*(2*B*a*b + A*b^2)*d^2*e + 3*(B*a^2 + 2*A*a*b)*d*e^2 +
10*(3*B*b^2*d*e^2 + 2*(2*B*a*b + A*b^2)*e^3)*x^2 + 5*(3*B*b^2*d^2*e + 2*(2*B*a*b + A*b^2)*d*e^2 + 3*(B*a^2 +
2*A*a*b)*e^3)*x)/(e^9*x^5 + 5*d*e^8*x^4 + 10*d^2*e^7*x^3 + 10*d^3*e^6*x^2 + 5*d^4*e^5*x + d^5*e^4)

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Fricas [A]  time = 1.74552, size = 432, normalized size = 3.6 \begin{align*} -\frac{30 \, B b^{2} e^{3} x^{3} + 3 \, B b^{2} d^{3} + 12 \, A a^{2} e^{3} + 2 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e + 3 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 10 \,{\left (3 \, B b^{2} d e^{2} + 2 \,{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 5 \,{\left (3 \, B b^{2} d^{2} e + 2 \,{\left (2 \, B a b + A b^{2}\right )} d e^{2} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{60 \,{\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)/(e*x+d)^6,x, algorithm="fricas")

[Out]

-1/60*(30*B*b^2*e^3*x^3 + 3*B*b^2*d^3 + 12*A*a^2*e^3 + 2*(2*B*a*b + A*b^2)*d^2*e + 3*(B*a^2 + 2*A*a*b)*d*e^2 +
10*(3*B*b^2*d*e^2 + 2*(2*B*a*b + A*b^2)*e^3)*x^2 + 5*(3*B*b^2*d^2*e + 2*(2*B*a*b + A*b^2)*d*e^2 + 3*(B*a^2 +
2*A*a*b)*e^3)*x)/(e^9*x^5 + 5*d*e^8*x^4 + 10*d^2*e^7*x^3 + 10*d^3*e^6*x^2 + 5*d^4*e^5*x + d^5*e^4)

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Sympy [B]  time = 44.4286, size = 236, normalized size = 1.97 \begin{align*} - \frac{12 A a^{2} e^{3} + 6 A a b d e^{2} + 2 A b^{2} d^{2} e + 3 B a^{2} d e^{2} + 4 B a b d^{2} e + 3 B b^{2} d^{3} + 30 B b^{2} e^{3} x^{3} + x^{2} \left (20 A b^{2} e^{3} + 40 B a b e^{3} + 30 B b^{2} d e^{2}\right ) + x \left (30 A a b e^{3} + 10 A b^{2} d e^{2} + 15 B a^{2} e^{3} + 20 B a b d e^{2} + 15 B b^{2} d^{2} e\right )}{60 d^{5} e^{4} + 300 d^{4} e^{5} x + 600 d^{3} e^{6} x^{2} + 600 d^{2} e^{7} x^{3} + 300 d e^{8} x^{4} + 60 e^{9} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2*(B*x+A)/(e*x+d)**6,x)

[Out]

-(12*A*a**2*e**3 + 6*A*a*b*d*e**2 + 2*A*b**2*d**2*e + 3*B*a**2*d*e**2 + 4*B*a*b*d**2*e + 3*B*b**2*d**3 + 30*B*
b**2*e**3*x**3 + x**2*(20*A*b**2*e**3 + 40*B*a*b*e**3 + 30*B*b**2*d*e**2) + x*(30*A*a*b*e**3 + 10*A*b**2*d*e**
2 + 15*B*a**2*e**3 + 20*B*a*b*d*e**2 + 15*B*b**2*d**2*e))/(60*d**5*e**4 + 300*d**4*e**5*x + 600*d**3*e**6*x**2
+ 600*d**2*e**7*x**3 + 300*d*e**8*x**4 + 60*e**9*x**5)

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Giac [A]  time = 1.6578, size = 216, normalized size = 1.8 \begin{align*} -\frac{{\left (30 \, B b^{2} x^{3} e^{3} + 30 \, B b^{2} d x^{2} e^{2} + 15 \, B b^{2} d^{2} x e + 3 \, B b^{2} d^{3} + 40 \, B a b x^{2} e^{3} + 20 \, A b^{2} x^{2} e^{3} + 20 \, B a b d x e^{2} + 10 \, A b^{2} d x e^{2} + 4 \, B a b d^{2} e + 2 \, A b^{2} d^{2} e + 15 \, B a^{2} x e^{3} + 30 \, A a b x e^{3} + 3 \, B a^{2} d e^{2} + 6 \, A a b d e^{2} + 12 \, A a^{2} e^{3}\right )} e^{\left (-4\right )}}{60 \,{\left (x e + d\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)/(e*x+d)^6,x, algorithm="giac")

[Out]

-1/60*(30*B*b^2*x^3*e^3 + 30*B*b^2*d*x^2*e^2 + 15*B*b^2*d^2*x*e + 3*B*b^2*d^3 + 40*B*a*b*x^2*e^3 + 20*A*b^2*x^
2*e^3 + 20*B*a*b*d*x*e^2 + 10*A*b^2*d*x*e^2 + 4*B*a*b*d^2*e + 2*A*b^2*d^2*e + 15*B*a^2*x*e^3 + 30*A*a*b*x*e^3
+ 3*B*a^2*d*e^2 + 6*A*a*b*d*e^2 + 12*A*a^2*e^3)*e^(-4)/(x*e + d)^5