3.1033 \(\int \frac{(a+b x)^2 (A+B x)}{(d+e x)^7} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0761238, 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} \[ \frac{b (-2 a B e-A b e+3 b B d)}{4 e^4 (d+e x)^4}-\frac{(b d-a e) (-a B e-2 A b e+3 b B d)}{5 e^4 (d+e x)^5}+\frac{(b d-a e)^2 (B d-A e)}{6 e^4 (d+e x)^6}-\frac{b^2 B}{3 e^4 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^2*(A + B*x))/(d + e*x)^7,x]

[Out]

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

Mathematica [A]  time = 0.0608497, size = 126, normalized size = 1.05 \[ -\frac{2 a^2 e^2 (5 A e+B (d+6 e x))+2 a b e \left (2 A e (d+6 e x)+B \left (d^2+6 d e x+15 e^2 x^2\right )\right )+b^2 \left (A e \left (d^2+6 d e x+15 e^2 x^2\right )+B \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )\right )}{60 e^4 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^2*(A + B*x))/(d + e*x)^7,x]

[Out]

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

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Maple [A]  time = 0.006, size = 166, normalized size = 1.4 \begin{align*} -{\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}}{6\,{e}^{4} \left ( ex+d \right ) ^{6}}}-{\frac{B{b}^{2}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{2\,Aba{e}^{2}-2\,Ad{b}^{2}e+B{a}^{2}{e}^{2}-4\,Bdabe+3\,{b}^{2}B{d}^{2}}{5\,{e}^{4} \left ( ex+d \right ) ^{5}}}-{\frac{b \left ( Abe+2\,Bae-3\,Bbd \right ) }{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)^7,x)

[Out]

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

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Maxima [A]  time = 1.52274, size = 281, normalized size = 2.34 \begin{align*} -\frac{20 \, B b^{2} e^{3} x^{3} + B b^{2} d^{3} + 10 \, A a^{2} e^{3} +{\left (2 \, B a b + A b^{2}\right )} d^{2} e + 2 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 15 \,{\left (B b^{2} d e^{2} +{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 6 \,{\left (B b^{2} d^{2} e +{\left (2 \, B a b + A b^{2}\right )} d e^{2} + 2 \,{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{60 \,{\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} 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)^7,x, algorithm="maxima")

[Out]

-1/60*(20*B*b^2*e^3*x^3 + B*b^2*d^3 + 10*A*a^2*e^3 + (2*B*a*b + A*b^2)*d^2*e + 2*(B*a^2 + 2*A*a*b)*d*e^2 + 15*
(B*b^2*d*e^2 + (2*B*a*b + A*b^2)*e^3)*x^2 + 6*(B*b^2*d^2*e + (2*B*a*b + A*b^2)*d*e^2 + 2*(B*a^2 + 2*A*a*b)*e^3
)*x)/(e^10*x^6 + 6*d*e^9*x^5 + 15*d^2*e^8*x^4 + 20*d^3*e^7*x^3 + 15*d^4*e^6*x^2 + 6*d^5*e^5*x + d^6*e^4)

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Fricas [A]  time = 1.7932, size = 440, normalized size = 3.67 \begin{align*} -\frac{20 \, B b^{2} e^{3} x^{3} + B b^{2} d^{3} + 10 \, A a^{2} e^{3} +{\left (2 \, B a b + A b^{2}\right )} d^{2} e + 2 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 15 \,{\left (B b^{2} d e^{2} +{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 6 \,{\left (B b^{2} d^{2} e +{\left (2 \, B a b + A b^{2}\right )} d e^{2} + 2 \,{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{60 \,{\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} 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)^7,x, algorithm="fricas")

[Out]

-1/60*(20*B*b^2*e^3*x^3 + B*b^2*d^3 + 10*A*a^2*e^3 + (2*B*a*b + A*b^2)*d^2*e + 2*(B*a^2 + 2*A*a*b)*d*e^2 + 15*
(B*b^2*d*e^2 + (2*B*a*b + A*b^2)*e^3)*x^2 + 6*(B*b^2*d^2*e + (2*B*a*b + A*b^2)*d*e^2 + 2*(B*a^2 + 2*A*a*b)*e^3
)*x)/(e^10*x^6 + 6*d*e^9*x^5 + 15*d^2*e^8*x^4 + 20*d^3*e^7*x^3 + 15*d^4*e^6*x^2 + 6*d^5*e^5*x + d^6*e^4)

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Sympy [B]  time = 90.8951, size = 245, normalized size = 2.04 \begin{align*} - \frac{10 A a^{2} e^{3} + 4 A a b d e^{2} + A b^{2} d^{2} e + 2 B a^{2} d e^{2} + 2 B a b d^{2} e + B b^{2} d^{3} + 20 B b^{2} e^{3} x^{3} + x^{2} \left (15 A b^{2} e^{3} + 30 B a b e^{3} + 15 B b^{2} d e^{2}\right ) + x \left (24 A a b e^{3} + 6 A b^{2} d e^{2} + 12 B a^{2} e^{3} + 12 B a b d e^{2} + 6 B b^{2} d^{2} e\right )}{60 d^{6} e^{4} + 360 d^{5} e^{5} x + 900 d^{4} e^{6} x^{2} + 1200 d^{3} e^{7} x^{3} + 900 d^{2} e^{8} x^{4} + 360 d e^{9} x^{5} + 60 e^{10} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2*(B*x+A)/(e*x+d)**7,x)

[Out]

-(10*A*a**2*e**3 + 4*A*a*b*d*e**2 + A*b**2*d**2*e + 2*B*a**2*d*e**2 + 2*B*a*b*d**2*e + B*b**2*d**3 + 20*B*b**2
*e**3*x**3 + x**2*(15*A*b**2*e**3 + 30*B*a*b*e**3 + 15*B*b**2*d*e**2) + x*(24*A*a*b*e**3 + 6*A*b**2*d*e**2 + 1
2*B*a**2*e**3 + 12*B*a*b*d*e**2 + 6*B*b**2*d**2*e))/(60*d**6*e**4 + 360*d**5*e**5*x + 900*d**4*e**6*x**2 + 120
0*d**3*e**7*x**3 + 900*d**2*e**8*x**4 + 360*d*e**9*x**5 + 60*e**10*x**6)

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Giac [A]  time = 2.99092, size = 213, normalized size = 1.78 \begin{align*} -\frac{{\left (20 \, B b^{2} x^{3} e^{3} + 15 \, B b^{2} d x^{2} e^{2} + 6 \, B b^{2} d^{2} x e + B b^{2} d^{3} + 30 \, B a b x^{2} e^{3} + 15 \, A b^{2} x^{2} e^{3} + 12 \, B a b d x e^{2} + 6 \, A b^{2} d x e^{2} + 2 \, B a b d^{2} e + A b^{2} d^{2} e + 12 \, B a^{2} x e^{3} + 24 \, A a b x e^{3} + 2 \, B a^{2} d e^{2} + 4 \, A a b d e^{2} + 10 \, A a^{2} e^{3}\right )} e^{\left (-4\right )}}{60 \,{\left (x e + d\right )}^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)/(e*x+d)^7,x, algorithm="giac")

[Out]

-1/60*(20*B*b^2*x^3*e^3 + 15*B*b^2*d*x^2*e^2 + 6*B*b^2*d^2*x*e + B*b^2*d^3 + 30*B*a*b*x^2*e^3 + 15*A*b^2*x^2*e
^3 + 12*B*a*b*d*x*e^2 + 6*A*b^2*d*x*e^2 + 2*B*a*b*d^2*e + A*b^2*d^2*e + 12*B*a^2*x*e^3 + 24*A*a*b*x*e^3 + 2*B*
a^2*d*e^2 + 4*A*a*b*d*e^2 + 10*A*a^2*e^3)*e^(-4)/(x*e + d)^6