3.1687 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^2}{(d+e x)^7} \, dx\)

Optimal. Leaf size=86 \[ \frac{(a+b x)^5 (-6 a B e+A b e+5 b B d)}{30 e (d+e x)^5 (b d-a e)^2}-\frac{(a+b x)^5 (B d-A e)}{6 e (d+e x)^6 (b d-a e)} \]

[Out]

-((B*d - A*e)*(a + b*x)^5)/(6*e*(b*d - a*e)*(d + e*x)^6) + ((5*b*B*d + A*b*e - 6*a*B*e)*(a + b*x)^5)/(30*e*(b*
d - a*e)^2*(d + e*x)^5)

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Rubi [A]  time = 0.0336874, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {27, 78, 37} \[ \frac{(a+b x)^5 (-6 a B e+A b e+5 b B d)}{30 e (d+e x)^5 (b d-a e)^2}-\frac{(a+b x)^5 (B d-A e)}{6 e (d+e x)^6 (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((B*d - A*e)*(a + b*x)^5)/(6*e*(b*d - a*e)*(d + e*x)^6) + ((5*b*B*d + A*b*e - 6*a*B*e)*(a + b*x)^5)/(30*e*(b*
d - a*e)^2*(d + e*x)^5)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^7} \, dx &=\int \frac{(a+b x)^4 (A+B x)}{(d+e x)^7} \, dx\\ &=-\frac{(B d-A e) (a+b x)^5}{6 e (b d-a e) (d+e x)^6}+\frac{(5 b B d+A b e-6 a B e) \int \frac{(a+b x)^4}{(d+e x)^6} \, dx}{6 e (b d-a e)}\\ &=-\frac{(B d-A e) (a+b x)^5}{6 e (b d-a e) (d+e x)^6}+\frac{(5 b B d+A b e-6 a B e) (a+b x)^5}{30 e (b d-a e)^2 (d+e x)^5}\\ \end{align*}

Mathematica [B]  time = 0.141783, size = 317, normalized size = 3.69 \[ -\frac{3 a^2 b^2 e^2 \left (A e \left (d^2+6 d e x+15 e^2 x^2\right )+B \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )\right )+2 a^3 b e^3 \left (2 A e (d+6 e x)+B \left (d^2+6 d e x+15 e^2 x^2\right )\right )+a^4 e^4 (5 A e+B (d+6 e x))+2 a b^3 e \left (A e \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )+2 B \left (15 d^2 e^2 x^2+6 d^3 e x+d^4+20 d e^3 x^3+15 e^4 x^4\right )\right )+b^4 \left (A e \left (15 d^2 e^2 x^2+6 d^3 e x+d^4+20 d e^3 x^3+15 e^4 x^4\right )+5 B \left (15 d^3 e^2 x^2+20 d^2 e^3 x^3+6 d^4 e x+d^5+15 d e^4 x^4+6 e^5 x^5\right )\right )}{30 e^6 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.007, size = 430, normalized size = 5. \begin{align*} -{\frac{{b}^{3} \left ( Abe+4\,aBe-5\,Bbd \right ) }{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}-{\frac{b \left ( 3\,A{a}^{2}b{e}^{3}-6\,Aa{b}^{2}d{e}^{2}+3\,A{b}^{3}{d}^{2}e+2\,B{e}^{3}{a}^{3}-9\,B{a}^{2}bd{e}^{2}+12\,Ba{b}^{2}{d}^{2}e-5\,B{b}^{3}{d}^{3} \right ) }{2\,{e}^{6} \left ( ex+d \right ) ^{4}}}-{\frac{A{a}^{4}{e}^{5}-4\,Ad{a}^{3}b{e}^{4}+6\,A{d}^{2}{a}^{2}{b}^{2}{e}^{3}-4\,A{d}^{3}a{b}^{3}{e}^{2}+A{d}^{4}{b}^{4}e-Bd{a}^{4}{e}^{4}+4\,B{d}^{2}{a}^{3}b{e}^{3}-6\,B{d}^{3}{a}^{2}{b}^{2}{e}^{2}+4\,B{d}^{4}a{b}^{3}e-{b}^{4}B{d}^{5}}{6\,{e}^{6} \left ( ex+d \right ) ^{6}}}-{\frac{4\,A{a}^{3}b{e}^{4}-12\,Ad{a}^{2}{b}^{2}{e}^{3}+12\,A{d}^{2}a{b}^{3}{e}^{2}-4\,A{d}^{3}{b}^{4}e+B{e}^{4}{a}^{4}-8\,Bd{a}^{3}b{e}^{3}+18\,B{d}^{2}{a}^{2}{b}^{2}{e}^{2}-16\,B{d}^{3}a{b}^{3}e+5\,{b}^{4}B{d}^{4}}{5\,{e}^{6} \left ( ex+d \right ) ^{5}}}-{\frac{{b}^{4}B}{{e}^{6} \left ( ex+d \right ) }}-{\frac{2\,{b}^{2} \left ( 2\,Aab{e}^{2}-2\,Ad{b}^{2}e+3\,{a}^{2}B{e}^{2}-8\,Bdabe+5\,B{b}^{2}{d}^{2} \right ) }{3\,{e}^{6} \left ( ex+d \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b^3*(A*b*e+4*B*a*e-5*B*b*d)/e^6/(e*x+d)^2-1/2*b*(3*A*a^2*b*e^3-6*A*a*b^2*d*e^2+3*A*b^3*d^2*e+2*B*a^3*e^3-
9*B*a^2*b*d*e^2+12*B*a*b^2*d^2*e-5*B*b^3*d^3)/e^6/(e*x+d)^4-1/6*(A*a^4*e^5-4*A*a^3*b*d*e^4+6*A*a^2*b^2*d^2*e^3
-4*A*a*b^3*d^3*e^2+A*b^4*d^4*e-B*a^4*d*e^4+4*B*a^3*b*d^2*e^3-6*B*a^2*b^2*d^3*e^2+4*B*a*b^3*d^4*e-B*b^4*d^5)/e^
6/(e*x+d)^6-1/5*(4*A*a^3*b*e^4-12*A*a^2*b^2*d*e^3+12*A*a*b^3*d^2*e^2-4*A*b^4*d^3*e+B*a^4*e^4-8*B*a^3*b*d*e^3+1
8*B*a^2*b^2*d^2*e^2-16*B*a*b^3*d^3*e+5*B*b^4*d^4)/e^6/(e*x+d)^5-b^4*B/e^6/(e*x+d)-2/3*b^2*(2*A*a*b*e^2-2*A*b^2
*d*e+3*B*a^2*e^2-8*B*a*b*d*e+5*B*b^2*d^2)/e^6/(e*x+d)^3

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Maxima [B]  time = 1.22702, size = 612, normalized size = 7.12 \begin{align*} -\frac{30 \, B b^{4} e^{5} x^{5} + 5 \, B b^{4} d^{5} + 5 \, A a^{4} e^{5} +{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} +{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} +{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 15 \,{\left (5 \, B b^{4} d e^{4} +{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 20 \,{\left (5 \, B b^{4} d^{2} e^{3} +{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 15 \,{\left (5 \, B b^{4} d^{3} e^{2} +{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} +{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 6 \,{\left (5 \, B b^{4} d^{4} e +{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} +{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} +{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{30 \,{\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(30*B*b^4*e^5*x^5 + 5*B*b^4*d^5 + 5*A*a^4*e^5 + (4*B*a*b^3 + A*b^4)*d^4*e + (3*B*a^2*b^2 + 2*A*a*b^3)*d^
3*e^2 + (2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 + (B*a^4 + 4*A*a^3*b)*d*e^4 + 15*(5*B*b^4*d*e^4 + (4*B*a*b^3 + A*b^4
)*e^5)*x^4 + 20*(5*B*b^4*d^2*e^3 + (4*B*a*b^3 + A*b^4)*d*e^4 + (3*B*a^2*b^2 + 2*A*a*b^3)*e^5)*x^3 + 15*(5*B*b^
4*d^3*e^2 + (4*B*a*b^3 + A*b^4)*d^2*e^3 + (3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 + (2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2
+ 6*(5*B*b^4*d^4*e + (4*B*a*b^3 + A*b^4)*d^3*e^2 + (3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 + (2*B*a^3*b + 3*A*a^2*b
^2)*d*e^4 + (B*a^4 + 4*A*a^3*b)*e^5)*x)/(e^12*x^6 + 6*d*e^11*x^5 + 15*d^2*e^10*x^4 + 20*d^3*e^9*x^3 + 15*d^4*e
^8*x^2 + 6*d^5*e^7*x + d^6*e^6)

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Fricas [B]  time = 1.50836, size = 936, normalized size = 10.88 \begin{align*} -\frac{30 \, B b^{4} e^{5} x^{5} + 5 \, B b^{4} d^{5} + 5 \, A a^{4} e^{5} +{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} +{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} +{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 15 \,{\left (5 \, B b^{4} d e^{4} +{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 20 \,{\left (5 \, B b^{4} d^{2} e^{3} +{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 15 \,{\left (5 \, B b^{4} d^{3} e^{2} +{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} +{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 6 \,{\left (5 \, B b^{4} d^{4} e +{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} +{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} +{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{30 \,{\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(30*B*b^4*e^5*x^5 + 5*B*b^4*d^5 + 5*A*a^4*e^5 + (4*B*a*b^3 + A*b^4)*d^4*e + (3*B*a^2*b^2 + 2*A*a*b^3)*d^
3*e^2 + (2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 + (B*a^4 + 4*A*a^3*b)*d*e^4 + 15*(5*B*b^4*d*e^4 + (4*B*a*b^3 + A*b^4
)*e^5)*x^4 + 20*(5*B*b^4*d^2*e^3 + (4*B*a*b^3 + A*b^4)*d*e^4 + (3*B*a^2*b^2 + 2*A*a*b^3)*e^5)*x^3 + 15*(5*B*b^
4*d^3*e^2 + (4*B*a*b^3 + A*b^4)*d^2*e^3 + (3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 + (2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2
+ 6*(5*B*b^4*d^4*e + (4*B*a*b^3 + A*b^4)*d^3*e^2 + (3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 + (2*B*a^3*b + 3*A*a^2*b
^2)*d*e^4 + (B*a^4 + 4*A*a^3*b)*e^5)*x)/(e^12*x^6 + 6*d*e^11*x^5 + 15*d^2*e^10*x^4 + 20*d^3*e^9*x^3 + 15*d^4*e
^8*x^2 + 6*d^5*e^7*x + d^6*e^6)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.16094, size = 591, normalized size = 6.87 \begin{align*} -\frac{{\left (30 \, B b^{4} x^{5} e^{5} + 75 \, B b^{4} d x^{4} e^{4} + 100 \, B b^{4} d^{2} x^{3} e^{3} + 75 \, B b^{4} d^{3} x^{2} e^{2} + 30 \, B b^{4} d^{4} x e + 5 \, B b^{4} d^{5} + 60 \, B a b^{3} x^{4} e^{5} + 15 \, A b^{4} x^{4} e^{5} + 80 \, B a b^{3} d x^{3} e^{4} + 20 \, A b^{4} d x^{3} e^{4} + 60 \, B a b^{3} d^{2} x^{2} e^{3} + 15 \, A b^{4} d^{2} x^{2} e^{3} + 24 \, B a b^{3} d^{3} x e^{2} + 6 \, A b^{4} d^{3} x e^{2} + 4 \, B a b^{3} d^{4} e + A b^{4} d^{4} e + 60 \, B a^{2} b^{2} x^{3} e^{5} + 40 \, A a b^{3} x^{3} e^{5} + 45 \, B a^{2} b^{2} d x^{2} e^{4} + 30 \, A a b^{3} d x^{2} e^{4} + 18 \, B a^{2} b^{2} d^{2} x e^{3} + 12 \, A a b^{3} d^{2} x e^{3} + 3 \, B a^{2} b^{2} d^{3} e^{2} + 2 \, A a b^{3} d^{3} e^{2} + 30 \, B a^{3} b x^{2} e^{5} + 45 \, A a^{2} b^{2} x^{2} e^{5} + 12 \, B a^{3} b d x e^{4} + 18 \, A a^{2} b^{2} d x e^{4} + 2 \, B a^{3} b d^{2} e^{3} + 3 \, A a^{2} b^{2} d^{2} e^{3} + 6 \, B a^{4} x e^{5} + 24 \, A a^{3} b x e^{5} + B a^{4} d e^{4} + 4 \, A a^{3} b d e^{4} + 5 \, A a^{4} e^{5}\right )} e^{\left (-6\right )}}{30 \,{\left (x e + d\right )}^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(30*B*b^4*x^5*e^5 + 75*B*b^4*d*x^4*e^4 + 100*B*b^4*d^2*x^3*e^3 + 75*B*b^4*d^3*x^2*e^2 + 30*B*b^4*d^4*x*e
+ 5*B*b^4*d^5 + 60*B*a*b^3*x^4*e^5 + 15*A*b^4*x^4*e^5 + 80*B*a*b^3*d*x^3*e^4 + 20*A*b^4*d*x^3*e^4 + 60*B*a*b^
3*d^2*x^2*e^3 + 15*A*b^4*d^2*x^2*e^3 + 24*B*a*b^3*d^3*x*e^2 + 6*A*b^4*d^3*x*e^2 + 4*B*a*b^3*d^4*e + A*b^4*d^4*
e + 60*B*a^2*b^2*x^3*e^5 + 40*A*a*b^3*x^3*e^5 + 45*B*a^2*b^2*d*x^2*e^4 + 30*A*a*b^3*d*x^2*e^4 + 18*B*a^2*b^2*d
^2*x*e^3 + 12*A*a*b^3*d^2*x*e^3 + 3*B*a^2*b^2*d^3*e^2 + 2*A*a*b^3*d^3*e^2 + 30*B*a^3*b*x^2*e^5 + 45*A*a^2*b^2*
x^2*e^5 + 12*B*a^3*b*d*x*e^4 + 18*A*a^2*b^2*d*x*e^4 + 2*B*a^3*b*d^2*e^3 + 3*A*a^2*b^2*d^2*e^3 + 6*B*a^4*x*e^5
+ 24*A*a^3*b*x*e^5 + B*a^4*d*e^4 + 4*A*a^3*b*d*e^4 + 5*A*a^4*e^5)*e^(-6)/(x*e + d)^6