3.1047 \(\int \frac{(a+b x)^3 (A+B x)}{(d+e x)^7} \, dx\)

Optimal. Leaf size=133 \[ \frac{b (a+b x)^4 (-3 a B e+A b e+2 b B d)}{60 e (d+e x)^4 (b d-a e)^3}+\frac{(a+b x)^4 (-3 a B e+A b e+2 b B d)}{15 e (d+e x)^5 (b d-a e)^2}-\frac{(a+b x)^4 (B d-A e)}{6 e (d+e x)^6 (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.0802214, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {78, 45, 37} \[ \frac{b (a+b x)^4 (-3 a B e+A b e+2 b B d)}{60 e (d+e x)^4 (b d-a e)^3}+\frac{(a+b x)^4 (-3 a B e+A b e+2 b B d)}{15 e (d+e x)^5 (b d-a e)^2}-\frac{(a+b x)^4 (B d-A e)}{6 e (d+e x)^6 (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^3*(A + B*x))/(d + e*x)^7,x]

[Out]

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

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 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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)^3 (A+B x)}{(d+e x)^7} \, dx &=-\frac{(B d-A e) (a+b x)^4}{6 e (b d-a e) (d+e x)^6}+\frac{(2 b B d+A b e-3 a B e) \int \frac{(a+b x)^3}{(d+e x)^6} \, dx}{3 e (b d-a e)}\\ &=-\frac{(B d-A e) (a+b x)^4}{6 e (b d-a e) (d+e x)^6}+\frac{(2 b B d+A b e-3 a B e) (a+b x)^4}{15 e (b d-a e)^2 (d+e x)^5}+\frac{(b (2 b B d+A b e-3 a B e)) \int \frac{(a+b x)^3}{(d+e x)^5} \, dx}{15 e (b d-a e)^2}\\ &=-\frac{(B d-A e) (a+b x)^4}{6 e (b d-a e) (d+e x)^6}+\frac{(2 b B d+A b e-3 a B e) (a+b x)^4}{15 e (b d-a e)^2 (d+e x)^5}+\frac{b (2 b B d+A b e-3 a B e) (a+b x)^4}{60 e (b d-a e)^3 (d+e x)^4}\\ \end{align*}

Mathematica [A]  time = 0.0962353, size = 211, normalized size = 1.59 \[ -\frac{3 a^2 b e^2 \left (2 A e (d+6 e x)+B \left (d^2+6 d e x+15 e^2 x^2\right )\right )+2 a^3 e^3 (5 A e+B (d+6 e x))+3 a b^2 e \left (A e \left (d^2+6 d e x+15 e^2 x^2\right )+B \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )\right )+b^3 \left (A e \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )+2 B \left (15 d^2 e^2 x^2+6 d^3 e x+d^4+20 d e^3 x^3+15 e^4 x^4\right )\right )}{60 e^5 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^3*(A + B*x))/(d + e*x)^7,x]

[Out]

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

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Maple [B]  time = 0.005, size = 281, normalized size = 2.1 \begin{align*} -{\frac{{a}^{3}A{e}^{4}-3\,Ad{a}^{2}b{e}^{3}+3\,A{d}^{2}a{b}^{2}{e}^{2}-A{d}^{3}{b}^{3}e-Bd{a}^{3}{e}^{3}+3\,B{d}^{2}{a}^{2}b{e}^{2}-3\,B{d}^{3}a{b}^{2}e+{b}^{3}B{d}^{4}}{6\,{e}^{5} \left ( ex+d \right ) ^{6}}}-{\frac{{b}^{2} \left ( Abe+3\,Bae-4\,Bbd \right ) }{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{B{b}^{3}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{3\,Ab{a}^{2}{e}^{3}-6\,Ada{b}^{2}{e}^{2}+3\,A{d}^{2}{b}^{3}e+B{a}^{3}{e}^{3}-6\,Bd{a}^{2}b{e}^{2}+9\,B{d}^{2}a{b}^{2}e-4\,{b}^{3}B{d}^{3}}{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}-{\frac{3\,b \left ( Aba{e}^{2}-A{b}^{2}de+B{a}^{2}{e}^{2}-3\,Bdabe+2\,{b}^{2}B{d}^{2} \right ) }{4\,{e}^{5} \left ( ex+d \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(B*x+A)/(e*x+d)^7,x)

[Out]

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

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Maxima [B]  time = 1.17627, size = 428, normalized size = 3.22 \begin{align*} -\frac{30 \, B b^{3} e^{4} x^{4} + 2 \, B b^{3} d^{4} + 10 \, A a^{3} e^{4} +{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 20 \,{\left (2 \, B b^{3} d e^{3} +{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 15 \,{\left (2 \, B b^{3} d^{2} e^{2} +{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 6 \,{\left (2 \, B b^{3} d^{3} e +{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{60 \,{\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^7,x, algorithm="maxima")

[Out]

-1/60*(30*B*b^3*e^4*x^4 + 2*B*b^3*d^4 + 10*A*a^3*e^4 + (3*B*a*b^2 + A*b^3)*d^3*e + 3*(B*a^2*b + A*a*b^2)*d^2*e
^2 + 2*(B*a^3 + 3*A*a^2*b)*d*e^3 + 20*(2*B*b^3*d*e^3 + (3*B*a*b^2 + A*b^3)*e^4)*x^3 + 15*(2*B*b^3*d^2*e^2 + (3
*B*a*b^2 + A*b^3)*d*e^3 + 3*(B*a^2*b + A*a*b^2)*e^4)*x^2 + 6*(2*B*b^3*d^3*e + (3*B*a*b^2 + A*b^3)*d^2*e^2 + 3*
(B*a^2*b + A*a*b^2)*d*e^3 + 2*(B*a^3 + 3*A*a^2*b)*e^4)*x)/(e^11*x^6 + 6*d*e^10*x^5 + 15*d^2*e^9*x^4 + 20*d^3*e
^8*x^3 + 15*d^4*e^7*x^2 + 6*d^5*e^6*x + d^6*e^5)

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Fricas [B]  time = 1.73926, size = 662, normalized size = 4.98 \begin{align*} -\frac{30 \, B b^{3} e^{4} x^{4} + 2 \, B b^{3} d^{4} + 10 \, A a^{3} e^{4} +{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 20 \,{\left (2 \, B b^{3} d e^{3} +{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 15 \,{\left (2 \, B b^{3} d^{2} e^{2} +{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 6 \,{\left (2 \, B b^{3} d^{3} e +{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{60 \,{\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^7,x, algorithm="fricas")

[Out]

-1/60*(30*B*b^3*e^4*x^4 + 2*B*b^3*d^4 + 10*A*a^3*e^4 + (3*B*a*b^2 + A*b^3)*d^3*e + 3*(B*a^2*b + A*a*b^2)*d^2*e
^2 + 2*(B*a^3 + 3*A*a^2*b)*d*e^3 + 20*(2*B*b^3*d*e^3 + (3*B*a*b^2 + A*b^3)*e^4)*x^3 + 15*(2*B*b^3*d^2*e^2 + (3
*B*a*b^2 + A*b^3)*d*e^3 + 3*(B*a^2*b + A*a*b^2)*e^4)*x^2 + 6*(2*B*b^3*d^3*e + (3*B*a*b^2 + A*b^3)*d^2*e^2 + 3*
(B*a^2*b + A*a*b^2)*d*e^3 + 2*(B*a^3 + 3*A*a^2*b)*e^4)*x)/(e^11*x^6 + 6*d*e^10*x^5 + 15*d^2*e^9*x^4 + 20*d^3*e
^8*x^3 + 15*d^4*e^7*x^2 + 6*d^5*e^6*x + d^6*e^5)

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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)**3*(B*x+A)/(e*x+d)**7,x)

[Out]

Timed out

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Giac [B]  time = 3.02076, size = 381, normalized size = 2.86 \begin{align*} -\frac{{\left (30 \, B b^{3} x^{4} e^{4} + 40 \, B b^{3} d x^{3} e^{3} + 30 \, B b^{3} d^{2} x^{2} e^{2} + 12 \, B b^{3} d^{3} x e + 2 \, B b^{3} d^{4} + 60 \, B a b^{2} x^{3} e^{4} + 20 \, A b^{3} x^{3} e^{4} + 45 \, B a b^{2} d x^{2} e^{3} + 15 \, A b^{3} d x^{2} e^{3} + 18 \, B a b^{2} d^{2} x e^{2} + 6 \, A b^{3} d^{2} x e^{2} + 3 \, B a b^{2} d^{3} e + A b^{3} d^{3} e + 45 \, B a^{2} b x^{2} e^{4} + 45 \, A a b^{2} x^{2} e^{4} + 18 \, B a^{2} b d x e^{3} + 18 \, A a b^{2} d x e^{3} + 3 \, B a^{2} b d^{2} e^{2} + 3 \, A a b^{2} d^{2} e^{2} + 12 \, B a^{3} x e^{4} + 36 \, A a^{2} b x e^{4} + 2 \, B a^{3} d e^{3} + 6 \, A a^{2} b d e^{3} + 10 \, A a^{3} e^{4}\right )} e^{\left (-5\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)^3*(B*x+A)/(e*x+d)^7,x, algorithm="giac")

[Out]

-1/60*(30*B*b^3*x^4*e^4 + 40*B*b^3*d*x^3*e^3 + 30*B*b^3*d^2*x^2*e^2 + 12*B*b^3*d^3*x*e + 2*B*b^3*d^4 + 60*B*a*
b^2*x^3*e^4 + 20*A*b^3*x^3*e^4 + 45*B*a*b^2*d*x^2*e^3 + 15*A*b^3*d*x^2*e^3 + 18*B*a*b^2*d^2*x*e^2 + 6*A*b^3*d^
2*x*e^2 + 3*B*a*b^2*d^3*e + A*b^3*d^3*e + 45*B*a^2*b*x^2*e^4 + 45*A*a*b^2*x^2*e^4 + 18*B*a^2*b*d*x*e^3 + 18*A*
a*b^2*d*x*e^3 + 3*B*a^2*b*d^2*e^2 + 3*A*a*b^2*d^2*e^2 + 12*B*a^3*x*e^4 + 36*A*a^2*b*x*e^4 + 2*B*a^3*d*e^3 + 6*
A*a^2*b*d*e^3 + 10*A*a^3*e^4)*e^(-5)/(x*e + d)^6