∫ (a+bx)6(A+Bx)___________

3.1068 \(\int \frac{(a+b x)^6 (A+B x)}{(d+e x)^9} \, dx\)

Optimal. Leaf size=86 \[ \frac{(a+b x)^7 (-8 a B e+A b e+7 b B d)}{56 e (d+e x)^7 (b d-a e)^2}-\frac{(a+b x)^7 (B d-A e)}{8 e (d+e x)^8 (b d-a e)} \]

[Out]

-((B*d - A*e)*(a + b*x)^7)/(8*e*(b*d - a*e)*(d + e*x)^8) + ((7*b*B*d + A*b*e - 8*a*B*e)*(a + b*x)^7)/(56*e*(b*

d - a*e)^2*(d + e*x)^7)

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Rubi [A] time = 0.0347568, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {78, 37} \[ \frac{(a+b x)^7 (-8 a B e+A b e+7 b B d)}{56 e (d+e x)^7 (b d-a e)^2}-\frac{(a+b x)^7 (B d-A e)}{8 e (d+e x)^8 (b d-a e)} \]

Antiderivative was successfully veriﬁed.

[In]

Int[((a + b*x)^6*(A + B*x))/(d + e*x)^9,x]

[Out]

-((B*d - A*e)*(a + b*x)^7)/(8*e*(b*d - a*e)*(d + e*x)^8) + ((7*b*B*d + A*b*e - 8*a*B*e)*(a + b*x)^7)/(56*e*(b*

d - a*e)^2*(d + e*x)^7)

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)^6 (A+B x)}{(d+e x)^9} \, dx &=-\frac{(B d-A e) (a+b x)^7}{8 e (b d-a e) (d+e x)^8}+\frac{(7 b B d+A b e-8 a B e) \int \frac{(a+b x)^6}{(d+e x)^8} \, dx}{8 e (b d-a e)}\\ &=-\frac{(B d-A e) (a+b x)^7}{8 e (b d-a e) (d+e x)^8}+\frac{(7 b B d+A b e-8 a B e) (a+b x)^7}{56 e (b d-a e)^2 (d+e x)^7}\\ \end{align*}

Mathematica [B] time = 0.285688, size = 597, normalized size = 6.94 \[ -\frac{a^2 b^4 e^2 \left (3 A e \left (28 d^2 e^2 x^2+8 d^3 e x+d^4+56 d e^3 x^3+70 e^4 x^4\right )+5 B \left (28 d^3 e^2 x^2+56 d^2 e^3 x^3+8 d^4 e x+d^5+70 d e^4 x^4+56 e^5 x^5\right )\right )+4 a^3 b^3 e^3 \left (A e \left (8 d^2 e x+d^3+28 d e^2 x^2+56 e^3 x^3\right )+B \left (28 d^2 e^2 x^2+8 d^3 e x+d^4+56 d e^3 x^3+70 e^4 x^4\right )\right )+a^4 b^2 e^4 \left (5 A e \left (d^2+8 d e x+28 e^2 x^2\right )+3 B \left (8 d^2 e x+d^3+28 d e^2 x^2+56 e^3 x^3\right )\right )+2 a^5 b e^5 \left (3 A e (d+8 e x)+B \left (d^2+8 d e x+28 e^2 x^2\right )\right )+a^6 e^6 (7 A e+B (d+8 e x))+2 a b^5 e \left (A e \left (28 d^3 e^2 x^2+56 d^2 e^3 x^3+8 d^4 e x+d^5+70 d e^4 x^4+56 e^5 x^5\right )+3 B \left (28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+8 d^5 e x+d^6+56 d e^5 x^5+28 e^6 x^6\right )\right )+b^6 \left (A e \left (28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+8 d^5 e x+d^6+56 d e^5 x^5+28 e^6 x^6\right )+7 B \left (28 d^5 e^2 x^2+56 d^4 e^3 x^3+70 d^3 e^4 x^4+56 d^2 e^5 x^5+8 d^6 e x+d^7+28 d e^6 x^6+8 e^7 x^7\right )\right )}{56 e^8 (d+e x)^8} \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^9,x]

[Out]

-(a^6*e^6*(7*A*e + B*(d + 8*e*x)) + 2*a^5*b*e^5*(3*A*e*(d + 8*e*x) + B*(d^2 + 8*d*e*x + 28*e^2*x^2)) + a^4*b^2

*e^4*(5*A*e*(d^2 + 8*d*e*x + 28*e^2*x^2) + 3*B*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3)) + 4*a^3*b^3*e^3*

(A*e*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + B*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e

^4*x^4)) + a^2*b^4*e^2*(3*A*e*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4) + 5*B*(d^5 + 8*d^

4*e*x + 28*d^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*x^5)) + 2*a*b^5*e*(A*e*(d^5 + 8*d^4*e*x + 28*d

^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*x^5) + 3*B*(d^6 + 8*d^5*e*x + 28*d^4*e^2*x^2 + 56*d^3*e^3*

x^3 + 70*d^2*e^4*x^4 + 56*d*e^5*x^5 + 28*e^6*x^6)) + b^6*(A*e*(d^6 + 8*d^5*e*x + 28*d^4*e^2*x^2 + 56*d^3*e^3*x

^3 + 70*d^2*e^4*x^4 + 56*d*e^5*x^5 + 28*e^6*x^6) + 7*B*(d^7 + 8*d^6*e*x + 28*d^5*e^2*x^2 + 56*d^4*e^3*x^3 + 70

*d^3*e^4*x^4 + 56*d^2*e^5*x^5 + 28*d*e^6*x^6 + 8*e^7*x^7)))/(56*e^8*(d + e*x)^8)

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Maple [B] time = 0.009, size = 814, normalized size = 9.5 \begin{align*} -{\frac{b \left ( 5\,A{a}^{4}b{e}^{5}-20\,A{a}^{3}{b}^{2}d{e}^{4}+30\,A{a}^{2}{b}^{3}{d}^{2}{e}^{3}-20\,Aa{b}^{4}{d}^{3}{e}^{2}+5\,A{b}^{5}{d}^{4}e+2\,B{a}^{5}{e}^{5}-15\,B{a}^{4}bd{e}^{4}+40\,B{a}^{3}{b}^{2}{d}^{2}{e}^{3}-50\,B{a}^{2}{b}^{3}{d}^{3}{e}^{2}+30\,Ba{b}^{4}{d}^{4}e-7\,B{b}^{5}{d}^{5} \right ) }{2\,{e}^{8} \left ( ex+d \right ) ^{6}}}-{\frac{{b}^{4} \left ( 2\,Aab{e}^{2}-2\,A{b}^{2}de+5\,B{a}^{2}{e}^{2}-12\,Babde+7\,{b}^{2}B{d}^{2} \right ) }{{e}^{8} \left ( ex+d \right ) ^{3}}}-{\frac{6\,{a}^{5}bA{e}^{6}-30\,Ad{a}^{4}{b}^{2}{e}^{5}+60\,A{d}^{2}{a}^{3}{b}^{3}{e}^{4}-60\,A{d}^{3}{a}^{2}{b}^{4}{e}^{3}+30\,A{d}^{4}a{b}^{5}{e}^{2}-6\,A{d}^{5}{b}^{6}e+B{a}^{6}{e}^{6}-12\,Bd{a}^{5}b{e}^{5}+45\,B{d}^{2}{a}^{4}{b}^{2}{e}^{4}-80\,B{d}^{3}{a}^{3}{b}^{3}{e}^{3}+75\,B{d}^{4}{a}^{2}{b}^{4}{e}^{2}-36\,B{d}^{5}a{b}^{5}e+7\,{b}^{6}B{d}^{6}}{7\,{e}^{8} \left ( ex+d \right ) ^{7}}}-{\frac{{b}^{5} \left ( Abe+6\,Bae-7\,Bbd \right ) }{2\,{e}^{8} \left ( ex+d \right ) ^{2}}}-{\frac{B{b}^{6}}{{e}^{8} \left ( ex+d \right ) }}-{\frac{{b}^{2} \left ( 4\,A{a}^{3}b{e}^{4}-12\,A{a}^{2}{b}^{2}d{e}^{3}+12\,Aa{b}^{3}{d}^{2}{e}^{2}-4\,A{b}^{4}{d}^{3}e+3\,B{a}^{4}{e}^{4}-16\,B{a}^{3}bd{e}^{3}+30\,B{a}^{2}{b}^{2}{d}^{2}{e}^{2}-24\,Ba{b}^{3}{d}^{3}e+7\,B{b}^{4}{d}^{4} \right ) }{{e}^{8} \left ( ex+d \right ) ^{5}}}-{\frac{{a}^{6}A{e}^{7}-6\,Ad{a}^{5}b{e}^{6}+15\,A{d}^{2}{a}^{4}{b}^{2}{e}^{5}-20\,A{d}^{3}{a}^{3}{b}^{3}{e}^{4}+15\,A{d}^{4}{a}^{2}{b}^{4}{e}^{3}-6\,A{d}^{5}a{b}^{5}{e}^{2}+A{d}^{6}{b}^{6}e-Bd{a}^{6}{e}^{6}+6\,B{d}^{2}{a}^{5}b{e}^{5}-15\,B{d}^{3}{a}^{4}{b}^{2}{e}^{4}+20\,B{d}^{4}{a}^{3}{b}^{3}{e}^{3}-15\,B{d}^{5}{a}^{2}{b}^{4}{e}^{2}+6\,B{d}^{6}a{b}^{5}e-{b}^{6}B{d}^{7}}{8\,{e}^{8} \left ( ex+d \right ) ^{8}}}-{\frac{5\,{b}^{3} \left ( 3\,A{a}^{2}b{e}^{3}-6\,Aa{b}^{2}d{e}^{2}+3\,A{b}^{3}{d}^{2}e+4\,B{a}^{3}{e}^{3}-15\,B{a}^{2}bd{e}^{2}+18\,Ba{b}^{2}{d}^{2}e-7\,{b}^{3}B{d}^{3} \right ) }{4\,{e}^{8} \left ( ex+d \right ) ^{4}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^6*(B*x+A)/(e*x+d)^9,x)

[Out]

-1/2*b*(5*A*a^4*b*e^5-20*A*a^3*b^2*d*e^4+30*A*a^2*b^3*d^2*e^3-20*A*a*b^4*d^3*e^2+5*A*b^5*d^4*e+2*B*a^5*e^5-15*

B*a^4*b*d*e^4+40*B*a^3*b^2*d^2*e^3-50*B*a^2*b^3*d^3*e^2+30*B*a*b^4*d^4*e-7*B*b^5*d^5)/e^8/(e*x+d)^6-b^4*(2*A*a

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

0*A*a^3*b^3*d^2*e^4-60*A*a^2*b^4*d^3*e^3+30*A*a*b^5*d^4*e^2-6*A*b^6*d^5*e+B*a^6*e^6-12*B*a^5*b*d*e^5+45*B*a^4*

b^2*d^2*e^4-80*B*a^3*b^3*d^3*e^3+75*B*a^2*b^4*d^4*e^2-36*B*a*b^5*d^5*e+7*B*b^6*d^6)/e^8/(e*x+d)^7-1/2*b^5*(A*b

*e+6*B*a*e-7*B*b*d)/e^8/(e*x+d)^2-B*b^6/e^8/(e*x+d)-b^2*(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+3*B*a^4*e^4-16*B*a^3*b*d*e^3+30*B*a^2*b^2*d^2*e^2-24*B*a*b^3*d^3*e+7*B*b^4*d^4)/e^8/(e*x+d)^5-1/8

*(A*a^6*e^7-6*A*a^5*b*d*e^6+15*A*a^4*b^2*d^2*e^5-20*A*a^3*b^3*d^3*e^4+15*A*a^2*b^4*d^4*e^3-6*A*a*b^5*d^5*e^2+A

*b^6*d^6*e-B*a^6*d*e^6+6*B*a^5*b*d^2*e^5-15*B*a^4*b^2*d^3*e^4+20*B*a^3*b^3*d^4*e^3-15*B*a^2*b^4*d^5*e^2+6*B*a*

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

*d*e^2+18*B*a*b^2*d^2*e-7*B*b^3*d^3)/e^8/(e*x+d)^4

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Maxima [B] time = 1.28023, size = 1111, normalized size = 12.92 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^9,x, algorithm="maxima")

[Out]

-1/56*(56*B*b^6*e^7*x^7 + 7*B*b^6*d^7 + 7*A*a^6*e^7 + (6*B*a*b^5 + A*b^6)*d^6*e + (5*B*a^2*b^4 + 2*A*a*b^5)*d^

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

d^2*e^5 + (B*a^6 + 6*A*a^5*b)*d*e^6 + 28*(7*B*b^6*d*e^6 + (6*B*a*b^5 + A*b^6)*e^7)*x^6 + 56*(7*B*b^6*d^2*e^5 +

(6*B*a*b^5 + A*b^6)*d*e^6 + (5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 + 70*(7*B*b^6*d^3*e^4 + (6*B*a*b^5 + A*b^6)*d^

2*e^5 + (5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 + (4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 56*(7*B*b^6*d^4*e^3 + (6*B*a*

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

4*A*a^3*b^3)*e^7)*x^3 + 28*(7*B*b^6*d^5*e^2 + (6*B*a*b^5 + A*b^6)*d^4*e^3 + (5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4

+ (4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 + (2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2

+ 8*(7*B*b^6*d^6*e + (6*B*a*b^5 + A*b^6)*d^5*e^2 + (5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 + (4*B*a^3*b^3 + 3*A*a^2

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

7)*x)/(e^16*x^8 + 8*d*e^15*x^7 + 28*d^2*e^14*x^6 + 56*d^3*e^13*x^5 + 70*d^4*e^12*x^4 + 56*d^5*e^11*x^3 + 28*d^

6*e^10*x^2 + 8*d^7*e^9*x + d^8*e^8)

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Fricas [B] time = 2.04363, size = 1683, normalized size = 19.57 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^9,x, algorithm="fricas")