3.1071 \(\int \frac{(a+b x)^6 (A+B x)}{(d+e x)^{12}} \, dx\)

Optimal. Leaf size=235 \[ \frac{b^3 (a+b x)^7 (-11 a B e+4 A b e+7 b B d)}{9240 e (d+e x)^7 (b d-a e)^5}+\frac{b^2 (a+b x)^7 (-11 a B e+4 A b e+7 b B d)}{1320 e (d+e x)^8 (b d-a e)^4}+\frac{b (a+b x)^7 (-11 a B e+4 A b e+7 b B d)}{330 e (d+e x)^9 (b d-a e)^3}+\frac{(a+b x)^7 (-11 a B e+4 A b e+7 b B d)}{110 e (d+e x)^{10} (b d-a e)^2}-\frac{(a+b x)^7 (B d-A e)}{11 e (d+e x)^{11} (b d-a e)} \]

[Out]

-((B*d - A*e)*(a + b*x)^7)/(11*e*(b*d - a*e)*(d + e*x)^11) + ((7*b*B*d + 4*A*b*e - 11*a*B*e)*(a + b*x)^7)/(110
*e*(b*d - a*e)^2*(d + e*x)^10) + (b*(7*b*B*d + 4*A*b*e - 11*a*B*e)*(a + b*x)^7)/(330*e*(b*d - a*e)^3*(d + e*x)
^9) + (b^2*(7*b*B*d + 4*A*b*e - 11*a*B*e)*(a + b*x)^7)/(1320*e*(b*d - a*e)^4*(d + e*x)^8) + (b^3*(7*b*B*d + 4*
A*b*e - 11*a*B*e)*(a + b*x)^7)/(9240*e*(b*d - a*e)^5*(d + e*x)^7)

________________________________________________________________________________________

Rubi [A]  time = 0.109263, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {78, 45, 37} \[ \frac{b^3 (a+b x)^7 (-11 a B e+4 A b e+7 b B d)}{9240 e (d+e x)^7 (b d-a e)^5}+\frac{b^2 (a+b x)^7 (-11 a B e+4 A b e+7 b B d)}{1320 e (d+e x)^8 (b d-a e)^4}+\frac{b (a+b x)^7 (-11 a B e+4 A b e+7 b B d)}{330 e (d+e x)^9 (b d-a e)^3}+\frac{(a+b x)^7 (-11 a B e+4 A b e+7 b B d)}{110 e (d+e x)^{10} (b d-a e)^2}-\frac{(a+b x)^7 (B d-A e)}{11 e (d+e x)^{11} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^6*(A + B*x))/(d + e*x)^12,x]

[Out]

-((B*d - A*e)*(a + b*x)^7)/(11*e*(b*d - a*e)*(d + e*x)^11) + ((7*b*B*d + 4*A*b*e - 11*a*B*e)*(a + b*x)^7)/(110
*e*(b*d - a*e)^2*(d + e*x)^10) + (b*(7*b*B*d + 4*A*b*e - 11*a*B*e)*(a + b*x)^7)/(330*e*(b*d - a*e)^3*(d + e*x)
^9) + (b^2*(7*b*B*d + 4*A*b*e - 11*a*B*e)*(a + b*x)^7)/(1320*e*(b*d - a*e)^4*(d + e*x)^8) + (b^3*(7*b*B*d + 4*
A*b*e - 11*a*B*e)*(a + b*x)^7)/(9240*e*(b*d - a*e)^5*(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 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)^6 (A+B x)}{(d+e x)^{12}} \, dx &=-\frac{(B d-A e) (a+b x)^7}{11 e (b d-a e) (d+e x)^{11}}+\frac{(7 b B d+4 A b e-11 a B e) \int \frac{(a+b x)^6}{(d+e x)^{11}} \, dx}{11 e (b d-a e)}\\ &=-\frac{(B d-A e) (a+b x)^7}{11 e (b d-a e) (d+e x)^{11}}+\frac{(7 b B d+4 A b e-11 a B e) (a+b x)^7}{110 e (b d-a e)^2 (d+e x)^{10}}+\frac{(3 b (7 b B d+4 A b e-11 a B e)) \int \frac{(a+b x)^6}{(d+e x)^{10}} \, dx}{110 e (b d-a e)^2}\\ &=-\frac{(B d-A e) (a+b x)^7}{11 e (b d-a e) (d+e x)^{11}}+\frac{(7 b B d+4 A b e-11 a B e) (a+b x)^7}{110 e (b d-a e)^2 (d+e x)^{10}}+\frac{b (7 b B d+4 A b e-11 a B e) (a+b x)^7}{330 e (b d-a e)^3 (d+e x)^9}+\frac{\left (b^2 (7 b B d+4 A b e-11 a B e)\right ) \int \frac{(a+b x)^6}{(d+e x)^9} \, dx}{165 e (b d-a e)^3}\\ &=-\frac{(B d-A e) (a+b x)^7}{11 e (b d-a e) (d+e x)^{11}}+\frac{(7 b B d+4 A b e-11 a B e) (a+b x)^7}{110 e (b d-a e)^2 (d+e x)^{10}}+\frac{b (7 b B d+4 A b e-11 a B e) (a+b x)^7}{330 e (b d-a e)^3 (d+e x)^9}+\frac{b^2 (7 b B d+4 A b e-11 a B e) (a+b x)^7}{1320 e (b d-a e)^4 (d+e x)^8}+\frac{\left (b^3 (7 b B d+4 A b e-11 a B e)\right ) \int \frac{(a+b x)^6}{(d+e x)^8} \, dx}{1320 e (b d-a e)^4}\\ &=-\frac{(B d-A e) (a+b x)^7}{11 e (b d-a e) (d+e x)^{11}}+\frac{(7 b B d+4 A b e-11 a B e) (a+b x)^7}{110 e (b d-a e)^2 (d+e x)^{10}}+\frac{b (7 b B d+4 A b e-11 a B e) (a+b x)^7}{330 e (b d-a e)^3 (d+e x)^9}+\frac{b^2 (7 b B d+4 A b e-11 a B e) (a+b x)^7}{1320 e (b d-a e)^4 (d+e x)^8}+\frac{b^3 (7 b B d+4 A b e-11 a B e) (a+b x)^7}{9240 e (b d-a e)^5 (d+e x)^7}\\ \end{align*}

Mathematica [B]  time = 0.274275, size = 605, normalized size = 2.57 \[ -\frac{10 a^2 b^4 e^2 \left (6 A e \left (55 d^2 e^2 x^2+11 d^3 e x+d^4+165 d e^3 x^3+330 e^4 x^4\right )+5 B \left (55 d^3 e^2 x^2+165 d^2 e^3 x^3+11 d^4 e x+d^5+330 d e^4 x^4+462 e^5 x^5\right )\right )+20 a^3 b^3 e^3 \left (7 A e \left (11 d^2 e x+d^3+55 d e^2 x^2+165 e^3 x^3\right )+4 B \left (55 d^2 e^2 x^2+11 d^3 e x+d^4+165 d e^3 x^3+330 e^4 x^4\right )\right )+35 a^4 b^2 e^4 \left (8 A e \left (d^2+11 d e x+55 e^2 x^2\right )+3 B \left (11 d^2 e x+d^3+55 d e^2 x^2+165 e^3 x^3\right )\right )+56 a^5 b e^5 \left (9 A e (d+11 e x)+2 B \left (d^2+11 d e x+55 e^2 x^2\right )\right )+84 a^6 e^6 (10 A e+B (d+11 e x))+4 a b^5 e \left (5 A e \left (55 d^3 e^2 x^2+165 d^2 e^3 x^3+11 d^4 e x+d^5+330 d e^4 x^4+462 e^5 x^5\right )+6 B \left (55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+11 d^5 e x+d^6+462 d e^5 x^5+462 e^6 x^6\right )\right )+b^6 \left (4 A e \left (55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+11 d^5 e x+d^6+462 d e^5 x^5+462 e^6 x^6\right )+7 B \left (55 d^5 e^2 x^2+165 d^4 e^3 x^3+330 d^3 e^4 x^4+462 d^2 e^5 x^5+11 d^6 e x+d^7+462 d e^6 x^6+330 e^7 x^7\right )\right )}{9240 e^8 (d+e x)^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^12,x]

[Out]

-(84*a^6*e^6*(10*A*e + B*(d + 11*e*x)) + 56*a^5*b*e^5*(9*A*e*(d + 11*e*x) + 2*B*(d^2 + 11*d*e*x + 55*e^2*x^2))
+ 35*a^4*b^2*e^4*(8*A*e*(d^2 + 11*d*e*x + 55*e^2*x^2) + 3*B*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e^3*x^3))
+ 20*a^3*b^3*e^3*(7*A*e*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e^3*x^3) + 4*B*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x
^2 + 165*d*e^3*x^3 + 330*e^4*x^4)) + 10*a^2*b^4*e^2*(6*A*e*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x^2 + 165*d*e^3*x^3
+ 330*e^4*x^4) + 5*B*(d^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 + 165*d^2*e^3*x^3 + 330*d*e^4*x^4 + 462*e^5*x^5)) + 4*
a*b^5*e*(5*A*e*(d^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 + 165*d^2*e^3*x^3 + 330*d*e^4*x^4 + 462*e^5*x^5) + 6*B*(d^6
+ 11*d^5*e*x + 55*d^4*e^2*x^2 + 165*d^3*e^3*x^3 + 330*d^2*e^4*x^4 + 462*d*e^5*x^5 + 462*e^6*x^6)) + b^6*(4*A*e
*(d^6 + 11*d^5*e*x + 55*d^4*e^2*x^2 + 165*d^3*e^3*x^3 + 330*d^2*e^4*x^4 + 462*d*e^5*x^5 + 462*e^6*x^6) + 7*B*(
d^7 + 11*d^6*e*x + 55*d^5*e^2*x^2 + 165*d^4*e^3*x^3 + 330*d^3*e^4*x^4 + 462*d^2*e^5*x^5 + 462*d*e^6*x^6 + 330*
e^7*x^7)))/(9240*e^8*(d + e*x)^11)

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Maple [B]  time = 0.01, size = 814, normalized size = 3.5 \begin{align*} -{\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 ) }{2\,{e}^{8} \left ( ex+d \right ) ^{6}}}-{\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 ) }{7\,{e}^{8} \left ( ex+d \right ) ^{7}}}-{\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 ) }{3\,{e}^{8} \left ( ex+d \right ) ^{9}}}-{\frac{{b}^{5} \left ( Abe+6\,Bae-7\,Bbd \right ) }{5\,{e}^{8} \left ( ex+d \right ) ^{5}}}-{\frac{5\,{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 ) }{8\,{e}^{8} \left ( ex+d \right ) ^{8}}}-{\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}}{10\,{e}^{8} \left ( ex+d \right ) ^{10}}}-{\frac{B{b}^{6}}{4\,{e}^{8} \left ( ex+d \right ) ^{4}}}-{\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}}{11\,{e}^{8} \left ( ex+d \right ) ^{11}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^6*(B*x+A)/(e*x+d)^12,x)

[Out]

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

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Maxima [B]  time = 1.34352, size = 1192, normalized size = 5.07 \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)^12,x, algorithm="maxima")

[Out]

-1/9240*(2310*B*b^6*e^7*x^7 + 7*B*b^6*d^7 + 840*A*a^6*e^7 + 4*(6*B*a*b^5 + A*b^6)*d^6*e + 10*(5*B*a^2*b^4 + 2*
A*a*b^5)*d^5*e^2 + 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 35*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 56*(2*B*a
^5*b + 5*A*a^4*b^2)*d^2*e^5 + 84*(B*a^6 + 6*A*a^5*b)*d*e^6 + 462*(7*B*b^6*d*e^6 + 4*(6*B*a*b^5 + A*b^6)*e^7)*x
^6 + 462*(7*B*b^6*d^2*e^5 + 4*(6*B*a*b^5 + A*b^6)*d*e^6 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 + 330*(7*B*b^6
*d^3*e^4 + 4*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 + 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)
*e^7)*x^4 + 165*(7*B*b^6*d^4*e^3 + 4*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 + 20*(
4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 35*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 55*(7*B*b^6*d^5*e^2 + 4*(6*B*a*b^
5 + A*b^6)*d^4*e^3 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 + 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 35*(3*B*a
^4*b^2 + 4*A*a^3*b^3)*d*e^6 + 56*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 11*(7*B*b^6*d^6*e + 4*(6*B*a*b^5 + A*b^6
)*d^5*e^2 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 + 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 35*(3*B*a^4*b^2 +
4*A*a^3*b^3)*d^2*e^5 + 56*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + 84*(B*a^6 + 6*A*a^5*b)*e^7)*x)/(e^19*x^11 + 11*d*e
^18*x^10 + 55*d^2*e^17*x^9 + 165*d^3*e^16*x^8 + 330*d^4*e^15*x^7 + 462*d^5*e^14*x^6 + 462*d^6*e^13*x^5 + 330*d
^7*e^12*x^4 + 165*d^8*e^11*x^3 + 55*d^9*e^10*x^2 + 11*d^10*e^9*x + d^11*e^8)

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Fricas [B]  time = 1.82459, size = 1887, normalized size = 8.03 \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)^12,x, algorithm="fricas")

[Out]

-1/9240*(2310*B*b^6*e^7*x^7 + 7*B*b^6*d^7 + 840*A*a^6*e^7 + 4*(6*B*a*b^5 + A*b^6)*d^6*e + 10*(5*B*a^2*b^4 + 2*
A*a*b^5)*d^5*e^2 + 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 35*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 56*(2*B*a
^5*b + 5*A*a^4*b^2)*d^2*e^5 + 84*(B*a^6 + 6*A*a^5*b)*d*e^6 + 462*(7*B*b^6*d*e^6 + 4*(6*B*a*b^5 + A*b^6)*e^7)*x
^6 + 462*(7*B*b^6*d^2*e^5 + 4*(6*B*a*b^5 + A*b^6)*d*e^6 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 + 330*(7*B*b^6
*d^3*e^4 + 4*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 + 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)
*e^7)*x^4 + 165*(7*B*b^6*d^4*e^3 + 4*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 + 20*(
4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 35*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 55*(7*B*b^6*d^5*e^2 + 4*(6*B*a*b^
5 + A*b^6)*d^4*e^3 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 + 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 35*(3*B*a
^4*b^2 + 4*A*a^3*b^3)*d*e^6 + 56*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 11*(7*B*b^6*d^6*e + 4*(6*B*a*b^5 + A*b^6
)*d^5*e^2 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 + 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 35*(3*B*a^4*b^2 +
4*A*a^3*b^3)*d^2*e^5 + 56*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + 84*(B*a^6 + 6*A*a^5*b)*e^7)*x)/(e^19*x^11 + 11*d*e
^18*x^10 + 55*d^2*e^17*x^9 + 165*d^3*e^16*x^8 + 330*d^4*e^15*x^7 + 462*d^5*e^14*x^6 + 462*d^6*e^13*x^5 + 330*d
^7*e^12*x^4 + 165*d^8*e^11*x^3 + 55*d^9*e^10*x^2 + 11*d^10*e^9*x + d^11*e^8)

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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)**6*(B*x+A)/(e*x+d)**12,x)

[Out]

Timed out

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Giac [B]  time = 1.54279, size = 1156, normalized size = 4.92 \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)^12,x, algorithm="giac")

[Out]

-1/9240*(2310*B*b^6*x^7*e^7 + 3234*B*b^6*d*x^6*e^6 + 3234*B*b^6*d^2*x^5*e^5 + 2310*B*b^6*d^3*x^4*e^4 + 1155*B*
b^6*d^4*x^3*e^3 + 385*B*b^6*d^5*x^2*e^2 + 77*B*b^6*d^6*x*e + 7*B*b^6*d^7 + 11088*B*a*b^5*x^6*e^7 + 1848*A*b^6*
x^6*e^7 + 11088*B*a*b^5*d*x^5*e^6 + 1848*A*b^6*d*x^5*e^6 + 7920*B*a*b^5*d^2*x^4*e^5 + 1320*A*b^6*d^2*x^4*e^5 +
3960*B*a*b^5*d^3*x^3*e^4 + 660*A*b^6*d^3*x^3*e^4 + 1320*B*a*b^5*d^4*x^2*e^3 + 220*A*b^6*d^4*x^2*e^3 + 264*B*a
*b^5*d^5*x*e^2 + 44*A*b^6*d^5*x*e^2 + 24*B*a*b^5*d^6*e + 4*A*b^6*d^6*e + 23100*B*a^2*b^4*x^5*e^7 + 9240*A*a*b^
5*x^5*e^7 + 16500*B*a^2*b^4*d*x^4*e^6 + 6600*A*a*b^5*d*x^4*e^6 + 8250*B*a^2*b^4*d^2*x^3*e^5 + 3300*A*a*b^5*d^2
*x^3*e^5 + 2750*B*a^2*b^4*d^3*x^2*e^4 + 1100*A*a*b^5*d^3*x^2*e^4 + 550*B*a^2*b^4*d^4*x*e^3 + 220*A*a*b^5*d^4*x
*e^3 + 50*B*a^2*b^4*d^5*e^2 + 20*A*a*b^5*d^5*e^2 + 26400*B*a^3*b^3*x^4*e^7 + 19800*A*a^2*b^4*x^4*e^7 + 13200*B
*a^3*b^3*d*x^3*e^6 + 9900*A*a^2*b^4*d*x^3*e^6 + 4400*B*a^3*b^3*d^2*x^2*e^5 + 3300*A*a^2*b^4*d^2*x^2*e^5 + 880*
B*a^3*b^3*d^3*x*e^4 + 660*A*a^2*b^4*d^3*x*e^4 + 80*B*a^3*b^3*d^4*e^3 + 60*A*a^2*b^4*d^4*e^3 + 17325*B*a^4*b^2*
x^3*e^7 + 23100*A*a^3*b^3*x^3*e^7 + 5775*B*a^4*b^2*d*x^2*e^6 + 7700*A*a^3*b^3*d*x^2*e^6 + 1155*B*a^4*b^2*d^2*x
*e^5 + 1540*A*a^3*b^3*d^2*x*e^5 + 105*B*a^4*b^2*d^3*e^4 + 140*A*a^3*b^3*d^3*e^4 + 6160*B*a^5*b*x^2*e^7 + 15400
*A*a^4*b^2*x^2*e^7 + 1232*B*a^5*b*d*x*e^6 + 3080*A*a^4*b^2*d*x*e^6 + 112*B*a^5*b*d^2*e^5 + 280*A*a^4*b^2*d^2*e
^5 + 924*B*a^6*x*e^7 + 5544*A*a^5*b*x*e^7 + 84*B*a^6*d*e^6 + 504*A*a^5*b*d*e^6 + 840*A*a^6*e^7)*e^(-8)/(x*e +
d)^11