3.11.67 \(\int \frac {(a+b x)^6 (A+B x)}{(d+e x)^8} \, dx\) [1067]

Optimal. Leaf size=213 \[ -\frac {(B d-A e) (a+b x)^7}{7 e (b d-a e) (d+e x)^7}-\frac {B (b d-a e)^6}{6 e^8 (d+e x)^6}+\frac {6 b B (b d-a e)^5}{5 e^8 (d+e x)^5}-\frac {15 b^2 B (b d-a e)^4}{4 e^8 (d+e x)^4}+\frac {20 b^3 B (b d-a e)^3}{3 e^8 (d+e x)^3}-\frac {15 b^4 B (b d-a e)^2}{2 e^8 (d+e x)^2}+\frac {6 b^5 B (b d-a e)}{e^8 (d+e x)}+\frac {b^6 B \log (d+e x)}{e^8} \]

[Out]

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

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Rubi [A]
time = 0.18, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {79, 45} \begin {gather*} -\frac {(a+b x)^7 (B d-A e)}{7 e (d+e x)^7 (b d-a e)}+\frac {6 b^5 B (b d-a e)}{e^8 (d+e x)}-\frac {15 b^4 B (b d-a e)^2}{2 e^8 (d+e x)^2}+\frac {20 b^3 B (b d-a e)^3}{3 e^8 (d+e x)^3}-\frac {15 b^2 B (b d-a e)^4}{4 e^8 (d+e x)^4}+\frac {6 b B (b d-a e)^5}{5 e^8 (d+e x)^5}-\frac {B (b d-a e)^6}{6 e^8 (d+e x)^6}+\frac {b^6 B \log (d+e x)}{e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^6*(A + B*x))/(d + e*x)^8,x]

[Out]

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 79

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] || L
tQ[p, n]))))

Rubi steps

\begin {align*} \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^8} \, dx &=-\frac {(B d-A e) (a+b x)^7}{7 e (b d-a e) (d+e x)^7}+\frac {B \int \frac {(a+b x)^6}{(d+e x)^7} \, dx}{e}\\ &=-\frac {(B d-A e) (a+b x)^7}{7 e (b d-a e) (d+e x)^7}+\frac {B \int \left (\frac {(-b d+a e)^6}{e^6 (d+e x)^7}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^6}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^5}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^4}+\frac {15 b^4 (b d-a e)^2}{e^6 (d+e x)^3}-\frac {6 b^5 (b d-a e)}{e^6 (d+e x)^2}+\frac {b^6}{e^6 (d+e x)}\right ) \, dx}{e}\\ &=-\frac {(B d-A e) (a+b x)^7}{7 e (b d-a e) (d+e x)^7}-\frac {B (b d-a e)^6}{6 e^8 (d+e x)^6}+\frac {6 b B (b d-a e)^5}{5 e^8 (d+e x)^5}-\frac {15 b^2 B (b d-a e)^4}{4 e^8 (d+e x)^4}+\frac {20 b^3 B (b d-a e)^3}{3 e^8 (d+e x)^3}-\frac {15 b^4 B (b d-a e)^2}{2 e^8 (d+e x)^2}+\frac {6 b^5 B (b d-a e)}{e^8 (d+e x)}+\frac {b^6 B \log (d+e x)}{e^8}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(615\) vs. \(2(213)=426\).
time = 0.32, size = 615, normalized size = 2.89 \begin {gather*} -\frac {10 a^6 e^6 (6 A e+B (d+7 e x))+12 a^5 b e^5 \left (5 A e (d+7 e x)+2 B \left (d^2+7 d e x+21 e^2 x^2\right )\right )+15 a^4 b^2 e^4 \left (4 A e \left (d^2+7 d e x+21 e^2 x^2\right )+3 B \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )\right )+20 a^3 b^3 e^3 \left (3 A e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+4 B \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )\right )+30 a^2 b^4 e^2 \left (2 A e \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+5 B \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )\right )+60 a b^5 e \left (A e \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )+6 B \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )\right )+b^6 \left (60 A e \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )-B d \left (1089 d^6+7203 d^5 e x+20139 d^4 e^2 x^2+30625 d^3 e^3 x^3+26950 d^2 e^4 x^4+13230 d e^5 x^5+2940 e^6 x^6\right )\right )-420 b^6 B (d+e x)^7 \log (d+e x)}{420 e^8 (d+e x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^8,x]

[Out]

-1/420*(10*a^6*e^6*(6*A*e + B*(d + 7*e*x)) + 12*a^5*b*e^5*(5*A*e*(d + 7*e*x) + 2*B*(d^2 + 7*d*e*x + 21*e^2*x^2
)) + 15*a^4*b^2*e^4*(4*A*e*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3*B*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3)) +
20*a^3*b^3*e^3*(3*A*e*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 4*B*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 +
35*d*e^3*x^3 + 35*e^4*x^4)) + 30*a^2*b^4*e^2*(2*A*e*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4
*x^4) + 5*B*(d^5 + 7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5)) + 60*a*b^5*e*(A*e
*(d^5 + 7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5) + 6*B*(d^6 + 7*d^5*e*x + 21*d
^4*e^2*x^2 + 35*d^3*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^5 + 7*e^6*x^6)) + b^6*(60*A*e*(d^6 + 7*d^5*e*x + 21*
d^4*e^2*x^2 + 35*d^3*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^5 + 7*e^6*x^6) - B*d*(1089*d^6 + 7203*d^5*e*x + 201
39*d^4*e^2*x^2 + 30625*d^3*e^3*x^3 + 26950*d^2*e^4*x^4 + 13230*d*e^5*x^5 + 2940*e^6*x^6)) - 420*b^6*B*(d + e*x
)^7*Log[d + e*x])/(e^8*(d + e*x)^7)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(811\) vs. \(2(201)=402\).
time = 0.07, size = 812, normalized size = 3.81 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^6*(B*x+A)/(e*x+d)^8,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 842 vs. \(2 (209) = 418\).
time = 0.35, size = 842, normalized size = 3.95 \begin {gather*} B b^{6} e^{\left (-8\right )} \log \left (x e + d\right ) + \frac {1089 \, B b^{6} d^{7} - 60 \, A a^{6} e^{7} - 60 \, {\left (6 \, B a b^{5} e + A b^{6} e\right )} d^{6} + 420 \, {\left (7 \, B b^{6} d e^{6} - 6 \, B a b^{5} e^{7} - A b^{6} e^{7}\right )} x^{6} - 30 \, {\left (5 \, B a^{2} b^{4} e^{2} + 2 \, A a b^{5} e^{2}\right )} d^{5} + 630 \, {\left (21 \, B b^{6} d^{2} e^{5} - 5 \, B a^{2} b^{4} e^{7} - 2 \, A a b^{5} e^{7} - 2 \, {\left (6 \, B a b^{5} e^{6} + A b^{6} e^{6}\right )} d\right )} x^{5} - 20 \, {\left (4 \, B a^{3} b^{3} e^{3} + 3 \, A a^{2} b^{4} e^{3}\right )} d^{4} + 350 \, {\left (77 \, B b^{6} d^{3} e^{4} - 8 \, B a^{3} b^{3} e^{7} - 6 \, A a^{2} b^{4} e^{7} - 6 \, {\left (6 \, B a b^{5} e^{5} + A b^{6} e^{5}\right )} d^{2} - 3 \, {\left (5 \, B a^{2} b^{4} e^{6} + 2 \, A a b^{5} e^{6}\right )} d\right )} x^{4} - 15 \, {\left (3 \, B a^{4} b^{2} e^{4} + 4 \, A a^{3} b^{3} e^{4}\right )} d^{3} + 175 \, {\left (175 \, B b^{6} d^{4} e^{3} - 9 \, B a^{4} b^{2} e^{7} - 12 \, A a^{3} b^{3} e^{7} - 12 \, {\left (6 \, B a b^{5} e^{4} + A b^{6} e^{4}\right )} d^{3} - 6 \, {\left (5 \, B a^{2} b^{4} e^{5} + 2 \, A a b^{5} e^{5}\right )} d^{2} - 4 \, {\left (4 \, B a^{3} b^{3} e^{6} + 3 \, A a^{2} b^{4} e^{6}\right )} d\right )} x^{3} - 12 \, {\left (2 \, B a^{5} b e^{5} + 5 \, A a^{4} b^{2} e^{5}\right )} d^{2} + 21 \, {\left (959 \, B b^{6} d^{5} e^{2} - 24 \, B a^{5} b e^{7} - 60 \, A a^{4} b^{2} e^{7} - 60 \, {\left (6 \, B a b^{5} e^{3} + A b^{6} e^{3}\right )} d^{4} - 30 \, {\left (5 \, B a^{2} b^{4} e^{4} + 2 \, A a b^{5} e^{4}\right )} d^{3} - 20 \, {\left (4 \, B a^{3} b^{3} e^{5} + 3 \, A a^{2} b^{4} e^{5}\right )} d^{2} - 15 \, {\left (3 \, B a^{4} b^{2} e^{6} + 4 \, A a^{3} b^{3} e^{6}\right )} d\right )} x^{2} - 10 \, {\left (B a^{6} e^{6} + 6 \, A a^{5} b e^{6}\right )} d + 7 \, {\left (1029 \, B b^{6} d^{6} e - 10 \, B a^{6} e^{7} - 60 \, A a^{5} b e^{7} - 60 \, {\left (6 \, B a b^{5} e^{2} + A b^{6} e^{2}\right )} d^{5} - 30 \, {\left (5 \, B a^{2} b^{4} e^{3} + 2 \, A a b^{5} e^{3}\right )} d^{4} - 20 \, {\left (4 \, B a^{3} b^{3} e^{4} + 3 \, A a^{2} b^{4} e^{4}\right )} d^{3} - 15 \, {\left (3 \, B a^{4} b^{2} e^{5} + 4 \, A a^{3} b^{3} e^{5}\right )} d^{2} - 12 \, {\left (2 \, B a^{5} b e^{6} + 5 \, A a^{4} b^{2} e^{6}\right )} d\right )} x}{420 \, {\left (x^{7} e^{15} + 7 \, d x^{6} e^{14} + 21 \, d^{2} x^{5} e^{13} + 35 \, d^{3} x^{4} e^{12} + 35 \, d^{4} x^{3} e^{11} + 21 \, d^{5} x^{2} e^{10} + 7 \, d^{6} x e^{9} + d^{7} e^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^8,x, algorithm="maxima")

[Out]

B*b^6*e^(-8)*log(x*e + d) + 1/420*(1089*B*b^6*d^7 - 60*A*a^6*e^7 - 60*(6*B*a*b^5*e + A*b^6*e)*d^6 + 420*(7*B*b
^6*d*e^6 - 6*B*a*b^5*e^7 - A*b^6*e^7)*x^6 - 30*(5*B*a^2*b^4*e^2 + 2*A*a*b^5*e^2)*d^5 + 630*(21*B*b^6*d^2*e^5 -
5*B*a^2*b^4*e^7 - 2*A*a*b^5*e^7 - 2*(6*B*a*b^5*e^6 + A*b^6*e^6)*d)*x^5 - 20*(4*B*a^3*b^3*e^3 + 3*A*a^2*b^4*e^
3)*d^4 + 350*(77*B*b^6*d^3*e^4 - 8*B*a^3*b^3*e^7 - 6*A*a^2*b^4*e^7 - 6*(6*B*a*b^5*e^5 + A*b^6*e^5)*d^2 - 3*(5*
B*a^2*b^4*e^6 + 2*A*a*b^5*e^6)*d)*x^4 - 15*(3*B*a^4*b^2*e^4 + 4*A*a^3*b^3*e^4)*d^3 + 175*(175*B*b^6*d^4*e^3 -
9*B*a^4*b^2*e^7 - 12*A*a^3*b^3*e^7 - 12*(6*B*a*b^5*e^4 + A*b^6*e^4)*d^3 - 6*(5*B*a^2*b^4*e^5 + 2*A*a*b^5*e^5)*
d^2 - 4*(4*B*a^3*b^3*e^6 + 3*A*a^2*b^4*e^6)*d)*x^3 - 12*(2*B*a^5*b*e^5 + 5*A*a^4*b^2*e^5)*d^2 + 21*(959*B*b^6*
d^5*e^2 - 24*B*a^5*b*e^7 - 60*A*a^4*b^2*e^7 - 60*(6*B*a*b^5*e^3 + A*b^6*e^3)*d^4 - 30*(5*B*a^2*b^4*e^4 + 2*A*a
*b^5*e^4)*d^3 - 20*(4*B*a^3*b^3*e^5 + 3*A*a^2*b^4*e^5)*d^2 - 15*(3*B*a^4*b^2*e^6 + 4*A*a^3*b^3*e^6)*d)*x^2 - 1
0*(B*a^6*e^6 + 6*A*a^5*b*e^6)*d + 7*(1029*B*b^6*d^6*e - 10*B*a^6*e^7 - 60*A*a^5*b*e^7 - 60*(6*B*a*b^5*e^2 + A*
b^6*e^2)*d^5 - 30*(5*B*a^2*b^4*e^3 + 2*A*a*b^5*e^3)*d^4 - 20*(4*B*a^3*b^3*e^4 + 3*A*a^2*b^4*e^4)*d^3 - 15*(3*B
*a^4*b^2*e^5 + 4*A*a^3*b^3*e^5)*d^2 - 12*(2*B*a^5*b*e^6 + 5*A*a^4*b^2*e^6)*d)*x)/(x^7*e^15 + 7*d*x^6*e^14 + 21
*d^2*x^5*e^13 + 35*d^3*x^4*e^12 + 35*d^4*x^3*e^11 + 21*d^5*x^2*e^10 + 7*d^6*x*e^9 + d^7*e^8)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 896 vs. \(2 (209) = 418\).
time = 0.87, size = 896, normalized size = 4.21 \begin {gather*} \frac {1089 \, B b^{6} d^{7} - {\left (60 \, A a^{6} + 420 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 630 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 700 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 525 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 252 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 70 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )} e^{7} + {\left (2940 \, B b^{6} d x^{6} - 1260 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d x^{5} - 1050 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d x^{4} - 700 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d x^{3} - 315 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d x^{2} - 84 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d x - 10 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d\right )} e^{6} + 3 \, {\left (4410 \, B b^{6} d^{2} x^{5} - 700 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} x^{4} - 350 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} x^{3} - 140 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} x^{2} - 35 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} x - 4 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2}\right )} e^{5} + 5 \, {\left (5390 \, B b^{6} d^{3} x^{4} - 420 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} x^{3} - 126 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} x^{2} - 28 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} x - 3 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3}\right )} e^{4} + 5 \, {\left (6125 \, B b^{6} d^{4} x^{3} - 252 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} x^{2} - 42 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} x - 4 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4}\right )} e^{3} + 3 \, {\left (6713 \, B b^{6} d^{5} x^{2} - 140 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} x - 10 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5}\right )} e^{2} + 3 \, {\left (2401 \, B b^{6} d^{6} x - 20 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6}\right )} e + 420 \, {\left (B b^{6} x^{7} e^{7} + 7 \, B b^{6} d x^{6} e^{6} + 21 \, B b^{6} d^{2} x^{5} e^{5} + 35 \, B b^{6} d^{3} x^{4} e^{4} + 35 \, B b^{6} d^{4} x^{3} e^{3} + 21 \, B b^{6} d^{5} x^{2} e^{2} + 7 \, B b^{6} d^{6} x e + B b^{6} d^{7}\right )} \log \left (x e + d\right )}{420 \, {\left (x^{7} e^{15} + 7 \, d x^{6} e^{14} + 21 \, d^{2} x^{5} e^{13} + 35 \, d^{3} x^{4} e^{12} + 35 \, d^{4} x^{3} e^{11} + 21 \, d^{5} x^{2} e^{10} + 7 \, d^{6} x e^{9} + d^{7} e^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^8,x, algorithm="fricas")

[Out]

1/420*(1089*B*b^6*d^7 - (60*A*a^6 + 420*(6*B*a*b^5 + A*b^6)*x^6 + 630*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 700*(4*B
*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 525*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 252*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 70*(B*
a^6 + 6*A*a^5*b)*x)*e^7 + (2940*B*b^6*d*x^6 - 1260*(6*B*a*b^5 + A*b^6)*d*x^5 - 1050*(5*B*a^2*b^4 + 2*A*a*b^5)*
d*x^4 - 700*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*x^3 - 315*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*x^2 - 84*(2*B*a^5*b + 5*A*a^
4*b^2)*d*x - 10*(B*a^6 + 6*A*a^5*b)*d)*e^6 + 3*(4410*B*b^6*d^2*x^5 - 700*(6*B*a*b^5 + A*b^6)*d^2*x^4 - 350*(5*
B*a^2*b^4 + 2*A*a*b^5)*d^2*x^3 - 140*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*x^2 - 35*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*
x - 4*(2*B*a^5*b + 5*A*a^4*b^2)*d^2)*e^5 + 5*(5390*B*b^6*d^3*x^4 - 420*(6*B*a*b^5 + A*b^6)*d^3*x^3 - 126*(5*B*
a^2*b^4 + 2*A*a*b^5)*d^3*x^2 - 28*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*x - 3*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3)*e^4 +
5*(6125*B*b^6*d^4*x^3 - 252*(6*B*a*b^5 + A*b^6)*d^4*x^2 - 42*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*x - 4*(4*B*a^3*b^3
+ 3*A*a^2*b^4)*d^4)*e^3 + 3*(6713*B*b^6*d^5*x^2 - 140*(6*B*a*b^5 + A*b^6)*d^5*x - 10*(5*B*a^2*b^4 + 2*A*a*b^5
)*d^5)*e^2 + 3*(2401*B*b^6*d^6*x - 20*(6*B*a*b^5 + A*b^6)*d^6)*e + 420*(B*b^6*x^7*e^7 + 7*B*b^6*d*x^6*e^6 + 21
*B*b^6*d^2*x^5*e^5 + 35*B*b^6*d^3*x^4*e^4 + 35*B*b^6*d^4*x^3*e^3 + 21*B*b^6*d^5*x^2*e^2 + 7*B*b^6*d^6*x*e + B*
b^6*d^7)*log(x*e + d))/(x^7*e^15 + 7*d*x^6*e^14 + 21*d^2*x^5*e^13 + 35*d^3*x^4*e^12 + 35*d^4*x^3*e^11 + 21*d^5
*x^2*e^10 + 7*d^6*x*e^9 + d^7*e^8)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**6*(B*x+A)/(e*x+d)**8,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 779 vs. \(2 (209) = 418\).
time = 1.90, size = 779, normalized size = 3.66 \begin {gather*} B b^{6} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {{\left (420 \, {\left (7 \, B b^{6} d e^{5} - 6 \, B a b^{5} e^{6} - A b^{6} e^{6}\right )} x^{6} + 630 \, {\left (21 \, B b^{6} d^{2} e^{4} - 12 \, B a b^{5} d e^{5} - 2 \, A b^{6} d e^{5} - 5 \, B a^{2} b^{4} e^{6} - 2 \, A a b^{5} e^{6}\right )} x^{5} + 350 \, {\left (77 \, B b^{6} d^{3} e^{3} - 36 \, B a b^{5} d^{2} e^{4} - 6 \, A b^{6} d^{2} e^{4} - 15 \, B a^{2} b^{4} d e^{5} - 6 \, A a b^{5} d e^{5} - 8 \, B a^{3} b^{3} e^{6} - 6 \, A a^{2} b^{4} e^{6}\right )} x^{4} + 175 \, {\left (175 \, B b^{6} d^{4} e^{2} - 72 \, B a b^{5} d^{3} e^{3} - 12 \, A b^{6} d^{3} e^{3} - 30 \, B a^{2} b^{4} d^{2} e^{4} - 12 \, A a b^{5} d^{2} e^{4} - 16 \, B a^{3} b^{3} d e^{5} - 12 \, A a^{2} b^{4} d e^{5} - 9 \, B a^{4} b^{2} e^{6} - 12 \, A a^{3} b^{3} e^{6}\right )} x^{3} + 21 \, {\left (959 \, B b^{6} d^{5} e - 360 \, B a b^{5} d^{4} e^{2} - 60 \, A b^{6} d^{4} e^{2} - 150 \, B a^{2} b^{4} d^{3} e^{3} - 60 \, A a b^{5} d^{3} e^{3} - 80 \, B a^{3} b^{3} d^{2} e^{4} - 60 \, A a^{2} b^{4} d^{2} e^{4} - 45 \, B a^{4} b^{2} d e^{5} - 60 \, A a^{3} b^{3} d e^{5} - 24 \, B a^{5} b e^{6} - 60 \, A a^{4} b^{2} e^{6}\right )} x^{2} + 7 \, {\left (1029 \, B b^{6} d^{6} - 360 \, B a b^{5} d^{5} e - 60 \, A b^{6} d^{5} e - 150 \, B a^{2} b^{4} d^{4} e^{2} - 60 \, A a b^{5} d^{4} e^{2} - 80 \, B a^{3} b^{3} d^{3} e^{3} - 60 \, A a^{2} b^{4} d^{3} e^{3} - 45 \, B a^{4} b^{2} d^{2} e^{4} - 60 \, A a^{3} b^{3} d^{2} e^{4} - 24 \, B a^{5} b d e^{5} - 60 \, A a^{4} b^{2} d e^{5} - 10 \, B a^{6} e^{6} - 60 \, A a^{5} b e^{6}\right )} x + {\left (1089 \, B b^{6} d^{7} - 360 \, B a b^{5} d^{6} e - 60 \, A b^{6} d^{6} e - 150 \, B a^{2} b^{4} d^{5} e^{2} - 60 \, A a b^{5} d^{5} e^{2} - 80 \, B a^{3} b^{3} d^{4} e^{3} - 60 \, A a^{2} b^{4} d^{4} e^{3} - 45 \, B a^{4} b^{2} d^{3} e^{4} - 60 \, A a^{3} b^{3} d^{3} e^{4} - 24 \, B a^{5} b d^{2} e^{5} - 60 \, A a^{4} b^{2} d^{2} e^{5} - 10 \, B a^{6} d e^{6} - 60 \, A a^{5} b d e^{6} - 60 \, A a^{6} e^{7}\right )} e^{\left (-1\right )}\right )} e^{\left (-7\right )}}{420 \, {\left (x e + d\right )}^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^8,x, algorithm="giac")

[Out]

B*b^6*e^(-8)*log(abs(x*e + d)) + 1/420*(420*(7*B*b^6*d*e^5 - 6*B*a*b^5*e^6 - A*b^6*e^6)*x^6 + 630*(21*B*b^6*d^
2*e^4 - 12*B*a*b^5*d*e^5 - 2*A*b^6*d*e^5 - 5*B*a^2*b^4*e^6 - 2*A*a*b^5*e^6)*x^5 + 350*(77*B*b^6*d^3*e^3 - 36*B
*a*b^5*d^2*e^4 - 6*A*b^6*d^2*e^4 - 15*B*a^2*b^4*d*e^5 - 6*A*a*b^5*d*e^5 - 8*B*a^3*b^3*e^6 - 6*A*a^2*b^4*e^6)*x
^4 + 175*(175*B*b^6*d^4*e^2 - 72*B*a*b^5*d^3*e^3 - 12*A*b^6*d^3*e^3 - 30*B*a^2*b^4*d^2*e^4 - 12*A*a*b^5*d^2*e^
4 - 16*B*a^3*b^3*d*e^5 - 12*A*a^2*b^4*d*e^5 - 9*B*a^4*b^2*e^6 - 12*A*a^3*b^3*e^6)*x^3 + 21*(959*B*b^6*d^5*e -
360*B*a*b^5*d^4*e^2 - 60*A*b^6*d^4*e^2 - 150*B*a^2*b^4*d^3*e^3 - 60*A*a*b^5*d^3*e^3 - 80*B*a^3*b^3*d^2*e^4 - 6
0*A*a^2*b^4*d^2*e^4 - 45*B*a^4*b^2*d*e^5 - 60*A*a^3*b^3*d*e^5 - 24*B*a^5*b*e^6 - 60*A*a^4*b^2*e^6)*x^2 + 7*(10
29*B*b^6*d^6 - 360*B*a*b^5*d^5*e - 60*A*b^6*d^5*e - 150*B*a^2*b^4*d^4*e^2 - 60*A*a*b^5*d^4*e^2 - 80*B*a^3*b^3*
d^3*e^3 - 60*A*a^2*b^4*d^3*e^3 - 45*B*a^4*b^2*d^2*e^4 - 60*A*a^3*b^3*d^2*e^4 - 24*B*a^5*b*d*e^5 - 60*A*a^4*b^2
*d*e^5 - 10*B*a^6*e^6 - 60*A*a^5*b*e^6)*x + (1089*B*b^6*d^7 - 360*B*a*b^5*d^6*e - 60*A*b^6*d^6*e - 150*B*a^2*b
^4*d^5*e^2 - 60*A*a*b^5*d^5*e^2 - 80*B*a^3*b^3*d^4*e^3 - 60*A*a^2*b^4*d^4*e^3 - 45*B*a^4*b^2*d^3*e^4 - 60*A*a^
3*b^3*d^3*e^4 - 24*B*a^5*b*d^2*e^5 - 60*A*a^4*b^2*d^2*e^5 - 10*B*a^6*d*e^6 - 60*A*a^5*b*d*e^6 - 60*A*a^6*e^7)*
e^(-1))*e^(-7)/(x*e + d)^7

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Mupad [B]
time = 1.32, size = 1046, normalized size = 4.91 \begin {gather*} -\frac {\frac {A\,a^6\,e^7}{7}-\frac {363\,B\,b^6\,d^7}{140}+\frac {A\,b^6\,d^6\,e}{7}+\frac {B\,a^6\,d\,e^6}{42}-B\,b^6\,d^7\,\ln \left (d+e\,x\right )+\frac {B\,a^6\,e^7\,x}{6}+A\,b^6\,e^7\,x^6-\frac {343\,B\,b^6\,d^6\,e\,x}{20}+\frac {A\,a\,b^5\,d^5\,e^2}{7}+\frac {2\,B\,a^5\,b\,d^2\,e^5}{35}+3\,A\,a\,b^5\,e^7\,x^5+\frac {6\,B\,a^5\,b\,e^7\,x^2}{5}+6\,B\,a\,b^5\,e^7\,x^6+A\,b^6\,d^5\,e^2\,x+3\,A\,b^6\,d\,e^6\,x^5-7\,B\,b^6\,d\,e^6\,x^6-B\,b^6\,e^7\,x^7\,\ln \left (d+e\,x\right )+\frac {A\,a^2\,b^4\,d^4\,e^3}{7}+\frac {A\,a^3\,b^3\,d^3\,e^4}{7}+\frac {A\,a^4\,b^2\,d^2\,e^5}{7}+\frac {5\,B\,a^2\,b^4\,d^5\,e^2}{14}+\frac {4\,B\,a^3\,b^3\,d^4\,e^3}{21}+\frac {3\,B\,a^4\,b^2\,d^3\,e^4}{28}+3\,A\,a^4\,b^2\,e^7\,x^2+5\,A\,a^3\,b^3\,e^7\,x^3+5\,A\,a^2\,b^4\,e^7\,x^4+\frac {15\,B\,a^4\,b^2\,e^7\,x^3}{4}+\frac {20\,B\,a^3\,b^3\,e^7\,x^4}{3}+\frac {15\,B\,a^2\,b^4\,e^7\,x^5}{2}+3\,A\,b^6\,d^4\,e^3\,x^2+5\,A\,b^6\,d^3\,e^4\,x^3+5\,A\,b^6\,d^2\,e^5\,x^4-\frac {959\,B\,b^6\,d^5\,e^2\,x^2}{20}-\frac {875\,B\,b^6\,d^4\,e^3\,x^3}{12}-\frac {385\,B\,b^6\,d^3\,e^4\,x^4}{6}-\frac {63\,B\,b^6\,d^2\,e^5\,x^5}{2}+\frac {A\,a^5\,b\,d\,e^6}{7}+\frac {6\,B\,a\,b^5\,d^6\,e}{7}+A\,a^5\,b\,e^7\,x+A\,a^2\,b^4\,d^3\,e^4\,x+A\,a^3\,b^3\,d^2\,e^5\,x+3\,A\,a\,b^5\,d^3\,e^4\,x^2+3\,A\,a^3\,b^3\,d\,e^6\,x^2+5\,A\,a\,b^5\,d^2\,e^5\,x^3+5\,A\,a^2\,b^4\,d\,e^6\,x^3+\frac {5\,B\,a^2\,b^4\,d^4\,e^3\,x}{2}+\frac {4\,B\,a^3\,b^3\,d^3\,e^4\,x}{3}+\frac {3\,B\,a^4\,b^2\,d^2\,e^5\,x}{4}+18\,B\,a\,b^5\,d^4\,e^3\,x^2+\frac {9\,B\,a^4\,b^2\,d\,e^6\,x^2}{4}+30\,B\,a\,b^5\,d^3\,e^4\,x^3+\frac {20\,B\,a^3\,b^3\,d\,e^6\,x^3}{3}+30\,B\,a\,b^5\,d^2\,e^5\,x^4+\frac {25\,B\,a^2\,b^4\,d\,e^6\,x^4}{2}-21\,B\,b^6\,d^5\,e^2\,x^2\,\ln \left (d+e\,x\right )-35\,B\,b^6\,d^4\,e^3\,x^3\,\ln \left (d+e\,x\right )-35\,B\,b^6\,d^3\,e^4\,x^4\,\ln \left (d+e\,x\right )-21\,B\,b^6\,d^2\,e^5\,x^5\,\ln \left (d+e\,x\right )+\frac {2\,B\,a^5\,b\,d\,e^6\,x}{5}-7\,B\,b^6\,d^6\,e\,x\,\ln \left (d+e\,x\right )+3\,A\,a^2\,b^4\,d^2\,e^5\,x^2+\frac {15\,B\,a^2\,b^4\,d^3\,e^4\,x^2}{2}+4\,B\,a^3\,b^3\,d^2\,e^5\,x^2+\frac {25\,B\,a^2\,b^4\,d^2\,e^5\,x^3}{2}+A\,a\,b^5\,d^4\,e^3\,x+A\,a^4\,b^2\,d\,e^6\,x+5\,A\,a\,b^5\,d\,e^6\,x^4+6\,B\,a\,b^5\,d^5\,e^2\,x+18\,B\,a\,b^5\,d\,e^6\,x^5-7\,B\,b^6\,d\,e^6\,x^6\,\ln \left (d+e\,x\right )}{e^8\,{\left (d+e\,x\right )}^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^6)/(d + e*x)^8,x)

[Out]

-((A*a^6*e^7)/7 - (363*B*b^6*d^7)/140 + (A*b^6*d^6*e)/7 + (B*a^6*d*e^6)/42 - B*b^6*d^7*log(d + e*x) + (B*a^6*e
^7*x)/6 + A*b^6*e^7*x^6 - (343*B*b^6*d^6*e*x)/20 + (A*a*b^5*d^5*e^2)/7 + (2*B*a^5*b*d^2*e^5)/35 + 3*A*a*b^5*e^
7*x^5 + (6*B*a^5*b*e^7*x^2)/5 + 6*B*a*b^5*e^7*x^6 + A*b^6*d^5*e^2*x + 3*A*b^6*d*e^6*x^5 - 7*B*b^6*d*e^6*x^6 -
B*b^6*e^7*x^7*log(d + e*x) + (A*a^2*b^4*d^4*e^3)/7 + (A*a^3*b^3*d^3*e^4)/7 + (A*a^4*b^2*d^2*e^5)/7 + (5*B*a^2*
b^4*d^5*e^2)/14 + (4*B*a^3*b^3*d^4*e^3)/21 + (3*B*a^4*b^2*d^3*e^4)/28 + 3*A*a^4*b^2*e^7*x^2 + 5*A*a^3*b^3*e^7*
x^3 + 5*A*a^2*b^4*e^7*x^4 + (15*B*a^4*b^2*e^7*x^3)/4 + (20*B*a^3*b^3*e^7*x^4)/3 + (15*B*a^2*b^4*e^7*x^5)/2 + 3
*A*b^6*d^4*e^3*x^2 + 5*A*b^6*d^3*e^4*x^3 + 5*A*b^6*d^2*e^5*x^4 - (959*B*b^6*d^5*e^2*x^2)/20 - (875*B*b^6*d^4*e
^3*x^3)/12 - (385*B*b^6*d^3*e^4*x^4)/6 - (63*B*b^6*d^2*e^5*x^5)/2 + (A*a^5*b*d*e^6)/7 + (6*B*a*b^5*d^6*e)/7 +
A*a^5*b*e^7*x + A*a^2*b^4*d^3*e^4*x + A*a^3*b^3*d^2*e^5*x + 3*A*a*b^5*d^3*e^4*x^2 + 3*A*a^3*b^3*d*e^6*x^2 + 5*
A*a*b^5*d^2*e^5*x^3 + 5*A*a^2*b^4*d*e^6*x^3 + (5*B*a^2*b^4*d^4*e^3*x)/2 + (4*B*a^3*b^3*d^3*e^4*x)/3 + (3*B*a^4
*b^2*d^2*e^5*x)/4 + 18*B*a*b^5*d^4*e^3*x^2 + (9*B*a^4*b^2*d*e^6*x^2)/4 + 30*B*a*b^5*d^3*e^4*x^3 + (20*B*a^3*b^
3*d*e^6*x^3)/3 + 30*B*a*b^5*d^2*e^5*x^4 + (25*B*a^2*b^4*d*e^6*x^4)/2 - 21*B*b^6*d^5*e^2*x^2*log(d + e*x) - 35*
B*b^6*d^4*e^3*x^3*log(d + e*x) - 35*B*b^6*d^3*e^4*x^4*log(d + e*x) - 21*B*b^6*d^2*e^5*x^5*log(d + e*x) + (2*B*
a^5*b*d*e^6*x)/5 - 7*B*b^6*d^6*e*x*log(d + e*x) + 3*A*a^2*b^4*d^2*e^5*x^2 + (15*B*a^2*b^4*d^3*e^4*x^2)/2 + 4*B
*a^3*b^3*d^2*e^5*x^2 + (25*B*a^2*b^4*d^2*e^5*x^3)/2 + A*a*b^5*d^4*e^3*x + A*a^4*b^2*d*e^6*x + 5*A*a*b^5*d*e^6*
x^4 + 6*B*a*b^5*d^5*e^2*x + 18*B*a*b^5*d*e^6*x^5 - 7*B*b^6*d*e^6*x^6*log(d + e*x))/(e^8*(d + e*x)^7)

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