3.10.23 \(\int \frac {(a+b x)^6 (A+B x)}{(d+e x)^3} \, dx\)

Optimal. Leaf size=276 \[ -\frac {b^5 (d+e x)^4 (-6 a B e-A b e+7 b B d)}{4 e^8}+\frac {b^4 (d+e x)^3 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{e^8}-\frac {5 b^3 (d+e x)^2 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{2 e^8}+\frac {5 b^2 x (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{e^7}-\frac {(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{e^8 (d+e x)}+\frac {(b d-a e)^6 (B d-A e)}{2 e^8 (d+e x)^2}-\frac {3 b (b d-a e)^4 \log (d+e x) (-2 a B e-5 A b e+7 b B d)}{e^8}+\frac {b^6 B (d+e x)^5}{5 e^8} \]

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Rubi [A]  time = 0.50, antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \begin {gather*} -\frac {b^5 (d+e x)^4 (-6 a B e-A b e+7 b B d)}{4 e^8}+\frac {b^4 (d+e x)^3 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{e^8}-\frac {5 b^3 (d+e x)^2 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{2 e^8}+\frac {5 b^2 x (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{e^7}-\frac {(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{e^8 (d+e x)}+\frac {(b d-a e)^6 (B d-A e)}{2 e^8 (d+e x)^2}-\frac {3 b (b d-a e)^4 \log (d+e x) (-2 a B e-5 A b e+7 b B d)}{e^8}+\frac {b^6 B (d+e x)^5}{5 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^6*(A + B*x))/(d + e*x)^3,x]

[Out]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

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Mathematica [A]  time = 0.19, size = 352, normalized size = 1.28 \begin {gather*} \frac {-20 b^4 e^3 x^3 \left (-5 a^2 B e^2-2 a b e (A e-3 B d)+b^2 d (A e-2 B d)\right )+10 b^3 e^2 x^2 \left (20 a^3 B e^3+15 a^2 b e^2 (A e-3 B d)+18 a b^2 d e (2 B d-A e)+2 b^3 d^2 (3 A e-5 B d)\right )-20 b^2 e x \left (-15 a^4 B e^4-20 a^3 b e^3 (A e-3 B d)+45 a^2 b^2 d e^2 (A e-2 B d)+12 a b^3 d^2 e (5 B d-3 A e)-5 b^4 d^3 (3 B d-2 A e)\right )+5 b^5 e^4 x^4 (6 a B e+A b e-3 b B d)-\frac {20 (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{d+e x}+\frac {10 (b d-a e)^6 (B d-A e)}{(d+e x)^2}-60 b (b d-a e)^4 \log (d+e x) (-2 a B e-5 A b e+7 b B d)+4 b^6 B e^5 x^5}{20 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^3,x]

[Out]

(-20*b^2*e*(-15*a^4*B*e^4 + 12*a*b^3*d^2*e*(5*B*d - 3*A*e) - 5*b^4*d^3*(3*B*d - 2*A*e) - 20*a^3*b*e^3*(-3*B*d
+ A*e) + 45*a^2*b^2*d*e^2*(-2*B*d + A*e))*x + 10*b^3*e^2*(20*a^3*B*e^3 + 18*a*b^2*d*e*(2*B*d - A*e) + 15*a^2*b
*e^2*(-3*B*d + A*e) + 2*b^3*d^2*(-5*B*d + 3*A*e))*x^2 - 20*b^4*e^3*(-5*a^2*B*e^2 - 2*a*b*e*(-3*B*d + A*e) + b^
2*d*(-2*B*d + A*e))*x^3 + 5*b^5*e^4*(-3*b*B*d + A*b*e + 6*a*B*e)*x^4 + 4*b^6*B*e^5*x^5 + (10*(b*d - a*e)^6*(B*
d - A*e))/(d + e*x)^2 - (20*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e))/(d + e*x) - 60*b*(b*d - a*e)^4*(7*b*B*d
- 5*A*b*e - 2*a*B*e)*Log[d + e*x])/(20*e^8)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)^6*(A + B*x))/(d + e*x)^3,x]

[Out]

IntegrateAlgebraic[((a + b*x)^6*(A + B*x))/(d + e*x)^3, x]

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fricas [B]  time = 1.95, size = 1177, normalized size = 4.26

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^3,x, algorithm="fricas")

[Out]

1/20*(4*B*b^6*e^7*x^7 - 130*B*b^6*d^7 - 10*A*a^6*e^7 + 110*(6*B*a*b^5 + A*b^6)*d^6*e - 270*(5*B*a^2*b^4 + 2*A*
a*b^5)*d^5*e^2 + 350*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 - 250*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 90*(2*B*a
^5*b + 5*A*a^4*b^2)*d^2*e^5 - 10*(B*a^6 + 6*A*a^5*b)*d*e^6 - (7*B*b^6*d*e^6 - 5*(6*B*a*b^5 + A*b^6)*e^7)*x^6 +
2*(7*B*b^6*d^2*e^5 - 5*(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 - 5*(7*B*b^6*d^3*e^4
- 5*(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 - 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^
4 + 20*(7*B*b^6*d^4*e^3 - 5*(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 - 10*(4*B*a^3*b
^3 + 3*A*a^2*b^4)*d*e^6 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 10*(50*B*b^6*d^5*e^2 - 34*(6*B*a*b^5 + A*b^
6)*d^4*e^3 + 63*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 - 55*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 20*(3*B*a^4*b^2 +
4*A*a^3*b^3)*d*e^6)*x^2 + 20*(8*B*b^6*d^6*e - 4*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4
*e^3 + 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 - 10*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 + 6*(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 - 60*(7*B*b^6*d^7 - 5*(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 - 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 5*(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 + (7*B*b^6*d^5*e^2 - 5*(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 - 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 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 + 2*(7*B*b^6*d^6*e - 5*(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 - 10*(4*B
*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 5*(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)*x)*
log(e*x + d))/(e^10*x^2 + 2*d*e^9*x + d^2*e^8)

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giac [B]  time = 1.20, size = 810, normalized size = 2.93 \begin {gather*} -3 \, {\left (7 \, B b^{6} d^{5} - 30 \, B a b^{5} d^{4} e - 5 \, A b^{6} d^{4} e + 50 \, B a^{2} b^{4} d^{3} e^{2} + 20 \, A a b^{5} d^{3} e^{2} - 40 \, B a^{3} b^{3} d^{2} e^{3} - 30 \, A a^{2} b^{4} d^{2} e^{3} + 15 \, B a^{4} b^{2} d e^{4} + 20 \, A a^{3} b^{3} d e^{4} - 2 \, B a^{5} b e^{5} - 5 \, A a^{4} b^{2} e^{5}\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{20} \, {\left (4 \, B b^{6} x^{5} e^{12} - 15 \, B b^{6} d x^{4} e^{11} + 40 \, B b^{6} d^{2} x^{3} e^{10} - 100 \, B b^{6} d^{3} x^{2} e^{9} + 300 \, B b^{6} d^{4} x e^{8} + 30 \, B a b^{5} x^{4} e^{12} + 5 \, A b^{6} x^{4} e^{12} - 120 \, B a b^{5} d x^{3} e^{11} - 20 \, A b^{6} d x^{3} e^{11} + 360 \, B a b^{5} d^{2} x^{2} e^{10} + 60 \, A b^{6} d^{2} x^{2} e^{10} - 1200 \, B a b^{5} d^{3} x e^{9} - 200 \, A b^{6} d^{3} x e^{9} + 100 \, B a^{2} b^{4} x^{3} e^{12} + 40 \, A a b^{5} x^{3} e^{12} - 450 \, B a^{2} b^{4} d x^{2} e^{11} - 180 \, A a b^{5} d x^{2} e^{11} + 1800 \, B a^{2} b^{4} d^{2} x e^{10} + 720 \, A a b^{5} d^{2} x e^{10} + 200 \, B a^{3} b^{3} x^{2} e^{12} + 150 \, A a^{2} b^{4} x^{2} e^{12} - 1200 \, B a^{3} b^{3} d x e^{11} - 900 \, A a^{2} b^{4} d x e^{11} + 300 \, B a^{4} b^{2} x e^{12} + 400 \, A a^{3} b^{3} x e^{12}\right )} e^{\left (-15\right )} - \frac {{\left (13 \, B b^{6} d^{7} - 66 \, B a b^{5} d^{6} e - 11 \, A b^{6} d^{6} e + 135 \, B a^{2} b^{4} d^{5} e^{2} + 54 \, A a b^{5} d^{5} e^{2} - 140 \, B a^{3} b^{3} d^{4} e^{3} - 105 \, A a^{2} b^{4} d^{4} e^{3} + 75 \, B a^{4} b^{2} d^{3} e^{4} + 100 \, A a^{3} b^{3} d^{3} e^{4} - 18 \, B a^{5} b d^{2} e^{5} - 45 \, A a^{4} b^{2} d^{2} e^{5} + B a^{6} d e^{6} + 6 \, A a^{5} b d e^{6} + A a^{6} e^{7} + 2 \, {\left (7 \, B b^{6} d^{6} e - 36 \, B a b^{5} d^{5} e^{2} - 6 \, A b^{6} d^{5} e^{2} + 75 \, B a^{2} b^{4} d^{4} e^{3} + 30 \, A a b^{5} d^{4} e^{3} - 80 \, B a^{3} b^{3} d^{3} e^{4} - 60 \, A a^{2} b^{4} d^{3} e^{4} + 45 \, B a^{4} b^{2} d^{2} e^{5} + 60 \, A a^{3} b^{3} d^{2} e^{5} - 12 \, B a^{5} b d e^{6} - 30 \, A a^{4} b^{2} d e^{6} + B a^{6} e^{7} + 6 \, A a^{5} b e^{7}\right )} x\right )} e^{\left (-8\right )}}{2 \, {\left (x e + d\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^3,x, algorithm="giac")

[Out]

-3*(7*B*b^6*d^5 - 30*B*a*b^5*d^4*e - 5*A*b^6*d^4*e + 50*B*a^2*b^4*d^3*e^2 + 20*A*a*b^5*d^3*e^2 - 40*B*a^3*b^3*
d^2*e^3 - 30*A*a^2*b^4*d^2*e^3 + 15*B*a^4*b^2*d*e^4 + 20*A*a^3*b^3*d*e^4 - 2*B*a^5*b*e^5 - 5*A*a^4*b^2*e^5)*e^
(-8)*log(abs(x*e + d)) + 1/20*(4*B*b^6*x^5*e^12 - 15*B*b^6*d*x^4*e^11 + 40*B*b^6*d^2*x^3*e^10 - 100*B*b^6*d^3*
x^2*e^9 + 300*B*b^6*d^4*x*e^8 + 30*B*a*b^5*x^4*e^12 + 5*A*b^6*x^4*e^12 - 120*B*a*b^5*d*x^3*e^11 - 20*A*b^6*d*x
^3*e^11 + 360*B*a*b^5*d^2*x^2*e^10 + 60*A*b^6*d^2*x^2*e^10 - 1200*B*a*b^5*d^3*x*e^9 - 200*A*b^6*d^3*x*e^9 + 10
0*B*a^2*b^4*x^3*e^12 + 40*A*a*b^5*x^3*e^12 - 450*B*a^2*b^4*d*x^2*e^11 - 180*A*a*b^5*d*x^2*e^11 + 1800*B*a^2*b^
4*d^2*x*e^10 + 720*A*a*b^5*d^2*x*e^10 + 200*B*a^3*b^3*x^2*e^12 + 150*A*a^2*b^4*x^2*e^12 - 1200*B*a^3*b^3*d*x*e
^11 - 900*A*a^2*b^4*d*x*e^11 + 300*B*a^4*b^2*x*e^12 + 400*A*a^3*b^3*x*e^12)*e^(-15) - 1/2*(13*B*b^6*d^7 - 66*B
*a*b^5*d^6*e - 11*A*b^6*d^6*e + 135*B*a^2*b^4*d^5*e^2 + 54*A*a*b^5*d^5*e^2 - 140*B*a^3*b^3*d^4*e^3 - 105*A*a^2
*b^4*d^4*e^3 + 75*B*a^4*b^2*d^3*e^4 + 100*A*a^3*b^3*d^3*e^4 - 18*B*a^5*b*d^2*e^5 - 45*A*a^4*b^2*d^2*e^5 + B*a^
6*d*e^6 + 6*A*a^5*b*d*e^6 + A*a^6*e^7 + 2*(7*B*b^6*d^6*e - 36*B*a*b^5*d^5*e^2 - 6*A*b^6*d^5*e^2 + 75*B*a^2*b^4
*d^4*e^3 + 30*A*a*b^5*d^4*e^3 - 80*B*a^3*b^3*d^3*e^4 - 60*A*a^2*b^4*d^3*e^4 + 45*B*a^4*b^2*d^2*e^5 + 60*A*a^3*
b^3*d^2*e^5 - 12*B*a^5*b*d*e^6 - 30*A*a^4*b^2*d*e^6 + B*a^6*e^7 + 6*A*a^5*b*e^7)*x)*e^(-8)/(x*e + d)^2

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maple [B]  time = 0.02, size = 1101, normalized size = 3.99

result 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)^3,x)

[Out]

1/5*b^6/e^3*B*x^5+1/4*b^6/e^3*A*x^4-1/e^2/(e*x+d)*B*a^6-1/2/e/(e*x+d)^2*A*a^6+15*b^2/e^3*B*a^4*x+15*b^6/e^7*B*
d^4*x+15*b^2/e^3*ln(e*x+d)*A*a^4+15*b^6/e^7*ln(e*x+d)*A*d^4+6*b/e^3*ln(e*x+d)*B*a^5-21*b^6/e^8*ln(e*x+d)*B*d^5
-1/2/e^7/(e*x+d)^2*A*b^6*d^6+1/2/e^2/(e*x+d)^2*B*d*a^6+1/2/e^8/(e*x+d)^2*B*b^6*d^7-b^6/e^4*A*x^3*d+5*b^4/e^3*B
*x^3*a^2+2*b^6/e^5*B*x^3*d^2+20*b^3/e^3*A*a^3*x-10*b^6/e^6*A*d^3*x+15/2*b^4/e^3*A*x^2*a^2+3*b^6/e^5*A*x^2*d^2+
10*b^3/e^3*B*x^2*a^3+2*b^5/e^3*A*x^3*a-5*b^6/e^6*B*x^2*d^3+3/2*b^5/e^3*B*x^4*a-3/4*b^6/e^4*B*x^4*d-6/e^2/(e*x+
d)*A*a^5*b+6/e^7/(e*x+d)*A*b^6*d^5-7/e^8/(e*x+d)*B*b^6*d^6+36*b^5/e^5*A*a*d^2*x-6*b^5/e^4*B*x^3*a*d+120*b^3/e^
5*ln(e*x+d)*B*a^3*d^2-150*b^4/e^6*ln(e*x+d)*B*a^2*d^3+90*b^5/e^7*ln(e*x+d)*B*a*d^4+3/e^2/(e*x+d)^2*A*d*a^5*b-1
5/2/e^3/(e*x+d)^2*A*d^2*a^4*b^2+10/e^4/(e*x+d)^2*A*a^3*b^3*d^3-15/2/e^5/(e*x+d)^2*A*a^2*b^4*d^4+3/e^6/(e*x+d)^
2*A*a*b^5*d^5-3/e^3/(e*x+d)^2*B*d^2*a^5*b+15/2/e^4/(e*x+d)^2*B*a^4*b^2*d^3-10/e^5/(e*x+d)^2*B*a^3*b^3*d^4+15/2
/e^6/(e*x+d)^2*B*a^2*b^4*d^5-3/e^7/(e*x+d)^2*B*a*b^5*d^6-9*b^5/e^4*A*x^2*a*d-45/2*b^4/e^4*B*x^2*a^2*d-60*b^3/e
^4*B*a^3*d*x+90*b^4/e^5*B*a^2*d^2*x-60*b^5/e^6*B*a*d^3*x+18*b^5/e^5*B*x^2*a*d^2-45*b^4/e^4*A*a^2*d*x+30/e^3/(e
*x+d)*A*a^4*b^2*d-60/e^4/(e*x+d)*A*a^3*b^3*d^2+60/e^5/(e*x+d)*A*a^2*b^4*d^3-30/e^6/(e*x+d)*A*a*b^5*d^4+12/e^3/
(e*x+d)*B*a^5*b*d-45/e^4/(e*x+d)*B*a^4*b^2*d^2+80/e^5/(e*x+d)*B*a^3*b^3*d^3-75/e^6/(e*x+d)*B*a^2*b^4*d^4+36/e^
7/(e*x+d)*B*a*b^5*d^5-60*b^3/e^4*ln(e*x+d)*A*a^3*d+90*b^4/e^5*ln(e*x+d)*A*a^2*d^2-60*b^5/e^6*ln(e*x+d)*A*a*d^3
-45*b^2/e^4*ln(e*x+d)*B*a^4*d

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maxima [B]  time = 0.74, size = 779, normalized size = 2.82 \begin {gather*} -\frac {13 \, B b^{6} d^{7} + A a^{6} e^{7} - 11 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 27 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} - 35 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 25 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} - 9 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} + 2 \, {\left (7 \, B b^{6} d^{6} e - 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + 15 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} - 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} + 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} - 6 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} + {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x}{2 \, {\left (e^{10} x^{2} + 2 \, d e^{9} x + d^{2} e^{8}\right )}} + \frac {4 \, B b^{6} e^{4} x^{5} - 5 \, {\left (3 \, B b^{6} d e^{3} - {\left (6 \, B a b^{5} + A b^{6}\right )} e^{4}\right )} x^{4} + 20 \, {\left (2 \, B b^{6} d^{2} e^{2} - {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{3} + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{4}\right )} x^{3} - 10 \, {\left (10 \, B b^{6} d^{3} e - 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{2} + 9 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{3} - 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{4}\right )} x^{2} + 20 \, {\left (15 \, B b^{6} d^{4} - 10 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e + 18 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{2} - 15 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{4}\right )} x}{20 \, e^{7}} - \frac {3 \, {\left (7 \, B b^{6} d^{5} - 5 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e + 10 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{2} - 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{4} - {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{5}\right )} \log \left (e x + d\right )}{e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^3,x, algorithm="maxima")

[Out]

-1/2*(13*B*b^6*d^7 + A*a^6*e^7 - 11*(6*B*a*b^5 + A*b^6)*d^6*e + 27*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 35*(4*B
*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 25*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 - 9*(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 + 2*(7*B*b^6*d^6*e - 6*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 15*(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 + 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 - 6*(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^10*x^2 + 2*d*e^9*x + d^2*e^8) + 1/20*(4*B*b^6*e^4*x^5 - 5*(3
*B*b^6*d*e^3 - (6*B*a*b^5 + A*b^6)*e^4)*x^4 + 20*(2*B*b^6*d^2*e^2 - (6*B*a*b^5 + A*b^6)*d*e^3 + (5*B*a^2*b^4 +
2*A*a*b^5)*e^4)*x^3 - 10*(10*B*b^6*d^3*e - 6*(6*B*a*b^5 + A*b^6)*d^2*e^2 + 9*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^3
- 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^4)*x^2 + 20*(15*B*b^6*d^4 - 10*(6*B*a*b^5 + A*b^6)*d^3*e + 18*(5*B*a^2*b^4 +
2*A*a*b^5)*d^2*e^2 - 15*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^4)*x)/e^7 - 3*(7*
B*b^6*d^5 - 5*(6*B*a*b^5 + A*b^6)*d^4*e + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^2 - 10*(4*B*a^3*b^3 + 3*A*a^2*b^4
)*d^2*e^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^4 - (2*B*a^5*b + 5*A*a^4*b^2)*e^5)*log(e*x + d)/e^8

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mupad [B]  time = 0.17, size = 1053, normalized size = 3.82 \begin {gather*} x\,\left (\frac {3\,d\,\left (\frac {3\,d^2\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e^3}-\frac {3\,B\,b^6\,d}{e^4}\right )}{e^2}-\frac {3\,d\,\left (\frac {3\,d\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e^3}-\frac {3\,B\,b^6\,d}{e^4}\right )}{e}-\frac {3\,a\,b^4\,\left (2\,A\,b+5\,B\,a\right )}{e^3}+\frac {3\,B\,b^6\,d^2}{e^5}\right )}{e}-\frac {5\,a^2\,b^3\,\left (3\,A\,b+4\,B\,a\right )}{e^3}+\frac {B\,b^6\,d^3}{e^6}\right )}{e}-\frac {d^3\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e^3}-\frac {3\,B\,b^6\,d}{e^4}\right )}{e^3}+\frac {3\,d^2\,\left (\frac {3\,d\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e^3}-\frac {3\,B\,b^6\,d}{e^4}\right )}{e}-\frac {3\,a\,b^4\,\left (2\,A\,b+5\,B\,a\right )}{e^3}+\frac {3\,B\,b^6\,d^2}{e^5}\right )}{e^2}+\frac {5\,a^3\,b^2\,\left (4\,A\,b+3\,B\,a\right )}{e^3}\right )-x^3\,\left (\frac {d\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e^3}-\frac {3\,B\,b^6\,d}{e^4}\right )}{e}-\frac {a\,b^4\,\left (2\,A\,b+5\,B\,a\right )}{e^3}+\frac {B\,b^6\,d^2}{e^5}\right )-x^2\,\left (\frac {3\,d^2\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e^3}-\frac {3\,B\,b^6\,d}{e^4}\right )}{2\,e^2}-\frac {3\,d\,\left (\frac {3\,d\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e^3}-\frac {3\,B\,b^6\,d}{e^4}\right )}{e}-\frac {3\,a\,b^4\,\left (2\,A\,b+5\,B\,a\right )}{e^3}+\frac {3\,B\,b^6\,d^2}{e^5}\right )}{2\,e}-\frac {5\,a^2\,b^3\,\left (3\,A\,b+4\,B\,a\right )}{2\,e^3}+\frac {B\,b^6\,d^3}{2\,e^6}\right )-\frac {\frac {B\,a^6\,d\,e^6+A\,a^6\,e^7-18\,B\,a^5\,b\,d^2\,e^5+6\,A\,a^5\,b\,d\,e^6+75\,B\,a^4\,b^2\,d^3\,e^4-45\,A\,a^4\,b^2\,d^2\,e^5-140\,B\,a^3\,b^3\,d^4\,e^3+100\,A\,a^3\,b^3\,d^3\,e^4+135\,B\,a^2\,b^4\,d^5\,e^2-105\,A\,a^2\,b^4\,d^4\,e^3-66\,B\,a\,b^5\,d^6\,e+54\,A\,a\,b^5\,d^5\,e^2+13\,B\,b^6\,d^7-11\,A\,b^6\,d^6\,e}{2\,e}+x\,\left (B\,a^6\,e^6-12\,B\,a^5\,b\,d\,e^5+6\,A\,a^5\,b\,e^6+45\,B\,a^4\,b^2\,d^2\,e^4-30\,A\,a^4\,b^2\,d\,e^5-80\,B\,a^3\,b^3\,d^3\,e^3+60\,A\,a^3\,b^3\,d^2\,e^4+75\,B\,a^2\,b^4\,d^4\,e^2-60\,A\,a^2\,b^4\,d^3\,e^3-36\,B\,a\,b^5\,d^5\,e+30\,A\,a\,b^5\,d^4\,e^2+7\,B\,b^6\,d^6-6\,A\,b^6\,d^5\,e\right )}{d^2\,e^7+2\,d\,e^8\,x+e^9\,x^2}+x^4\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{4\,e^3}-\frac {3\,B\,b^6\,d}{4\,e^4}\right )+\frac {\ln \left (d+e\,x\right )\,\left (6\,B\,a^5\,b\,e^5-45\,B\,a^4\,b^2\,d\,e^4+15\,A\,a^4\,b^2\,e^5+120\,B\,a^3\,b^3\,d^2\,e^3-60\,A\,a^3\,b^3\,d\,e^4-150\,B\,a^2\,b^4\,d^3\,e^2+90\,A\,a^2\,b^4\,d^2\,e^3+90\,B\,a\,b^5\,d^4\,e-60\,A\,a\,b^5\,d^3\,e^2-21\,B\,b^6\,d^5+15\,A\,b^6\,d^4\,e\right )}{e^8}+\frac {B\,b^6\,x^5}{5\,e^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^6)/(d + e*x)^3,x)

[Out]

x*((3*d*((3*d^2*((A*b^6 + 6*B*a*b^5)/e^3 - (3*B*b^6*d)/e^4))/e^2 - (3*d*((3*d*((A*b^6 + 6*B*a*b^5)/e^3 - (3*B*
b^6*d)/e^4))/e - (3*a*b^4*(2*A*b + 5*B*a))/e^3 + (3*B*b^6*d^2)/e^5))/e - (5*a^2*b^3*(3*A*b + 4*B*a))/e^3 + (B*
b^6*d^3)/e^6))/e - (d^3*((A*b^6 + 6*B*a*b^5)/e^3 - (3*B*b^6*d)/e^4))/e^3 + (3*d^2*((3*d*((A*b^6 + 6*B*a*b^5)/e
^3 - (3*B*b^6*d)/e^4))/e - (3*a*b^4*(2*A*b + 5*B*a))/e^3 + (3*B*b^6*d^2)/e^5))/e^2 + (5*a^3*b^2*(4*A*b + 3*B*a
))/e^3) - x^3*((d*((A*b^6 + 6*B*a*b^5)/e^3 - (3*B*b^6*d)/e^4))/e - (a*b^4*(2*A*b + 5*B*a))/e^3 + (B*b^6*d^2)/e
^5) - x^2*((3*d^2*((A*b^6 + 6*B*a*b^5)/e^3 - (3*B*b^6*d)/e^4))/(2*e^2) - (3*d*((3*d*((A*b^6 + 6*B*a*b^5)/e^3 -
(3*B*b^6*d)/e^4))/e - (3*a*b^4*(2*A*b + 5*B*a))/e^3 + (3*B*b^6*d^2)/e^5))/(2*e) - (5*a^2*b^3*(3*A*b + 4*B*a))
/(2*e^3) + (B*b^6*d^3)/(2*e^6)) - ((A*a^6*e^7 + 13*B*b^6*d^7 - 11*A*b^6*d^6*e + B*a^6*d*e^6 + 54*A*a*b^5*d^5*e
^2 - 18*B*a^5*b*d^2*e^5 - 105*A*a^2*b^4*d^4*e^3 + 100*A*a^3*b^3*d^3*e^4 - 45*A*a^4*b^2*d^2*e^5 + 135*B*a^2*b^4
*d^5*e^2 - 140*B*a^3*b^3*d^4*e^3 + 75*B*a^4*b^2*d^3*e^4 + 6*A*a^5*b*d*e^6 - 66*B*a*b^5*d^6*e)/(2*e) + x*(B*a^6
*e^6 + 7*B*b^6*d^6 + 6*A*a^5*b*e^6 - 6*A*b^6*d^5*e + 30*A*a*b^5*d^4*e^2 - 30*A*a^4*b^2*d*e^5 - 60*A*a^2*b^4*d^
3*e^3 + 60*A*a^3*b^3*d^2*e^4 + 75*B*a^2*b^4*d^4*e^2 - 80*B*a^3*b^3*d^3*e^3 + 45*B*a^4*b^2*d^2*e^4 - 36*B*a*b^5
*d^5*e - 12*B*a^5*b*d*e^5))/(d^2*e^7 + e^9*x^2 + 2*d*e^8*x) + x^4*((A*b^6 + 6*B*a*b^5)/(4*e^3) - (3*B*b^6*d)/(
4*e^4)) + (log(d + e*x)*(6*B*a^5*b*e^5 - 21*B*b^6*d^5 + 15*A*b^6*d^4*e + 15*A*a^4*b^2*e^5 - 60*A*a*b^5*d^3*e^2
- 60*A*a^3*b^3*d*e^4 - 45*B*a^4*b^2*d*e^4 + 90*A*a^2*b^4*d^2*e^3 - 150*B*a^2*b^4*d^3*e^2 + 120*B*a^3*b^3*d^2*
e^3 + 90*B*a*b^5*d^4*e))/e^8 + (B*b^6*x^5)/(5*e^3)

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sympy [B]  time = 15.37, size = 821, normalized size = 2.97 \begin {gather*} \frac {B b^{6} x^{5}}{5 e^{3}} + \frac {3 b \left (a e - b d\right )^{4} \left (5 A b e + 2 B a e - 7 B b d\right ) \log {\left (d + e x \right )}}{e^{8}} + x^{4} \left (\frac {A b^{6}}{4 e^{3}} + \frac {3 B a b^{5}}{2 e^{3}} - \frac {3 B b^{6} d}{4 e^{4}}\right ) + x^{3} \left (\frac {2 A a b^{5}}{e^{3}} - \frac {A b^{6} d}{e^{4}} + \frac {5 B a^{2} b^{4}}{e^{3}} - \frac {6 B a b^{5} d}{e^{4}} + \frac {2 B b^{6} d^{2}}{e^{5}}\right ) + x^{2} \left (\frac {15 A a^{2} b^{4}}{2 e^{3}} - \frac {9 A a b^{5} d}{e^{4}} + \frac {3 A b^{6} d^{2}}{e^{5}} + \frac {10 B a^{3} b^{3}}{e^{3}} - \frac {45 B a^{2} b^{4} d}{2 e^{4}} + \frac {18 B a b^{5} d^{2}}{e^{5}} - \frac {5 B b^{6} d^{3}}{e^{6}}\right ) + x \left (\frac {20 A a^{3} b^{3}}{e^{3}} - \frac {45 A a^{2} b^{4} d}{e^{4}} + \frac {36 A a b^{5} d^{2}}{e^{5}} - \frac {10 A b^{6} d^{3}}{e^{6}} + \frac {15 B a^{4} b^{2}}{e^{3}} - \frac {60 B a^{3} b^{3} d}{e^{4}} + \frac {90 B a^{2} b^{4} d^{2}}{e^{5}} - \frac {60 B a b^{5} d^{3}}{e^{6}} + \frac {15 B b^{6} d^{4}}{e^{7}}\right ) + \frac {- A a^{6} e^{7} - 6 A a^{5} b d e^{6} + 45 A a^{4} b^{2} d^{2} e^{5} - 100 A a^{3} b^{3} d^{3} e^{4} + 105 A a^{2} b^{4} d^{4} e^{3} - 54 A a b^{5} d^{5} e^{2} + 11 A b^{6} d^{6} e - B a^{6} d e^{6} + 18 B a^{5} b d^{2} e^{5} - 75 B a^{4} b^{2} d^{3} e^{4} + 140 B a^{3} b^{3} d^{4} e^{3} - 135 B a^{2} b^{4} d^{5} e^{2} + 66 B a b^{5} d^{6} e - 13 B b^{6} d^{7} + x \left (- 12 A a^{5} b e^{7} + 60 A a^{4} b^{2} d e^{6} - 120 A a^{3} b^{3} d^{2} e^{5} + 120 A a^{2} b^{4} d^{3} e^{4} - 60 A a b^{5} d^{4} e^{3} + 12 A b^{6} d^{5} e^{2} - 2 B a^{6} e^{7} + 24 B a^{5} b d e^{6} - 90 B a^{4} b^{2} d^{2} e^{5} + 160 B a^{3} b^{3} d^{3} e^{4} - 150 B a^{2} b^{4} d^{4} e^{3} + 72 B a b^{5} d^{5} e^{2} - 14 B b^{6} d^{6} e\right )}{2 d^{2} e^{8} + 4 d e^{9} x + 2 e^{10} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**6*(B*x+A)/(e*x+d)**3,x)

[Out]

B*b**6*x**5/(5*e**3) + 3*b*(a*e - b*d)**4*(5*A*b*e + 2*B*a*e - 7*B*b*d)*log(d + e*x)/e**8 + x**4*(A*b**6/(4*e*
*3) + 3*B*a*b**5/(2*e**3) - 3*B*b**6*d/(4*e**4)) + x**3*(2*A*a*b**5/e**3 - A*b**6*d/e**4 + 5*B*a**2*b**4/e**3
- 6*B*a*b**5*d/e**4 + 2*B*b**6*d**2/e**5) + x**2*(15*A*a**2*b**4/(2*e**3) - 9*A*a*b**5*d/e**4 + 3*A*b**6*d**2/
e**5 + 10*B*a**3*b**3/e**3 - 45*B*a**2*b**4*d/(2*e**4) + 18*B*a*b**5*d**2/e**5 - 5*B*b**6*d**3/e**6) + x*(20*A
*a**3*b**3/e**3 - 45*A*a**2*b**4*d/e**4 + 36*A*a*b**5*d**2/e**5 - 10*A*b**6*d**3/e**6 + 15*B*a**4*b**2/e**3 -
60*B*a**3*b**3*d/e**4 + 90*B*a**2*b**4*d**2/e**5 - 60*B*a*b**5*d**3/e**6 + 15*B*b**6*d**4/e**7) + (-A*a**6*e**
7 - 6*A*a**5*b*d*e**6 + 45*A*a**4*b**2*d**2*e**5 - 100*A*a**3*b**3*d**3*e**4 + 105*A*a**2*b**4*d**4*e**3 - 54*
A*a*b**5*d**5*e**2 + 11*A*b**6*d**6*e - B*a**6*d*e**6 + 18*B*a**5*b*d**2*e**5 - 75*B*a**4*b**2*d**3*e**4 + 140
*B*a**3*b**3*d**4*e**3 - 135*B*a**2*b**4*d**5*e**2 + 66*B*a*b**5*d**6*e - 13*B*b**6*d**7 + x*(-12*A*a**5*b*e**
7 + 60*A*a**4*b**2*d*e**6 - 120*A*a**3*b**3*d**2*e**5 + 120*A*a**2*b**4*d**3*e**4 - 60*A*a*b**5*d**4*e**3 + 12
*A*b**6*d**5*e**2 - 2*B*a**6*e**7 + 24*B*a**5*b*d*e**6 - 90*B*a**4*b**2*d**2*e**5 + 160*B*a**3*b**3*d**3*e**4
- 150*B*a**2*b**4*d**4*e**3 + 72*B*a*b**5*d**5*e**2 - 14*B*b**6*d**6*e))/(2*d**2*e**8 + 4*d*e**9*x + 2*e**10*x
**2)

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