3.1122 \(\int \frac {(A+B x) (d+e x)^5}{(a+b x)^2} \, dx\)

Optimal. Leaf size=227 \[ \frac {e^4 (a+b x)^4 (-6 a B e+A b e+5 b B d)}{4 b^7}+\frac {5 e^3 (a+b x)^3 (b d-a e) (-3 a B e+A b e+2 b B d)}{3 b^7}+\frac {5 e^2 (a+b x)^2 (b d-a e)^2 (-2 a B e+A b e+b B d)}{b^7}-\frac {(A b-a B) (b d-a e)^5}{b^7 (a+b x)}+\frac {(b d-a e)^4 \log (a+b x) (-6 a B e+5 A b e+b B d)}{b^7}+\frac {5 e x (b d-a e)^3 (-3 a B e+2 A b e+b B d)}{b^6}+\frac {B e^5 (a+b x)^5}{5 b^7} \]

[Out]

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

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Rubi [A]  time = 0.39, antiderivative size = 227, 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} \[ \frac {5 e^2 (a+b x)^2 (b d-a e)^2 (-2 a B e+A b e+b B d)}{b^7}+\frac {5 e^3 (a+b x)^3 (b d-a e) (-3 a B e+A b e+2 b B d)}{3 b^7}+\frac {e^4 (a+b x)^4 (-6 a B e+A b e+5 b B d)}{4 b^7}-\frac {(A b-a B) (b d-a e)^5}{b^7 (a+b x)}+\frac {5 e x (b d-a e)^3 (-3 a B e+2 A b e+b B d)}{b^6}+\frac {(b d-a e)^4 \log (a+b x) (-6 a B e+5 A b e+b B d)}{b^7}+\frac {B e^5 (a+b x)^5}{5 b^7} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^5)/(a + b*x)^2,x]

[Out]

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

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) (d+e x)^5}{(a+b x)^2} \, dx &=\int \left (\frac {5 e (b d-a e)^3 (b B d+2 A b e-3 a B e)}{b^6}+\frac {(A b-a B) (b d-a e)^5}{b^6 (a+b x)^2}+\frac {(b d-a e)^4 (b B d+5 A b e-6 a B e)}{b^6 (a+b x)}+\frac {10 e^2 (b d-a e)^2 (b B d+A b e-2 a B e) (a+b x)}{b^6}+\frac {5 e^3 (b d-a e) (2 b B d+A b e-3 a B e) (a+b x)^2}{b^6}+\frac {e^4 (5 b B d+A b e-6 a B e) (a+b x)^3}{b^6}+\frac {B e^5 (a+b x)^4}{b^6}\right ) \, dx\\ &=\frac {5 e (b d-a e)^3 (b B d+2 A b e-3 a B e) x}{b^6}-\frac {(A b-a B) (b d-a e)^5}{b^7 (a+b x)}+\frac {5 e^2 (b d-a e)^2 (b B d+A b e-2 a B e) (a+b x)^2}{b^7}+\frac {5 e^3 (b d-a e) (2 b B d+A b e-3 a B e) (a+b x)^3}{3 b^7}+\frac {e^4 (5 b B d+A b e-6 a B e) (a+b x)^4}{4 b^7}+\frac {B e^5 (a+b x)^5}{5 b^7}+\frac {(b d-a e)^4 (b B d+5 A b e-6 a B e) \log (a+b x)}{b^7}\\ \end {align*}

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Mathematica [B]  time = 0.30, size = 500, normalized size = 2.20 \[ \frac {-5 A b \left (-12 a^5 e^5+12 a^4 b e^4 (5 d+4 e x)+30 a^3 b^2 e^3 \left (-4 d^2-6 d e x+e^2 x^2\right )-10 a^2 b^3 e^2 \left (-12 d^3-24 d^2 e x+12 d e^2 x^2+e^3 x^3\right )+5 a b^4 e \left (-12 d^4-24 d^3 e x+36 d^2 e^2 x^2+8 d e^3 x^3+e^4 x^4\right )+b^5 \left (12 d^5-120 d^3 e^2 x^2-60 d^2 e^3 x^3-20 d e^4 x^4-3 e^5 x^5\right )\right )+B \left (-60 a^6 e^5+300 a^5 b e^4 (d+e x)+60 a^4 b^2 e^3 \left (-10 d^2-20 d e x+3 e^2 x^2\right )+30 a^3 b^3 e^2 \left (20 d^3+60 d^2 e x-25 d e^2 x^2-2 e^3 x^3\right )+10 a^2 b^4 e \left (-30 d^4-120 d^3 e x+120 d^2 e^2 x^2+25 d e^3 x^3+3 e^4 x^4\right )+a b^5 \left (60 d^5+300 d^4 e x-900 d^3 e^2 x^2-400 d^2 e^3 x^3-125 d e^4 x^4-18 e^5 x^5\right )+b^6 e x^2 \left (300 d^4+300 d^3 e x+200 d^2 e^2 x^2+75 d e^3 x^3+12 e^4 x^4\right )\right )+60 (a+b x) (b d-a e)^4 \log (a+b x) (-6 a B e+5 A b e+b B d)}{60 b^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^5)/(a + b*x)^2,x]

[Out]

(B*(-60*a^6*e^5 + 300*a^5*b*e^4*(d + e*x) + 60*a^4*b^2*e^3*(-10*d^2 - 20*d*e*x + 3*e^2*x^2) + 30*a^3*b^3*e^2*(
20*d^3 + 60*d^2*e*x - 25*d*e^2*x^2 - 2*e^3*x^3) + 10*a^2*b^4*e*(-30*d^4 - 120*d^3*e*x + 120*d^2*e^2*x^2 + 25*d
*e^3*x^3 + 3*e^4*x^4) + b^6*e*x^2*(300*d^4 + 300*d^3*e*x + 200*d^2*e^2*x^2 + 75*d*e^3*x^3 + 12*e^4*x^4) + a*b^
5*(60*d^5 + 300*d^4*e*x - 900*d^3*e^2*x^2 - 400*d^2*e^3*x^3 - 125*d*e^4*x^4 - 18*e^5*x^5)) - 5*A*b*(-12*a^5*e^
5 + 12*a^4*b*e^4*(5*d + 4*e*x) + 30*a^3*b^2*e^3*(-4*d^2 - 6*d*e*x + e^2*x^2) - 10*a^2*b^3*e^2*(-12*d^3 - 24*d^
2*e*x + 12*d*e^2*x^2 + e^3*x^3) + 5*a*b^4*e*(-12*d^4 - 24*d^3*e*x + 36*d^2*e^2*x^2 + 8*d*e^3*x^3 + e^4*x^4) +
b^5*(12*d^5 - 120*d^3*e^2*x^2 - 60*d^2*e^3*x^3 - 20*d*e^4*x^4 - 3*e^5*x^5)) + 60*(b*d - a*e)^4*(b*B*d + 5*A*b*
e - 6*a*B*e)*(a + b*x)*Log[a + b*x])/(60*b^7*(a + b*x))

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fricas [B]  time = 0.90, size = 832, normalized size = 3.67 \[ \frac {12 \, B b^{6} e^{5} x^{6} + 60 \, {\left (B a b^{5} - A b^{6}\right )} d^{5} - 300 \, {\left (B a^{2} b^{4} - A a b^{5}\right )} d^{4} e + 600 \, {\left (B a^{3} b^{3} - A a^{2} b^{4}\right )} d^{3} e^{2} - 600 \, {\left (B a^{4} b^{2} - A a^{3} b^{3}\right )} d^{2} e^{3} + 300 \, {\left (B a^{5} b - A a^{4} b^{2}\right )} d e^{4} - 60 \, {\left (B a^{6} - A a^{5} b\right )} e^{5} + 3 \, {\left (25 \, B b^{6} d e^{4} - {\left (6 \, B a b^{5} - 5 \, A b^{6}\right )} e^{5}\right )} x^{5} + 5 \, {\left (40 \, B b^{6} d^{2} e^{3} - 5 \, {\left (5 \, B a b^{5} - 4 \, A b^{6}\right )} d e^{4} + {\left (6 \, B a^{2} b^{4} - 5 \, A a b^{5}\right )} e^{5}\right )} x^{4} + 10 \, {\left (30 \, B b^{6} d^{3} e^{2} - 10 \, {\left (4 \, B a b^{5} - 3 \, A b^{6}\right )} d^{2} e^{3} + 5 \, {\left (5 \, B a^{2} b^{4} - 4 \, A a b^{5}\right )} d e^{4} - {\left (6 \, B a^{3} b^{3} - 5 \, A a^{2} b^{4}\right )} e^{5}\right )} x^{3} + 30 \, {\left (10 \, B b^{6} d^{4} e - 10 \, {\left (3 \, B a b^{5} - 2 \, A b^{6}\right )} d^{3} e^{2} + 10 \, {\left (4 \, B a^{2} b^{4} - 3 \, A a b^{5}\right )} d^{2} e^{3} - 5 \, {\left (5 \, B a^{3} b^{3} - 4 \, A a^{2} b^{4}\right )} d e^{4} + {\left (6 \, B a^{4} b^{2} - 5 \, A a^{3} b^{3}\right )} e^{5}\right )} x^{2} + 60 \, {\left (5 \, B a b^{5} d^{4} e - 10 \, {\left (2 \, B a^{2} b^{4} - A a b^{5}\right )} d^{3} e^{2} + 10 \, {\left (3 \, B a^{3} b^{3} - 2 \, A a^{2} b^{4}\right )} d^{2} e^{3} - 5 \, {\left (4 \, B a^{4} b^{2} - 3 \, A a^{3} b^{3}\right )} d e^{4} + {\left (5 \, B a^{5} b - 4 \, A a^{4} b^{2}\right )} e^{5}\right )} x + 60 \, {\left (B a b^{5} d^{5} - 5 \, {\left (2 \, B a^{2} b^{4} - A a b^{5}\right )} d^{4} e + 10 \, {\left (3 \, B a^{3} b^{3} - 2 \, A a^{2} b^{4}\right )} d^{3} e^{2} - 10 \, {\left (4 \, B a^{4} b^{2} - 3 \, A a^{3} b^{3}\right )} d^{2} e^{3} + 5 \, {\left (5 \, B a^{5} b - 4 \, A a^{4} b^{2}\right )} d e^{4} - {\left (6 \, B a^{6} - 5 \, A a^{5} b\right )} e^{5} + {\left (B b^{6} d^{5} - 5 \, {\left (2 \, B a b^{5} - A b^{6}\right )} d^{4} e + 10 \, {\left (3 \, 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 (5 \, B a^{4} b^{2} - 4 \, A a^{3} b^{3}\right )} d e^{4} - {\left (6 \, B a^{5} b - 5 \, A a^{4} b^{2}\right )} e^{5}\right )} x\right )} \log \left (b x + a\right )}{60 \, {\left (b^{8} x + a b^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^5/(b*x+a)^2,x, algorithm="fricas")

[Out]

1/60*(12*B*b^6*e^5*x^6 + 60*(B*a*b^5 - A*b^6)*d^5 - 300*(B*a^2*b^4 - A*a*b^5)*d^4*e + 600*(B*a^3*b^3 - A*a^2*b
^4)*d^3*e^2 - 600*(B*a^4*b^2 - A*a^3*b^3)*d^2*e^3 + 300*(B*a^5*b - A*a^4*b^2)*d*e^4 - 60*(B*a^6 - A*a^5*b)*e^5
+ 3*(25*B*b^6*d*e^4 - (6*B*a*b^5 - 5*A*b^6)*e^5)*x^5 + 5*(40*B*b^6*d^2*e^3 - 5*(5*B*a*b^5 - 4*A*b^6)*d*e^4 +
(6*B*a^2*b^4 - 5*A*a*b^5)*e^5)*x^4 + 10*(30*B*b^6*d^3*e^2 - 10*(4*B*a*b^5 - 3*A*b^6)*d^2*e^3 + 5*(5*B*a^2*b^4
- 4*A*a*b^5)*d*e^4 - (6*B*a^3*b^3 - 5*A*a^2*b^4)*e^5)*x^3 + 30*(10*B*b^6*d^4*e - 10*(3*B*a*b^5 - 2*A*b^6)*d^3*
e^2 + 10*(4*B*a^2*b^4 - 3*A*a*b^5)*d^2*e^3 - 5*(5*B*a^3*b^3 - 4*A*a^2*b^4)*d*e^4 + (6*B*a^4*b^2 - 5*A*a^3*b^3)
*e^5)*x^2 + 60*(5*B*a*b^5*d^4*e - 10*(2*B*a^2*b^4 - A*a*b^5)*d^3*e^2 + 10*(3*B*a^3*b^3 - 2*A*a^2*b^4)*d^2*e^3
- 5*(4*B*a^4*b^2 - 3*A*a^3*b^3)*d*e^4 + (5*B*a^5*b - 4*A*a^4*b^2)*e^5)*x + 60*(B*a*b^5*d^5 - 5*(2*B*a^2*b^4 -
A*a*b^5)*d^4*e + 10*(3*B*a^3*b^3 - 2*A*a^2*b^4)*d^3*e^2 - 10*(4*B*a^4*b^2 - 3*A*a^3*b^3)*d^2*e^3 + 5*(5*B*a^5*
b - 4*A*a^4*b^2)*d*e^4 - (6*B*a^6 - 5*A*a^5*b)*e^5 + (B*b^6*d^5 - 5*(2*B*a*b^5 - A*b^6)*d^4*e + 10*(3*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*(5*B*a^4*b^2 - 4*A*a^3*b^3)*d*e^4 - (6*B*a
^5*b - 5*A*a^4*b^2)*e^5)*x)*log(b*x + a))/(b^8*x + a*b^7)

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giac [B]  time = 1.35, size = 706, normalized size = 3.11 \[ \frac {{\left (b x + a\right )}^{5} {\left (12 \, B e^{5} + \frac {15 \, {\left (5 \, B b^{2} d e^{4} - 6 \, B a b e^{5} + A b^{2} e^{5}\right )}}{{\left (b x + a\right )} b} + \frac {100 \, {\left (2 \, B b^{4} d^{2} e^{3} - 5 \, B a b^{3} d e^{4} + A b^{4} d e^{4} + 3 \, B a^{2} b^{2} e^{5} - A a b^{3} e^{5}\right )}}{{\left (b x + a\right )}^{2} b^{2}} + \frac {300 \, {\left (B b^{6} d^{3} e^{2} - 4 \, B a b^{5} d^{2} e^{3} + A b^{6} d^{2} e^{3} + 5 \, B a^{2} b^{4} d e^{4} - 2 \, A a b^{5} d e^{4} - 2 \, B a^{3} b^{3} e^{5} + A a^{2} b^{4} e^{5}\right )}}{{\left (b x + a\right )}^{3} b^{3}} + \frac {300 \, {\left (B b^{8} d^{4} e - 6 \, B a b^{7} d^{3} e^{2} + 2 \, A b^{8} d^{3} e^{2} + 12 \, B a^{2} b^{6} d^{2} e^{3} - 6 \, A a b^{7} d^{2} e^{3} - 10 \, B a^{3} b^{5} d e^{4} + 6 \, A a^{2} b^{6} d e^{4} + 3 \, B a^{4} b^{4} e^{5} - 2 \, A a^{3} b^{5} e^{5}\right )}}{{\left (b x + a\right )}^{4} b^{4}}\right )}}{60 \, b^{7}} - \frac {{\left (B b^{5} d^{5} - 10 \, B a b^{4} d^{4} e + 5 \, A b^{5} d^{4} e + 30 \, B a^{2} b^{3} d^{3} e^{2} - 20 \, A a b^{4} d^{3} e^{2} - 40 \, B a^{3} b^{2} d^{2} e^{3} + 30 \, A a^{2} b^{3} d^{2} e^{3} + 25 \, B a^{4} b d e^{4} - 20 \, A a^{3} b^{2} d e^{4} - 6 \, B a^{5} e^{5} + 5 \, A a^{4} b e^{5}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{7}} + \frac {\frac {B a b^{10} d^{5}}{b x + a} - \frac {A b^{11} d^{5}}{b x + a} - \frac {5 \, B a^{2} b^{9} d^{4} e}{b x + a} + \frac {5 \, A a b^{10} d^{4} e}{b x + a} + \frac {10 \, B a^{3} b^{8} d^{3} e^{2}}{b x + a} - \frac {10 \, A a^{2} b^{9} d^{3} e^{2}}{b x + a} - \frac {10 \, B a^{4} b^{7} d^{2} e^{3}}{b x + a} + \frac {10 \, A a^{3} b^{8} d^{2} e^{3}}{b x + a} + \frac {5 \, B a^{5} b^{6} d e^{4}}{b x + a} - \frac {5 \, A a^{4} b^{7} d e^{4}}{b x + a} - \frac {B a^{6} b^{5} e^{5}}{b x + a} + \frac {A a^{5} b^{6} e^{5}}{b x + a}}{b^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^5/(b*x+a)^2,x, algorithm="giac")

[Out]

1/60*(b*x + a)^5*(12*B*e^5 + 15*(5*B*b^2*d*e^4 - 6*B*a*b*e^5 + A*b^2*e^5)/((b*x + a)*b) + 100*(2*B*b^4*d^2*e^3
- 5*B*a*b^3*d*e^4 + A*b^4*d*e^4 + 3*B*a^2*b^2*e^5 - A*a*b^3*e^5)/((b*x + a)^2*b^2) + 300*(B*b^6*d^3*e^2 - 4*B
*a*b^5*d^2*e^3 + A*b^6*d^2*e^3 + 5*B*a^2*b^4*d*e^4 - 2*A*a*b^5*d*e^4 - 2*B*a^3*b^3*e^5 + A*a^2*b^4*e^5)/((b*x
+ a)^3*b^3) + 300*(B*b^8*d^4*e - 6*B*a*b^7*d^3*e^2 + 2*A*b^8*d^3*e^2 + 12*B*a^2*b^6*d^2*e^3 - 6*A*a*b^7*d^2*e^
3 - 10*B*a^3*b^5*d*e^4 + 6*A*a^2*b^6*d*e^4 + 3*B*a^4*b^4*e^5 - 2*A*a^3*b^5*e^5)/((b*x + a)^4*b^4))/b^7 - (B*b^
5*d^5 - 10*B*a*b^4*d^4*e + 5*A*b^5*d^4*e + 30*B*a^2*b^3*d^3*e^2 - 20*A*a*b^4*d^3*e^2 - 40*B*a^3*b^2*d^2*e^3 +
30*A*a^2*b^3*d^2*e^3 + 25*B*a^4*b*d*e^4 - 20*A*a^3*b^2*d*e^4 - 6*B*a^5*e^5 + 5*A*a^4*b*e^5)*log(abs(b*x + a)/(
(b*x + a)^2*abs(b)))/b^7 + (B*a*b^10*d^5/(b*x + a) - A*b^11*d^5/(b*x + a) - 5*B*a^2*b^9*d^4*e/(b*x + a) + 5*A*
a*b^10*d^4*e/(b*x + a) + 10*B*a^3*b^8*d^3*e^2/(b*x + a) - 10*A*a^2*b^9*d^3*e^2/(b*x + a) - 10*B*a^4*b^7*d^2*e^
3/(b*x + a) + 10*A*a^3*b^8*d^2*e^3/(b*x + a) + 5*B*a^5*b^6*d*e^4/(b*x + a) - 5*A*a^4*b^7*d*e^4/(b*x + a) - B*a
^6*b^5*e^5/(b*x + a) + A*a^5*b^6*e^5/(b*x + a))/b^12

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maple [B]  time = 0.01, size = 787, normalized size = 3.47 \[ \frac {B \,e^{5} x^{5}}{5 b^{2}}+\frac {A \,e^{5} x^{4}}{4 b^{2}}-\frac {B a \,e^{5} x^{4}}{2 b^{3}}+\frac {5 B d \,e^{4} x^{4}}{4 b^{2}}-\frac {2 A a \,e^{5} x^{3}}{3 b^{3}}+\frac {5 A d \,e^{4} x^{3}}{3 b^{2}}+\frac {B \,a^{2} e^{5} x^{3}}{b^{4}}-\frac {10 B a d \,e^{4} x^{3}}{3 b^{3}}+\frac {10 B \,d^{2} e^{3} x^{3}}{3 b^{2}}+\frac {3 A \,a^{2} e^{5} x^{2}}{2 b^{4}}-\frac {5 A a d \,e^{4} x^{2}}{b^{3}}+\frac {5 A \,d^{2} e^{3} x^{2}}{b^{2}}-\frac {2 B \,a^{3} e^{5} x^{2}}{b^{5}}+\frac {15 B \,a^{2} d \,e^{4} x^{2}}{2 b^{4}}-\frac {10 B a \,d^{2} e^{3} x^{2}}{b^{3}}+\frac {5 B \,d^{3} e^{2} x^{2}}{b^{2}}+\frac {A \,a^{5} e^{5}}{\left (b x +a \right ) b^{6}}-\frac {5 A \,a^{4} d \,e^{4}}{\left (b x +a \right ) b^{5}}+\frac {5 A \,a^{4} e^{5} \ln \left (b x +a \right )}{b^{6}}+\frac {10 A \,a^{3} d^{2} e^{3}}{\left (b x +a \right ) b^{4}}-\frac {20 A \,a^{3} d \,e^{4} \ln \left (b x +a \right )}{b^{5}}-\frac {4 A \,a^{3} e^{5} x}{b^{5}}-\frac {10 A \,a^{2} d^{3} e^{2}}{\left (b x +a \right ) b^{3}}+\frac {30 A \,a^{2} d^{2} e^{3} \ln \left (b x +a \right )}{b^{4}}+\frac {15 A \,a^{2} d \,e^{4} x}{b^{4}}+\frac {5 A a \,d^{4} e}{\left (b x +a \right ) b^{2}}-\frac {20 A a \,d^{3} e^{2} \ln \left (b x +a \right )}{b^{3}}-\frac {20 A a \,d^{2} e^{3} x}{b^{3}}-\frac {A \,d^{5}}{\left (b x +a \right ) b}+\frac {5 A \,d^{4} e \ln \left (b x +a \right )}{b^{2}}+\frac {10 A \,d^{3} e^{2} x}{b^{2}}-\frac {B \,a^{6} e^{5}}{\left (b x +a \right ) b^{7}}+\frac {5 B \,a^{5} d \,e^{4}}{\left (b x +a \right ) b^{6}}-\frac {6 B \,a^{5} e^{5} \ln \left (b x +a \right )}{b^{7}}-\frac {10 B \,a^{4} d^{2} e^{3}}{\left (b x +a \right ) b^{5}}+\frac {25 B \,a^{4} d \,e^{4} \ln \left (b x +a \right )}{b^{6}}+\frac {5 B \,a^{4} e^{5} x}{b^{6}}+\frac {10 B \,a^{3} d^{3} e^{2}}{\left (b x +a \right ) b^{4}}-\frac {40 B \,a^{3} d^{2} e^{3} \ln \left (b x +a \right )}{b^{5}}-\frac {20 B \,a^{3} d \,e^{4} x}{b^{5}}-\frac {5 B \,a^{2} d^{4} e}{\left (b x +a \right ) b^{3}}+\frac {30 B \,a^{2} d^{3} e^{2} \ln \left (b x +a \right )}{b^{4}}+\frac {30 B \,a^{2} d^{2} e^{3} x}{b^{4}}+\frac {B a \,d^{5}}{\left (b x +a \right ) b^{2}}-\frac {10 B a \,d^{4} e \ln \left (b x +a \right )}{b^{3}}-\frac {20 B a \,d^{3} e^{2} x}{b^{3}}+\frac {B \,d^{5} \ln \left (b x +a \right )}{b^{2}}+\frac {5 B \,d^{4} e x}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^5/(b*x+a)^2,x)

[Out]

1/5*e^5/b^2*B*x^5+1/4*e^5/b^2*A*x^4-1/b/(b*x+a)*A*d^5+1/b^2*ln(b*x+a)*B*d^5-1/b^7/(b*x+a)*B*a^6*e^5+1/b^2/(b*x
+a)*B*a*d^5+5/b^6*ln(b*x+a)*A*a^4*e^5+5/b^2*ln(b*x+a)*A*d^4*e-6/b^7*ln(b*x+a)*B*a^5*e^5+5*e^5/b^6*B*a^4*x+5*e/
b^2*B*d^4*x+5*e^2/b^2*B*x^2*d^3-4*e^5/b^5*A*a^3*x+10*e^2/b^2*A*d^3*x+3/2*e^5/b^4*A*x^2*a^2+5*e^3/b^2*A*x^2*d^2
-2*e^5/b^5*B*x^2*a^3-2/3*e^5/b^3*A*x^3*a+5/3*e^4/b^2*A*x^3*d+e^5/b^4*B*x^3*a^2-1/2*e^5/b^3*B*x^4*a+5/4*e^4/b^2
*B*x^4*d+10/3*e^3/b^2*B*x^3*d^2+1/b^6/(b*x+a)*A*a^5*e^5+15/2*e^4/b^4*B*x^2*a^2*d-10/3*e^4/b^3*B*x^3*a*d-20*e^3
/b^3*A*a*d^2*x-20*e^4/b^5*B*a^3*d*x+30*e^3/b^4*B*a^2*d^2*x-20*e^2/b^3*B*a*d^3*x-10*e^3/b^3*B*x^2*a*d^2+15*e^4/
b^4*A*a^2*d*x-5/b^5/(b*x+a)*A*a^4*d*e^4+10/b^4/(b*x+a)*A*a^3*d^2*e^3+30/b^4*ln(b*x+a)*B*a^2*d^3*e^2-10/b^3*ln(
b*x+a)*B*a*d^4*e-40/b^5*ln(b*x+a)*B*a^3*d^2*e^3-5*e^4/b^3*A*x^2*a*d-10/b^3/(b*x+a)*A*a^2*d^3*e^2+5/b^2/(b*x+a)
*a*A*d^4*e+5/b^6/(b*x+a)*B*a^5*d*e^4-10/b^5/(b*x+a)*B*a^4*d^2*e^3+10/b^4/(b*x+a)*B*a^3*d^3*e^2-5/b^3/(b*x+a)*B
*a^2*d^4*e-20/b^5*ln(b*x+a)*A*a^3*d*e^4+30/b^4*ln(b*x+a)*A*a^2*d^2*e^3-20/b^3*ln(b*x+a)*A*a*d^3*e^2+25/b^6*ln(
b*x+a)*B*a^4*d*e^4

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maxima [B]  time = 0.49, size = 579, normalized size = 2.55 \[ \frac {{\left (B a b^{5} - A b^{6}\right )} d^{5} - 5 \, {\left (B a^{2} b^{4} - A a b^{5}\right )} d^{4} e + 10 \, {\left (B a^{3} b^{3} - A a^{2} b^{4}\right )} d^{3} e^{2} - 10 \, {\left (B a^{4} b^{2} - A a^{3} b^{3}\right )} d^{2} e^{3} + 5 \, {\left (B a^{5} b - A a^{4} b^{2}\right )} d e^{4} - {\left (B a^{6} - A a^{5} b\right )} e^{5}}{b^{8} x + a b^{7}} + \frac {12 \, B b^{4} e^{5} x^{5} + 15 \, {\left (5 \, B b^{4} d e^{4} - {\left (2 \, B a b^{3} - A b^{4}\right )} e^{5}\right )} x^{4} + 20 \, {\left (10 \, B b^{4} d^{2} e^{3} - 5 \, {\left (2 \, B a b^{3} - A b^{4}\right )} d e^{4} + {\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 30 \, {\left (10 \, B b^{4} d^{3} e^{2} - 10 \, {\left (2 \, B a b^{3} - A b^{4}\right )} d^{2} e^{3} + 5 \, {\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d e^{4} - {\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 60 \, {\left (5 \, B b^{4} d^{4} e - 10 \, {\left (2 \, B a b^{3} - A b^{4}\right )} d^{3} e^{2} + 10 \, {\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d^{2} e^{3} - 5 \, {\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} d e^{4} + {\left (5 \, B a^{4} - 4 \, A a^{3} b\right )} e^{5}\right )} x}{60 \, b^{6}} + \frac {{\left (B b^{5} d^{5} - 5 \, {\left (2 \, B a b^{4} - A b^{5}\right )} d^{4} e + 10 \, {\left (3 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} d^{3} e^{2} - 10 \, {\left (4 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} d^{2} e^{3} + 5 \, {\left (5 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} d e^{4} - {\left (6 \, B a^{5} - 5 \, A a^{4} b\right )} e^{5}\right )} \log \left (b x + a\right )}{b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^5/(b*x+a)^2,x, algorithm="maxima")

[Out]

((B*a*b^5 - A*b^6)*d^5 - 5*(B*a^2*b^4 - A*a*b^5)*d^4*e + 10*(B*a^3*b^3 - A*a^2*b^4)*d^3*e^2 - 10*(B*a^4*b^2 -
A*a^3*b^3)*d^2*e^3 + 5*(B*a^5*b - A*a^4*b^2)*d*e^4 - (B*a^6 - A*a^5*b)*e^5)/(b^8*x + a*b^7) + 1/60*(12*B*b^4*e
^5*x^5 + 15*(5*B*b^4*d*e^4 - (2*B*a*b^3 - A*b^4)*e^5)*x^4 + 20*(10*B*b^4*d^2*e^3 - 5*(2*B*a*b^3 - A*b^4)*d*e^4
+ (3*B*a^2*b^2 - 2*A*a*b^3)*e^5)*x^3 + 30*(10*B*b^4*d^3*e^2 - 10*(2*B*a*b^3 - A*b^4)*d^2*e^3 + 5*(3*B*a^2*b^2
- 2*A*a*b^3)*d*e^4 - (4*B*a^3*b - 3*A*a^2*b^2)*e^5)*x^2 + 60*(5*B*b^4*d^4*e - 10*(2*B*a*b^3 - A*b^4)*d^3*e^2
+ 10*(3*B*a^2*b^2 - 2*A*a*b^3)*d^2*e^3 - 5*(4*B*a^3*b - 3*A*a^2*b^2)*d*e^4 + (5*B*a^4 - 4*A*a^3*b)*e^5)*x)/b^6
+ (B*b^5*d^5 - 5*(2*B*a*b^4 - A*b^5)*d^4*e + 10*(3*B*a^2*b^3 - 2*A*a*b^4)*d^3*e^2 - 10*(4*B*a^3*b^2 - 3*A*a^2
*b^3)*d^2*e^3 + 5*(5*B*a^4*b - 4*A*a^3*b^2)*d*e^4 - (6*B*a^5 - 5*A*a^4*b)*e^5)*log(b*x + a)/b^7

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mupad [B]  time = 1.14, size = 766, normalized size = 3.37 \[ x^2\,\left (\frac {a\,\left (\frac {2\,a\,\left (\frac {A\,e^5+5\,B\,d\,e^4}{b^2}-\frac {2\,B\,a\,e^5}{b^3}\right )}{b}-\frac {5\,d\,e^3\,\left (A\,e+2\,B\,d\right )}{b^2}+\frac {B\,a^2\,e^5}{b^4}\right )}{b}-\frac {a^2\,\left (\frac {A\,e^5+5\,B\,d\,e^4}{b^2}-\frac {2\,B\,a\,e^5}{b^3}\right )}{2\,b^2}+\frac {5\,d^2\,e^2\,\left (A\,e+B\,d\right )}{b^2}\right )-x^3\,\left (\frac {2\,a\,\left (\frac {A\,e^5+5\,B\,d\,e^4}{b^2}-\frac {2\,B\,a\,e^5}{b^3}\right )}{3\,b}-\frac {5\,d\,e^3\,\left (A\,e+2\,B\,d\right )}{3\,b^2}+\frac {B\,a^2\,e^5}{3\,b^4}\right )+x^4\,\left (\frac {A\,e^5+5\,B\,d\,e^4}{4\,b^2}-\frac {B\,a\,e^5}{2\,b^3}\right )+x\,\left (\frac {a^2\,\left (\frac {2\,a\,\left (\frac {A\,e^5+5\,B\,d\,e^4}{b^2}-\frac {2\,B\,a\,e^5}{b^3}\right )}{b}-\frac {5\,d\,e^3\,\left (A\,e+2\,B\,d\right )}{b^2}+\frac {B\,a^2\,e^5}{b^4}\right )}{b^2}-\frac {2\,a\,\left (\frac {2\,a\,\left (\frac {2\,a\,\left (\frac {A\,e^5+5\,B\,d\,e^4}{b^2}-\frac {2\,B\,a\,e^5}{b^3}\right )}{b}-\frac {5\,d\,e^3\,\left (A\,e+2\,B\,d\right )}{b^2}+\frac {B\,a^2\,e^5}{b^4}\right )}{b}-\frac {a^2\,\left (\frac {A\,e^5+5\,B\,d\,e^4}{b^2}-\frac {2\,B\,a\,e^5}{b^3}\right )}{b^2}+\frac {10\,d^2\,e^2\,\left (A\,e+B\,d\right )}{b^2}\right )}{b}+\frac {5\,d^3\,e\,\left (2\,A\,e+B\,d\right )}{b^2}\right )+\frac {\ln \left (a+b\,x\right )\,\left (-6\,B\,a^5\,e^5+25\,B\,a^4\,b\,d\,e^4+5\,A\,a^4\,b\,e^5-40\,B\,a^3\,b^2\,d^2\,e^3-20\,A\,a^3\,b^2\,d\,e^4+30\,B\,a^2\,b^3\,d^3\,e^2+30\,A\,a^2\,b^3\,d^2\,e^3-10\,B\,a\,b^4\,d^4\,e-20\,A\,a\,b^4\,d^3\,e^2+B\,b^5\,d^5+5\,A\,b^5\,d^4\,e\right )}{b^7}-\frac {B\,a^6\,e^5-5\,B\,a^5\,b\,d\,e^4-A\,a^5\,b\,e^5+10\,B\,a^4\,b^2\,d^2\,e^3+5\,A\,a^4\,b^2\,d\,e^4-10\,B\,a^3\,b^3\,d^3\,e^2-10\,A\,a^3\,b^3\,d^2\,e^3+5\,B\,a^2\,b^4\,d^4\,e+10\,A\,a^2\,b^4\,d^3\,e^2-B\,a\,b^5\,d^5-5\,A\,a\,b^5\,d^4\,e+A\,b^6\,d^5}{b\,\left (x\,b^7+a\,b^6\right )}+\frac {B\,e^5\,x^5}{5\,b^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^5)/(a + b*x)^2,x)

[Out]

x^2*((a*((2*a*((A*e^5 + 5*B*d*e^4)/b^2 - (2*B*a*e^5)/b^3))/b - (5*d*e^3*(A*e + 2*B*d))/b^2 + (B*a^2*e^5)/b^4))
/b - (a^2*((A*e^5 + 5*B*d*e^4)/b^2 - (2*B*a*e^5)/b^3))/(2*b^2) + (5*d^2*e^2*(A*e + B*d))/b^2) - x^3*((2*a*((A*
e^5 + 5*B*d*e^4)/b^2 - (2*B*a*e^5)/b^3))/(3*b) - (5*d*e^3*(A*e + 2*B*d))/(3*b^2) + (B*a^2*e^5)/(3*b^4)) + x^4*
((A*e^5 + 5*B*d*e^4)/(4*b^2) - (B*a*e^5)/(2*b^3)) + x*((a^2*((2*a*((A*e^5 + 5*B*d*e^4)/b^2 - (2*B*a*e^5)/b^3))
/b - (5*d*e^3*(A*e + 2*B*d))/b^2 + (B*a^2*e^5)/b^4))/b^2 - (2*a*((2*a*((2*a*((A*e^5 + 5*B*d*e^4)/b^2 - (2*B*a*
e^5)/b^3))/b - (5*d*e^3*(A*e + 2*B*d))/b^2 + (B*a^2*e^5)/b^4))/b - (a^2*((A*e^5 + 5*B*d*e^4)/b^2 - (2*B*a*e^5)
/b^3))/b^2 + (10*d^2*e^2*(A*e + B*d))/b^2))/b + (5*d^3*e*(2*A*e + B*d))/b^2) + (log(a + b*x)*(B*b^5*d^5 - 6*B*
a^5*e^5 + 5*A*a^4*b*e^5 + 5*A*b^5*d^4*e - 20*A*a*b^4*d^3*e^2 - 20*A*a^3*b^2*d*e^4 + 30*A*a^2*b^3*d^2*e^3 + 30*
B*a^2*b^3*d^3*e^2 - 40*B*a^3*b^2*d^2*e^3 - 10*B*a*b^4*d^4*e + 25*B*a^4*b*d*e^4))/b^7 - (A*b^6*d^5 + B*a^6*e^5
- A*a^5*b*e^5 - B*a*b^5*d^5 + 5*A*a^4*b^2*d*e^4 + 5*B*a^2*b^4*d^4*e + 10*A*a^2*b^4*d^3*e^2 - 10*A*a^3*b^3*d^2*
e^3 - 10*B*a^3*b^3*d^3*e^2 + 10*B*a^4*b^2*d^2*e^3 - 5*A*a*b^5*d^4*e - 5*B*a^5*b*d*e^4)/(b*(a*b^6 + b^7*x)) + (
B*e^5*x^5)/(5*b^2)

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sympy [B]  time = 3.21, size = 573, normalized size = 2.52 \[ \frac {B e^{5} x^{5}}{5 b^{2}} + x^{4} \left (\frac {A e^{5}}{4 b^{2}} - \frac {B a e^{5}}{2 b^{3}} + \frac {5 B d e^{4}}{4 b^{2}}\right ) + x^{3} \left (- \frac {2 A a e^{5}}{3 b^{3}} + \frac {5 A d e^{4}}{3 b^{2}} + \frac {B a^{2} e^{5}}{b^{4}} - \frac {10 B a d e^{4}}{3 b^{3}} + \frac {10 B d^{2} e^{3}}{3 b^{2}}\right ) + x^{2} \left (\frac {3 A a^{2} e^{5}}{2 b^{4}} - \frac {5 A a d e^{4}}{b^{3}} + \frac {5 A d^{2} e^{3}}{b^{2}} - \frac {2 B a^{3} e^{5}}{b^{5}} + \frac {15 B a^{2} d e^{4}}{2 b^{4}} - \frac {10 B a d^{2} e^{3}}{b^{3}} + \frac {5 B d^{3} e^{2}}{b^{2}}\right ) + x \left (- \frac {4 A a^{3} e^{5}}{b^{5}} + \frac {15 A a^{2} d e^{4}}{b^{4}} - \frac {20 A a d^{2} e^{3}}{b^{3}} + \frac {10 A d^{3} e^{2}}{b^{2}} + \frac {5 B a^{4} e^{5}}{b^{6}} - \frac {20 B a^{3} d e^{4}}{b^{5}} + \frac {30 B a^{2} d^{2} e^{3}}{b^{4}} - \frac {20 B a d^{3} e^{2}}{b^{3}} + \frac {5 B d^{4} e}{b^{2}}\right ) + \frac {A a^{5} b e^{5} - 5 A a^{4} b^{2} d e^{4} + 10 A a^{3} b^{3} d^{2} e^{3} - 10 A a^{2} b^{4} d^{3} e^{2} + 5 A a b^{5} d^{4} e - A b^{6} d^{5} - B a^{6} e^{5} + 5 B a^{5} b d e^{4} - 10 B a^{4} b^{2} d^{2} e^{3} + 10 B a^{3} b^{3} d^{3} e^{2} - 5 B a^{2} b^{4} d^{4} e + B a b^{5} d^{5}}{a b^{7} + b^{8} x} - \frac {\left (a e - b d\right )^{4} \left (- 5 A b e + 6 B a e - B b d\right ) \log {\left (a + b x \right )}}{b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**5/(b*x+a)**2,x)

[Out]

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

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