3.151 \(\int \frac {(a+b x)^{10} (A+B x)}{x^4} \, dx\)

Optimal. Leaf size=216 \[ -\frac {a^{10} A}{3 x^3}-\frac {a^9 (a B+10 A b)}{2 x^2}-\frac {5 a^8 b (2 a B+9 A b)}{x}+15 a^7 b^2 \log (x) (3 a B+8 A b)+30 a^6 b^3 x (4 a B+7 A b)+21 a^5 b^4 x^2 (5 a B+6 A b)+14 a^4 b^5 x^3 (6 a B+5 A b)+\frac {15}{2} a^3 b^6 x^4 (7 a B+4 A b)+3 a^2 b^7 x^5 (8 a B+3 A b)+\frac {1}{7} b^9 x^7 (10 a B+A b)+\frac {5}{6} a b^8 x^6 (9 a B+2 A b)+\frac {1}{8} b^{10} B x^8 \]

[Out]

-1/3*a^10*A/x^3-1/2*a^9*(10*A*b+B*a)/x^2-5*a^8*b*(9*A*b+2*B*a)/x+30*a^6*b^3*(7*A*b+4*B*a)*x+21*a^5*b^4*(6*A*b+
5*B*a)*x^2+14*a^4*b^5*(5*A*b+6*B*a)*x^3+15/2*a^3*b^6*(4*A*b+7*B*a)*x^4+3*a^2*b^7*(3*A*b+8*B*a)*x^5+5/6*a*b^8*(
2*A*b+9*B*a)*x^6+1/7*b^9*(A*b+10*B*a)*x^7+1/8*b^10*B*x^8+15*a^7*b^2*(8*A*b+3*B*a)*ln(x)

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Rubi [A]  time = 0.14, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {76} \[ 21 a^5 b^4 x^2 (5 a B+6 A b)+14 a^4 b^5 x^3 (6 a B+5 A b)+\frac {15}{2} a^3 b^6 x^4 (7 a B+4 A b)+3 a^2 b^7 x^5 (8 a B+3 A b)+30 a^6 b^3 x (4 a B+7 A b)+15 a^7 b^2 \log (x) (3 a B+8 A b)-\frac {a^9 (a B+10 A b)}{2 x^2}-\frac {5 a^8 b (2 a B+9 A b)}{x}-\frac {a^{10} A}{3 x^3}+\frac {5}{6} a b^8 x^6 (9 a B+2 A b)+\frac {1}{7} b^9 x^7 (10 a B+A b)+\frac {1}{8} b^{10} B x^8 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/x^4,x]

[Out]

-(a^10*A)/(3*x^3) - (a^9*(10*A*b + a*B))/(2*x^2) - (5*a^8*b*(9*A*b + 2*a*B))/x + 30*a^6*b^3*(7*A*b + 4*a*B)*x
+ 21*a^5*b^4*(6*A*b + 5*a*B)*x^2 + 14*a^4*b^5*(5*A*b + 6*a*B)*x^3 + (15*a^3*b^6*(4*A*b + 7*a*B)*x^4)/2 + 3*a^2
*b^7*(3*A*b + 8*a*B)*x^5 + (5*a*b^8*(2*A*b + 9*a*B)*x^6)/6 + (b^9*(A*b + 10*a*B)*x^7)/7 + (b^10*B*x^8)/8 + 15*
a^7*b^2*(8*A*b + 3*a*B)*Log[x]

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

\begin {align*} \int \frac {(a+b x)^{10} (A+B x)}{x^4} \, dx &=\int \left (30 a^6 b^3 (7 A b+4 a B)+\frac {a^{10} A}{x^4}+\frac {a^9 (10 A b+a B)}{x^3}+\frac {5 a^8 b (9 A b+2 a B)}{x^2}+\frac {15 a^7 b^2 (8 A b+3 a B)}{x}+42 a^5 b^4 (6 A b+5 a B) x+42 a^4 b^5 (5 A b+6 a B) x^2+30 a^3 b^6 (4 A b+7 a B) x^3+15 a^2 b^7 (3 A b+8 a B) x^4+5 a b^8 (2 A b+9 a B) x^5+b^9 (A b+10 a B) x^6+b^{10} B x^7\right ) \, dx\\ &=-\frac {a^{10} A}{3 x^3}-\frac {a^9 (10 A b+a B)}{2 x^2}-\frac {5 a^8 b (9 A b+2 a B)}{x}+30 a^6 b^3 (7 A b+4 a B) x+21 a^5 b^4 (6 A b+5 a B) x^2+14 a^4 b^5 (5 A b+6 a B) x^3+\frac {15}{2} a^3 b^6 (4 A b+7 a B) x^4+3 a^2 b^7 (3 A b+8 a B) x^5+\frac {5}{6} a b^8 (2 A b+9 a B) x^6+\frac {1}{7} b^9 (A b+10 a B) x^7+\frac {1}{8} b^{10} B x^8+15 a^7 b^2 (8 A b+3 a B) \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 208, normalized size = 0.96 \[ -\frac {a^{10} (2 A+3 B x)}{6 x^3}-\frac {5 a^9 b (A+2 B x)}{x^2}-\frac {45 a^8 A b^2}{x}+15 a^7 b^2 \log (x) (3 a B+8 A b)+120 a^7 b^3 B x+105 a^6 b^4 x (2 A+B x)+42 a^5 b^5 x^2 (3 A+2 B x)+\frac {35}{2} a^4 b^6 x^3 (4 A+3 B x)+6 a^3 b^7 x^4 (5 A+4 B x)+\frac {3}{2} a^2 b^8 x^5 (6 A+5 B x)+\frac {5}{21} a b^9 x^6 (7 A+6 B x)+\frac {1}{56} b^{10} x^7 (8 A+7 B x) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/x^4,x]

[Out]

(-45*a^8*A*b^2)/x + 120*a^7*b^3*B*x + 105*a^6*b^4*x*(2*A + B*x) - (5*a^9*b*(A + 2*B*x))/x^2 + 42*a^5*b^5*x^2*(
3*A + 2*B*x) - (a^10*(2*A + 3*B*x))/(6*x^3) + (35*a^4*b^6*x^3*(4*A + 3*B*x))/2 + 6*a^3*b^7*x^4*(5*A + 4*B*x) +
(3*a^2*b^8*x^5*(6*A + 5*B*x))/2 + (5*a*b^9*x^6*(7*A + 6*B*x))/21 + (b^10*x^7*(8*A + 7*B*x))/56 + 15*a^7*b^2*(
8*A*b + 3*a*B)*Log[x]

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fricas [A]  time = 0.68, size = 245, normalized size = 1.13 \[ \frac {21 \, B b^{10} x^{11} - 56 \, A a^{10} + 24 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 140 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 504 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 1260 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 2352 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 3528 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 5040 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 2520 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} \log \relax (x) - 840 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} - 84 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{168 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^4,x, algorithm="fricas")

[Out]

1/168*(21*B*b^10*x^11 - 56*A*a^10 + 24*(10*B*a*b^9 + A*b^10)*x^10 + 140*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 504*(8
*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 1260*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 2352*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 +
3528*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 5040*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 2520*(3*B*a^8*b^2 + 8*A*a^7*b^3)
*x^3*log(x) - 840*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 - 84*(B*a^10 + 10*A*a^9*b)*x)/x^3

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giac [A]  time = 0.85, size = 241, normalized size = 1.12 \[ \frac {1}{8} \, B b^{10} x^{8} + \frac {10}{7} \, B a b^{9} x^{7} + \frac {1}{7} \, A b^{10} x^{7} + \frac {15}{2} \, B a^{2} b^{8} x^{6} + \frac {5}{3} \, A a b^{9} x^{6} + 24 \, B a^{3} b^{7} x^{5} + 9 \, A a^{2} b^{8} x^{5} + \frac {105}{2} \, B a^{4} b^{6} x^{4} + 30 \, A a^{3} b^{7} x^{4} + 84 \, B a^{5} b^{5} x^{3} + 70 \, A a^{4} b^{6} x^{3} + 105 \, B a^{6} b^{4} x^{2} + 126 \, A a^{5} b^{5} x^{2} + 120 \, B a^{7} b^{3} x + 210 \, A a^{6} b^{4} x + 15 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} \log \left ({\left | x \right |}\right ) - \frac {2 \, A a^{10} + 30 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 3 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^4,x, algorithm="giac")

[Out]

1/8*B*b^10*x^8 + 10/7*B*a*b^9*x^7 + 1/7*A*b^10*x^7 + 15/2*B*a^2*b^8*x^6 + 5/3*A*a*b^9*x^6 + 24*B*a^3*b^7*x^5 +
9*A*a^2*b^8*x^5 + 105/2*B*a^4*b^6*x^4 + 30*A*a^3*b^7*x^4 + 84*B*a^5*b^5*x^3 + 70*A*a^4*b^6*x^3 + 105*B*a^6*b^
4*x^2 + 126*A*a^5*b^5*x^2 + 120*B*a^7*b^3*x + 210*A*a^6*b^4*x + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*log(abs(x)) - 1
/6*(2*A*a^10 + 30*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 3*(B*a^10 + 10*A*a^9*b)*x)/x^3

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maple [A]  time = 0.01, size = 240, normalized size = 1.11 \[ \frac {B \,b^{10} x^{8}}{8}+\frac {A \,b^{10} x^{7}}{7}+\frac {10 B a \,b^{9} x^{7}}{7}+\frac {5 A a \,b^{9} x^{6}}{3}+\frac {15 B \,a^{2} b^{8} x^{6}}{2}+9 A \,a^{2} b^{8} x^{5}+24 B \,a^{3} b^{7} x^{5}+30 A \,a^{3} b^{7} x^{4}+\frac {105 B \,a^{4} b^{6} x^{4}}{2}+70 A \,a^{4} b^{6} x^{3}+84 B \,a^{5} b^{5} x^{3}+126 A \,a^{5} b^{5} x^{2}+105 B \,a^{6} b^{4} x^{2}+120 A \,a^{7} b^{3} \ln \relax (x )+210 A \,a^{6} b^{4} x +45 B \,a^{8} b^{2} \ln \relax (x )+120 B \,a^{7} b^{3} x -\frac {45 A \,a^{8} b^{2}}{x}-\frac {10 B \,a^{9} b}{x}-\frac {5 A \,a^{9} b}{x^{2}}-\frac {B \,a^{10}}{2 x^{2}}-\frac {A \,a^{10}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10*(B*x+A)/x^4,x)

[Out]

1/8*b^10*B*x^8+1/7*A*x^7*b^10+10/7*B*x^7*a*b^9+5/3*A*x^6*a*b^9+15/2*B*x^6*a^2*b^8+9*A*x^5*a^2*b^8+24*B*x^5*a^3
*b^7+30*A*x^4*a^3*b^7+105/2*B*x^4*a^4*b^6+70*A*x^3*a^4*b^6+84*B*x^3*a^5*b^5+126*A*x^2*a^5*b^5+105*B*x^2*a^6*b^
4+210*a^6*b^4*A*x+120*a^7*b^3*B*x-45*a^8*b^2/x*A-10*a^9*b/x*B+120*A*ln(x)*a^7*b^3+45*B*ln(x)*a^8*b^2-5*a^9/x^2
*A*b-1/2*a^10/x^2*B-1/3*a^10*A/x^3

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maxima [A]  time = 1.04, size = 241, normalized size = 1.12 \[ \frac {1}{8} \, B b^{10} x^{8} + \frac {1}{7} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{7} + \frac {5}{6} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{6} + 3 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{5} + \frac {15}{2} \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{4} + 14 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{3} + 21 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{2} + 30 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x + 15 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} \log \relax (x) - \frac {2 \, A a^{10} + 30 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 3 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^4,x, algorithm="maxima")

[Out]

1/8*B*b^10*x^8 + 1/7*(10*B*a*b^9 + A*b^10)*x^7 + 5/6*(9*B*a^2*b^8 + 2*A*a*b^9)*x^6 + 3*(8*B*a^3*b^7 + 3*A*a^2*
b^8)*x^5 + 15/2*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^4 + 14*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^3 + 21*(5*B*a^6*b^4 + 6*A*a
^5*b^5)*x^2 + 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*log(x) - 1/6*(2*A*a^10 + 30*(2
*B*a^9*b + 9*A*a^8*b^2)*x^2 + 3*(B*a^10 + 10*A*a^9*b)*x)/x^3

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mupad [B]  time = 0.36, size = 215, normalized size = 1.00 \[ x^7\,\left (\frac {A\,b^{10}}{7}+\frac {10\,B\,a\,b^9}{7}\right )-\frac {x\,\left (\frac {B\,a^{10}}{2}+5\,A\,b\,a^9\right )+\frac {A\,a^{10}}{3}+x^2\,\left (10\,B\,a^9\,b+45\,A\,a^8\,b^2\right )}{x^3}+\ln \relax (x)\,\left (45\,B\,a^8\,b^2+120\,A\,a^7\,b^3\right )+\frac {B\,b^{10}\,x^8}{8}+21\,a^5\,b^4\,x^2\,\left (6\,A\,b+5\,B\,a\right )+14\,a^4\,b^5\,x^3\,\left (5\,A\,b+6\,B\,a\right )+\frac {15\,a^3\,b^6\,x^4\,\left (4\,A\,b+7\,B\,a\right )}{2}+3\,a^2\,b^7\,x^5\,\left (3\,A\,b+8\,B\,a\right )+30\,a^6\,b^3\,x\,\left (7\,A\,b+4\,B\,a\right )+\frac {5\,a\,b^8\,x^6\,\left (2\,A\,b+9\,B\,a\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^10)/x^4,x)

[Out]

x^7*((A*b^10)/7 + (10*B*a*b^9)/7) - (x*((B*a^10)/2 + 5*A*a^9*b) + (A*a^10)/3 + x^2*(45*A*a^8*b^2 + 10*B*a^9*b)
)/x^3 + log(x)*(120*A*a^7*b^3 + 45*B*a^8*b^2) + (B*b^10*x^8)/8 + 21*a^5*b^4*x^2*(6*A*b + 5*B*a) + 14*a^4*b^5*x
^3*(5*A*b + 6*B*a) + (15*a^3*b^6*x^4*(4*A*b + 7*B*a))/2 + 3*a^2*b^7*x^5*(3*A*b + 8*B*a) + 30*a^6*b^3*x*(7*A*b
+ 4*B*a) + (5*a*b^8*x^6*(2*A*b + 9*B*a))/6

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sympy [A]  time = 1.21, size = 250, normalized size = 1.16 \[ \frac {B b^{10} x^{8}}{8} + 15 a^{7} b^{2} \left (8 A b + 3 B a\right ) \log {\relax (x )} + x^{7} \left (\frac {A b^{10}}{7} + \frac {10 B a b^{9}}{7}\right ) + x^{6} \left (\frac {5 A a b^{9}}{3} + \frac {15 B a^{2} b^{8}}{2}\right ) + x^{5} \left (9 A a^{2} b^{8} + 24 B a^{3} b^{7}\right ) + x^{4} \left (30 A a^{3} b^{7} + \frac {105 B a^{4} b^{6}}{2}\right ) + x^{3} \left (70 A a^{4} b^{6} + 84 B a^{5} b^{5}\right ) + x^{2} \left (126 A a^{5} b^{5} + 105 B a^{6} b^{4}\right ) + x \left (210 A a^{6} b^{4} + 120 B a^{7} b^{3}\right ) + \frac {- 2 A a^{10} + x^{2} \left (- 270 A a^{8} b^{2} - 60 B a^{9} b\right ) + x \left (- 30 A a^{9} b - 3 B a^{10}\right )}{6 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10*(B*x+A)/x**4,x)

[Out]

B*b**10*x**8/8 + 15*a**7*b**2*(8*A*b + 3*B*a)*log(x) + x**7*(A*b**10/7 + 10*B*a*b**9/7) + x**6*(5*A*a*b**9/3 +
15*B*a**2*b**8/2) + x**5*(9*A*a**2*b**8 + 24*B*a**3*b**7) + x**4*(30*A*a**3*b**7 + 105*B*a**4*b**6/2) + x**3*
(70*A*a**4*b**6 + 84*B*a**5*b**5) + x**2*(126*A*a**5*b**5 + 105*B*a**6*b**4) + x*(210*A*a**6*b**4 + 120*B*a**7
*b**3) + (-2*A*a**10 + x**2*(-270*A*a**8*b**2 - 60*B*a**9*b) + x*(-30*A*a**9*b - 3*B*a**10))/(6*x**3)

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