3.2.40 \(\int \frac {(a+b x)^{10} (A+B x)}{x^7} \, dx\)

Optimal. Leaf size=218 \[ -\frac {a^{10} A}{6 x^6}-\frac {a^9 (a B+10 A b)}{5 x^5}-\frac {5 a^8 b (2 a B+9 A b)}{4 x^4}-\frac {5 a^7 b^2 (3 a B+8 A b)}{x^3}-\frac {15 a^6 b^3 (4 a B+7 A b)}{x^2}-\frac {42 a^5 b^4 (5 a B+6 A b)}{x}+42 a^4 b^5 \log (x) (6 a B+5 A b)+30 a^3 b^6 x (7 a B+4 A b)+\frac {15}{2} a^2 b^7 x^2 (8 a B+3 A b)+\frac {1}{4} b^9 x^4 (10 a B+A b)+\frac {5}{3} a b^8 x^3 (9 a B+2 A b)+\frac {1}{5} b^{10} B x^5 \]

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Rubi [A]  time = 0.14, antiderivative size = 218, 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} \begin {gather*} -\frac {5 a^7 b^2 (3 a B+8 A b)}{x^3}-\frac {15 a^6 b^3 (4 a B+7 A b)}{x^2}+\frac {15}{2} a^2 b^7 x^2 (8 a B+3 A b)-\frac {42 a^5 b^4 (5 a B+6 A b)}{x}+30 a^3 b^6 x (7 a B+4 A b)+42 a^4 b^5 \log (x) (6 a B+5 A b)-\frac {a^9 (a B+10 A b)}{5 x^5}-\frac {5 a^8 b (2 a B+9 A b)}{4 x^4}-\frac {a^{10} A}{6 x^6}+\frac {5}{3} a b^8 x^3 (9 a B+2 A b)+\frac {1}{4} b^9 x^4 (10 a B+A b)+\frac {1}{5} b^{10} B x^5 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/x^7,x]

[Out]

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

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Mathematica [A]  time = 0.12, size = 210, normalized size = 0.96 \begin {gather*} -\frac {a^{10} (5 A+6 B x)}{30 x^6}-\frac {a^9 b (4 A+5 B x)}{2 x^5}-\frac {15 a^8 b^2 (3 A+4 B x)}{4 x^4}-\frac {20 a^7 b^3 (2 A+3 B x)}{x^3}-\frac {105 a^6 b^4 (A+2 B x)}{x^2}-\frac {252 a^5 A b^5}{x}+42 a^4 b^5 \log (x) (6 a B+5 A b)+210 a^4 b^6 B x+60 a^3 b^7 x (2 A+B x)+\frac {15}{2} a^2 b^8 x^2 (3 A+2 B x)+\frac {5}{6} a b^9 x^3 (4 A+3 B x)+\frac {1}{20} b^{10} x^4 (5 A+4 B x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/x^7,x]

[Out]

(-252*a^5*A*b^5)/x + 210*a^4*b^6*B*x + 60*a^3*b^7*x*(2*A + B*x) - (105*a^6*b^4*(A + 2*B*x))/x^2 + (15*a^2*b^8*
x^2*(3*A + 2*B*x))/2 - (20*a^7*b^3*(2*A + 3*B*x))/x^3 + (5*a*b^9*x^3*(4*A + 3*B*x))/6 - (15*a^8*b^2*(3*A + 4*B
*x))/(4*x^4) + (b^10*x^4*(5*A + 4*B*x))/20 - (a^9*b*(4*A + 5*B*x))/(2*x^5) - (a^10*(5*A + 6*B*x))/(30*x^6) + 4
2*a^4*b^5*(5*A*b + 6*a*B)*Log[x]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^{10} (A+B x)}{x^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)^10*(A + B*x))/x^7,x]

[Out]

IntegrateAlgebraic[((a + b*x)^10*(A + B*x))/x^7, x]

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fricas [A]  time = 1.25, size = 245, normalized size = 1.12 \begin {gather*} \frac {12 \, B b^{10} x^{11} - 10 \, A a^{10} + 15 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 100 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 450 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 1800 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 2520 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} \log \relax (x) - 2520 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} - 900 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} - 300 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} - 75 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} - 12 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{60 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^7,x, algorithm="fricas")

[Out]

1/60*(12*B*b^10*x^11 - 10*A*a^10 + 15*(10*B*a*b^9 + A*b^10)*x^10 + 100*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 450*(8*
B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 1800*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 2520*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6*log
(x) - 2520*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 - 900*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 - 300*(3*B*a^8*b^2 + 8*A*a^7*
b^3)*x^3 - 75*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 - 12*(B*a^10 + 10*A*a^9*b)*x)/x^6

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giac [A]  time = 0.88, size = 241, normalized size = 1.11 \begin {gather*} \frac {1}{5} \, B b^{10} x^{5} + \frac {5}{2} \, B a b^{9} x^{4} + \frac {1}{4} \, A b^{10} x^{4} + 15 \, B a^{2} b^{8} x^{3} + \frac {10}{3} \, A a b^{9} x^{3} + 60 \, B a^{3} b^{7} x^{2} + \frac {45}{2} \, A a^{2} b^{8} x^{2} + 210 \, B a^{4} b^{6} x + 120 \, A a^{3} b^{7} x + 42 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} \log \left ({\left | x \right |}\right ) - \frac {10 \, A a^{10} + 2520 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 900 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 300 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 75 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 12 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{60 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^7,x, algorithm="giac")

[Out]

1/5*B*b^10*x^5 + 5/2*B*a*b^9*x^4 + 1/4*A*b^10*x^4 + 15*B*a^2*b^8*x^3 + 10/3*A*a*b^9*x^3 + 60*B*a^3*b^7*x^2 + 4
5/2*A*a^2*b^8*x^2 + 210*B*a^4*b^6*x + 120*A*a^3*b^7*x + 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*log(abs(x)) - 1/60*(10*
A*a^10 + 2520*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 900*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 300*(3*B*a^8*b^2 + 8*A*a
^7*b^3)*x^3 + 75*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 12*(B*a^10 + 10*A*a^9*b)*x)/x^6

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maple [A]  time = 0.01, size = 240, normalized size = 1.10 \begin {gather*} \frac {B \,b^{10} x^{5}}{5}+\frac {A \,b^{10} x^{4}}{4}+\frac {5 B a \,b^{9} x^{4}}{2}+\frac {10 A a \,b^{9} x^{3}}{3}+15 B \,a^{2} b^{8} x^{3}+\frac {45 A \,a^{2} b^{8} x^{2}}{2}+60 B \,a^{3} b^{7} x^{2}+210 A \,a^{4} b^{6} \ln \relax (x )+120 A \,a^{3} b^{7} x +252 B \,a^{5} b^{5} \ln \relax (x )+210 B \,a^{4} b^{6} x -\frac {252 A \,a^{5} b^{5}}{x}-\frac {210 B \,a^{6} b^{4}}{x}-\frac {105 A \,a^{6} b^{4}}{x^{2}}-\frac {60 B \,a^{7} b^{3}}{x^{2}}-\frac {40 A \,a^{7} b^{3}}{x^{3}}-\frac {15 B \,a^{8} b^{2}}{x^{3}}-\frac {45 A \,a^{8} b^{2}}{4 x^{4}}-\frac {5 B \,a^{9} b}{2 x^{4}}-\frac {2 A \,a^{9} b}{x^{5}}-\frac {B \,a^{10}}{5 x^{5}}-\frac {A \,a^{10}}{6 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10*(B*x+A)/x^7,x)

[Out]

1/5*b^10*B*x^5+1/4*A*x^4*b^10+5/2*B*x^4*a*b^9+10/3*A*x^3*a*b^9+15*B*x^3*a^2*b^8+45/2*A*x^2*a^2*b^8+60*B*x^2*a^
3*b^7+120*a^3*b^7*A*x+210*a^4*b^6*B*x-1/6*a^10*A/x^6-2*a^9/x^5*A*b-1/5*a^10/x^5*B-252*a^5*b^5/x*A-210*a^6*b^4/
x*B+210*A*ln(x)*a^4*b^6+252*B*ln(x)*a^5*b^5-105*a^6*b^4/x^2*A-60*a^7*b^3/x^2*B-40*a^7*b^3/x^3*A-15*a^8*b^2/x^3
*B-45/4*a^8*b^2/x^4*A-5/2*a^9*b/x^4*B

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maxima [A]  time = 1.10, size = 241, normalized size = 1.11 \begin {gather*} \frac {1}{5} \, B b^{10} x^{5} + \frac {1}{4} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{4} + \frac {5}{3} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{3} + \frac {15}{2} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{2} + 30 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x + 42 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} \log \relax (x) - \frac {10 \, A a^{10} + 2520 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 900 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 300 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 75 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 12 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{60 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^7,x, algorithm="maxima")

[Out]

1/5*B*b^10*x^5 + 1/4*(10*B*a*b^9 + A*b^10)*x^4 + 5/3*(9*B*a^2*b^8 + 2*A*a*b^9)*x^3 + 15/2*(8*B*a^3*b^7 + 3*A*a
^2*b^8)*x^2 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x + 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*log(x) - 1/60*(10*A*a^10 + 252
0*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 900*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 300*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3
+ 75*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 12*(B*a^10 + 10*A*a^9*b)*x)/x^6

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mupad [B]  time = 0.08, size = 224, normalized size = 1.03 \begin {gather*} x^4\,\left (\frac {A\,b^{10}}{4}+\frac {5\,B\,a\,b^9}{2}\right )-\frac {x\,\left (\frac {B\,a^{10}}{5}+2\,A\,b\,a^9\right )+\frac {A\,a^{10}}{6}+x^2\,\left (\frac {5\,B\,a^9\,b}{2}+\frac {45\,A\,a^8\,b^2}{4}\right )+x^3\,\left (15\,B\,a^8\,b^2+40\,A\,a^7\,b^3\right )+x^4\,\left (60\,B\,a^7\,b^3+105\,A\,a^6\,b^4\right )+x^5\,\left (210\,B\,a^6\,b^4+252\,A\,a^5\,b^5\right )}{x^6}+\ln \relax (x)\,\left (252\,B\,a^5\,b^5+210\,A\,a^4\,b^6\right )+\frac {B\,b^{10}\,x^5}{5}+\frac {15\,a^2\,b^7\,x^2\,\left (3\,A\,b+8\,B\,a\right )}{2}+30\,a^3\,b^6\,x\,\left (4\,A\,b+7\,B\,a\right )+\frac {5\,a\,b^8\,x^3\,\left (2\,A\,b+9\,B\,a\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^10)/x^7,x)

[Out]

x^4*((A*b^10)/4 + (5*B*a*b^9)/2) - (x*((B*a^10)/5 + 2*A*a^9*b) + (A*a^10)/6 + x^2*((45*A*a^8*b^2)/4 + (5*B*a^9
*b)/2) + x^3*(40*A*a^7*b^3 + 15*B*a^8*b^2) + x^4*(105*A*a^6*b^4 + 60*B*a^7*b^3) + x^5*(252*A*a^5*b^5 + 210*B*a
^6*b^4))/x^6 + log(x)*(210*A*a^4*b^6 + 252*B*a^5*b^5) + (B*b^10*x^5)/5 + (15*a^2*b^7*x^2*(3*A*b + 8*B*a))/2 +
30*a^3*b^6*x*(4*A*b + 7*B*a) + (5*a*b^8*x^3*(2*A*b + 9*B*a))/3

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sympy [A]  time = 4.56, size = 253, normalized size = 1.16 \begin {gather*} \frac {B b^{10} x^{5}}{5} + 42 a^{4} b^{5} \left (5 A b + 6 B a\right ) \log {\relax (x )} + x^{4} \left (\frac {A b^{10}}{4} + \frac {5 B a b^{9}}{2}\right ) + x^{3} \left (\frac {10 A a b^{9}}{3} + 15 B a^{2} b^{8}\right ) + x^{2} \left (\frac {45 A a^{2} b^{8}}{2} + 60 B a^{3} b^{7}\right ) + x \left (120 A a^{3} b^{7} + 210 B a^{4} b^{6}\right ) + \frac {- 10 A a^{10} + x^{5} \left (- 15120 A a^{5} b^{5} - 12600 B a^{6} b^{4}\right ) + x^{4} \left (- 6300 A a^{6} b^{4} - 3600 B a^{7} b^{3}\right ) + x^{3} \left (- 2400 A a^{7} b^{3} - 900 B a^{8} b^{2}\right ) + x^{2} \left (- 675 A a^{8} b^{2} - 150 B a^{9} b\right ) + x \left (- 120 A a^{9} b - 12 B a^{10}\right )}{60 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10*(B*x+A)/x**7,x)

[Out]

B*b**10*x**5/5 + 42*a**4*b**5*(5*A*b + 6*B*a)*log(x) + x**4*(A*b**10/4 + 5*B*a*b**9/2) + x**3*(10*A*a*b**9/3 +
15*B*a**2*b**8) + x**2*(45*A*a**2*b**8/2 + 60*B*a**3*b**7) + x*(120*A*a**3*b**7 + 210*B*a**4*b**6) + (-10*A*a
**10 + x**5*(-15120*A*a**5*b**5 - 12600*B*a**6*b**4) + x**4*(-6300*A*a**6*b**4 - 3600*B*a**7*b**3) + x**3*(-24
00*A*a**7*b**3 - 900*B*a**8*b**2) + x**2*(-675*A*a**8*b**2 - 150*B*a**9*b) + x*(-120*A*a**9*b - 12*B*a**10))/(
60*x**6)

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