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

Optimal. Leaf size=215 \[ -\frac {a^{10} A}{9 x^9}-\frac {a^9 (a B+10 A b)}{8 x^8}-\frac {5 a^8 b (2 a B+9 A b)}{7 x^7}-\frac {5 a^7 b^2 (3 a B+8 A b)}{2 x^6}-\frac {6 a^6 b^3 (4 a B+7 A b)}{x^5}-\frac {21 a^5 b^4 (5 a B+6 A b)}{2 x^4}-\frac {14 a^4 b^5 (6 a B+5 A b)}{x^3}-\frac {15 a^3 b^6 (7 a B+4 A b)}{x^2}-\frac {15 a^2 b^7 (8 a B+3 A b)}{x}+b^9 x (10 a B+A b)+5 a b^8 \log (x) (9 a B+2 A b)+\frac {1}{2} b^{10} B x^2 \]

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Rubi [A]  time = 0.16, antiderivative size = 215, 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)}{2 x^6}-\frac {6 a^6 b^3 (4 a B+7 A b)}{x^5}-\frac {21 a^5 b^4 (5 a B+6 A b)}{2 x^4}-\frac {14 a^4 b^5 (6 a B+5 A b)}{x^3}-\frac {15 a^3 b^6 (7 a B+4 A b)}{x^2}-\frac {15 a^2 b^7 (8 a B+3 A b)}{x}-\frac {a^9 (a B+10 A b)}{8 x^8}-\frac {5 a^8 b (2 a B+9 A b)}{7 x^7}-\frac {a^{10} A}{9 x^9}+b^9 x (10 a B+A b)+5 a b^8 \log (x) (9 a B+2 A b)+\frac {1}{2} b^{10} B x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/x^10,x]

[Out]

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

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Mathematica [A]  time = 0.12, size = 206, normalized size = 0.96 \begin {gather*} -\frac {a^{10} (8 A+9 B x)}{72 x^9}-\frac {5 a^9 b (7 A+8 B x)}{28 x^8}-\frac {15 a^8 b^2 (6 A+7 B x)}{14 x^7}-\frac {4 a^7 b^3 (5 A+6 B x)}{x^6}-\frac {21 a^6 b^4 (4 A+5 B x)}{2 x^5}-\frac {21 a^5 b^5 (3 A+4 B x)}{x^4}-\frac {35 a^4 b^6 (2 A+3 B x)}{x^3}-\frac {60 a^3 b^7 (A+2 B x)}{x^2}-\frac {45 a^2 A b^8}{x}+5 a b^8 \log (x) (9 a B+2 A b)+10 a b^9 B x+\frac {1}{2} b^{10} x (2 A+B x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-45*a^2*A*b^8)/x + 10*a*b^9*B*x + (b^10*x*(2*A + B*x))/2 - (60*a^3*b^7*(A + 2*B*x))/x^2 - (35*a^4*b^6*(2*A +
3*B*x))/x^3 - (21*a^5*b^5*(3*A + 4*B*x))/x^4 - (21*a^6*b^4*(4*A + 5*B*x))/(2*x^5) - (4*a^7*b^3*(5*A + 6*B*x))/
x^6 - (15*a^8*b^2*(6*A + 7*B*x))/(14*x^7) - (5*a^9*b*(7*A + 8*B*x))/(28*x^8) - (a^10*(8*A + 9*B*x))/(72*x^9) +
5*a*b^8*(2*A*b + 9*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^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.48, size = 245, normalized size = 1.14 \begin {gather*} \frac {252 \, B b^{10} x^{11} - 56 \, A a^{10} + 504 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 2520 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} \log \relax (x) - 7560 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} - 7560 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} - 7056 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} - 5292 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} - 3024 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} - 1260 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} - 360 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} - 63 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{504 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/504*(252*B*b^10*x^11 - 56*A*a^10 + 504*(10*B*a*b^9 + A*b^10)*x^10 + 2520*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9*log(x
) - 7560*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 - 7560*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 - 7056*(6*B*a^5*b^5 + 5*A*a^4*
b^6)*x^6 - 5292*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 - 3024*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 - 1260*(3*B*a^8*b^2 + 8
*A*a^7*b^3)*x^3 - 360*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 - 63*(B*a^10 + 10*A*a^9*b)*x)/x^9

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giac [A]  time = 0.83, size = 240, normalized size = 1.12 \begin {gather*} \frac {1}{2} \, B b^{10} x^{2} + 10 \, B a b^{9} x + A b^{10} x + 5 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} \log \left ({\left | x \right |}\right ) - \frac {56 \, A a^{10} + 7560 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 7560 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 7056 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 5292 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 3024 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 1260 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 360 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 63 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{504 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*B*b^10*x^2 + 10*B*a*b^9*x + A*b^10*x + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*log(abs(x)) - 1/504*(56*A*a^10 + 7560*(
8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 7560*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 7056*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 +
5292*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 3024*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 1260*(3*B*a^8*b^2 + 8*A*a^7*b^3
)*x^3 + 360*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 63*(B*a^10 + 10*A*a^9*b)*x)/x^9

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maple [A]  time = 0.01, size = 239, normalized size = 1.11 \begin {gather*} \frac {B \,b^{10} x^{2}}{2}+10 A a \,b^{9} \ln \relax (x )+A \,b^{10} x +45 B \,a^{2} b^{8} \ln \relax (x )+10 B a \,b^{9} x -\frac {45 A \,a^{2} b^{8}}{x}-\frac {120 B \,a^{3} b^{7}}{x}-\frac {60 A \,a^{3} b^{7}}{x^{2}}-\frac {105 B \,a^{4} b^{6}}{x^{2}}-\frac {70 A \,a^{4} b^{6}}{x^{3}}-\frac {84 B \,a^{5} b^{5}}{x^{3}}-\frac {63 A \,a^{5} b^{5}}{x^{4}}-\frac {105 B \,a^{6} b^{4}}{2 x^{4}}-\frac {42 A \,a^{6} b^{4}}{x^{5}}-\frac {24 B \,a^{7} b^{3}}{x^{5}}-\frac {20 A \,a^{7} b^{3}}{x^{6}}-\frac {15 B \,a^{8} b^{2}}{2 x^{6}}-\frac {45 A \,a^{8} b^{2}}{7 x^{7}}-\frac {10 B \,a^{9} b}{7 x^{7}}-\frac {5 A \,a^{9} b}{4 x^{8}}-\frac {B \,a^{10}}{8 x^{8}}-\frac {A \,a^{10}}{9 x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^10*B*x^2+b^10*A*x+10*a*b^9*B*x-20*a^7*b^3/x^6*A-15/2*a^8*b^2/x^6*B-42*a^6*b^4/x^5*A-24*a^7*b^3/x^5*B-45*
a^2*b^8/x*A-120*a^3*b^7/x*B+10*A*ln(x)*a*b^9+45*B*ln(x)*a^2*b^8-60*a^3*b^7/x^2*A-105*a^4*b^6/x^2*B-5/4*a^9/x^8
*A*b-1/8*a^10/x^8*B-1/9*a^10*A/x^9-70*a^4*b^6/x^3*A-84*a^5*b^5/x^3*B-45/7*a^8*b^2/x^7*A-10/7*a^9*b/x^7*B-63*a^
5*b^5/x^4*A-105/2*a^6*b^4/x^4*B

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maxima [A]  time = 1.13, size = 240, normalized size = 1.12 \begin {gather*} \frac {1}{2} \, B b^{10} x^{2} + {\left (10 \, B a b^{9} + A b^{10}\right )} x + 5 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} \log \relax (x) - \frac {56 \, A a^{10} + 7560 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 7560 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 7056 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 5292 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 3024 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 1260 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 360 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 63 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{504 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*B*b^10*x^2 + (10*B*a*b^9 + A*b^10)*x + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*log(x) - 1/504*(56*A*a^10 + 7560*(8*B*a
^3*b^7 + 3*A*a^2*b^8)*x^8 + 7560*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 7056*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 5292
*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 3024*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 1260*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3
+ 360*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 63*(B*a^10 + 10*A*a^9*b)*x)/x^9

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mupad [B]  time = 0.36, size = 232, normalized size = 1.08 \begin {gather*} x\,\left (A\,b^{10}+10\,B\,a\,b^9\right )+\ln \relax (x)\,\left (45\,B\,a^2\,b^8+10\,A\,a\,b^9\right )-\frac {x\,\left (\frac {B\,a^{10}}{8}+\frac {5\,A\,b\,a^9}{4}\right )+\frac {A\,a^{10}}{9}+x^2\,\left (\frac {10\,B\,a^9\,b}{7}+\frac {45\,A\,a^8\,b^2}{7}\right )+x^3\,\left (\frac {15\,B\,a^8\,b^2}{2}+20\,A\,a^7\,b^3\right )+x^4\,\left (24\,B\,a^7\,b^3+42\,A\,a^6\,b^4\right )+x^6\,\left (84\,B\,a^5\,b^5+70\,A\,a^4\,b^6\right )+x^7\,\left (105\,B\,a^4\,b^6+60\,A\,a^3\,b^7\right )+x^8\,\left (120\,B\,a^3\,b^7+45\,A\,a^2\,b^8\right )+x^5\,\left (\frac {105\,B\,a^6\,b^4}{2}+63\,A\,a^5\,b^5\right )}{x^9}+\frac {B\,b^{10}\,x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^10)/x^10,x)

[Out]

x*(A*b^10 + 10*B*a*b^9) + log(x)*(45*B*a^2*b^8 + 10*A*a*b^9) - (x*((B*a^10)/8 + (5*A*a^9*b)/4) + (A*a^10)/9 +
x^2*((45*A*a^8*b^2)/7 + (10*B*a^9*b)/7) + x^3*(20*A*a^7*b^3 + (15*B*a^8*b^2)/2) + x^4*(42*A*a^6*b^4 + 24*B*a^7
*b^3) + x^6*(70*A*a^4*b^6 + 84*B*a^5*b^5) + x^7*(60*A*a^3*b^7 + 105*B*a^4*b^6) + x^8*(45*A*a^2*b^8 + 120*B*a^3
*b^7) + x^5*(63*A*a^5*b^5 + (105*B*a^6*b^4)/2))/x^9 + (B*b^10*x^2)/2

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sympy [A]  time = 11.43, size = 252, normalized size = 1.17 \begin {gather*} \frac {B b^{10} x^{2}}{2} + 5 a b^{8} \left (2 A b + 9 B a\right ) \log {\relax (x )} + x \left (A b^{10} + 10 B a b^{9}\right ) + \frac {- 56 A a^{10} + x^{8} \left (- 22680 A a^{2} b^{8} - 60480 B a^{3} b^{7}\right ) + x^{7} \left (- 30240 A a^{3} b^{7} - 52920 B a^{4} b^{6}\right ) + x^{6} \left (- 35280 A a^{4} b^{6} - 42336 B a^{5} b^{5}\right ) + x^{5} \left (- 31752 A a^{5} b^{5} - 26460 B a^{6} b^{4}\right ) + x^{4} \left (- 21168 A a^{6} b^{4} - 12096 B a^{7} b^{3}\right ) + x^{3} \left (- 10080 A a^{7} b^{3} - 3780 B a^{8} b^{2}\right ) + x^{2} \left (- 3240 A a^{8} b^{2} - 720 B a^{9} b\right ) + x \left (- 630 A a^{9} b - 63 B a^{10}\right )}{504 x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*b**10*x**2/2 + 5*a*b**8*(2*A*b + 9*B*a)*log(x) + x*(A*b**10 + 10*B*a*b**9) + (-56*A*a**10 + x**8*(-22680*A*a
**2*b**8 - 60480*B*a**3*b**7) + x**7*(-30240*A*a**3*b**7 - 52920*B*a**4*b**6) + x**6*(-35280*A*a**4*b**6 - 423
36*B*a**5*b**5) + x**5*(-31752*A*a**5*b**5 - 26460*B*a**6*b**4) + x**4*(-21168*A*a**6*b**4 - 12096*B*a**7*b**3
) + x**3*(-10080*A*a**7*b**3 - 3780*B*a**8*b**2) + x**2*(-3240*A*a**8*b**2 - 720*B*a**9*b) + x*(-630*A*a**9*b
- 63*B*a**10))/(504*x**9)

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