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

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

[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]

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Rubi [A]  time = 0.155233, antiderivative size = 215, normalized size of antiderivative = 1., 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} \[ -\frac{5 a^7 b^2 (3 a B+8 A b)}{2 x^6}-\frac{6 a^6 b^3 (4 a B+7 A b)}{x^5}-\frac{21 a^5 b^4 (5 a B+6 A b)}{2 x^4}-\frac{14 a^4 b^5 (6 a B+5 A b)}{x^3}-\frac{15 a^3 b^6 (7 a B+4 A b)}{x^2}-\frac{15 a^2 b^7 (8 a B+3 A b)}{x}-\frac{a^9 (a B+10 A b)}{8 x^8}-\frac{5 a^8 b (2 a B+9 A b)}{7 x^7}-\frac{a^{10} A}{9 x^9}+b^9 x (10 a B+A b)+5 a b^8 \log (x) (9 a B+2 A b)+\frac{1}{2} b^{10} B x^2 \]

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*}

Mathematica [A]  time = 0.0889648, size = 206, normalized size = 0.96 \[ -\frac{15 a^8 b^2 (6 A+7 B x)}{14 x^7}-\frac{4 a^7 b^3 (5 A+6 B x)}{x^6}-\frac{21 a^6 b^4 (4 A+5 B x)}{2 x^5}-\frac{21 a^5 b^5 (3 A+4 B x)}{x^4}-\frac{35 a^4 b^6 (2 A+3 B x)}{x^3}-\frac{60 a^3 b^7 (A+2 B x)}{x^2}-\frac{45 a^2 A b^8}{x}-\frac{5 a^9 b (7 A+8 B x)}{28 x^8}-\frac{a^{10} (8 A+9 B x)}{72 x^9}+5 a b^8 \log (x) (9 a B+2 A b)+10 a b^9 B x+\frac{1}{2} b^{10} x (2 A+B x) \]

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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Maple [A]  time = 0.009, size = 239, normalized size = 1.1 \begin{align*}{\frac{{b}^{10}B{x}^{2}}{2}}+{b}^{10}Ax+10\,a{b}^{9}Bx+10\,A\ln \left ( x \right ) a{b}^{9}+45\,B\ln \left ( x \right ){a}^{2}{b}^{8}-70\,{\frac{{a}^{4}{b}^{6}A}{{x}^{3}}}-84\,{\frac{{a}^{5}{b}^{5}B}{{x}^{3}}}-42\,{\frac{{a}^{6}{b}^{4}A}{{x}^{5}}}-24\,{\frac{{a}^{7}{b}^{3}B}{{x}^{5}}}-63\,{\frac{{a}^{5}{b}^{5}A}{{x}^{4}}}-{\frac{105\,{a}^{6}{b}^{4}B}{2\,{x}^{4}}}-{\frac{5\,{a}^{9}bA}{4\,{x}^{8}}}-{\frac{{a}^{10}B}{8\,{x}^{8}}}-60\,{\frac{{a}^{3}{b}^{7}A}{{x}^{2}}}-105\,{\frac{{a}^{4}{b}^{6}B}{{x}^{2}}}-20\,{\frac{{a}^{7}{b}^{3}A}{{x}^{6}}}-{\frac{15\,{a}^{8}{b}^{2}B}{2\,{x}^{6}}}-{\frac{45\,{a}^{8}{b}^{2}A}{7\,{x}^{7}}}-{\frac{10\,{a}^{9}bB}{7\,{x}^{7}}}-45\,{\frac{{a}^{2}{b}^{8}A}{x}}-120\,{\frac{{a}^{3}{b}^{7}B}{x}}-{\frac{A{a}^{10}}{9\,{x}^{9}}} \end{align*}

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

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Maxima [A]  time = 1.03486, size = 324, normalized size = 1.51 \begin{align*} \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 \left (x\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{align*}

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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Fricas [A]  time = 1.33895, size = 563, normalized size = 2.62 \begin{align*} \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 \left (x\right ) - 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{align*}

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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Sympy [A]  time = 14.9016, size = 238, normalized size = 1.11 \begin{align*} \frac{B b^{10} x^{2}}{2} + 5 a b^{8} \left (2 A b + 9 B a\right ) \log{\left (x \right )} + 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{align*}

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 + 42336*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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Giac [A]  time = 1.21809, size = 324, normalized size = 1.51 \begin{align*} \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{align*}

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