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

Optimal. Leaf size=148 \[ \frac{45}{8} a^2 A b^8 x^8+\frac{120}{7} a^3 A b^7 x^7+35 a^4 A b^6 x^6+\frac{252}{5} a^5 A b^5 x^5+\frac{105}{2} a^6 A b^4 x^4+40 a^7 A b^3 x^3+\frac{45}{2} a^8 A b^2 x^2+10 a^9 A b x+a^{10} A \log (x)+\frac{10}{9} a A b^9 x^9+\frac{B (a+b x)^{11}}{11 b}+\frac{1}{10} A b^{10} x^{10} \]

[Out]

10*a^9*A*b*x + (45*a^8*A*b^2*x^2)/2 + 40*a^7*A*b^3*x^3 + (105*a^6*A*b^4*x^4)/2 + (252*a^5*A*b^5*x^5)/5 + 35*a^
4*A*b^6*x^6 + (120*a^3*A*b^7*x^7)/7 + (45*a^2*A*b^8*x^8)/8 + (10*a*A*b^9*x^9)/9 + (A*b^10*x^10)/10 + (B*(a + b
*x)^11)/(11*b) + a^10*A*Log[x]

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Rubi [A]  time = 0.0635937, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {80, 43} \[ \frac{45}{8} a^2 A b^8 x^8+\frac{120}{7} a^3 A b^7 x^7+35 a^4 A b^6 x^6+\frac{252}{5} a^5 A b^5 x^5+\frac{105}{2} a^6 A b^4 x^4+40 a^7 A b^3 x^3+\frac{45}{2} a^8 A b^2 x^2+10 a^9 A b x+a^{10} A \log (x)+\frac{10}{9} a A b^9 x^9+\frac{B (a+b x)^{11}}{11 b}+\frac{1}{10} A b^{10} x^{10} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/x,x]

[Out]

10*a^9*A*b*x + (45*a^8*A*b^2*x^2)/2 + 40*a^7*A*b^3*x^3 + (105*a^6*A*b^4*x^4)/2 + (252*a^5*A*b^5*x^5)/5 + 35*a^
4*A*b^6*x^6 + (120*a^3*A*b^7*x^7)/7 + (45*a^2*A*b^8*x^8)/8 + (10*a*A*b^9*x^9)/9 + (A*b^10*x^10)/10 + (B*(a + b
*x)^11)/(11*b) + a^10*A*Log[x]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{(a+b x)^{10} (A+B x)}{x} \, dx &=\frac{B (a+b x)^{11}}{11 b}+A \int \frac{(a+b x)^{10}}{x} \, dx\\ &=\frac{B (a+b x)^{11}}{11 b}+A \int \left (10 a^9 b+\frac{a^{10}}{x}+45 a^8 b^2 x+120 a^7 b^3 x^2+210 a^6 b^4 x^3+252 a^5 b^5 x^4+210 a^4 b^6 x^5+120 a^3 b^7 x^6+45 a^2 b^8 x^7+10 a b^9 x^8+b^{10} x^9\right ) \, dx\\ &=10 a^9 A b x+\frac{45}{2} a^8 A b^2 x^2+40 a^7 A b^3 x^3+\frac{105}{2} a^6 A b^4 x^4+\frac{252}{5} a^5 A b^5 x^5+35 a^4 A b^6 x^6+\frac{120}{7} a^3 A b^7 x^7+\frac{45}{8} a^2 A b^8 x^8+\frac{10}{9} a A b^9 x^9+\frac{1}{10} A b^{10} x^{10}+\frac{B (a+b x)^{11}}{11 b}+a^{10} A \log (x)\\ \end{align*}

Mathematica [A]  time = 0.049723, size = 208, normalized size = 1.41 \[ \frac{15}{2} a^8 b^2 x^2 (3 A+2 B x)+10 a^7 b^3 x^3 (4 A+3 B x)+\frac{21}{2} a^6 b^4 x^4 (5 A+4 B x)+\frac{42}{5} a^5 b^5 x^5 (6 A+5 B x)+5 a^4 b^6 x^6 (7 A+6 B x)+\frac{15}{7} a^3 b^7 x^7 (8 A+7 B x)+\frac{5}{8} a^2 b^8 x^8 (9 A+8 B x)+5 a^9 b x (2 A+B x)+a^{10} A \log (x)+a^{10} B x+\frac{1}{9} a b^9 x^9 (10 A+9 B x)+\frac{1}{110} b^{10} x^{10} (11 A+10 B x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^10*B*x + 5*a^9*b*x*(2*A + B*x) + (15*a^8*b^2*x^2*(3*A + 2*B*x))/2 + 10*a^7*b^3*x^3*(4*A + 3*B*x) + (21*a^6*b
^4*x^4*(5*A + 4*B*x))/2 + (42*a^5*b^5*x^5*(6*A + 5*B*x))/5 + 5*a^4*b^6*x^6*(7*A + 6*B*x) + (15*a^3*b^7*x^7*(8*
A + 7*B*x))/7 + (5*a^2*b^8*x^8*(9*A + 8*B*x))/8 + (a*b^9*x^9*(10*A + 9*B*x))/9 + (b^10*x^10*(11*A + 10*B*x))/1
10 + a^10*A*Log[x]

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Maple [A]  time = 0.004, size = 238, normalized size = 1.6 \begin{align*}{\frac{B{b}^{10}{x}^{11}}{11}}+{\frac{A{b}^{10}{x}^{10}}{10}}+B{x}^{10}a{b}^{9}+{\frac{10\,aA{b}^{9}{x}^{9}}{9}}+5\,B{x}^{9}{a}^{2}{b}^{8}+{\frac{45\,{a}^{2}A{b}^{8}{x}^{8}}{8}}+15\,B{x}^{8}{a}^{3}{b}^{7}+{\frac{120\,{a}^{3}A{b}^{7}{x}^{7}}{7}}+30\,B{x}^{7}{a}^{4}{b}^{6}+35\,{a}^{4}A{b}^{6}{x}^{6}+42\,B{x}^{6}{a}^{5}{b}^{5}+{\frac{252\,{a}^{5}A{b}^{5}{x}^{5}}{5}}+42\,B{x}^{5}{a}^{6}{b}^{4}+{\frac{105\,{a}^{6}A{b}^{4}{x}^{4}}{2}}+30\,B{x}^{4}{a}^{7}{b}^{3}+40\,{a}^{7}A{b}^{3}{x}^{3}+15\,B{x}^{3}{a}^{8}{b}^{2}+{\frac{45\,{a}^{8}A{b}^{2}{x}^{2}}{2}}+5\,B{x}^{2}{a}^{9}b+10\,{a}^{9}Abx+{a}^{10}Bx+{a}^{10}A\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*B*b^10*x^11+1/10*A*b^10*x^10+B*x^10*a*b^9+10/9*a*A*b^9*x^9+5*B*x^9*a^2*b^8+45/8*a^2*A*b^8*x^8+15*B*x^8*a^
3*b^7+120/7*a^3*A*b^7*x^7+30*B*x^7*a^4*b^6+35*a^4*A*b^6*x^6+42*B*x^6*a^5*b^5+252/5*a^5*A*b^5*x^5+42*B*x^5*a^6*
b^4+105/2*a^6*A*b^4*x^4+30*B*x^4*a^7*b^3+40*a^7*A*b^3*x^3+15*B*x^3*a^8*b^2+45/2*a^8*A*b^2*x^2+5*B*x^2*a^9*b+10
*a^9*A*b*x+a^10*B*x+a^10*A*ln(x)

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Maxima [A]  time = 0.997698, size = 321, normalized size = 2.17 \begin{align*} \frac{1}{11} \, B b^{10} x^{11} + A a^{10} \log \left (x\right ) + \frac{1}{10} \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + \frac{5}{9} \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + \frac{15}{8} \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + \frac{30}{7} \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 7 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + \frac{42}{5} \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + \frac{15}{2} \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 5 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + \frac{5}{2} \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} +{\left (B a^{10} + 10 \, A a^{9} b\right )} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*B*b^10*x^11 + A*a^10*log(x) + 1/10*(10*B*a*b^9 + A*b^10)*x^10 + 5/9*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 15/8*
(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 30/7*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 7*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 4
2/5*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 15/2*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 5*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3
+ 5/2*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + (B*a^10 + 10*A*a^9*b)*x

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Fricas [A]  time = 1.482, size = 532, normalized size = 3.59 \begin{align*} \frac{1}{11} \, B b^{10} x^{11} + A a^{10} \log \left (x\right ) + \frac{1}{10} \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + \frac{5}{9} \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + \frac{15}{8} \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + \frac{30}{7} \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 7 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + \frac{42}{5} \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + \frac{15}{2} \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 5 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + \frac{5}{2} \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} +{\left (B a^{10} + 10 \, A a^{9} b\right )} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*B*b^10*x^11 + A*a^10*log(x) + 1/10*(10*B*a*b^9 + A*b^10)*x^10 + 5/9*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 15/8*
(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 30/7*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 7*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 4
2/5*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 15/2*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 5*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3
+ 5/2*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + (B*a^10 + 10*A*a^9*b)*x

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Sympy [A]  time = 0.69903, size = 246, normalized size = 1.66 \begin{align*} A a^{10} \log{\left (x \right )} + \frac{B b^{10} x^{11}}{11} + x^{10} \left (\frac{A b^{10}}{10} + B a b^{9}\right ) + x^{9} \left (\frac{10 A a b^{9}}{9} + 5 B a^{2} b^{8}\right ) + x^{8} \left (\frac{45 A a^{2} b^{8}}{8} + 15 B a^{3} b^{7}\right ) + x^{7} \left (\frac{120 A a^{3} b^{7}}{7} + 30 B a^{4} b^{6}\right ) + x^{6} \left (35 A a^{4} b^{6} + 42 B a^{5} b^{5}\right ) + x^{5} \left (\frac{252 A a^{5} b^{5}}{5} + 42 B a^{6} b^{4}\right ) + x^{4} \left (\frac{105 A a^{6} b^{4}}{2} + 30 B a^{7} b^{3}\right ) + x^{3} \left (40 A a^{7} b^{3} + 15 B a^{8} b^{2}\right ) + x^{2} \left (\frac{45 A a^{8} b^{2}}{2} + 5 B a^{9} b\right ) + x \left (10 A a^{9} b + B a^{10}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**10*log(x) + B*b**10*x**11/11 + x**10*(A*b**10/10 + B*a*b**9) + x**9*(10*A*a*b**9/9 + 5*B*a**2*b**8) + x**
8*(45*A*a**2*b**8/8 + 15*B*a**3*b**7) + x**7*(120*A*a**3*b**7/7 + 30*B*a**4*b**6) + x**6*(35*A*a**4*b**6 + 42*
B*a**5*b**5) + x**5*(252*A*a**5*b**5/5 + 42*B*a**6*b**4) + x**4*(105*A*a**6*b**4/2 + 30*B*a**7*b**3) + x**3*(4
0*A*a**7*b**3 + 15*B*a**8*b**2) + x**2*(45*A*a**8*b**2/2 + 5*B*a**9*b) + x*(10*A*a**9*b + B*a**10)

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Giac [A]  time = 1.19042, size = 321, normalized size = 2.17 \begin{align*} \frac{1}{11} \, B b^{10} x^{11} + B a b^{9} x^{10} + \frac{1}{10} \, A b^{10} x^{10} + 5 \, B a^{2} b^{8} x^{9} + \frac{10}{9} \, A a b^{9} x^{9} + 15 \, B a^{3} b^{7} x^{8} + \frac{45}{8} \, A a^{2} b^{8} x^{8} + 30 \, B a^{4} b^{6} x^{7} + \frac{120}{7} \, A a^{3} b^{7} x^{7} + 42 \, B a^{5} b^{5} x^{6} + 35 \, A a^{4} b^{6} x^{6} + 42 \, B a^{6} b^{4} x^{5} + \frac{252}{5} \, A a^{5} b^{5} x^{5} + 30 \, B a^{7} b^{3} x^{4} + \frac{105}{2} \, A a^{6} b^{4} x^{4} + 15 \, B a^{8} b^{2} x^{3} + 40 \, A a^{7} b^{3} x^{3} + 5 \, B a^{9} b x^{2} + \frac{45}{2} \, A a^{8} b^{2} x^{2} + B a^{10} x + 10 \, A a^{9} b x + A a^{10} \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*B*b^10*x^11 + B*a*b^9*x^10 + 1/10*A*b^10*x^10 + 5*B*a^2*b^8*x^9 + 10/9*A*a*b^9*x^9 + 15*B*a^3*b^7*x^8 + 4
5/8*A*a^2*b^8*x^8 + 30*B*a^4*b^6*x^7 + 120/7*A*a^3*b^7*x^7 + 42*B*a^5*b^5*x^6 + 35*A*a^4*b^6*x^6 + 42*B*a^6*b^
4*x^5 + 252/5*A*a^5*b^5*x^5 + 30*B*a^7*b^3*x^4 + 105/2*A*a^6*b^4*x^4 + 15*B*a^8*b^2*x^3 + 40*A*a^7*b^3*x^3 + 5
*B*a^9*b*x^2 + 45/2*A*a^8*b^2*x^2 + B*a^10*x + 10*A*a^9*b*x + A*a^10*log(abs(x))