3.545 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^3}{x} \, dx\)

Optimal. Leaf size=96 \[ a^6 A \log (x)+6 a^5 A b x+\frac {15}{2} a^4 A b^2 x^2+\frac {20}{3} a^3 A b^3 x^3+\frac {15}{4} a^2 A b^4 x^4+\frac {6}{5} a A b^5 x^5+\frac {B (a+b x)^7}{7 b}+\frac {1}{6} A b^6 x^6 \]

[Out]

6*a^5*A*b*x+15/2*a^4*A*b^2*x^2+20/3*a^3*A*b^3*x^3+15/4*a^2*A*b^4*x^4+6/5*a*A*b^5*x^5+1/6*A*b^6*x^6+1/7*B*(b*x+
a)^7/b+a^6*A*ln(x)

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {27, 80, 43} \[ \frac {15}{4} a^2 A b^4 x^4+\frac {20}{3} a^3 A b^3 x^3+\frac {15}{2} a^4 A b^2 x^2+6 a^5 A b x+a^6 A \log (x)+\frac {6}{5} a A b^5 x^5+\frac {B (a+b x)^7}{7 b}+\frac {1}{6} A b^6 x^6 \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x,x]

[Out]

6*a^5*A*b*x + (15*a^4*A*b^2*x^2)/2 + (20*a^3*A*b^3*x^3)/3 + (15*a^2*A*b^4*x^4)/4 + (6*a*A*b^5*x^5)/5 + (A*b^6*
x^6)/6 + (B*(a + b*x)^7)/(7*b) + a^6*A*Log[x]

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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])

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]

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x} \, dx &=\int \frac {(a+b x)^6 (A+B x)}{x} \, dx\\ &=\frac {B (a+b x)^7}{7 b}+A \int \frac {(a+b x)^6}{x} \, dx\\ &=\frac {B (a+b x)^7}{7 b}+A \int \left (6 a^5 b+\frac {a^6}{x}+15 a^4 b^2 x+20 a^3 b^3 x^2+15 a^2 b^4 x^3+6 a b^5 x^4+b^6 x^5\right ) \, dx\\ &=6 a^5 A b x+\frac {15}{2} a^4 A b^2 x^2+\frac {20}{3} a^3 A b^3 x^3+\frac {15}{4} a^2 A b^4 x^4+\frac {6}{5} a A b^5 x^5+\frac {1}{6} A b^6 x^6+\frac {B (a+b x)^7}{7 b}+a^6 A \log (x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 128, normalized size = 1.33 \[ a^6 A \log (x)+a^6 B x+3 a^5 b x (2 A+B x)+\frac {5}{2} a^4 b^2 x^2 (3 A+2 B x)+\frac {5}{3} a^3 b^3 x^3 (4 A+3 B x)+\frac {3}{4} a^2 b^4 x^4 (5 A+4 B x)+\frac {1}{5} a b^5 x^5 (6 A+5 B x)+\frac {1}{42} b^6 x^6 (7 A+6 B x) \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x,x]

[Out]

a^6*B*x + 3*a^5*b*x*(2*A + B*x) + (5*a^4*b^2*x^2*(3*A + 2*B*x))/2 + (5*a^3*b^3*x^3*(4*A + 3*B*x))/3 + (3*a^2*b
^4*x^4*(5*A + 4*B*x))/4 + (a*b^5*x^5*(6*A + 5*B*x))/5 + (b^6*x^6*(7*A + 6*B*x))/42 + a^6*A*Log[x]

________________________________________________________________________________________

fricas [A]  time = 0.71, size = 142, normalized size = 1.48 \[ \frac {1}{7} \, B b^{6} x^{7} + A a^{6} \log \relax (x) + \frac {1}{6} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + \frac {3}{5} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + \frac {5}{4} \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + \frac {5}{3} \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + \frac {3}{2} \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + {\left (B a^{6} + 6 \, A a^{5} b\right )} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x,x, algorithm="fricas")

[Out]

1/7*B*b^6*x^7 + A*a^6*log(x) + 1/6*(6*B*a*b^5 + A*b^6)*x^6 + 3/5*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 5/4*(4*B*a^3*
b^3 + 3*A*a^2*b^4)*x^4 + 5/3*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 3/2*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + (B*a^6 + 6*
A*a^5*b)*x

________________________________________________________________________________________

giac [A]  time = 0.17, size = 142, normalized size = 1.48 \[ \frac {1}{7} \, B b^{6} x^{7} + B a b^{5} x^{6} + \frac {1}{6} \, A b^{6} x^{6} + 3 \, B a^{2} b^{4} x^{5} + \frac {6}{5} \, A a b^{5} x^{5} + 5 \, B a^{3} b^{3} x^{4} + \frac {15}{4} \, A a^{2} b^{4} x^{4} + 5 \, B a^{4} b^{2} x^{3} + \frac {20}{3} \, A a^{3} b^{3} x^{3} + 3 \, B a^{5} b x^{2} + \frac {15}{2} \, A a^{4} b^{2} x^{2} + B a^{6} x + 6 \, A a^{5} b x + A a^{6} \log \left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x,x, algorithm="giac")

[Out]

1/7*B*b^6*x^7 + B*a*b^5*x^6 + 1/6*A*b^6*x^6 + 3*B*a^2*b^4*x^5 + 6/5*A*a*b^5*x^5 + 5*B*a^3*b^3*x^4 + 15/4*A*a^2
*b^4*x^4 + 5*B*a^4*b^2*x^3 + 20/3*A*a^3*b^3*x^3 + 3*B*a^5*b*x^2 + 15/2*A*a^4*b^2*x^2 + B*a^6*x + 6*A*a^5*b*x +
A*a^6*log(abs(x))

________________________________________________________________________________________

maple [A]  time = 0.04, size = 142, normalized size = 1.48 \[ \frac {B \,b^{6} x^{7}}{7}+\frac {A \,b^{6} x^{6}}{6}+B a \,b^{5} x^{6}+\frac {6 A a \,b^{5} x^{5}}{5}+3 B \,a^{2} b^{4} x^{5}+\frac {15 A \,a^{2} b^{4} x^{4}}{4}+5 B \,a^{3} b^{3} x^{4}+\frac {20 A \,a^{3} b^{3} x^{3}}{3}+5 B \,a^{4} b^{2} x^{3}+\frac {15 A \,a^{4} b^{2} x^{2}}{2}+3 B \,a^{5} b \,x^{2}+A \,a^{6} \ln \relax (x )+6 A \,a^{5} b x +B \,a^{6} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x,x)

[Out]

1/7*B*b^6*x^7+1/6*A*b^6*x^6+B*x^6*a*b^5+6/5*a*A*b^5*x^5+3*B*x^5*a^2*b^4+15/4*a^2*A*b^4*x^4+5*B*x^4*a^3*b^3+20/
3*a^3*A*b^3*x^3+5*B*x^3*a^4*b^2+15/2*a^4*A*b^2*x^2+3*B*x^2*a^5*b+6*a^5*A*b*x+B*a^6*x+a^6*A*ln(x)

________________________________________________________________________________________

maxima [A]  time = 0.51, size = 142, normalized size = 1.48 \[ \frac {1}{7} \, B b^{6} x^{7} + A a^{6} \log \relax (x) + \frac {1}{6} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + \frac {3}{5} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + \frac {5}{4} \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + \frac {5}{3} \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + \frac {3}{2} \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + {\left (B a^{6} + 6 \, A a^{5} b\right )} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x,x, algorithm="maxima")

[Out]

1/7*B*b^6*x^7 + A*a^6*log(x) + 1/6*(6*B*a*b^5 + A*b^6)*x^6 + 3/5*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 5/4*(4*B*a^3*
b^3 + 3*A*a^2*b^4)*x^4 + 5/3*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 3/2*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + (B*a^6 + 6*
A*a^5*b)*x

________________________________________________________________________________________

mupad [B]  time = 1.08, size = 125, normalized size = 1.30 \[ x\,\left (B\,a^6+6\,A\,b\,a^5\right )+x^6\,\left (\frac {A\,b^6}{6}+B\,a\,b^5\right )+\frac {B\,b^6\,x^7}{7}+A\,a^6\,\ln \relax (x)+\frac {5\,a^3\,b^2\,x^3\,\left (4\,A\,b+3\,B\,a\right )}{3}+\frac {5\,a^2\,b^3\,x^4\,\left (3\,A\,b+4\,B\,a\right )}{4}+\frac {3\,a^4\,b\,x^2\,\left (5\,A\,b+2\,B\,a\right )}{2}+\frac {3\,a\,b^4\,x^5\,\left (2\,A\,b+5\,B\,a\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^3)/x,x)

[Out]

x*(B*a^6 + 6*A*a^5*b) + x^6*((A*b^6)/6 + B*a*b^5) + (B*b^6*x^7)/7 + A*a^6*log(x) + (5*a^3*b^2*x^3*(4*A*b + 3*B
*a))/3 + (5*a^2*b^3*x^4*(3*A*b + 4*B*a))/4 + (3*a^4*b*x^2*(5*A*b + 2*B*a))/2 + (3*a*b^4*x^5*(2*A*b + 5*B*a))/5

________________________________________________________________________________________

sympy [A]  time = 0.28, size = 148, normalized size = 1.54 \[ A a^{6} \log {\relax (x )} + \frac {B b^{6} x^{7}}{7} + x^{6} \left (\frac {A b^{6}}{6} + B a b^{5}\right ) + x^{5} \left (\frac {6 A a b^{5}}{5} + 3 B a^{2} b^{4}\right ) + x^{4} \left (\frac {15 A a^{2} b^{4}}{4} + 5 B a^{3} b^{3}\right ) + x^{3} \left (\frac {20 A a^{3} b^{3}}{3} + 5 B a^{4} b^{2}\right ) + x^{2} \left (\frac {15 A a^{4} b^{2}}{2} + 3 B a^{5} b\right ) + x \left (6 A a^{5} b + B a^{6}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**3/x,x)

[Out]

A*a**6*log(x) + B*b**6*x**7/7 + x**6*(A*b**6/6 + B*a*b**5) + x**5*(6*A*a*b**5/5 + 3*B*a**2*b**4) + x**4*(15*A*
a**2*b**4/4 + 5*B*a**3*b**3) + x**3*(20*A*a**3*b**3/3 + 5*B*a**4*b**2) + x**2*(15*A*a**4*b**2/2 + 3*B*a**5*b)
+ x*(6*A*a**5*b + B*a**6)

________________________________________________________________________________________