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

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

[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]

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Rubi [A]  time = 0.0365966, antiderivative size = 96, normalized size of antiderivative = 1., 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 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) \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 = 0.0316995, size = 128, normalized size = 1.33 \[ \frac{5}{2} a^4 b^2 x^2 (3 A+2 B x)+\frac{5}{3} a^3 b^3 x^3 (4 A+3 B x)+\frac{3}{4} a^2 b^4 x^4 (5 A+4 B x)+3 a^5 b x (2 A+B x)+a^6 A \log (x)+a^6 B x+\frac{1}{5} a b^5 x^5 (6 A+5 B x)+\frac{1}{42} b^6 x^6 (7 A+6 B x) \]

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]

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Maple [A]  time = 0.001, size = 142, normalized size = 1.5 \begin{align*}{\frac{B{b}^{6}{x}^{7}}{7}}+{\frac{A{b}^{6}{x}^{6}}{6}}+B{x}^{6}a{b}^{5}+{\frac{6\,aA{b}^{5}{x}^{5}}{5}}+3\,B{x}^{5}{a}^{2}{b}^{4}+{\frac{15\,{a}^{2}A{b}^{4}{x}^{4}}{4}}+5\,B{x}^{4}{a}^{3}{b}^{3}+{\frac{20\,{a}^{3}A{b}^{3}{x}^{3}}{3}}+5\,B{x}^{3}{a}^{4}{b}^{2}+{\frac{15\,{a}^{4}A{b}^{2}{x}^{2}}{2}}+3\,B{x}^{2}{a}^{5}b+6\,{a}^{5}Abx+B{a}^{6}x+{a}^{6}A\ln \left ( x \right ) \end{align*}

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)

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Maxima [A]  time = 0.986948, size = 192, normalized size = 2. \begin{align*} \frac{1}{7} \, B b^{6} x^{7} + A a^{6} \log \left (x\right ) + \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 \end{align*}

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

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Fricas [A]  time = 1.47715, size = 313, normalized size = 3.26 \begin{align*} \frac{1}{7} \, B b^{6} x^{7} + A a^{6} \log \left (x\right ) + \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 \end{align*}

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

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Sympy [A]  time = 0.462634, size = 148, normalized size = 1.54 \begin{align*} A a^{6} \log{\left (x \right )} + \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 ) \end{align*}

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)

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Giac [A]  time = 1.13604, size = 192, normalized size = 2. \begin{align*} \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 ) \end{align*}

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))