3.876 \(\int \frac{(A+B x) (a+b x+c x^2)^3}{x^7} \, dx\)

Optimal. Leaf size=155 \[ -\frac{a^2 (a B+3 A b)}{5 x^5}-\frac{a^3 A}{6 x^6}-\frac{3 a A c^2+6 a b B c+3 A b^2 c+b^3 B}{2 x^2}-\frac{3 a \left (A \left (a c+b^2\right )+a b B\right )}{4 x^4}-\frac{A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )}{3 x^3}-\frac{3 c \left (a B c+A b c+b^2 B\right )}{x}+c^2 \log (x) (A c+3 b B)+B c^3 x \]

[Out]

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

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Rubi [A]  time = 0.109169, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {765} \[ -\frac{a^2 (a B+3 A b)}{5 x^5}-\frac{a^3 A}{6 x^6}-\frac{3 a A c^2+6 a b B c+3 A b^2 c+b^3 B}{2 x^2}-\frac{3 a \left (A \left (a c+b^2\right )+a b B\right )}{4 x^4}-\frac{A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )}{3 x^3}-\frac{3 c \left (a B c+A b c+b^2 B\right )}{x}+c^2 \log (x) (A c+3 b B)+B c^3 x \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^3)/x^7,x]

[Out]

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

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [A]  time = 0.0909048, size = 169, normalized size = 1.09 \[ -\frac{3 a^2 x (3 A (4 b+5 c x)+5 B x (3 b+4 c x))+2 a^3 (5 A+6 B x)+15 a x^2 \left (A \left (3 b^2+8 b c x+6 c^2 x^2\right )+4 B x \left (b^2+3 b c x+3 c^2 x^2\right )\right )+10 x^3 \left (A b \left (2 b^2+9 b c x+18 c^2 x^2\right )+3 B x \left (6 b^2 c x+b^3-2 c^3 x^3\right )\right )-60 c^2 x^6 \log (x) (A c+3 b B)}{60 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/x^7,x]

[Out]

-(2*a^3*(5*A + 6*B*x) + 3*a^2*x*(5*B*x*(3*b + 4*c*x) + 3*A*(4*b + 5*c*x)) + 15*a*x^2*(4*B*x*(b^2 + 3*b*c*x + 3
*c^2*x^2) + A*(3*b^2 + 8*b*c*x + 6*c^2*x^2)) + 10*x^3*(A*b*(2*b^2 + 9*b*c*x + 18*c^2*x^2) + 3*B*x*(b^3 + 6*b^2
*c*x - 2*c^3*x^3)) - 60*c^2*(3*b*B + A*c)*x^6*Log[x])/(60*x^6)

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Maple [A]  time = 0.007, size = 188, normalized size = 1.2 \begin{align*} B{c}^{3}x+A\ln \left ( x \right ){c}^{3}+3\,B\ln \left ( x \right ) b{c}^{2}-2\,{\frac{Aabc}{{x}^{3}}}-{\frac{A{b}^{3}}{3\,{x}^{3}}}-{\frac{B{a}^{2}c}{{x}^{3}}}-{\frac{Ba{b}^{2}}{{x}^{3}}}-{\frac{3\,aA{c}^{2}}{2\,{x}^{2}}}-{\frac{3\,A{b}^{2}c}{2\,{x}^{2}}}-3\,{\frac{abBc}{{x}^{2}}}-{\frac{{b}^{3}B}{2\,{x}^{2}}}-3\,{\frac{Ab{c}^{2}}{x}}-3\,{\frac{Ba{c}^{2}}{x}}-3\,{\frac{B{b}^{2}c}{x}}-{\frac{3\,Ab{a}^{2}}{5\,{x}^{5}}}-{\frac{B{a}^{3}}{5\,{x}^{5}}}-{\frac{3\,A{a}^{2}c}{4\,{x}^{4}}}-{\frac{3\,Aa{b}^{2}}{4\,{x}^{4}}}-{\frac{3\,B{a}^{2}b}{4\,{x}^{4}}}-{\frac{A{a}^{3}}{6\,{x}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x+a)^3/x^7,x)

[Out]

B*c^3*x+A*ln(x)*c^3+3*B*ln(x)*b*c^2-2/x^3*A*a*b*c-1/3*A*b^3/x^3-1/x^3*B*a^2*c-1/x^3*B*a*b^2-3/2/x^2*a*A*c^2-3/
2*b^2/x^2*A*c-3/x^2*a*b*B*c-1/2*b^3/x^2*B-3*b*c^2/x*A-3*c^2/x*a*B-3*b^2*c/x*B-3/5*a^2/x^5*A*b-1/5*a^3/x^5*B-3/
4*a^2/x^4*A*c-3/4*a/x^4*A*b^2-3/4*a^2/x^4*b*B-1/6*a^3*A/x^6

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Maxima [A]  time = 1.09071, size = 219, normalized size = 1.41 \begin{align*} B c^{3} x +{\left (3 \, B b c^{2} + A c^{3}\right )} \log \left (x\right ) - \frac{180 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{5} + 30 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 10 \, A a^{3} + 20 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 45 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 12 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{60 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^3/x^7,x, algorithm="maxima")

[Out]

B*c^3*x + (3*B*b*c^2 + A*c^3)*log(x) - 1/60*(180*(B*b^2*c + (B*a + A*b)*c^2)*x^5 + 30*(B*b^3 + 3*A*a*c^2 + 3*(
2*B*a*b + A*b^2)*c)*x^4 + 10*A*a^3 + 20*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 + 45*(B*a^2*b + A*a*b^
2 + A*a^2*c)*x^2 + 12*(B*a^3 + 3*A*a^2*b)*x)/x^6

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Fricas [A]  time = 1.30756, size = 381, normalized size = 2.46 \begin{align*} \frac{60 \, B c^{3} x^{7} + 60 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} \log \left (x\right ) - 180 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{5} - 30 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} - 10 \, A a^{3} - 20 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} - 45 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} - 12 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{60 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^3/x^7,x, algorithm="fricas")

[Out]

1/60*(60*B*c^3*x^7 + 60*(3*B*b*c^2 + A*c^3)*x^6*log(x) - 180*(B*b^2*c + (B*a + A*b)*c^2)*x^5 - 30*(B*b^3 + 3*A
*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 - 10*A*a^3 - 20*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 - 45*(B*a^
2*b + A*a*b^2 + A*a^2*c)*x^2 - 12*(B*a^3 + 3*A*a^2*b)*x)/x^6

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Sympy [A]  time = 39.2824, size = 178, normalized size = 1.15 \begin{align*} B c^{3} x + c^{2} \left (A c + 3 B b\right ) \log{\left (x \right )} - \frac{10 A a^{3} + x^{5} \left (180 A b c^{2} + 180 B a c^{2} + 180 B b^{2} c\right ) + x^{4} \left (90 A a c^{2} + 90 A b^{2} c + 180 B a b c + 30 B b^{3}\right ) + x^{3} \left (120 A a b c + 20 A b^{3} + 60 B a^{2} c + 60 B a b^{2}\right ) + x^{2} \left (45 A a^{2} c + 45 A a b^{2} + 45 B a^{2} b\right ) + x \left (36 A a^{2} b + 12 B a^{3}\right )}{60 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x+a)**3/x**7,x)

[Out]

B*c**3*x + c**2*(A*c + 3*B*b)*log(x) - (10*A*a**3 + x**5*(180*A*b*c**2 + 180*B*a*c**2 + 180*B*b**2*c) + x**4*(
90*A*a*c**2 + 90*A*b**2*c + 180*B*a*b*c + 30*B*b**3) + x**3*(120*A*a*b*c + 20*A*b**3 + 60*B*a**2*c + 60*B*a*b*
*2) + x**2*(45*A*a**2*c + 45*A*a*b**2 + 45*B*a**2*b) + x*(36*A*a**2*b + 12*B*a**3))/(60*x**6)

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Giac [A]  time = 1.37135, size = 219, normalized size = 1.41 \begin{align*} B c^{3} x +{\left (3 \, B b c^{2} + A c^{3}\right )} \log \left ({\left | x \right |}\right ) - \frac{180 \,{\left (B b^{2} c + B a c^{2} + A b c^{2}\right )} x^{5} + 30 \,{\left (B b^{3} + 6 \, B a b c + 3 \, A b^{2} c + 3 \, A a c^{2}\right )} x^{4} + 10 \, A a^{3} + 20 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \, B a^{2} c + 6 \, A a b c\right )} x^{3} + 45 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 12 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{60 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^3/x^7,x, algorithm="giac")

[Out]

B*c^3*x + (3*B*b*c^2 + A*c^3)*log(abs(x)) - 1/60*(180*(B*b^2*c + B*a*c^2 + A*b*c^2)*x^5 + 30*(B*b^3 + 6*B*a*b*
c + 3*A*b^2*c + 3*A*a*c^2)*x^4 + 10*A*a^3 + 20*(3*B*a*b^2 + A*b^3 + 3*B*a^2*c + 6*A*a*b*c)*x^3 + 45*(B*a^2*b +
A*a*b^2 + A*a^2*c)*x^2 + 12*(B*a^3 + 3*A*a^2*b)*x)/x^6