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

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

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Rubi [A]  time = 0.11, antiderivative size = 155, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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 \end {gather*}

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*}

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Mathematica [A]  time = 0.09, size = 169, normalized size = 1.09 \begin {gather*} -\frac {2 a^3 (5 A+6 B x)+3 a^2 x (3 A (4 b+5 c x)+5 B x (3 b+4 c 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 (b^3+6 b^2 c x-2 c^3 x^3\right )\right )-60 c^2 x^6 \log (x) (A c+3 b B)}{60 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/60*(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])/x^6

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 168, normalized size = 1.08 \begin {gather*} \frac {60 \, B c^{3} x^{7} + 60 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} \log \relax (x) - 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 {gather*}

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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giac [A]  time = 0.15, size = 162, normalized size = 1.05 \begin {gather*} 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 {gather*}

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

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maple [A]  time = 0.06, size = 188, normalized size = 1.21 \begin {gather*} A \,c^{3} \ln \relax (x )+3 B b \,c^{2} \ln \relax (x )+B \,c^{3} x -\frac {3 A b \,c^{2}}{x}-\frac {3 B a \,c^{2}}{x}-\frac {3 B \,b^{2} c}{x}-\frac {3 A a \,c^{2}}{2 x^{2}}-\frac {3 A \,b^{2} c}{2 x^{2}}-\frac {3 B a b c}{x^{2}}-\frac {B \,b^{3}}{2 x^{2}}-\frac {2 A a b c}{x^{3}}-\frac {A \,b^{3}}{3 x^{3}}-\frac {B \,a^{2} c}{x^{3}}-\frac {B a \,b^{2}}{x^{3}}-\frac {3 A \,a^{2} c}{4 x^{4}}-\frac {3 A a \,b^{2}}{4 x^{4}}-\frac {3 B \,a^{2} b}{4 x^{4}}-\frac {3 A \,a^{2} b}{5 x^{5}}-\frac {B \,a^{3}}{5 x^{5}}-\frac {A \,a^{3}}{6 x^{6}} \end {gather*}

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

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maxima [A]  time = 0.54, size = 162, normalized size = 1.05 \begin {gather*} B c^{3} x + {\left (3 \, B b c^{2} + A c^{3}\right )} \log \relax (x) - \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 {gather*}

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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mupad [B]  time = 0.09, size = 163, normalized size = 1.05 \begin {gather*} \ln \relax (x)\,\left (A\,c^3+3\,B\,b\,c^2\right )-\frac {x^3\,\left (B\,c\,a^2+B\,a\,b^2+2\,A\,c\,a\,b+\frac {A\,b^3}{3}\right )+x^4\,\left (\frac {B\,b^3}{2}+\frac {3\,A\,b^2\,c}{2}+3\,B\,a\,b\,c+\frac {3\,A\,a\,c^2}{2}\right )+x\,\left (\frac {B\,a^3}{5}+\frac {3\,A\,b\,a^2}{5}\right )+\frac {A\,a^3}{6}+x^2\,\left (\frac {3\,B\,a^2\,b}{4}+\frac {3\,A\,c\,a^2}{4}+\frac {3\,A\,a\,b^2}{4}\right )+x^5\,\left (3\,B\,b^2\,c+3\,A\,b\,c^2+3\,B\,a\,c^2\right )}{x^6}+B\,c^3\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 28.65, size = 187, normalized size = 1.21 \begin {gather*} B c^{3} x + c^{2} \left (A c + 3 B b\right ) \log {\relax (x )} + \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 {gather*}

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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