3.380 \(\int \frac {(a+b x^3) (c+d x+e x^2+f x^3+g x^4+h x^5)}{x^3} \, dx\)

Optimal. Leaf size=86 \[ x (a f+b c)+\frac {1}{2} x^2 (a g+b d)+\frac {1}{3} x^3 (a h+b e)-\frac {a c}{2 x^2}-\frac {a d}{x}+a e \log (x)+\frac {1}{4} b f x^4+\frac {1}{5} b g x^5+\frac {1}{6} b h x^6 \]

[Out]

-1/2*a*c/x^2-a*d/x+(a*f+b*c)*x+1/2*(a*g+b*d)*x^2+1/3*(a*h+b*e)*x^3+1/4*b*f*x^4+1/5*b*g*x^5+1/6*b*h*x^6+a*e*ln(
x)

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Rubi [A]  time = 0.07, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {1820} \[ x (a f+b c)+\frac {1}{2} x^2 (a g+b d)+\frac {1}{3} x^3 (a h+b e)-\frac {a c}{2 x^2}-\frac {a d}{x}+a e \log (x)+\frac {1}{4} b f x^4+\frac {1}{5} b g x^5+\frac {1}{6} b h x^6 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/x^3,x]

[Out]

-(a*c)/(2*x^2) - (a*d)/x + (b*c + a*f)*x + ((b*d + a*g)*x^2)/2 + ((b*e + a*h)*x^3)/3 + (b*f*x^4)/4 + (b*g*x^5)
/5 + (b*h*x^6)/6 + a*e*Log[x]

Rule 1820

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a +
b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin {align*} \int \frac {\left (a+b x^3\right ) \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{x^3} \, dx &=\int \left (b c \left (1+\frac {a f}{b c}\right )+\frac {a c}{x^3}+\frac {a d}{x^2}+\frac {a e}{x}+(b d+a g) x+(b e+a h) x^2+b f x^3+b g x^4+b h x^5\right ) \, dx\\ &=-\frac {a c}{2 x^2}-\frac {a d}{x}+(b c+a f) x+\frac {1}{2} (b d+a g) x^2+\frac {1}{3} (b e+a h) x^3+\frac {1}{4} b f x^4+\frac {1}{5} b g x^5+\frac {1}{6} b h x^6+a e \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 78, normalized size = 0.91 \[ \frac {a \left (-3 c-6 d x+6 f x^3+3 g x^4+2 h x^5\right )}{6 x^2}+a e \log (x)+b c x+\frac {1}{60} b x^2 \left (30 d+x \left (20 e+15 f x+12 g x^2+10 h x^3\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/x^3,x]

[Out]

b*c*x + (a*(-3*c - 6*d*x + 6*f*x^3 + 3*g*x^4 + 2*h*x^5))/(6*x^2) + (b*x^2*(30*d + x*(20*e + 15*f*x + 12*g*x^2
+ 10*h*x^3)))/60 + a*e*Log[x]

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fricas [A]  time = 0.59, size = 81, normalized size = 0.94 \[ \frac {10 \, b h x^{8} + 12 \, b g x^{7} + 15 \, b f x^{6} + 20 \, {\left (b e + a h\right )} x^{5} + 30 \, {\left (b d + a g\right )} x^{4} + 60 \, a e x^{2} \log \relax (x) + 60 \, {\left (b c + a f\right )} x^{3} - 60 \, a d x - 30 \, a c}{60 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x^3,x, algorithm="fricas")

[Out]

1/60*(10*b*h*x^8 + 12*b*g*x^7 + 15*b*f*x^6 + 20*(b*e + a*h)*x^5 + 30*(b*d + a*g)*x^4 + 60*a*e*x^2*log(x) + 60*
(b*c + a*f)*x^3 - 60*a*d*x - 30*a*c)/x^2

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giac [A]  time = 0.16, size = 80, normalized size = 0.93 \[ \frac {1}{6} \, b h x^{6} + \frac {1}{5} \, b g x^{5} + \frac {1}{4} \, b f x^{4} + \frac {1}{3} \, a h x^{3} + \frac {1}{3} \, b x^{3} e + \frac {1}{2} \, b d x^{2} + \frac {1}{2} \, a g x^{2} + b c x + a f x + a e \log \left ({\left | x \right |}\right ) - \frac {2 \, a d x + a c}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x^3,x, algorithm="giac")

[Out]

1/6*b*h*x^6 + 1/5*b*g*x^5 + 1/4*b*f*x^4 + 1/3*a*h*x^3 + 1/3*b*x^3*e + 1/2*b*d*x^2 + 1/2*a*g*x^2 + b*c*x + a*f*
x + a*e*log(abs(x)) - 1/2*(2*a*d*x + a*c)/x^2

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maple [A]  time = 0.05, size = 78, normalized size = 0.91 \[ \frac {b h \,x^{6}}{6}+\frac {b g \,x^{5}}{5}+\frac {b f \,x^{4}}{4}+\frac {a h \,x^{3}}{3}+\frac {b e \,x^{3}}{3}+\frac {a g \,x^{2}}{2}+\frac {b d \,x^{2}}{2}+a e \ln \relax (x )+a f x +b c x -\frac {a d}{x}-\frac {a c}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x^3,x)

[Out]

1/6*b*h*x^6+1/5*b*g*x^5+1/4*b*f*x^4+1/3*x^3*a*h+1/3*b*e*x^3+1/2*x^2*a*g+1/2*b*d*x^2+a*f*x+b*c*x-1/2*a*c/x^2-a*
d/x+a*e*ln(x)

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maxima [A]  time = 1.34, size = 74, normalized size = 0.86 \[ \frac {1}{6} \, b h x^{6} + \frac {1}{5} \, b g x^{5} + \frac {1}{4} \, b f x^{4} + \frac {1}{3} \, {\left (b e + a h\right )} x^{3} + \frac {1}{2} \, {\left (b d + a g\right )} x^{2} + a e \log \relax (x) + {\left (b c + a f\right )} x - \frac {2 \, a d x + a c}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x^3,x, algorithm="maxima")

[Out]

1/6*b*h*x^6 + 1/5*b*g*x^5 + 1/4*b*f*x^4 + 1/3*(b*e + a*h)*x^3 + 1/2*(b*d + a*g)*x^2 + a*e*log(x) + (b*c + a*f)
*x - 1/2*(2*a*d*x + a*c)/x^2

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mupad [B]  time = 0.04, size = 76, normalized size = 0.88 \[ x\,\left (b\,c+a\,f\right )-\frac {\frac {a\,c}{2}+a\,d\,x}{x^2}+x^2\,\left (\frac {b\,d}{2}+\frac {a\,g}{2}\right )+x^3\,\left (\frac {b\,e}{3}+\frac {a\,h}{3}\right )+\frac {b\,h\,x^6}{6}+a\,e\,\ln \relax (x)+\frac {b\,f\,x^4}{4}+\frac {b\,g\,x^5}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x^3)*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/x^3,x)

[Out]

x*(b*c + a*f) - ((a*c)/2 + a*d*x)/x^2 + x^2*((b*d)/2 + (a*g)/2) + x^3*((b*e)/3 + (a*h)/3) + (b*h*x^6)/6 + a*e*
log(x) + (b*f*x^4)/4 + (b*g*x^5)/5

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sympy [A]  time = 0.31, size = 83, normalized size = 0.97 \[ a e \log {\relax (x )} + \frac {b f x^{4}}{4} + \frac {b g x^{5}}{5} + \frac {b h x^{6}}{6} + x^{3} \left (\frac {a h}{3} + \frac {b e}{3}\right ) + x^{2} \left (\frac {a g}{2} + \frac {b d}{2}\right ) + x \left (a f + b c\right ) + \frac {- a c - 2 a d x}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)*(h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/x**3,x)

[Out]

a*e*log(x) + b*f*x**4/4 + b*g*x**5/5 + b*h*x**6/6 + x**3*(a*h/3 + b*e/3) + x**2*(a*g/2 + b*d/2) + x*(a*f + b*c
) + (-a*c - 2*a*d*x)/(2*x**2)

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