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]

-(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]

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Rubi [A]  time = 0.0724331, antiderivative size = 86, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0597417, 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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Maple [A]  time = 0.005, size = 78, normalized size = 0.9 \begin{align*}{\frac{bh{x}^{6}}{6}}+{\frac{bg{x}^{5}}{5}}+{\frac{bf{x}^{4}}{4}}+{\frac{{x}^{3}ah}{3}}+{\frac{be{x}^{3}}{3}}+{\frac{{x}^{2}ag}{2}}+{\frac{bd{x}^{2}}{2}}+afx+bcx+ae\ln \left ( x \right ) -{\frac{ac}{2\,{x}^{2}}}-{\frac{ad}{x}} \end{align*}

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+a*e*ln(x)-1/2*
a*c/x^2-a*d/x

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Maxima [A]  time = 0.940463, size = 100, normalized size = 1.16 \begin{align*} \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 \left (x\right ) +{\left (b c + a f\right )} x - \frac{2 \, a d x + a c}{2 \, x^{2}} \end{align*}

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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Fricas [A]  time = 1.11308, size = 205, normalized size = 2.38 \begin{align*} \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 \left (x\right ) + 60 \,{\left (b c + a f\right )} x^{3} - 60 \, a d x - 30 \, a c}{60 \, x^{2}} \end{align*}

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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Sympy [A]  time = 0.456412, size = 82, normalized size = 0.95 \begin{align*} a e \log{\left (x \right )} + \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}} \end{align*}

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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Giac [A]  time = 1.06629, size = 108, normalized size = 1.26 \begin{align*} \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}} \end{align*}

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