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

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

[Out]

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

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Rubi [A]  time = 0.070949, 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} \[ \log (x) (a f+b c)+x (a g+b d)+\frac{1}{2} x^2 (a h+b e)-\frac{a c}{3 x^3}-\frac{a d}{2 x^2}-\frac{a e}{x}+\frac{1}{3} b f x^3+\frac{1}{4} b g x^4+\frac{1}{5} b h x^5 \]

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^4,x]

[Out]

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

Mathematica [A]  time = 0.0605947, size = 76, normalized size = 0.88 \[ \log (x) (a f+b c)-\frac{a \left (2 c+3 x \left (d+2 e x+x^3 (-(2 g+h x))\right )\right )}{6 x^3}+\frac{1}{60} b x \left (60 d+x \left (30 e+x \left (20 f+15 g x+12 h x^2\right )\right )\right ) \]

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^4,x]

[Out]

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

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Maple [A]  time = 0.006, size = 76, normalized size = 0.9 \begin{align*}{\frac{bh{x}^{5}}{5}}+{\frac{bg{x}^{4}}{4}}+{\frac{bf{x}^{3}}{3}}+{\frac{{x}^{2}ah}{2}}+{\frac{be{x}^{2}}{2}}+agx+bdx+\ln \left ( x \right ) af+\ln \left ( x \right ) bc-{\frac{ac}{3\,{x}^{3}}}-{\frac{ad}{2\,{x}^{2}}}-{\frac{ae}{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^4,x)

[Out]

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

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Maxima [A]  time = 0.942655, size = 101, normalized size = 1.17 \begin{align*} \frac{1}{5} \, b h x^{5} + \frac{1}{4} \, b g x^{4} + \frac{1}{3} \, b f x^{3} + \frac{1}{2} \,{\left (b e + a h\right )} x^{2} +{\left (b d + a g\right )} x +{\left (b c + a f\right )} \log \left (x\right ) - \frac{6 \, a e x^{2} + 3 \, a d x + 2 \, a c}{6 \, x^{3}} \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^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.24613, size = 205, normalized size = 2.38 \begin{align*} \frac{12 \, b h x^{8} + 15 \, b g x^{7} + 20 \, b f x^{6} + 30 \,{\left (b e + a h\right )} x^{5} + 60 \,{\left (b d + a g\right )} x^{4} + 60 \,{\left (b c + a f\right )} x^{3} \log \left (x\right ) - 60 \, a e x^{2} - 30 \, a d x - 20 \, a c}{60 \, x^{3}} \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^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.732298, size = 82, normalized size = 0.95 \begin{align*} \frac{b f x^{3}}{3} + \frac{b g x^{4}}{4} + \frac{b h x^{5}}{5} + x^{2} \left (\frac{a h}{2} + \frac{b e}{2}\right ) + x \left (a g + b d\right ) + \left (a f + b c\right ) \log{\left (x \right )} - \frac{2 a c + 3 a d x + 6 a e x^{2}}{6 x^{3}} \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**4,x)

[Out]

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

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Giac [A]  time = 1.05191, size = 107, normalized size = 1.24 \begin{align*} \frac{1}{5} \, b h x^{5} + \frac{1}{4} \, b g x^{4} + \frac{1}{3} \, b f x^{3} + \frac{1}{2} \, a h x^{2} + \frac{1}{2} \, b x^{2} e + b d x + a g x +{\left (b c + a f\right )} \log \left ({\left | x \right |}\right ) - \frac{6 \, a x^{2} e + 3 \, a d x + 2 \, a c}{6 \, x^{3}} \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^4,x, algorithm="giac")

[Out]

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