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

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

[Out]

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

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Rubi [A]  time = 0.0729936, 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} \[ -\frac{a f+b c}{x}+\log (x) (a g+b d)+x (a h+b e)-\frac{a c}{4 x^4}-\frac{a d}{3 x^3}-\frac{a e}{2 x^2}+\frac{1}{2} b f x^2+\frac{1}{3} b g x^3+\frac{1}{4} b h x^4 \]

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^5,x]

[Out]

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

Mathematica [A]  time = 0.0607921, size = 77, normalized size = 0.9 \[ \log (x) (a g+b d)-\frac{a \left (3 c+4 d x+6 x^2 \left (e+2 f x-2 h x^3\right )\right )}{12 x^4}+b \left (-\frac{c}{x}+e x+\frac{1}{12} x^2 \left (6 f+4 g x+3 h x^2\right )\right ) \]

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^5,x]

[Out]

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

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Maple [A]  time = 0.005, size = 76, normalized size = 0.9 \begin{align*}{\frac{bh{x}^{4}}{4}}+{\frac{bg{x}^{3}}{3}}+{\frac{bf{x}^{2}}{2}}+ahx+bxe+\ln \left ( x \right ) ag+\ln \left ( x \right ) bd-{\frac{ad}{3\,{x}^{3}}}-{\frac{ac}{4\,{x}^{4}}}-{\frac{ae}{2\,{x}^{2}}}-{\frac{af}{x}}-{\frac{bc}{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^5,x)

[Out]

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

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Maxima [A]  time = 1.20498, size = 101, normalized size = 1.17 \begin{align*} \frac{1}{4} \, b h x^{4} + \frac{1}{3} \, b g x^{3} + \frac{1}{2} \, b f x^{2} +{\left (b e + a h\right )} x +{\left (b d + a g\right )} \log \left (x\right ) - \frac{6 \, a e x^{2} + 12 \,{\left (b c + a f\right )} x^{3} + 4 \, a d x + 3 \, a c}{12 \, x^{4}} \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^5,x, algorithm="maxima")

[Out]

1/4*b*h*x^4 + 1/3*b*g*x^3 + 1/2*b*f*x^2 + (b*e + a*h)*x + (b*d + a*g)*log(x) - 1/12*(6*a*e*x^2 + 12*(b*c + a*f
)*x^3 + 4*a*d*x + 3*a*c)/x^4

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Fricas [A]  time = 1.23532, size = 197, normalized size = 2.29 \begin{align*} \frac{3 \, b h x^{8} + 4 \, b g x^{7} + 6 \, b f x^{6} + 12 \,{\left (b e + a h\right )} x^{5} + 12 \,{\left (b d + a g\right )} x^{4} \log \left (x\right ) - 6 \, a e x^{2} - 12 \,{\left (b c + a f\right )} x^{3} - 4 \, a d x - 3 \, a c}{12 \, x^{4}} \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^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.88505, size = 82, normalized size = 0.95 \begin{align*} \frac{b f x^{2}}{2} + \frac{b g x^{3}}{3} + \frac{b h x^{4}}{4} + x \left (a h + b e\right ) + \left (a g + b d\right ) \log{\left (x \right )} - \frac{3 a c + 4 a d x + 6 a e x^{2} + x^{3} \left (12 a f + 12 b c\right )}{12 x^{4}} \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**5,x)

[Out]

b*f*x**2/2 + b*g*x**3/3 + b*h*x**4/4 + x*(a*h + b*e) + (a*g + b*d)*log(x) - (3*a*c + 4*a*d*x + 6*a*e*x**2 + x*
*3*(12*a*f + 12*b*c))/(12*x**4)

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Giac [A]  time = 1.07568, size = 104, normalized size = 1.21 \begin{align*} \frac{1}{4} \, b h x^{4} + \frac{1}{3} \, b g x^{3} + \frac{1}{2} \, b f x^{2} + a h x + b x e +{\left (b d + a g\right )} \log \left ({\left | x \right |}\right ) - \frac{12 \,{\left (b c + a f\right )} x^{3} + 6 \, a x^{2} e + 4 \, a d x + 3 \, a c}{12 \, x^{4}} \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^5,x, algorithm="giac")

[Out]

1/4*b*h*x^4 + 1/3*b*g*x^3 + 1/2*b*f*x^2 + a*h*x + b*x*e + (b*d + a*g)*log(abs(x)) - 1/12*(12*(b*c + a*f)*x^3 +
6*a*x^2*e + 4*a*d*x + 3*a*c)/x^4