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]

-1/4*a*c/x^4-1/3*a*d/x^3-1/2*a*e/x^2+(-a*f-b*c)/x+(a*h+b*e)*x+1/2*b*f*x^2+1/3*b*g*x^3+1/4*b*h*x^4+(a*g+b*d)*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} \[ -\frac {a f+b c}{x}+\log (x) (a g+b d)+x (a h+b e)-\frac {a c}{4 x^4}-\frac {a d}{3 x^3}-\frac {a e}{2 x^2}+\frac {1}{2} b f x^2+\frac {1}{3} b g x^3+\frac {1}{4} b h x^4 \]

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*}

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Mathematica [A]  time = 0.08, size = 77, normalized size = 0.90 \[ \log (x) (a g+b d)-\frac {a \left (3 c+4 d x+6 x^2 \left (e+2 f x-2 h x^3\right )\right )}{12 x^4}+b \left (-\frac {c}{x}+e x+\frac {1}{12} x^2 \left (6 f+4 g x+3 h x^2\right )\right ) \]

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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fricas [A]  time = 0.42, size = 81, normalized size = 0.94 \[ \frac {3 \, b h x^{8} + 4 \, b g x^{7} + 6 \, b f x^{6} + 12 \, {\left (b e + a h\right )} x^{5} + 12 \, {\left (b d + a g\right )} x^{4} \log \relax (x) - 6 \, a e x^{2} - 12 \, {\left (b c + a f\right )} x^{3} - 4 \, a d x - 3 \, a c}{12 \, x^{4}} \]

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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giac [A]  time = 0.15, size = 77, normalized size = 0.90 \[ \frac {1}{4} \, b h x^{4} + \frac {1}{3} \, b g x^{3} + \frac {1}{2} \, b f x^{2} + a h x + b x e + {\left (b d + a g\right )} \log \left ({\left | x \right |}\right ) - \frac {12 \, {\left (b c + a f\right )} x^{3} + 6 \, a x^{2} e + 4 \, a d x + 3 \, a c}{12 \, x^{4}} \]

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

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maple [A]  time = 0.05, size = 76, normalized size = 0.88 \[ \frac {b h \,x^{4}}{4}+\frac {b g \,x^{3}}{3}+\frac {b f \,x^{2}}{2}+a g \ln \relax (x )+a h x +b d \ln \relax (x )+b e x -\frac {a f}{x}-\frac {b c}{x}-\frac {a e}{2 x^{2}}-\frac {a d}{3 x^{3}}-\frac {a c}{4 x^{4}} \]

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

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maxima [A]  time = 1.34, size = 75, normalized size = 0.87 \[ \frac {1}{4} \, b h x^{4} + \frac {1}{3} \, b g x^{3} + \frac {1}{2} \, b f x^{2} + {\left (b e + a h\right )} x + {\left (b d + a g\right )} \log \relax (x) - \frac {6 \, a e x^{2} + 12 \, {\left (b c + a f\right )} x^{3} + 4 \, a d x + 3 \, a c}{12 \, x^{4}} \]

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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mupad [B]  time = 4.98, size = 74, normalized size = 0.86 \[ x\,\left (b\,e+a\,h\right )-\frac {\left (b\,c+a\,f\right )\,x^3+\frac {a\,e\,x^2}{2}+\frac {a\,d\,x}{3}+\frac {a\,c}{4}}{x^4}+\ln \relax (x)\,\left (b\,d+a\,g\right )+\frac {b\,h\,x^4}{4}+\frac {b\,f\,x^2}{2}+\frac {b\,g\,x^3}{3} \]

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^5,x)

[Out]

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

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sympy [A]  time = 2.57, size = 83, normalized size = 0.97 \[ \frac {b f x^{2}}{2} + \frac {b g x^{3}}{3} + \frac {b h x^{4}}{4} + x \left (a h + b e\right ) + \left (a g + b d\right ) \log {\relax (x )} + \frac {- 3 a c - 4 a d x - 6 a e x^{2} + x^{3} \left (- 12 a f - 12 b c\right )}{12 x^{4}} \]

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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