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

Optimal. Leaf size=88 \[ \frac{1}{3} x^3 (a f+b c)+\frac{1}{4} x^4 (a g+b d)+\frac{1}{5} x^5 (a h+b e)+a c \log (x)+a d x+\frac{1}{2} a e x^2+\frac{1}{6} b f x^6+\frac{1}{7} b g x^7+\frac{1}{8} b h x^8 \]

[Out]

a*d*x + (a*e*x^2)/2 + ((b*c + a*f)*x^3)/3 + ((b*d + a*g)*x^4)/4 + ((b*e + a*h)*x^5)/5 + (b*f*x^6)/6 + (b*g*x^7
)/7 + (b*h*x^8)/8 + a*c*Log[x]

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Rubi [A]  time = 0.0578865, antiderivative size = 88, 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} \[ \frac{1}{3} x^3 (a f+b c)+\frac{1}{4} x^4 (a g+b d)+\frac{1}{5} x^5 (a h+b e)+a c \log (x)+a d x+\frac{1}{2} a e x^2+\frac{1}{6} b f x^6+\frac{1}{7} b g x^7+\frac{1}{8} b h x^8 \]

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,x]

[Out]

a*d*x + (a*e*x^2)/2 + ((b*c + a*f)*x^3)/3 + ((b*d + a*g)*x^4)/4 + ((b*e + a*h)*x^5)/5 + (b*f*x^6)/6 + (b*g*x^7
)/7 + (b*h*x^8)/8 + a*c*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} \, dx &=\int \left (a d+\frac{a c}{x}+a e x+(b c+a f) x^2+(b d+a g) x^3+(b e+a h) x^4+b f x^5+b g x^6+b h x^7\right ) \, dx\\ &=a d x+\frac{1}{2} a e x^2+\frac{1}{3} (b c+a f) x^3+\frac{1}{4} (b d+a g) x^4+\frac{1}{5} (b e+a h) x^5+\frac{1}{6} b f x^6+\frac{1}{7} b g x^7+\frac{1}{8} b h x^8+a c \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0287393, size = 88, normalized size = 1. \[ \frac{1}{3} x^3 (a f+b c)+\frac{1}{4} x^4 (a g+b d)+\frac{1}{5} x^5 (a h+b e)+a c \log (x)+a d x+\frac{1}{2} a e x^2+\frac{1}{6} b f x^6+\frac{1}{7} b g x^7+\frac{1}{8} b h x^8 \]

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,x]

[Out]

a*d*x + (a*e*x^2)/2 + ((b*c + a*f)*x^3)/3 + ((b*d + a*g)*x^4)/4 + ((b*e + a*h)*x^5)/5 + (b*f*x^6)/6 + (b*g*x^7
)/7 + (b*h*x^8)/8 + a*c*Log[x]

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Maple [A]  time = 0.003, size = 81, normalized size = 0.9 \begin{align*}{\frac{bh{x}^{8}}{8}}+{\frac{bg{x}^{7}}{7}}+{\frac{bf{x}^{6}}{6}}+{\frac{{x}^{5}ah}{5}}+{\frac{be{x}^{5}}{5}}+{\frac{{x}^{4}ag}{4}}+{\frac{bd{x}^{4}}{4}}+{\frac{{x}^{3}af}{3}}+{\frac{bc{x}^{3}}{3}}+{\frac{ae{x}^{2}}{2}}+adx+ac\ln \left ( x \right ) \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,x)

[Out]

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

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Maxima [A]  time = 0.943872, size = 100, normalized size = 1.14 \begin{align*} \frac{1}{8} \, b h x^{8} + \frac{1}{7} \, b g x^{7} + \frac{1}{6} \, b f x^{6} + \frac{1}{5} \,{\left (b e + a h\right )} x^{5} + \frac{1}{4} \,{\left (b d + a g\right )} x^{4} + \frac{1}{2} \, a e x^{2} + \frac{1}{3} \,{\left (b c + a f\right )} x^{3} + a d x + a c \log \left (x\right ) \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,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.992809, size = 192, normalized size = 2.18 \begin{align*} \frac{1}{8} \, b h x^{8} + \frac{1}{7} \, b g x^{7} + \frac{1}{6} \, b f x^{6} + \frac{1}{5} \,{\left (b e + a h\right )} x^{5} + \frac{1}{4} \,{\left (b d + a g\right )} x^{4} + \frac{1}{2} \, a e x^{2} + \frac{1}{3} \,{\left (b c + a f\right )} x^{3} + a d x + a c \log \left (x\right ) \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,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.339668, size = 85, normalized size = 0.97 \begin{align*} a c \log{\left (x \right )} + a d x + \frac{a e x^{2}}{2} + \frac{b f x^{6}}{6} + \frac{b g x^{7}}{7} + \frac{b h x^{8}}{8} + x^{5} \left (\frac{a h}{5} + \frac{b e}{5}\right ) + x^{4} \left (\frac{a g}{4} + \frac{b d}{4}\right ) + x^{3} \left (\frac{a f}{3} + \frac{b c}{3}\right ) \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,x)

[Out]

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

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Giac [A]  time = 1.05718, size = 112, normalized size = 1.27 \begin{align*} \frac{1}{8} \, b h x^{8} + \frac{1}{7} \, b g x^{7} + \frac{1}{6} \, b f x^{6} + \frac{1}{5} \, a h x^{5} + \frac{1}{5} \, b x^{5} e + \frac{1}{4} \, b d x^{4} + \frac{1}{4} \, a g x^{4} + \frac{1}{3} \, b c x^{3} + \frac{1}{3} \, a f x^{3} + \frac{1}{2} \, a x^{2} e + a d x + a c \log \left ({\left | x \right |}\right ) \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,x, algorithm="giac")

[Out]

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