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

Optimal. Leaf size=86 \[ \frac {1}{2} x^2 (a f+b c)+\frac {1}{3} x^3 (a g+b d)+\frac {1}{4} x^4 (a h+b e)-\frac {a c}{x}+a d \log (x)+a e x+\frac {1}{5} b f x^5+\frac {1}{6} b g x^6+\frac {1}{7} b h x^7 \]

[Out]

-a*c/x+a*e*x+1/2*(a*f+b*c)*x^2+1/3*(a*g+b*d)*x^3+1/4*(a*h+b*e)*x^4+1/5*b*f*x^5+1/6*b*g*x^6+1/7*b*h*x^7+a*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} \[ \frac {1}{2} x^2 (a f+b c)+\frac {1}{3} x^3 (a g+b d)+\frac {1}{4} x^4 (a h+b e)-\frac {a c}{x}+a d \log (x)+a e x+\frac {1}{5} b f x^5+\frac {1}{6} b g x^6+\frac {1}{7} b h x^7 \]

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^2,x]

[Out]

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

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Mathematica [A]  time = 0.07, size = 86, normalized size = 1.00 \[ \frac {1}{2} x^2 (a f+b c)+\frac {1}{3} x^3 (a g+b d)+\frac {1}{4} x^4 (a h+b e)-\frac {a c}{x}+a d \log (x)+a e x+\frac {1}{5} b f x^5+\frac {1}{6} b g x^6+\frac {1}{7} b h x^7 \]

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^2,x]

[Out]

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

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fricas [A]  time = 0.52, size = 81, normalized size = 0.94 \[ \frac {60 \, b h x^{8} + 70 \, b g x^{7} + 84 \, b f x^{6} + 105 \, {\left (b e + a h\right )} x^{5} + 140 \, {\left (b d + a g\right )} x^{4} + 420 \, a e x^{2} + 210 \, {\left (b c + a f\right )} x^{3} + 420 \, a d x \log \relax (x) - 420 \, a c}{420 \, x} \]

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^2,x, algorithm="fricas")

[Out]

1/420*(60*b*h*x^8 + 70*b*g*x^7 + 84*b*f*x^6 + 105*(b*e + a*h)*x^5 + 140*(b*d + a*g)*x^4 + 420*a*e*x^2 + 210*(b
*c + a*f)*x^3 + 420*a*d*x*log(x) - 420*a*c)/x

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giac [A]  time = 0.16, size = 83, normalized size = 0.97 \[ \frac {1}{7} \, b h x^{7} + \frac {1}{6} \, b g x^{6} + \frac {1}{5} \, b f x^{5} + \frac {1}{4} \, a h x^{4} + \frac {1}{4} \, b x^{4} e + \frac {1}{3} \, b d x^{3} + \frac {1}{3} \, a g x^{3} + \frac {1}{2} \, b c x^{2} + \frac {1}{2} \, a f x^{2} + a x e + a d \log \left ({\left | x \right |}\right ) - \frac {a c}{x} \]

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^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 81, normalized size = 0.94 \[ \frac {b h \,x^{7}}{7}+\frac {b g \,x^{6}}{6}+\frac {b f \,x^{5}}{5}+\frac {a h \,x^{4}}{4}+\frac {b e \,x^{4}}{4}+\frac {a g \,x^{3}}{3}+\frac {b d \,x^{3}}{3}+\frac {a f \,x^{2}}{2}+\frac {b c \,x^{2}}{2}+a d \ln \relax (x )+a e x -\frac {a c}{x} \]

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^2,x)

[Out]

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

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maxima [A]  time = 1.35, size = 74, normalized size = 0.86 \[ \frac {1}{7} \, b h x^{7} + \frac {1}{6} \, b g x^{6} + \frac {1}{5} \, b f x^{5} + \frac {1}{4} \, {\left (b e + a h\right )} x^{4} + \frac {1}{3} \, {\left (b d + a g\right )} x^{3} + a e x + \frac {1}{2} \, {\left (b c + a f\right )} x^{2} + a d \log \relax (x) - \frac {a c}{x} \]

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^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.05, size = 77, normalized size = 0.90 \[ x^2\,\left (\frac {b\,c}{2}+\frac {a\,f}{2}\right )+x^3\,\left (\frac {b\,d}{3}+\frac {a\,g}{3}\right )+x^4\,\left (\frac {b\,e}{4}+\frac {a\,h}{4}\right )+\frac {b\,h\,x^7}{7}+a\,d\,\ln \relax (x)+a\,e\,x-\frac {a\,c}{x}+\frac {b\,f\,x^5}{5}+\frac {b\,g\,x^6}{6} \]

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^2,x)

[Out]

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

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sympy [A]  time = 0.23, size = 82, normalized size = 0.95 \[ - \frac {a c}{x} + a d \log {\relax (x )} + a e x + \frac {b f x^{5}}{5} + \frac {b g x^{6}}{6} + \frac {b h x^{7}}{7} + x^{4} \left (\frac {a h}{4} + \frac {b e}{4}\right ) + x^{3} \left (\frac {a g}{3} + \frac {b d}{3}\right ) + x^{2} \left (\frac {a f}{2} + \frac {b c}{2}\right ) \]

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**2,x)

[Out]

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

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