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

Optimal. Leaf size=152 \[ -\frac {a^2 c}{3 x^3}-\frac {a^2 d}{2 x^2}-\frac {a^2 e}{x}+\frac {1}{3} b x^3 (2 a f+b c)+a \log (x) (a f+2 b c)+\frac {1}{4} b x^4 (2 a g+b d)+a x (a g+2 b d)+\frac {1}{5} b x^5 (2 a h+b e)+\frac {1}{2} a x^2 (a h+2 b e)+\frac {1}{6} b^2 f x^6+\frac {1}{7} b^2 g x^7+\frac {1}{8} b^2 h x^8 \]

[Out]

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

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Rubi [A]  time = 0.12, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {1820} \[ -\frac {a^2 c}{3 x^3}-\frac {a^2 d}{2 x^2}-\frac {a^2 e}{x}+\frac {1}{3} b x^3 (2 a f+b c)+a \log (x) (a f+2 b c)+\frac {1}{4} b x^4 (2 a g+b d)+a x (a g+2 b d)+\frac {1}{5} b x^5 (2 a h+b e)+\frac {1}{2} a x^2 (a h+2 b e)+\frac {1}{6} b^2 f x^6+\frac {1}{7} b^2 g x^7+\frac {1}{8} b^2 h x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.10, size = 123, normalized size = 0.81 \[ -\frac {a^2 \left (2 c+3 x \left (d+2 e x-\left (x^3 (2 g+h x)\right )\right )\right )}{6 x^3}+a \log (x) (a f+2 b c)+\frac {1}{30} a b x \left (60 d+x \left (30 e+x \left (20 f+15 g x+12 h x^2\right )\right )\right )+\frac {1}{840} b^2 x^3 \left (280 c+x \left (210 d+x \left (168 e+140 f x+120 g x^2+105 h x^3\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(a^2*(2*c + 3*x*(d + 2*e*x - x^3*(2*g + h*x))))/x^3 + (a*b*x*(60*d + x*(30*e + x*(20*f + 15*g*x + 12*h*x^
2))))/30 + (b^2*x^3*(280*c + x*(210*d + x*(168*e + 140*f*x + 120*g*x^2 + 105*h*x^3))))/840 + a*(2*b*c + a*f)*L
og[x]

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fricas [A]  time = 0.44, size = 153, normalized size = 1.01 \[ \frac {105 \, b^{2} h x^{11} + 120 \, b^{2} g x^{10} + 140 \, b^{2} f x^{9} + 168 \, {\left (b^{2} e + 2 \, a b h\right )} x^{8} + 210 \, {\left (b^{2} d + 2 \, a b g\right )} x^{7} + 280 \, {\left (b^{2} c + 2 \, a b f\right )} x^{6} + 420 \, {\left (2 \, a b e + a^{2} h\right )} x^{5} - 840 \, a^{2} e x^{2} + 840 \, {\left (2 \, a b d + a^{2} g\right )} x^{4} + 840 \, {\left (2 \, a b c + a^{2} f\right )} x^{3} \log \relax (x) - 420 \, a^{2} d x - 280 \, a^{2} c}{840 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x^4,x, algorithm="fricas")

[Out]

1/840*(105*b^2*h*x^11 + 120*b^2*g*x^10 + 140*b^2*f*x^9 + 168*(b^2*e + 2*a*b*h)*x^8 + 210*(b^2*d + 2*a*b*g)*x^7
+ 280*(b^2*c + 2*a*b*f)*x^6 + 420*(2*a*b*e + a^2*h)*x^5 - 840*a^2*e*x^2 + 840*(2*a*b*d + a^2*g)*x^4 + 840*(2*
a*b*c + a^2*f)*x^3*log(x) - 420*a^2*d*x - 280*a^2*c)/x^3

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giac [A]  time = 0.17, size = 153, normalized size = 1.01 \[ \frac {1}{8} \, b^{2} h x^{8} + \frac {1}{7} \, b^{2} g x^{7} + \frac {1}{6} \, b^{2} f x^{6} + \frac {2}{5} \, a b h x^{5} + \frac {1}{5} \, b^{2} x^{5} e + \frac {1}{4} \, b^{2} d x^{4} + \frac {1}{2} \, a b g x^{4} + \frac {1}{3} \, b^{2} c x^{3} + \frac {2}{3} \, a b f x^{3} + \frac {1}{2} \, a^{2} h x^{2} + a b x^{2} e + 2 \, a b d x + a^{2} g x + {\left (2 \, a b c + a^{2} f\right )} \log \left ({\left | x \right |}\right ) - \frac {6 \, a^{2} x^{2} e + 3 \, a^{2} d x + 2 \, a^{2} c}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x^4,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 149, normalized size = 0.98 \[ \frac {b^{2} h \,x^{8}}{8}+\frac {b^{2} g \,x^{7}}{7}+\frac {b^{2} f \,x^{6}}{6}+\frac {2 a b h \,x^{5}}{5}+\frac {b^{2} e \,x^{5}}{5}+\frac {a b g \,x^{4}}{2}+\frac {b^{2} d \,x^{4}}{4}+\frac {2 a b f \,x^{3}}{3}+\frac {b^{2} c \,x^{3}}{3}+\frac {a^{2} h \,x^{2}}{2}+a b e \,x^{2}+a^{2} f \ln \relax (x )+a^{2} g x +2 a b c \ln \relax (x )+2 a b d x -\frac {a^{2} e}{x}-\frac {a^{2} d}{2 x^{2}}-\frac {a^{2} c}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^2*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x^4,x)

[Out]

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

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maxima [A]  time = 1.32, size = 147, normalized size = 0.97 \[ \frac {1}{8} \, b^{2} h x^{8} + \frac {1}{7} \, b^{2} g x^{7} + \frac {1}{6} \, b^{2} f x^{6} + \frac {1}{5} \, {\left (b^{2} e + 2 \, a b h\right )} x^{5} + \frac {1}{4} \, {\left (b^{2} d + 2 \, a b g\right )} x^{4} + \frac {1}{3} \, {\left (b^{2} c + 2 \, a b f\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b e + a^{2} h\right )} x^{2} + {\left (2 \, a b d + a^{2} g\right )} x + {\left (2 \, a b c + a^{2} f\right )} \log \relax (x) - \frac {6 \, a^{2} e x^{2} + 3 \, a^{2} d x + 2 \, a^{2} c}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x^4,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.08, size = 145, normalized size = 0.95 \[ x\,\left (g\,a^2+2\,b\,d\,a\right )-\frac {e\,a^2\,x^2+\frac {d\,a^2\,x}{2}+\frac {c\,a^2}{3}}{x^3}+x^3\,\left (\frac {c\,b^2}{3}+\frac {2\,a\,f\,b}{3}\right )+x^4\,\left (\frac {d\,b^2}{4}+\frac {a\,g\,b}{2}\right )+x^2\,\left (\frac {h\,a^2}{2}+b\,e\,a\right )+x^5\,\left (\frac {e\,b^2}{5}+\frac {2\,a\,h\,b}{5}\right )+\ln \relax (x)\,\left (f\,a^2+2\,b\,c\,a\right )+\frac {b^2\,f\,x^6}{6}+\frac {b^2\,g\,x^7}{7}+\frac {b^2\,h\,x^8}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x^3)^2*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/x^4,x)

[Out]

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

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sympy [A]  time = 0.88, size = 158, normalized size = 1.04 \[ a \left (a f + 2 b c\right ) \log {\relax (x )} + \frac {b^{2} f x^{6}}{6} + \frac {b^{2} g x^{7}}{7} + \frac {b^{2} h x^{8}}{8} + x^{5} \left (\frac {2 a b h}{5} + \frac {b^{2} e}{5}\right ) + x^{4} \left (\frac {a b g}{2} + \frac {b^{2} d}{4}\right ) + x^{3} \left (\frac {2 a b f}{3} + \frac {b^{2} c}{3}\right ) + x^{2} \left (\frac {a^{2} h}{2} + a b e\right ) + x \left (a^{2} g + 2 a b d\right ) + \frac {- 2 a^{2} c - 3 a^{2} d x - 6 a^{2} e x^{2}}{6 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**2*(h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/x**4,x)

[Out]

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

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