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

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

[Out]

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

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Rubi [A]  time = 0.117274, antiderivative size = 152, normalized size of antiderivative = 1., 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} \[ -\frac{a^2 c}{4 x^4}-\frac{a^2 d}{3 x^3}-\frac{a^2 e}{2 x^2}+\frac{1}{2} b x^2 (2 a f+b c)-\frac{a (a f+2 b c)}{x}+\frac{1}{3} b x^3 (2 a g+b d)+a \log (x) (a g+2 b d)+\frac{1}{4} b x^4 (2 a h+b e)+a x (a h+2 b e)+\frac{1}{5} b^2 f x^5+\frac{1}{6} b^2 g x^6+\frac{1}{7} b^2 h x^7 \]

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^5,x]

[Out]

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

Mathematica [A]  time = 0.0725245, size = 125, normalized size = 0.82 \[ -\frac{a^2 \left (3 c+4 d x+6 x^2 \left (e+2 f x-2 h x^3\right )\right )}{12 x^4}-\frac{2 a b c}{x}+a \log (x) (a g+2 b d)+\frac{1}{6} a b x (12 e+x (6 f+x (4 g+3 h x)))+\frac{1}{420} b^2 x^2 \left (210 c+x \left (140 d+x \left (105 e+84 f x+70 g x^2+60 h x^3\right )\right )\right ) \]

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^5,x]

[Out]

(-2*a*b*c)/x - (a^2*(3*c + 4*d*x + 6*x^2*(e + 2*f*x - 2*h*x^3)))/(12*x^4) + (a*b*x*(12*e + x*(6*f + x*(4*g + 3
*h*x))))/6 + (b^2*x^2*(210*c + x*(140*d + x*(105*e + 84*f*x + 70*g*x^2 + 60*h*x^3))))/420 + a*(2*b*d + a*g)*Lo
g[x]

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Maple [A]  time = 0.006, size = 149, normalized size = 1. \begin{align*}{\frac{{b}^{2}h{x}^{7}}{7}}+{\frac{{b}^{2}g{x}^{6}}{6}}+{\frac{{b}^{2}f{x}^{5}}{5}}+{\frac{{x}^{4}abh}{2}}+{\frac{{x}^{4}{b}^{2}e}{4}}+{\frac{2\,{x}^{3}abg}{3}}+{\frac{{b}^{2}d{x}^{3}}{3}}+{x}^{2}abf+{\frac{{x}^{2}{b}^{2}c}{2}}+{a}^{2}hx+2\,aebx+\ln \left ( x \right ){a}^{2}g+2\,\ln \left ( x \right ) abd-{\frac{{a}^{2}d}{3\,{x}^{3}}}-{\frac{{a}^{2}c}{4\,{x}^{4}}}-{\frac{{a}^{2}e}{2\,{x}^{2}}}-{\frac{{a}^{2}f}{x}}-2\,{\frac{abc}{x}} \end{align*}

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^5,x)

[Out]

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

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Maxima [A]  time = 0.947725, size = 198, normalized size = 1.3 \begin{align*} \frac{1}{7} \, b^{2} h x^{7} + \frac{1}{6} \, b^{2} g x^{6} + \frac{1}{5} \, b^{2} f x^{5} + \frac{1}{4} \,{\left (b^{2} e + 2 \, a b h\right )} x^{4} + \frac{1}{3} \,{\left (b^{2} d + 2 \, a b g\right )} x^{3} + \frac{1}{2} \,{\left (b^{2} c + 2 \, a b f\right )} x^{2} +{\left (2 \, a b e + a^{2} h\right )} x +{\left (2 \, a b d + a^{2} g\right )} \log \left (x\right ) - \frac{6 \, a^{2} e x^{2} + 4 \, a^{2} d x + 12 \,{\left (2 \, a b c + a^{2} f\right )} x^{3} + 3 \, a^{2} c}{12 \, x^{4}} \end{align*}

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^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.990554, size = 371, normalized size = 2.44 \begin{align*} \frac{60 \, b^{2} h x^{11} + 70 \, b^{2} g x^{10} + 84 \, b^{2} f x^{9} + 105 \,{\left (b^{2} e + 2 \, a b h\right )} x^{8} + 140 \,{\left (b^{2} d + 2 \, a b g\right )} x^{7} + 210 \,{\left (b^{2} c + 2 \, a b f\right )} x^{6} + 420 \,{\left (2 \, a b e + a^{2} h\right )} x^{5} + 420 \,{\left (2 \, a b d + a^{2} g\right )} x^{4} \log \left (x\right ) - 210 \, a^{2} e x^{2} - 140 \, a^{2} d x - 420 \,{\left (2 \, a b c + a^{2} f\right )} x^{3} - 105 \, a^{2} c}{420 \, x^{4}} \end{align*}

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^5,x, algorithm="fricas")

[Out]

1/420*(60*b^2*h*x^11 + 70*b^2*g*x^10 + 84*b^2*f*x^9 + 105*(b^2*e + 2*a*b*h)*x^8 + 140*(b^2*d + 2*a*b*g)*x^7 +
210*(b^2*c + 2*a*b*f)*x^6 + 420*(2*a*b*e + a^2*h)*x^5 + 420*(2*a*b*d + a^2*g)*x^4*log(x) - 210*a^2*e*x^2 - 140
*a^2*d*x - 420*(2*a*b*c + a^2*f)*x^3 - 105*a^2*c)/x^4

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Sympy [A]  time = 2.7749, size = 155, normalized size = 1.02 \begin{align*} a \left (a g + 2 b d\right ) \log{\left (x \right )} + \frac{b^{2} f x^{5}}{5} + \frac{b^{2} g x^{6}}{6} + \frac{b^{2} h x^{7}}{7} + x^{4} \left (\frac{a b h}{2} + \frac{b^{2} e}{4}\right ) + x^{3} \left (\frac{2 a b g}{3} + \frac{b^{2} d}{3}\right ) + x^{2} \left (a b f + \frac{b^{2} c}{2}\right ) + x \left (a^{2} h + 2 a b e\right ) - \frac{3 a^{2} c + 4 a^{2} d x + 6 a^{2} e x^{2} + x^{3} \left (12 a^{2} f + 24 a b c\right )}{12 x^{4}} \end{align*}

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**5,x)

[Out]

a*(a*g + 2*b*d)*log(x) + b**2*f*x**5/5 + b**2*g*x**6/6 + b**2*h*x**7/7 + x**4*(a*b*h/2 + b**2*e/4) + x**3*(2*a
*b*g/3 + b**2*d/3) + x**2*(a*b*f + b**2*c/2) + x*(a**2*h + 2*a*b*e) - (3*a**2*c + 4*a**2*d*x + 6*a**2*e*x**2 +
x**3*(12*a**2*f + 24*a*b*c))/(12*x**4)

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Giac [A]  time = 1.06372, size = 205, normalized size = 1.35 \begin{align*} \frac{1}{7} \, b^{2} h x^{7} + \frac{1}{6} \, b^{2} g x^{6} + \frac{1}{5} \, b^{2} f x^{5} + \frac{1}{2} \, a b h x^{4} + \frac{1}{4} \, b^{2} x^{4} e + \frac{1}{3} \, b^{2} d x^{3} + \frac{2}{3} \, a b g x^{3} + \frac{1}{2} \, b^{2} c x^{2} + a b f x^{2} + a^{2} h x + 2 \, a b x e +{\left (2 \, a b d + a^{2} g\right )} \log \left ({\left | x \right |}\right ) - \frac{6 \, a^{2} x^{2} e + 4 \, a^{2} d x + 12 \,{\left (2 \, a b c + a^{2} f\right )} x^{3} + 3 \, a^{2} c}{12 \, x^{4}} \end{align*}

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^5,x, algorithm="giac")

[Out]

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