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]

-(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]

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Rubi [A]  time = 0.118, 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} \[ -\frac{a^2 c}{3 x^3}-\frac{a^2 d}{2 x^2}-\frac{a^2 e}{x}+\frac{1}{3} b x^3 (2 a f+b c)+a \log (x) (a f+2 b c)+\frac{1}{4} b x^4 (2 a g+b d)+a x (a g+2 b d)+\frac{1}{5} b x^5 (2 a h+b e)+\frac{1}{2} a x^2 (a h+2 b e)+\frac{1}{6} b^2 f x^6+\frac{1}{7} b^2 g x^7+\frac{1}{8} b^2 h x^8 \]

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*}

Mathematica [A]  time = 0.0806839, size = 123, normalized size = 0.81 \[ -\frac{a^2 \left (2 c+3 x \left (d+2 e x+x^3 (-(2 g+h x))\right )\right )}{6 x^3}+a \log (x) (a f+2 b c)+\frac{1}{30} a b x \left (60 d+x \left (30 e+x \left (20 f+15 g x+12 h x^2\right )\right )\right )+\frac{1}{840} b^2 x^3 \left (280 c+x \left (210 d+x \left (168 e+140 f x+120 g x^2+105 h x^3\right )\right )\right ) \]

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]

-(a^2*(2*c + 3*x*(d + 2*e*x - x^3*(2*g + h*x))))/(6*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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Maple [A]  time = 0.005, size = 149, normalized size = 1. \begin{align*}{\frac{{b}^{2}h{x}^{8}}{8}}+{\frac{{b}^{2}g{x}^{7}}{7}}+{\frac{{b}^{2}f{x}^{6}}{6}}+{\frac{2\,{x}^{5}abh}{5}}+{\frac{{x}^{5}{b}^{2}e}{5}}+{\frac{{x}^{4}abg}{2}}+{\frac{{b}^{2}d{x}^{4}}{4}}+{\frac{2\,{x}^{3}abf}{3}}+{\frac{{b}^{2}c{x}^{3}}{3}}+{\frac{{x}^{2}{a}^{2}h}{2}}+aeb{x}^{2}+{a}^{2}gx+2\,bdax+\ln \left ( x \right ){a}^{2}f+2\,\ln \left ( x \right ) abc-{\frac{{a}^{2}c}{3\,{x}^{3}}}-{\frac{{a}^{2}d}{2\,{x}^{2}}}-{\frac{{a}^{2}e}{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^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*b^2*d*x^4+2/3*x^3*a*b*
f+1/3*b^2*c*x^3+1/2*x^2*a^2*h+a*e*b*x^2+a^2*g*x+2*b*d*a*x+ln(x)*a^2*f+2*ln(x)*a*b*c-1/3*a^2*c/x^3-1/2*a^2*d/x^
2-a^2*e/x

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Maxima [A]  time = 0.9255, size = 198, normalized size = 1.3 \begin{align*} \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 \left (x\right ) - \frac{6 \, a^{2} e x^{2} + 3 \, a^{2} d x + 2 \, a^{2} c}{6 \, x^{3}} \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^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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Fricas [A]  time = 0.991457, size = 375, normalized size = 2.47 \begin{align*} \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 \left (x\right ) - 420 \, a^{2} d x - 280 \, a^{2} c}{840 \, x^{3}} \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^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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Sympy [A]  time = 0.872923, size = 156, normalized size = 1.03 \begin{align*} a \left (a f + 2 b c\right ) \log{\left (x \right )} + \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}} \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**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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Giac [A]  time = 1.07006, size = 207, normalized size = 1.36 \begin{align*} \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}} \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^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