3.5.1 \(\int \frac {(a+b x^3)^3 (c+d x+e x^2+f x^3+g x^4+h x^5)}{x^4} \, dx\) [401]

Optimal. Leaf size=209 \[ -\frac {a^3 c}{3 x^3}-\frac {a^3 d}{2 x^2}-\frac {a^3 e}{x}+a^2 (3 b d+a g) x+\frac {1}{2} a^2 (3 b e+a h) x^2+a b (b c+a f) x^3+\frac {3}{4} a b (b d+a g) x^4+\frac {3}{5} a b (b e+a h) x^5+\frac {1}{6} b^2 (b c+3 a f) x^6+\frac {1}{7} b^2 (b d+3 a g) x^7+\frac {1}{8} b^2 (b e+3 a h) x^8+\frac {1}{9} b^3 f x^9+\frac {1}{10} b^3 g x^{10}+\frac {1}{11} b^3 h x^{11}+a^2 (3 b c+a f) \log (x) \]

[Out]

-1/3*a^3*c/x^3-1/2*a^3*d/x^2-a^3*e/x+a^2*(a*g+3*b*d)*x+1/2*a^2*(a*h+3*b*e)*x^2+a*b*(a*f+b*c)*x^3+3/4*a*b*(a*g+
b*d)*x^4+3/5*a*b*(a*h+b*e)*x^5+1/6*b^2*(3*a*f+b*c)*x^6+1/7*b^2*(3*a*g+b*d)*x^7+1/8*b^2*(3*a*h+b*e)*x^8+1/9*b^3
*f*x^9+1/10*b^3*g*x^10+1/11*b^3*h*x^11+a^2*(a*f+3*b*c)*ln(x)

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Rubi [A]
time = 0.12, antiderivative size = 209, 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 = {1834} \begin {gather*} -\frac {a^3 c}{3 x^3}-\frac {a^3 d}{2 x^2}-\frac {a^3 e}{x}+a^2 \log (x) (a f+3 b c)+a^2 x (a g+3 b d)+\frac {1}{2} a^2 x^2 (a h+3 b e)+\frac {1}{6} b^2 x^6 (3 a f+b c)+\frac {1}{7} b^2 x^7 (3 a g+b d)+\frac {1}{8} b^2 x^8 (3 a h+b e)+a b x^3 (a f+b c)+\frac {3}{4} a b x^4 (a g+b d)+\frac {3}{5} a b x^5 (a h+b e)+\frac {1}{9} b^3 f x^9+\frac {1}{10} b^3 g x^{10}+\frac {1}{11} b^3 h x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(a^3*c)/x^3 - (a^3*d)/(2*x^2) - (a^3*e)/x + a^2*(3*b*d + a*g)*x + (a^2*(3*b*e + a*h)*x^2)/2 + a*b*(b*c +
a*f)*x^3 + (3*a*b*(b*d + a*g)*x^4)/4 + (3*a*b*(b*e + a*h)*x^5)/5 + (b^2*(b*c + 3*a*f)*x^6)/6 + (b^2*(b*d + 3*a
*g)*x^7)/7 + (b^2*(b*e + 3*a*h)*x^8)/8 + (b^3*f*x^9)/9 + (b^3*g*x^10)/10 + (b^3*h*x^11)/11 + a^2*(3*b*c + a*f)
*Log[x]

Rule 1834

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 )^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{x^4} \, dx &=\int \left (a^2 (3 b d+a g)+\frac {a^3 c}{x^4}+\frac {a^3 d}{x^3}+\frac {a^3 e}{x^2}+\frac {a^2 (3 b c+a f)}{x}+a^2 (3 b e+a h) x+3 a b (b c+a f) x^2+3 a b (b d+a g) x^3+3 a b (b e+a h) x^4+b^2 (b c+3 a f) x^5+b^2 (b d+3 a g) x^6+b^2 (b e+3 a h) x^7+b^3 f x^8+b^3 g x^9+b^3 h x^{10}\right ) \, dx\\ &=-\frac {a^3 c}{3 x^3}-\frac {a^3 d}{2 x^2}-\frac {a^3 e}{x}+a^2 (3 b d+a g) x+\frac {1}{2} a^2 (3 b e+a h) x^2+a b (b c+a f) x^3+\frac {3}{4} a b (b d+a g) x^4+\frac {3}{5} a b (b e+a h) x^5+\frac {1}{6} b^2 (b c+3 a f) x^6+\frac {1}{7} b^2 (b d+3 a g) x^7+\frac {1}{8} b^2 (b e+3 a h) x^8+\frac {1}{9} b^3 f x^9+\frac {1}{10} b^3 g x^{10}+\frac {1}{11} b^3 h x^{11}+a^2 (3 b c+a f) \log (x)\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 172, normalized size = 0.82 \begin {gather*} -\frac {a^3 \left (2 c+3 x \left (d+2 e x-x^3 (2 g+h x)\right )\right )}{6 x^3}+\frac {1}{20} a^2 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}{280} a 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 )+\frac {b^3 x^6 \left (4620 c+x \left (3960 d+7 x \left (495 e+4 x \left (110 f+99 g x+90 h x^2\right )\right )\right )\right )}{27720}+a^2 (3 b c+a f) \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(a^3*(2*c + 3*x*(d + 2*e*x - x^3*(2*g + h*x))))/x^3 + (a^2*b*x*(60*d + x*(30*e + x*(20*f + 15*g*x + 12*h*
x^2))))/20 + (a*b^2*x^3*(280*c + x*(210*d + x*(168*e + 140*f*x + 120*g*x^2 + 105*h*x^3))))/280 + (b^3*x^6*(462
0*c + x*(3960*d + 7*x*(495*e + 4*x*(110*f + 99*g*x + 90*h*x^2)))))/27720 + a^2*(3*b*c + a*f)*Log[x]

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Maple [A]
time = 0.36, size = 218, normalized size = 1.04

method	result	size
norman	\(\frac {\left (\frac {1}{2} a^{3} h +\frac {3}{2} a^{2} b e \right ) x^{5}+\left (\frac {1}{2} a \,b^{2} f +\frac {1}{6} b^{3} c \right ) x^{9}+\left (\frac {3}{7} a \,b^{2} g +\frac {1}{7} b^{3} d \right ) x^{10}+\left (\frac {3}{8} a \,b^{2} h +\frac {1}{8} e \,b^{3}\right ) x^{11}+\left (\frac {3}{4} a^{2} b g +\frac {3}{4} a \,b^{2} d \right ) x^{7}+\left (\frac {3}{5} a^{2} b h +\frac {3}{5} a \,b^{2} e \right ) x^{8}+\left (a^{2} b f +a c \,b^{2}\right ) x^{6}+\left (a^{3} g +3 d \,a^{2} b \right ) x^{4}-\frac {c \,a^{3}}{3}-\frac {a^{3} d x}{2}-a^{3} e \,x^{2}+\frac {b^{3} g \,x^{13}}{10}+\frac {b^{3} h \,x^{14}}{11}+\frac {f \,x^{12} b^{3}}{9}}{x^{3}}+\left (a^{3} f +3 c \,a^{2} b \right ) \ln \left (x \right )\)	\(216\)
default	\(\frac {b^{3} h \,x^{11}}{11}+\frac {b^{3} g \,x^{10}}{10}+\frac {b^{3} f \,x^{9}}{9}+\frac {3 a \,b^{2} h \,x^{8}}{8}+\frac {b^{3} e \,x^{8}}{8}+\frac {3 a \,b^{2} g \,x^{7}}{7}+\frac {b^{3} d \,x^{7}}{7}+\frac {a \,b^{2} f \,x^{6}}{2}+\frac {b^{3} c \,x^{6}}{6}+\frac {3 a^{2} b h \,x^{5}}{5}+\frac {3 a \,b^{2} e \,x^{5}}{5}+\frac {3 a^{2} b g \,x^{4}}{4}+\frac {3 a \,b^{2} d \,x^{4}}{4}+a^{2} b f \,x^{3}+a \,b^{2} c \,x^{3}+\frac {a^{3} h \,x^{2}}{2}+\frac {3 a^{2} b e \,x^{2}}{2}+a^{3} g x +3 a^{2} b d x -\frac {a^{3} d}{2 x^{2}}-\frac {a^{3} c}{3 x^{3}}+a^{2} \left (a f +3 b c \right ) \ln \left (x \right )-\frac {a^{3} e}{x}\)	\(218\)
risch	\(\frac {b^{3} h \,x^{11}}{11}+\frac {b^{3} g \,x^{10}}{10}+\frac {b^{3} f \,x^{9}}{9}+\frac {3 a \,b^{2} h \,x^{8}}{8}+\frac {b^{3} e \,x^{8}}{8}+\frac {3 a \,b^{2} g \,x^{7}}{7}+\frac {b^{3} d \,x^{7}}{7}+\frac {a \,b^{2} f \,x^{6}}{2}+\frac {b^{3} c \,x^{6}}{6}+\frac {3 a^{2} b h \,x^{5}}{5}+\frac {3 a \,b^{2} e \,x^{5}}{5}+\frac {3 a^{2} b g \,x^{4}}{4}+\frac {3 a \,b^{2} d \,x^{4}}{4}+a^{2} b f \,x^{3}+a \,b^{2} c \,x^{3}+\frac {a^{3} h \,x^{2}}{2}+\frac {3 a^{2} b e \,x^{2}}{2}+a^{3} g x +3 a^{2} b d x +\frac {-a^{3} e \,x^{2}-\frac {1}{2} a^{3} d x -\frac {1}{3} c \,a^{3}}{x^{3}}+\ln \left (x \right ) a^{3} f +3 \ln \left (x \right ) a^{2} b c\)	\(220\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*b^3*h*x^11+1/10*b^3*g*x^10+1/9*b^3*f*x^9+3/8*a*b^2*h*x^8+1/8*b^3*e*x^8+3/7*a*b^2*g*x^7+1/7*b^3*d*x^7+1/2*
a*b^2*f*x^6+1/6*b^3*c*x^6+3/5*a^2*b*h*x^5+3/5*a*b^2*e*x^5+3/4*a^2*b*g*x^4+3/4*a*b^2*d*x^4+a^2*b*f*x^3+a*b^2*c*
x^3+1/2*a^3*h*x^2+3/2*a^2*b*e*x^2+a^3*g*x+3*a^2*b*d*x-1/2*a^3*d/x^2-1/3*a^3*c/x^3+a^2*(a*f+3*b*c)*ln(x)-a^3*e/
x

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Maxima [A]
time = 0.28, size = 216, normalized size = 1.03 \begin {gather*} \frac {1}{11} \, b^{3} h x^{11} + \frac {1}{10} \, b^{3} g x^{10} + \frac {1}{9} \, b^{3} f x^{9} + \frac {1}{8} \, {\left (3 \, a b^{2} h + b^{3} e\right )} x^{8} + \frac {1}{7} \, {\left (b^{3} d + 3 \, a b^{2} g\right )} x^{7} + \frac {1}{6} \, {\left (b^{3} c + 3 \, a b^{2} f\right )} x^{6} + \frac {3}{5} \, {\left (a^{2} b h + a b^{2} e\right )} x^{5} + \frac {3}{4} \, {\left (a b^{2} d + a^{2} b g\right )} x^{4} + {\left (a b^{2} c + a^{2} b f\right )} x^{3} + \frac {1}{2} \, {\left (a^{3} h + 3 \, a^{2} b e\right )} x^{2} + {\left (3 \, a^{2} b d + a^{3} g\right )} x + {\left (3 \, a^{2} b c + a^{3} f\right )} \log \left (x\right ) - \frac {6 \, a^{3} x^{2} e + 3 \, a^{3} d x + 2 \, a^{3} c}{6 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.41, size = 219, normalized size = 1.05 \begin {gather*} \frac {2520 \, b^{3} h x^{14} + 2772 \, b^{3} g x^{13} + 3080 \, b^{3} f x^{12} + 3465 \, {\left (b^{3} e + 3 \, a b^{2} h\right )} x^{11} + 3960 \, {\left (b^{3} d + 3 \, a b^{2} g\right )} x^{10} + 4620 \, {\left (b^{3} c + 3 \, a b^{2} f\right )} x^{9} + 16632 \, {\left (a b^{2} e + a^{2} b h\right )} x^{8} + 20790 \, {\left (a b^{2} d + a^{2} b g\right )} x^{7} + 27720 \, {\left (a b^{2} c + a^{2} b f\right )} x^{6} - 27720 \, a^{3} e x^{2} + 13860 \, {\left (3 \, a^{2} b e + a^{3} h\right )} x^{5} - 13860 \, a^{3} d x + 27720 \, {\left (3 \, a^{2} b d + a^{3} g\right )} x^{4} + 27720 \, {\left (3 \, a^{2} b c + a^{3} f\right )} x^{3} \log \left (x\right ) - 9240 \, a^{3} c}{27720 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27720*(2520*b^3*h*x^14 + 2772*b^3*g*x^13 + 3080*b^3*f*x^12 + 3465*(b^3*e + 3*a*b^2*h)*x^11 + 3960*(b^3*d + 3
*a*b^2*g)*x^10 + 4620*(b^3*c + 3*a*b^2*f)*x^9 + 16632*(a*b^2*e + a^2*b*h)*x^8 + 20790*(a*b^2*d + a^2*b*g)*x^7
+ 27720*(a*b^2*c + a^2*b*f)*x^6 - 27720*a^3*e*x^2 + 13860*(3*a^2*b*e + a^3*h)*x^5 - 13860*a^3*d*x + 27720*(3*a
^2*b*d + a^3*g)*x^4 + 27720*(3*a^2*b*c + a^3*f)*x^3*log(x) - 9240*a^3*c)/x^3

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Sympy [A]
time = 0.46, size = 236, normalized size = 1.13 \begin {gather*} a^{2} \left (a f + 3 b c\right ) \log {\left (x \right )} + \frac {b^{3} f x^{9}}{9} + \frac {b^{3} g x^{10}}{10} + \frac {b^{3} h x^{11}}{11} + x^{8} \cdot \left (\frac {3 a b^{2} h}{8} + \frac {b^{3} e}{8}\right ) + x^{7} \cdot \left (\frac {3 a b^{2} g}{7} + \frac {b^{3} d}{7}\right ) + x^{6} \left (\frac {a b^{2} f}{2} + \frac {b^{3} c}{6}\right ) + x^{5} \cdot \left (\frac {3 a^{2} b h}{5} + \frac {3 a b^{2} e}{5}\right ) + x^{4} \cdot \left (\frac {3 a^{2} b g}{4} + \frac {3 a b^{2} d}{4}\right ) + x^{3} \left (a^{2} b f + a b^{2} c\right ) + x^{2} \left (\frac {a^{3} h}{2} + \frac {3 a^{2} b e}{2}\right ) + x \left (a^{3} g + 3 a^{2} b d\right ) + \frac {- 2 a^{3} c - 3 a^{3} d x - 6 a^{3} e x^{2}}{6 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*(a*f + 3*b*c)*log(x) + b**3*f*x**9/9 + b**3*g*x**10/10 + b**3*h*x**11/11 + x**8*(3*a*b**2*h/8 + b**3*e/8)
+ x**7*(3*a*b**2*g/7 + b**3*d/7) + x**6*(a*b**2*f/2 + b**3*c/6) + x**5*(3*a**2*b*h/5 + 3*a*b**2*e/5) + x**4*(
3*a**2*b*g/4 + 3*a*b**2*d/4) + x**3*(a**2*b*f + a*b**2*c) + x**2*(a**3*h/2 + 3*a**2*b*e/2) + x*(a**3*g + 3*a**
2*b*d) + (-2*a**3*c - 3*a**3*d*x - 6*a**3*e*x**2)/(6*x**3)

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Giac [A]
time = 0.50, size = 225, normalized size = 1.08 \begin {gather*} \frac {1}{11} \, b^{3} h x^{11} + \frac {1}{10} \, b^{3} g x^{10} + \frac {1}{9} \, b^{3} f x^{9} + \frac {3}{8} \, a b^{2} h x^{8} + \frac {1}{8} \, b^{3} x^{8} e + \frac {1}{7} \, b^{3} d x^{7} + \frac {3}{7} \, a b^{2} g x^{7} + \frac {1}{6} \, b^{3} c x^{6} + \frac {1}{2} \, a b^{2} f x^{6} + \frac {3}{5} \, a^{2} b h x^{5} + \frac {3}{5} \, a b^{2} x^{5} e + \frac {3}{4} \, a b^{2} d x^{4} + \frac {3}{4} \, a^{2} b g x^{4} + a b^{2} c x^{3} + a^{2} b f x^{3} + \frac {1}{2} \, a^{3} h x^{2} + \frac {3}{2} \, a^{2} b x^{2} e + 3 \, a^{2} b d x + a^{3} g x + {\left (3 \, a^{2} b c + a^{3} f\right )} \log \left ({\left | x \right |}\right ) - \frac {6 \, a^{3} x^{2} e + 3 \, a^{3} d x + 2 \, a^{3} c}{6 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*b^3*h*x^11 + 1/10*b^3*g*x^10 + 1/9*b^3*f*x^9 + 3/8*a*b^2*h*x^8 + 1/8*b^3*x^8*e + 1/7*b^3*d*x^7 + 3/7*a*b^
2*g*x^7 + 1/6*b^3*c*x^6 + 1/2*a*b^2*f*x^6 + 3/5*a^2*b*h*x^5 + 3/5*a*b^2*x^5*e + 3/4*a*b^2*d*x^4 + 3/4*a^2*b*g*
x^4 + a*b^2*c*x^3 + a^2*b*f*x^3 + 1/2*a^3*h*x^2 + 3/2*a^2*b*x^2*e + 3*a^2*b*d*x + a^3*g*x + (3*a^2*b*c + a^3*f
)*log(abs(x)) - 1/6*(6*a^3*x^2*e + 3*a^3*d*x + 2*a^3*c)/x^3

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Mupad [B]
time = 0.12, size = 199, normalized size = 0.95 \begin {gather*} x^6\,\left (\frac {c\,b^3}{6}+\frac {a\,f\,b^2}{2}\right )+x^7\,\left (\frac {d\,b^3}{7}+\frac {3\,a\,g\,b^2}{7}\right )+x^2\,\left (\frac {h\,a^3}{2}+\frac {3\,b\,e\,a^2}{2}\right )+x^8\,\left (\frac {e\,b^3}{8}+\frac {3\,a\,h\,b^2}{8}\right )+\ln \left (x\right )\,\left (f\,a^3+3\,b\,c\,a^2\right )-\frac {e\,a^3\,x^2+\frac {d\,a^3\,x}{2}+\frac {c\,a^3}{3}}{x^3}+x\,\left (g\,a^3+3\,b\,d\,a^2\right )+\frac {b^3\,f\,x^9}{9}+\frac {b^3\,g\,x^{10}}{10}+\frac {b^3\,h\,x^{11}}{11}+a\,b\,x^3\,\left (b\,c+a\,f\right )+\frac {3\,a\,b\,x^4\,\left (b\,d+a\,g\right )}{4}+\frac {3\,a\,b\,x^5\,\left (b\,e+a\,h\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^6*((b^3*c)/6 + (a*b^2*f)/2) + x^7*((b^3*d)/7 + (3*a*b^2*g)/7) + x^2*((a^3*h)/2 + (3*a^2*b*e)/2) + x^8*((b^3*
e)/8 + (3*a*b^2*h)/8) + log(x)*(a^3*f + 3*a^2*b*c) - ((a^3*c)/3 + a^3*e*x^2 + (a^3*d*x)/2)/x^3 + x*(a^3*g + 3*
a^2*b*d) + (b^3*f*x^9)/9 + (b^3*g*x^10)/10 + (b^3*h*x^11)/11 + a*b*x^3*(b*c + a*f) + (3*a*b*x^4*(b*d + a*g))/4
+ (3*a*b*x^5*(b*e + a*h))/5

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