3.5.2 \(\int \frac {(a+b x^3)^3 (c+d x+e x^2+f x^3+g x^4+h x^5)}{x^5} \, dx\) [402]

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

[Out]

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

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Rubi [A]
time = 0.11, 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}{4 x^4}-\frac {a^3 d}{3 x^3}-\frac {a^3 e}{2 x^2}-\frac {a^2 (a f+3 b c)}{x}+a^2 \log (x) (a g+3 b d)+a^2 x (a h+3 b e)+\frac {1}{5} b^2 x^5 (3 a f+b c)+\frac {1}{6} b^2 x^6 (3 a g+b d)+\frac {1}{7} b^2 x^7 (3 a h+b e)+\frac {3}{2} a b x^2 (a f+b c)+a b x^3 (a g+b d)+\frac {3}{4} a b x^4 (a h+b e)+\frac {1}{8} b^3 f x^8+\frac {1}{9} b^3 g x^9+\frac {1}{10} b^3 h x^{10} \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^5,x]

[Out]

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

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Mathematica [A]
time = 0.08, size = 170, normalized size = 0.81 \begin {gather*} \frac {-210 a^3 \left (3 c+4 d x+6 x^2 \left (e+2 f x-2 h x^3\right )\right )+630 a^2 b x^3 \left (-12 c+x^2 \left (12 e+6 f x+4 g x^2+3 h x^3\right )\right )+18 a b^2 x^6 \left (210 c+x \left (140 d+105 e x+84 f x^2+70 g x^3+60 h x^4\right )\right )+b^3 x^9 \left (504 c+x \left (420 d+360 e x+315 f x^2+280 g x^3+252 h x^4\right )\right )}{2520 x^4}+a^2 (3 b d+a g) \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^5,x]

[Out]

(-210*a^3*(3*c + 4*d*x + 6*x^2*(e + 2*f*x - 2*h*x^3)) + 630*a^2*b*x^3*(-12*c + x^2*(12*e + 6*f*x + 4*g*x^2 + 3
*h*x^3)) + 18*a*b^2*x^6*(210*c + x*(140*d + 105*e*x + 84*f*x^2 + 70*g*x^3 + 60*h*x^4)) + b^3*x^9*(504*c + x*(4
20*d + 360*e*x + 315*f*x^2 + 280*g*x^3 + 252*h*x^4)))/(2520*x^4) + a^2*(3*b*d + a*g)*Log[x]

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Maple [A]
time = 0.36, size = 215, normalized size = 1.03

method	result	size
default	\(\frac {b^{3} h \,x^{10}}{10}+\frac {b^{3} g \,x^{9}}{9}+\frac {b^{3} f \,x^{8}}{8}+\frac {3 a \,b^{2} h \,x^{7}}{7}+\frac {b^{3} e \,x^{7}}{7}+\frac {a \,b^{2} g \,x^{6}}{2}+\frac {b^{3} d \,x^{6}}{6}+\frac {3 a \,b^{2} f \,x^{5}}{5}+\frac {b^{3} c \,x^{5}}{5}+\frac {3 a^{2} b h \,x^{4}}{4}+\frac {3 a \,b^{2} e \,x^{4}}{4}+a^{2} b g \,x^{3}+a \,b^{2} d \,x^{3}+\frac {3 a^{2} b f \,x^{2}}{2}+\frac {3 a \,b^{2} c \,x^{2}}{2}+a^{3} h x +3 a^{2} b e x -\frac {a^{3} c}{4 x^{4}}-\frac {a^{3} e}{2 x^{2}}-\frac {a^{3} d}{3 x^{3}}+a^{2} \left (a g +3 b d \right ) \ln \left (x \right )-\frac {a^{2} \left (a f +3 b c \right )}{x}\)	\(215\)
norman	\(\frac {\left (\frac {3}{5} a \,b^{2} f +\frac {1}{5} b^{3} c \right ) x^{9}+\left (\frac {1}{2} a \,b^{2} g +\frac {1}{6} b^{3} d \right ) x^{10}+\left (\frac {3}{7} a \,b^{2} h +\frac {1}{7} e \,b^{3}\right ) x^{11}+\left (\frac {3}{2} a^{2} b f +\frac {3}{2} a c \,b^{2}\right ) x^{6}+\left (\frac {3}{4} a^{2} b h +\frac {3}{4} a \,b^{2} e \right ) x^{8}+\left (-a^{3} f -3 c \,a^{2} b \right ) x^{3}+\left (a^{2} b g +a \,b^{2} d \right ) x^{7}+\left (a^{3} h +3 a^{2} b e \right ) x^{5}-\frac {c \,a^{3}}{4}-\frac {a^{3} d x}{3}-\frac {a^{3} e \,x^{2}}{2}+\frac {b^{3} g \,x^{13}}{9}+\frac {b^{3} h \,x^{14}}{10}+\frac {f \,x^{12} b^{3}}{8}}{x^{4}}+\left (a^{3} g +3 d \,a^{2} b \right ) \ln \left (x \right )\)	\(216\)
risch	\(\frac {b^{3} h \,x^{10}}{10}+\frac {b^{3} g \,x^{9}}{9}+\frac {b^{3} f \,x^{8}}{8}+\frac {3 a \,b^{2} h \,x^{7}}{7}+\frac {b^{3} e \,x^{7}}{7}+\frac {a \,b^{2} g \,x^{6}}{2}+\frac {b^{3} d \,x^{6}}{6}+\frac {3 a \,b^{2} f \,x^{5}}{5}+\frac {b^{3} c \,x^{5}}{5}+\frac {3 a^{2} b h \,x^{4}}{4}+\frac {3 a \,b^{2} e \,x^{4}}{4}+a^{2} b g \,x^{3}+a \,b^{2} d \,x^{3}+\frac {3 a^{2} b f \,x^{2}}{2}+\frac {3 a \,b^{2} c \,x^{2}}{2}+a^{3} h x +3 a^{2} b e x +\frac {\left (-a^{3} f -3 c \,a^{2} b \right ) x^{3}-\frac {a^{3} e \,x^{2}}{2}-\frac {a^{3} d x}{3}-\frac {c \,a^{3}}{4}}{x^{4}}+\ln \left (x \right ) a^{3} g +3 \ln \left (x \right ) a^{2} b d\)	\(219\)

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^5,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 216, normalized size = 1.03 \begin {gather*} \frac {1}{10} \, b^{3} h x^{10} + \frac {1}{9} \, b^{3} g x^{9} + \frac {1}{8} \, b^{3} f x^{8} + \frac {1}{7} \, {\left (3 \, a b^{2} h + b^{3} e\right )} x^{7} + \frac {1}{6} \, {\left (b^{3} d + 3 \, a b^{2} g\right )} x^{6} + \frac {1}{5} \, {\left (b^{3} c + 3 \, a b^{2} f\right )} x^{5} + \frac {3}{4} \, {\left (a^{2} b h + a b^{2} e\right )} x^{4} + {\left (a b^{2} d + a^{2} b g\right )} x^{3} + \frac {3}{2} \, {\left (a b^{2} c + a^{2} b f\right )} x^{2} + {\left (a^{3} h + 3 \, a^{2} b e\right )} x + {\left (3 \, a^{2} b d + a^{3} g\right )} \log \left (x\right ) - \frac {6 \, a^{3} x^{2} e + 4 \, a^{3} d x + 3 \, a^{3} c + 12 \, {\left (3 \, a^{2} b c + a^{3} f\right )} x^{3}}{12 \, x^{4}} \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^5,x, algorithm="maxima")

[Out]

1/10*b^3*h*x^10 + 1/9*b^3*g*x^9 + 1/8*b^3*f*x^8 + 1/7*(3*a*b^2*h + b^3*e)*x^7 + 1/6*(b^3*d + 3*a*b^2*g)*x^6 +
1/5*(b^3*c + 3*a*b^2*f)*x^5 + 3/4*(a^2*b*h + a*b^2*e)*x^4 + (a*b^2*d + a^2*b*g)*x^3 + 3/2*(a*b^2*c + a^2*b*f)*
x^2 + (a^3*h + 3*a^2*b*e)*x + (3*a^2*b*d + a^3*g)*log(x) - 1/12*(6*a^3*x^2*e + 4*a^3*d*x + 3*a^3*c + 12*(3*a^2
*b*c + a^3*f)*x^3)/x^4

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Fricas [A]
time = 0.39, size = 219, normalized size = 1.05 \begin {gather*} \frac {252 \, b^{3} h x^{14} + 280 \, b^{3} g x^{13} + 315 \, b^{3} f x^{12} + 360 \, {\left (b^{3} e + 3 \, a b^{2} h\right )} x^{11} + 420 \, {\left (b^{3} d + 3 \, a b^{2} g\right )} x^{10} + 504 \, {\left (b^{3} c + 3 \, a b^{2} f\right )} x^{9} + 1890 \, {\left (a b^{2} e + a^{2} b h\right )} x^{8} + 2520 \, {\left (a b^{2} d + a^{2} b g\right )} x^{7} + 3780 \, {\left (a b^{2} c + a^{2} b f\right )} x^{6} - 1260 \, a^{3} e x^{2} + 2520 \, {\left (3 \, a^{2} b e + a^{3} h\right )} x^{5} + 2520 \, {\left (3 \, a^{2} b d + a^{3} g\right )} x^{4} \log \left (x\right ) - 840 \, a^{3} d x - 630 \, a^{3} c - 2520 \, {\left (3 \, a^{2} b c + a^{3} f\right )} x^{3}}{2520 \, x^{4}} \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^5,x, algorithm="fricas")

[Out]

1/2520*(252*b^3*h*x^14 + 280*b^3*g*x^13 + 315*b^3*f*x^12 + 360*(b^3*e + 3*a*b^2*h)*x^11 + 420*(b^3*d + 3*a*b^2
*g)*x^10 + 504*(b^3*c + 3*a*b^2*f)*x^9 + 1890*(a*b^2*e + a^2*b*h)*x^8 + 2520*(a*b^2*d + a^2*b*g)*x^7 + 3780*(a
*b^2*c + a^2*b*f)*x^6 - 1260*a^3*e*x^2 + 2520*(3*a^2*b*e + a^3*h)*x^5 + 2520*(3*a^2*b*d + a^3*g)*x^4*log(x) -
840*a^3*d*x - 630*a^3*c - 2520*(3*a^2*b*c + a^3*f)*x^3)/x^4

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Sympy [A]
time = 1.57, size = 235, normalized size = 1.12 \begin {gather*} a^{2} \left (a g + 3 b d\right ) \log {\left (x \right )} + \frac {b^{3} f x^{8}}{8} + \frac {b^{3} g x^{9}}{9} + \frac {b^{3} h x^{10}}{10} + x^{7} \cdot \left (\frac {3 a b^{2} h}{7} + \frac {b^{3} e}{7}\right ) + x^{6} \left (\frac {a b^{2} g}{2} + \frac {b^{3} d}{6}\right ) + x^{5} \cdot \left (\frac {3 a b^{2} f}{5} + \frac {b^{3} c}{5}\right ) + x^{4} \cdot \left (\frac {3 a^{2} b h}{4} + \frac {3 a b^{2} e}{4}\right ) + x^{3} \left (a^{2} b g + a b^{2} d\right ) + x^{2} \cdot \left (\frac {3 a^{2} b f}{2} + \frac {3 a b^{2} c}{2}\right ) + x \left (a^{3} h + 3 a^{2} b e\right ) + \frac {- 3 a^{3} c - 4 a^{3} d x - 6 a^{3} e x^{2} + x^{3} \left (- 12 a^{3} f - 36 a^{2} b c\right )}{12 x^{4}} \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**5,x)

[Out]

a**2*(a*g + 3*b*d)*log(x) + b**3*f*x**8/8 + b**3*g*x**9/9 + b**3*h*x**10/10 + x**7*(3*a*b**2*h/7 + b**3*e/7) +
x**6*(a*b**2*g/2 + b**3*d/6) + x**5*(3*a*b**2*f/5 + b**3*c/5) + x**4*(3*a**2*b*h/4 + 3*a*b**2*e/4) + x**3*(a*
*2*b*g + a*b**2*d) + x**2*(3*a**2*b*f/2 + 3*a*b**2*c/2) + x*(a**3*h + 3*a**2*b*e) + (-3*a**3*c - 4*a**3*d*x -
6*a**3*e*x**2 + x**3*(-12*a**3*f - 36*a**2*b*c))/(12*x**4)

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Giac [A]
time = 0.60, size = 224, normalized size = 1.07 \begin {gather*} \frac {1}{10} \, b^{3} h x^{10} + \frac {1}{9} \, b^{3} g x^{9} + \frac {1}{8} \, b^{3} f x^{8} + \frac {3}{7} \, a b^{2} h x^{7} + \frac {1}{7} \, b^{3} x^{7} e + \frac {1}{6} \, b^{3} d x^{6} + \frac {1}{2} \, a b^{2} g x^{6} + \frac {1}{5} \, b^{3} c x^{5} + \frac {3}{5} \, a b^{2} f x^{5} + \frac {3}{4} \, a^{2} b h x^{4} + \frac {3}{4} \, a b^{2} x^{4} e + a b^{2} d x^{3} + a^{2} b g x^{3} + \frac {3}{2} \, a b^{2} c x^{2} + \frac {3}{2} \, a^{2} b f x^{2} + a^{3} h x + 3 \, a^{2} b x e + {\left (3 \, a^{2} b d + a^{3} g\right )} \log \left ({\left | x \right |}\right ) - \frac {6 \, a^{3} x^{2} e + 4 \, a^{3} d x + 3 \, a^{3} c + 12 \, {\left (3 \, a^{2} b c + a^{3} f\right )} x^{3}}{12 \, x^{4}} \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^5,x, algorithm="giac")

[Out]

1/10*b^3*h*x^10 + 1/9*b^3*g*x^9 + 1/8*b^3*f*x^8 + 3/7*a*b^2*h*x^7 + 1/7*b^3*x^7*e + 1/6*b^3*d*x^6 + 1/2*a*b^2*
g*x^6 + 1/5*b^3*c*x^5 + 3/5*a*b^2*f*x^5 + 3/4*a^2*b*h*x^4 + 3/4*a*b^2*x^4*e + a*b^2*d*x^3 + a^2*b*g*x^3 + 3/2*
a*b^2*c*x^2 + 3/2*a^2*b*f*x^2 + a^3*h*x + 3*a^2*b*x*e + (3*a^2*b*d + a^3*g)*log(abs(x)) - 1/12*(6*a^3*x^2*e +
4*a^3*d*x + 3*a^3*c + 12*(3*a^2*b*c + a^3*f)*x^3)/x^4

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Mupad [B]
time = 5.03, size = 199, normalized size = 0.95 \begin {gather*} x^5\,\left (\frac {c\,b^3}{5}+\frac {3\,a\,f\,b^2}{5}\right )+x^6\,\left (\frac {d\,b^3}{6}+\frac {a\,g\,b^2}{2}\right )+x^7\,\left (\frac {e\,b^3}{7}+\frac {3\,a\,h\,b^2}{7}\right )+\ln \left (x\right )\,\left (g\,a^3+3\,b\,d\,a^2\right )-\frac {x^3\,\left (f\,a^3+3\,b\,c\,a^2\right )+\frac {a^3\,c}{4}+\frac {a^3\,e\,x^2}{2}+\frac {a^3\,d\,x}{3}}{x^4}+x\,\left (h\,a^3+3\,b\,e\,a^2\right )+\frac {b^3\,f\,x^8}{8}+\frac {b^3\,g\,x^9}{9}+\frac {b^3\,h\,x^{10}}{10}+\frac {3\,a\,b\,x^2\,\left (b\,c+a\,f\right )}{2}+a\,b\,x^3\,\left (b\,d+a\,g\right )+\frac {3\,a\,b\,x^4\,\left (b\,e+a\,h\right )}{4} \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^5,x)

[Out]

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

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