Integrand size = 38, antiderivative size = 345 \[ \int \frac {x^4 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=\frac {h x}{b^3}+\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 b^3 \left (a+b x^3\right )^2}-\frac {x \left (a (7 b e-13 a h)-2 b (b c-4 a f) x-3 b (b d-3 a g) x^2\right )}{18 a b^3 \left (a+b x^3\right )}-\frac {\left (b^{5/3} c+2 a^{2/3} b e+5 a b^{2/3} f-14 a^{5/3} h\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{4/3} b^{10/3}}-\frac {\left (b^{2/3} (b c+5 a f)-2 a^{2/3} (b e-7 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{10/3}}+\frac {\left (b^{2/3} (b c+5 a f)-2 a^{2/3} (b e-7 a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{10/3}}+\frac {g \log \left (a+b x^3\right )}{3 b^3} \] Output:
h*x/b^3+1/6*x*(a*(-a*h+b*e)-b*(-a*f+b*c)*x-b*(-a*g+b*d)*x^2)/b^3/(b*x^3+a) ^2-1/18*x*(a*(-13*a*h+7*b*e)-2*b*(-4*a*f+b*c)*x-3*b*(-3*a*g+b*d)*x^2)/a/b^ 3/(b*x^3+a)-1/27*(b^(5/3)*c+2*a^(2/3)*b*e+5*a*b^(2/3)*f-14*a^(5/3)*h)*arct an(1/3*(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a^(1/3))*3^(1/2)/a^(4/3)/b^(10/3)-1/2 7*(b^(2/3)*(5*a*f+b*c)-2*a^(2/3)*(-7*a*h+b*e))*ln(a^(1/3)+b^(1/3)*x)/a^(4/ 3)/b^(10/3)+1/54*(b^(2/3)*(5*a*f+b*c)-2*a^(2/3)*(-7*a*h+b*e))*ln(a^(2/3)-a ^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(4/3)/b^(10/3)+1/3*g*ln(b*x^3+a)/b^3
Time = 0.22 (sec) , antiderivative size = 342, normalized size of antiderivative = 0.99 \[ \int \frac {x^4 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=\frac {54 b^{2/3} h x-\frac {9 b^{2/3} \left (b^2 c x^2+a^2 (g+h x)-a b (d+x (e+f x))\right )}{\left (a+b x^3\right )^2}+\frac {3 b^{2/3} \left (2 b^2 c x^2+a^2 (12 g+13 h x)-a b (6 d+x (7 e+8 f x))\right )}{a \left (a+b x^3\right )}-\frac {2 \sqrt {3} \left (b^2 c+2 a^{2/3} b^{4/3} e+5 a b f-14 a^{5/3} \sqrt [3]{b} h\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{4/3}}-\frac {2 \left (b^2 c-2 a^{2/3} b^{4/3} e+5 a b f+14 a^{5/3} \sqrt [3]{b} h\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{4/3}}+\frac {\left (b^2 c-2 a^{2/3} b^{4/3} e+5 a b f+14 a^{5/3} \sqrt [3]{b} h\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{4/3}}+18 b^{2/3} g \log \left (a+b x^3\right )}{54 b^{11/3}} \] Input:
Integrate[(x^4*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3)^3,x]
Output:
(54*b^(2/3)*h*x - (9*b^(2/3)*(b^2*c*x^2 + a^2*(g + h*x) - a*b*(d + x*(e + f*x))))/(a + b*x^3)^2 + (3*b^(2/3)*(2*b^2*c*x^2 + a^2*(12*g + 13*h*x) - a* b*(6*d + x*(7*e + 8*f*x))))/(a*(a + b*x^3)) - (2*Sqrt[3]*(b^2*c + 2*a^(2/3 )*b^(4/3)*e + 5*a*b*f - 14*a^(5/3)*b^(1/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^ (1/3))/Sqrt[3]])/a^(4/3) - (2*(b^2*c - 2*a^(2/3)*b^(4/3)*e + 5*a*b*f + 14* a^(5/3)*b^(1/3)*h)*Log[a^(1/3) + b^(1/3)*x])/a^(4/3) + ((b^2*c - 2*a^(2/3) *b^(4/3)*e + 5*a*b*f + 14*a^(5/3)*b^(1/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3) *x + b^(2/3)*x^2])/a^(4/3) + 18*b^(2/3)*g*Log[a + b*x^3])/(54*b^(11/3))
Time = 1.84 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {2367, 2397, 27, 2426, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^4 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx\) |
\(\Big \downarrow \) 2367 |
\(\displaystyle \frac {x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 b^3 \left (a+b x^3\right )^2}-\frac {\int \frac {-6 a b^2 h x^6-6 a b^2 g x^5-6 a b^2 f x^4-6 a b (b e-a h) x^3-3 a b (b d-a g) x^2-2 a b (b c-a f) x+a^2 (b e-a h)}{\left (b x^3+a\right )^2}dx}{6 a b^3}\) |
\(\Big \downarrow \) 2397 |
\(\displaystyle \frac {x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 b^3 \left (a+b x^3\right )^2}-\frac {\frac {x \left (-2 b x (b c-4 a f)-3 b x^2 (b d-3 a g)+a (7 b e-13 a h)\right )}{3 \left (a+b x^3\right )}-\frac {\int \frac {2 \left (9 a^2 h x^3 b^3+9 a^2 g x^2 b^3+a (b c+5 a f) x b^3+a^2 (2 b e-5 a h) b^2\right )}{b x^3+a}dx}{3 a b^2}}{6 a b^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 b^3 \left (a+b x^3\right )^2}-\frac {\frac {x \left (-2 b x (b c-4 a f)-3 b x^2 (b d-3 a g)+a (7 b e-13 a h)\right )}{3 \left (a+b x^3\right )}-\frac {2 \int \frac {9 a^2 h x^3 b^3+9 a^2 g x^2 b^3+a (b c+5 a f) x b^3+a^2 (2 b e-5 a h) b^2}{b x^3+a}dx}{3 a b^2}}{6 a b^3}\) |
\(\Big \downarrow \) 2426 |
\(\displaystyle \frac {x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 b^3 \left (a+b x^3\right )^2}-\frac {\frac {x \left (-2 b x (b c-4 a f)-3 b x^2 (b d-3 a g)+a (7 b e-13 a h)\right )}{3 \left (a+b x^3\right )}-\frac {2 \int \left (9 a^2 h b^2+\frac {9 a^2 g x^2 b^3+a (b c+5 a f) x b^3+2 a^2 (b e-7 a h) b^2}{b x^3+a}\right )dx}{3 a b^2}}{6 a b^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 b^3 \left (a+b x^3\right )^2}-\frac {\frac {x \left (-2 b x (b c-4 a f)-3 b x^2 (b d-3 a g)+a (7 b e-13 a h)\right )}{3 \left (a+b x^3\right )}-\frac {2 \left (-\frac {a^{2/3} b^{5/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (2 a^{2/3} b e-14 a^{5/3} h+5 a b^{2/3} f+b^{5/3} c\right )}{\sqrt {3}}+\frac {1}{6} a^{2/3} b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (b^{2/3} (5 a f+b c)-2 a^{2/3} (b e-7 a h)\right )-\frac {1}{3} a^{2/3} b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (b^{2/3} (5 a f+b c)-2 a^{2/3} (b e-7 a h)\right )+3 a^2 b^2 g \log \left (a+b x^3\right )+9 a^2 b^2 h x\right )}{3 a b^2}}{6 a b^3}\) |
Input:
Int[(x^4*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3)^3,x]
Output:
(x*(a*(b*e - a*h) - b*(b*c - a*f)*x - b*(b*d - a*g)*x^2))/(6*b^3*(a + b*x^ 3)^2) - ((x*(a*(7*b*e - 13*a*h) - 2*b*(b*c - 4*a*f)*x - 3*b*(b*d - 3*a*g)* x^2))/(3*(a + b*x^3)) - (2*(9*a^2*b^2*h*x - (a^(2/3)*b^(5/3)*(b^(5/3)*c + 2*a^(2/3)*b*e + 5*a*b^(2/3)*f - 14*a^(5/3)*h)*ArcTan[(a^(1/3) - 2*b^(1/3)* x)/(Sqrt[3]*a^(1/3))])/Sqrt[3] - (a^(2/3)*b^(5/3)*(b^(2/3)*(b*c + 5*a*f) - 2*a^(2/3)*(b*e - 7*a*h))*Log[a^(1/3) + b^(1/3)*x])/3 + (a^(2/3)*b^(5/3)*( b^(2/3)*(b*c + 5*a*f) - 2*a^(2/3)*(b*e - 7*a*h))*Log[a^(2/3) - a^(1/3)*b^( 1/3)*x + b^(2/3)*x^2])/6 + 3*a^2*b^2*g*Log[a + b*x^3]))/(3*a*b^2))/(6*a*b^ 3)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = m + Expon[Pq, x]}, Module[{Q = PolynomialQuotient[b^(Floor[(q - 1)/n] + 1) *x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*x^ m*Pq, a + b*x^n, x]}, Simp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floo r[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)) I nt[(a + b*x^n)^(p + 1)*ExpandToSum[a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], x], x], x]] /; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0 ] && LtQ[p, -1] && IGtQ[m, 0]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x]}, S imp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)) Int[(a + b*x^n)^(p + 1)* ExpandToSum[a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], x], x], x]] /; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]
Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IntegerQ[n]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.14 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.49
method | result | size |
risch | \(\frac {h x}{b^{3}}+\frac {-\frac {b^{2} \left (4 a f -c b \right ) x^{5}}{9 a}+\left (\frac {13}{18} a b h -\frac {7}{18} e \,b^{2}\right ) x^{4}+\left (\frac {2}{3} a b g -\frac {1}{3} b^{2} d \right ) x^{3}-\frac {b \left (5 a f +c b \right ) x^{2}}{18}+\frac {a \left (5 a h -2 b e \right ) x}{9}+\frac {a^{2} g}{2}-\frac {d a b}{6}}{b^{3} \left (b \,x^{3}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\left (9 b g \,\textit {\_R}^{2}+\frac {b \left (5 a f +c b \right ) \textit {\_R}}{a}-14 a h +2 b e \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{27 b^{4}}\) | \(168\) |
default | \(\frac {h x}{b^{3}}-\frac {\frac {\frac {b^{2} \left (4 a f -c b \right ) x^{5}}{9 a}+\left (-\frac {13}{18} a b h +\frac {7}{18} e \,b^{2}\right ) x^{4}+\left (-\frac {2}{3} a b g +\frac {1}{3} b^{2} d \right ) x^{3}+\frac {b \left (5 a f +c b \right ) x^{2}}{18}-\frac {a \left (5 a h -2 b e \right ) x}{9}-\frac {a^{2} g}{2}+\frac {d a b}{6}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (14 a^{2} h -2 a b e \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\left (-5 a b f -b^{2} c \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )-3 a g \ln \left (b \,x^{3}+a \right )}{9 a}}{b^{3}}\) | \(339\) |
Input:
int(x^4*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^3,x,method=_RETURNVERBOS E)
Output:
h*x/b^3+(-1/9*b^2*(4*a*f-b*c)/a*x^5+(13/18*a*b*h-7/18*e*b^2)*x^4+(2/3*a*b* g-1/3*b^2*d)*x^3-1/18*b*(5*a*f+b*c)*x^2+1/9*a*(5*a*h-2*b*e)*x+1/2*a^2*g-1/ 6*d*a*b)/b^3/(b*x^3+a)^2+1/27/b^4*sum((9*b*g*_R^2+b*(5*a*f+b*c)/a*_R-14*a* h+2*b*e)/_R^2*ln(x-_R),_R=RootOf(_Z^3*b+a))
Result contains complex when optimal does not.
Time = 2.36 (sec) , antiderivative size = 12967, normalized size of antiderivative = 37.59 \[ \int \frac {x^4 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(x^4*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^3,x, algorithm="fr icas")
Output:
Too large to include
Timed out. \[ \int \frac {x^4 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=\text {Timed out} \] Input:
integrate(x**4*(h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(b*x**3+a)**3,x)
Output:
Timed out
Time = 0.13 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.13 \[ \int \frac {x^4 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=\frac {2 \, {\left (b^{3} c - 4 \, a b^{2} f\right )} x^{5} - {\left (7 \, a b^{2} e - 13 \, a^{2} b h\right )} x^{4} - 3 \, a^{2} b d + 9 \, a^{3} g - 6 \, {\left (a b^{2} d - 2 \, a^{2} b g\right )} x^{3} - {\left (a b^{2} c + 5 \, a^{2} b f\right )} x^{2} - 2 \, {\left (2 \, a^{2} b e - 5 \, a^{3} h\right )} x}{18 \, {\left (a b^{5} x^{6} + 2 \, a^{2} b^{4} x^{3} + a^{3} b^{3}\right )}} + \frac {h x}{b^{3}} + \frac {\sqrt {3} {\left (b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} + 5 \, a b f \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, a b e \left (\frac {a}{b}\right )^{\frac {1}{3}} - 14 \, a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{2} b^{3}} + \frac {{\left (18 \, a b g \left (\frac {a}{b}\right )^{\frac {2}{3}} + b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a b f \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a b e + 14 \, a^{2} h\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (9 \, a b g \left (\frac {a}{b}\right )^{\frac {2}{3}} - b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, a b f \left (\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a b e - 14 \, a^{2} h\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate(x^4*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^3,x, algorithm="ma xima")
Output:
1/18*(2*(b^3*c - 4*a*b^2*f)*x^5 - (7*a*b^2*e - 13*a^2*b*h)*x^4 - 3*a^2*b*d + 9*a^3*g - 6*(a*b^2*d - 2*a^2*b*g)*x^3 - (a*b^2*c + 5*a^2*b*f)*x^2 - 2*( 2*a^2*b*e - 5*a^3*h)*x)/(a*b^5*x^6 + 2*a^2*b^4*x^3 + a^3*b^3) + h*x/b^3 + 1/27*sqrt(3)*(b^2*c*(a/b)^(2/3) + 5*a*b*f*(a/b)^(2/3) + 2*a*b*e*(a/b)^(1/3 ) - 14*a^2*h*(a/b)^(1/3))*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/ 3))/(a^2*b^3) + 1/54*(18*a*b*g*(a/b)^(2/3) + b^2*c*(a/b)^(1/3) + 5*a*b*f*( a/b)^(1/3) - 2*a*b*e + 14*a^2*h)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a *b^4*(a/b)^(2/3)) + 1/27*(9*a*b*g*(a/b)^(2/3) - b^2*c*(a/b)^(1/3) - 5*a*b* f*(a/b)^(1/3) + 2*a*b*e - 14*a^2*h)*log(x + (a/b)^(1/3))/(a*b^4*(a/b)^(2/3 ))
Time = 0.14 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.10 \[ \int \frac {x^4 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=\frac {h x}{b^{3}} + \frac {g \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} - \frac {\sqrt {3} {\left (2 \, a b e - 14 \, a^{2} h - \left (-a b^{2}\right )^{\frac {1}{3}} b c - 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2}} - \frac {{\left (2 \, a b e - 14 \, a^{2} h + \left (-a b^{2}\right )^{\frac {1}{3}} b c + 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2}} + \frac {2 \, {\left (b^{3} c - 4 \, a b^{2} f\right )} x^{5} - {\left (7 \, a b^{2} e - 13 \, a^{2} b h\right )} x^{4} - 3 \, a^{2} b d + 9 \, a^{3} g - 6 \, {\left (a b^{2} d - 2 \, a^{2} b g\right )} x^{3} - {\left (a b^{2} c + 5 \, a^{2} b f\right )} x^{2} - 2 \, {\left (2 \, a^{2} b e - 5 \, a^{3} h\right )} x}{18 \, {\left (b x^{3} + a\right )}^{2} a b^{3}} - \frac {{\left (a b^{6} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a^{2} b^{5} f \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a^{2} b^{5} e - 14 \, a^{3} b^{4} h\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{3} b^{7}} \] Input:
integrate(x^4*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^3,x, algorithm="gi ac")
Output:
h*x/b^3 + 1/3*g*log(abs(b*x^3 + a))/b^3 - 1/27*sqrt(3)*(2*a*b*e - 14*a^2*h - (-a*b^2)^(1/3)*b*c - 5*(-a*b^2)^(1/3)*a*f)*arctan(1/3*sqrt(3)*(2*x + (- a/b)^(1/3))/(-a/b)^(1/3))/((-a*b^2)^(2/3)*a*b^2) - 1/54*(2*a*b*e - 14*a^2* h + (-a*b^2)^(1/3)*b*c + 5*(-a*b^2)^(1/3)*a*f)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/((-a*b^2)^(2/3)*a*b^2) + 1/18*(2*(b^3*c - 4*a*b^2*f)*x^5 - ( 7*a*b^2*e - 13*a^2*b*h)*x^4 - 3*a^2*b*d + 9*a^3*g - 6*(a*b^2*d - 2*a^2*b*g )*x^3 - (a*b^2*c + 5*a^2*b*f)*x^2 - 2*(2*a^2*b*e - 5*a^3*h)*x)/((b*x^3 + a )^2*a*b^3) - 1/27*(a*b^6*c*(-a/b)^(1/3) + 5*a^2*b^5*f*(-a/b)^(1/3) + 2*a^2 *b^5*e - 14*a^3*b^4*h)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^3*b^7)
Time = 0.62 (sec) , antiderivative size = 916, normalized size of antiderivative = 2.66 \[ \int \frac {x^4 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx =\text {Too large to display} \] Input:
int((x^4*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3)^3,x)
Output:
symsum(log(root(19683*a^4*b^10*z^3 - 19683*a^4*b^7*g*z^2 - 5670*a^4*b^4*f* h*z - 1134*a^3*b^5*c*h*z + 810*a^3*b^5*e*f*z + 162*a^2*b^6*c*e*z + 6561*a^ 4*b^4*g^2*z + 1890*a^4*b*f*g*h + 378*a^3*b^2*c*g*h - 270*a^3*b^2*e*f*g - 5 4*a^2*b^3*c*e*g - 1176*a^4*b*e*h^2 + 15*a*b^4*c^2*f + 168*a^3*b^2*e^2*h + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 - 8*a^2*b^3*e^3 - 729*a^4*b*g^3 + 2744* a^5*h^3 + b^5*c^3, z, k)*(9*root(19683*a^4*b^10*z^3 - 19683*a^4*b^7*g*z^2 - 5670*a^4*b^4*f*h*z - 1134*a^3*b^5*c*h*z + 810*a^3*b^5*e*f*z + 162*a^2*b^ 6*c*e*z + 6561*a^4*b^4*g^2*z + 1890*a^4*b*f*g*h + 378*a^3*b^2*c*g*h - 270* a^3*b^2*e*f*g - 54*a^2*b^3*c*e*g - 1176*a^4*b*e*h^2 + 15*a*b^4*c^2*f + 168 *a^3*b^2*e^2*h + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 - 8*a^2*b^3*e^3 - 729* a^4*b*g^3 + 2744*a^5*h^3 + b^5*c^3, z, k)*a*b^2 - (6*a*g)/b + (x*(54*a^2*b ^4*e - 378*a^3*b^3*h))/(81*a^2*b^4)) + (81*a^2*g^2 + 2*b^2*c*e - 70*a^2*f* h - 14*a*b*c*h + 10*a*b*e*f)/(81*a*b^4) + (x*(b^3*c^2 + 25*a^2*b*f^2 + 126 *a^3*g*h + 10*a*b^2*c*f - 18*a^2*b*e*g))/(81*a^2*b^4))*root(19683*a^4*b^10 *z^3 - 19683*a^4*b^7*g*z^2 - 5670*a^4*b^4*f*h*z - 1134*a^3*b^5*c*h*z + 810 *a^3*b^5*e*f*z + 162*a^2*b^6*c*e*z + 6561*a^4*b^4*g^2*z + 1890*a^4*b*f*g*h + 378*a^3*b^2*c*g*h - 270*a^3*b^2*e*f*g - 54*a^2*b^3*c*e*g - 1176*a^4*b*e *h^2 + 15*a*b^4*c^2*f + 168*a^3*b^2*e^2*h + 75*a^2*b^3*c*f^2 + 125*a^3*b^2 *f^3 - 8*a^2*b^3*e^3 - 729*a^4*b*g^3 + 2744*a^5*h^3 + b^5*c^3, z, k), k, 1 , 3) - (x^2*((b^2*c)/18 + (5*a*b*f)/18) - (a^2*g)/2 - x*((5*a^2*h)/9 - ...
Time = 0.18 (sec) , antiderivative size = 1316, normalized size of antiderivative = 3.81 \[ \int \frac {x^4 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx =\text {Too large to display} \] Input:
int(x^4*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^3,x)
Output:
(28*b**(1/3)*a**(2/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqr t(3)))*a**3*h - 4*b**(1/3)*a**(2/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x) /(a**(1/3)*sqrt(3)))*a**2*b*e + 56*b**(1/3)*a**(2/3)*sqrt(3)*atan((a**(1/3 ) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**2*b*h*x**3 - 8*b**(1/3)*a**(2/3)* sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a*b**2*e*x**3 + 28*b**(1/3)*a**(2/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqr t(3)))*a*b**2*h*x**6 - 4*b**(1/3)*a**(2/3)*sqrt(3)*atan((a**(1/3) - 2*b**( 1/3)*x)/(a**(1/3)*sqrt(3)))*b**3*e*x**6 - 10*sqrt(3)*atan((a**(1/3) - 2*b* *(1/3)*x)/(a**(1/3)*sqrt(3)))*a**3*b*f - 2*sqrt(3)*atan((a**(1/3) - 2*b**( 1/3)*x)/(a**(1/3)*sqrt(3)))*a**2*b**2*c - 20*sqrt(3)*atan((a**(1/3) - 2*b* *(1/3)*x)/(a**(1/3)*sqrt(3)))*a**2*b**2*f*x**3 - 4*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a*b**3*c*x**3 - 10*sqrt(3)*atan((a**(1 /3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a*b**3*f*x**6 - 2*sqrt(3)*atan((a* *(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*b**4*c*x**6 + 14*b**(1/3)*a**(2 /3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**3*h - 2*b**(1/3 )*a**(2/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**2*b*e + 28*b**(1/3)*a**(2/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a **2*b*h*x**3 - 4*b**(1/3)*a**(2/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b* *(2/3)*x**2)*a*b**2*e*x**3 + 14*b**(1/3)*a**(2/3)*log(a**(2/3) - b**(1/3)* a**(1/3)*x + b**(2/3)*x**2)*a*b**2*h*x**6 - 2*b**(1/3)*a**(2/3)*log(a**...