Integrand size = 36, antiderivative size = 323 \[ \int \frac {x \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a b^2 \left (a+b x^3\right )^2}+\frac {x \left (a (b e-7 a h)+2 b (2 b c+a f) x+3 b (b d+a g) x^2\right )}{18 a^2 b^2 \left (a+b x^3\right )}-\frac {\left (2 b^{5/3} c+a^{2/3} b e+a b^{2/3} f+2 a^{5/3} h\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{7/3} b^{7/3}}-\frac {\left (b^{2/3} (2 b c+a f)-a^{2/3} (b e+2 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{7/3}}+\frac {\left (b^{2/3} (2 b c+a f)-a^{2/3} (b e+2 a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{7/3} b^{7/3}} \] Output:
-1/6*x*(a*(-a*h+b*e)-b*(-a*f+b*c)*x-b*(-a*g+b*d)*x^2)/a/b^2/(b*x^3+a)^2+1/ 18*x*(a*(-7*a*h+b*e)+2*b*(a*f+2*b*c)*x+3*b*(a*g+b*d)*x^2)/a^2/b^2/(b*x^3+a )-1/27*(2*b^(5/3)*c+a^(2/3)*b*e+a*b^(2/3)*f+2*a^(5/3)*h)*arctan(1/3*(a^(1/ 3)-2*b^(1/3)*x)*3^(1/2)/a^(1/3))*3^(1/2)/a^(7/3)/b^(7/3)-1/27*(b^(2/3)*(a* f+2*b*c)-a^(2/3)*(2*a*h+b*e))*ln(a^(1/3)+b^(1/3)*x)/a^(7/3)/b^(7/3)+1/54*( b^(2/3)*(a*f+2*b*c)-a^(2/3)*(2*a*h+b*e))*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2 /3)*x^2)/a^(7/3)/b^(7/3)
Time = 0.24 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.92 \[ \int \frac {x \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=\frac {-\frac {3 \sqrt [3]{a} \sqrt [3]{b} \left (-4 b^2 c x^2-a b x (e+2 f x)+a^2 (6 g+7 h x)\right )}{a+b x^3}+\frac {9 a^{4/3} \sqrt [3]{b} \left (b^2 c x^2+a^2 (g+h x)-a b (d+x (e+f x))\right )}{\left (a+b x^3\right )^2}-2 \sqrt {3} \left (2 b^{5/3} c+a^{2/3} b e+a b^{2/3} f+2 a^{5/3} h\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 \left (-2 b^{5/3} c+a^{2/3} b e-a b^{2/3} f+2 a^{5/3} h\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+\left (2 b^{5/3} c-a^{2/3} b e+a b^{2/3} f-2 a^{5/3} h\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{7/3} b^{7/3}} \] Input:
Integrate[(x*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3)^3,x]
Output:
((-3*a^(1/3)*b^(1/3)*(-4*b^2*c*x^2 - a*b*x*(e + 2*f*x) + a^2*(6*g + 7*h*x) ))/(a + b*x^3) + (9*a^(4/3)*b^(1/3)*(b^2*c*x^2 + a^2*(g + h*x) - a*b*(d + x*(e + f*x))))/(a + b*x^3)^2 - 2*Sqrt[3]*(2*b^(5/3)*c + a^(2/3)*b*e + a*b^ (2/3)*f + 2*a^(5/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*(-2 *b^(5/3)*c + a^(2/3)*b*e - a*b^(2/3)*f + 2*a^(5/3)*h)*Log[a^(1/3) + b^(1/3 )*x] + (2*b^(5/3)*c - a^(2/3)*b*e + a*b^(2/3)*f - 2*a^(5/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(7/3)*b^(7/3))
Time = 1.45 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.361, Rules used = {2367, 25, 2397, 27, 2399, 16, 27, 1142, 25, 27, 1082, 217, 1103}
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.
| \(\displaystyle \int \frac {x \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 2367 |
| \(\displaystyle -\frac {\int -\frac {6 a b h x^3+3 b (b d+a g) x^2+2 b (2 b c+a f) x+a (b e-a h)}{\left (b x^3+a\right )^2}dx}{6 a b^2}-\frac {x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 a b^2 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {6 a b h x^3+3 b (b d+a g) x^2+2 b (2 b c+a f) x+a (b e-a h)}{\left (b x^3+a\right )^2}dx}{6 a b^2}-\frac {x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 a b^2 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2397 |
| \(\displaystyle \frac {\frac {x \left (2 b x (a f+2 b c)+3 b x^2 (a g+b d)+a (b e-7 a h)\right )}{3 a \left (a+b x^3\right )}-\frac {\int -\frac {2 b (a (b e+2 a h)+b (2 b c+a f) x)}{b x^3+a}dx}{3 a b}}{6 a b^2}-\frac {x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 a b^2 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \int \frac {a (b e+2 a h)+b (2 b c+a f) x}{b x^3+a}dx}{3 a}+\frac {x \left (2 b x (a f+2 b c)+3 b x^2 (a g+b d)+a (b e-7 a h)\right )}{3 a \left (a+b x^3\right )}}{6 a b^2}-\frac {x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 a b^2 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2399 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\int \frac {\sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a} \left (b^{2/3} (2 b c+a f)+2 a^{2/3} (b e+2 a h)\right )+\sqrt [3]{b} \left (b^{2/3} (2 b c+a f)-a^{2/3} (b e+2 a h)\right ) x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}-\frac {\left (b^{2/3} (a f+2 b c)-a^{2/3} (2 a h+b e)\right ) \int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 \sqrt [3]{a}}\right )}{3 a}+\frac {x \left (2 b x (a f+2 b c)+3 b x^2 (a g+b d)+a (b e-7 a h)\right )}{3 a \left (a+b x^3\right )}}{6 a b^2}-\frac {x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 a b^2 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\int \frac {\sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a} \left (b^{2/3} (2 b c+a f)+2 a^{2/3} (b e+2 a h)\right )+\sqrt [3]{b} \left (b^{2/3} (2 b c+a f)-a^{2/3} (b e+2 a h)\right ) x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (b^{2/3} (a f+2 b c)-a^{2/3} (2 a h+b e)\right )}{3 \sqrt [3]{a} \sqrt [3]{b}}\right )}{3 a}+\frac {x \left (2 b x (a f+2 b c)+3 b x^2 (a g+b d)+a (b e-7 a h)\right )}{3 a \left (a+b x^3\right )}}{6 a b^2}-\frac {x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 a b^2 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\int \frac {\sqrt [3]{a} \left (b^{2/3} (2 b c+a f)+2 a^{2/3} (b e+2 a h)\right )+\sqrt [3]{b} \left (b^{2/3} (2 b c+a f)-a^{2/3} (b e+2 a h)\right ) x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (b^{2/3} (a f+2 b c)-a^{2/3} (2 a h+b e)\right )}{3 \sqrt [3]{a} \sqrt [3]{b}}\right )}{3 a}+\frac {x \left (2 b x (a f+2 b c)+3 b x^2 (a g+b d)+a (b e-7 a h)\right )}{3 a \left (a+b x^3\right )}}{6 a b^2}-\frac {x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 a b^2 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \left (a^{2/3} b e+2 a^{5/3} h+a b^{2/3} f+2 b^{5/3} c\right ) \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\left (b^{2/3} (a f+2 b c)-a^{2/3} (2 a h+b e)\right ) \int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (b^{2/3} (a f+2 b c)-a^{2/3} (2 a h+b e)\right )}{3 \sqrt [3]{a} \sqrt [3]{b}}\right )}{3 a}+\frac {x \left (2 b x (a f+2 b c)+3 b x^2 (a g+b d)+a (b e-7 a h)\right )}{3 a \left (a+b x^3\right )}}{6 a b^2}-\frac {x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 a b^2 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \left (a^{2/3} b e+2 a^{5/3} h+a b^{2/3} f+2 b^{5/3} c\right ) \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\left (b^{2/3} (a f+2 b c)-a^{2/3} (2 a h+b e)\right ) \int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (b^{2/3} (a f+2 b c)-a^{2/3} (2 a h+b e)\right )}{3 \sqrt [3]{a} \sqrt [3]{b}}\right )}{3 a}+\frac {x \left (2 b x (a f+2 b c)+3 b x^2 (a g+b d)+a (b e-7 a h)\right )}{3 a \left (a+b x^3\right )}}{6 a b^2}-\frac {x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 a b^2 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \left (a^{2/3} b e+2 a^{5/3} h+a b^{2/3} f+2 b^{5/3} c\right ) \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {1}{2} \left (b^{2/3} (a f+2 b c)-a^{2/3} (2 a h+b e)\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (b^{2/3} (a f+2 b c)-a^{2/3} (2 a h+b e)\right )}{3 \sqrt [3]{a} \sqrt [3]{b}}\right )}{3 a}+\frac {x \left (2 b x (a f+2 b c)+3 b x^2 (a g+b d)+a (b e-7 a h)\right )}{3 a \left (a+b x^3\right )}}{6 a b^2}-\frac {x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 a b^2 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\frac {3 \left (a^{2/3} b e+2 a^{5/3} h+a b^{2/3} f+2 b^{5/3} c\right ) \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}-\frac {1}{2} \left (b^{2/3} (a f+2 b c)-a^{2/3} (2 a h+b e)\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (b^{2/3} (a f+2 b c)-a^{2/3} (2 a h+b e)\right )}{3 \sqrt [3]{a} \sqrt [3]{b}}\right )}{3 a}+\frac {x \left (2 b x (a f+2 b c)+3 b x^2 (a g+b d)+a (b e-7 a h)\right )}{3 a \left (a+b x^3\right )}}{6 a b^2}-\frac {x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 a b^2 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {2 \left (\frac {-\frac {1}{2} \left (b^{2/3} (a f+2 b c)-a^{2/3} (2 a h+b e)\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (a^{2/3} b e+2 a^{5/3} h+a b^{2/3} f+2 b^{5/3} c\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (b^{2/3} (a f+2 b c)-a^{2/3} (2 a h+b e)\right )}{3 \sqrt [3]{a} \sqrt [3]{b}}\right )}{3 a}+\frac {x \left (2 b x (a f+2 b c)+3 b x^2 (a g+b d)+a (b e-7 a h)\right )}{3 a \left (a+b x^3\right )}}{6 a b^2}-\frac {x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 a b^2 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (b^{2/3} (a f+2 b c)-a^{2/3} (2 a h+b e)\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (a^{2/3} b e+2 a^{5/3} h+a b^{2/3} f+2 b^{5/3} c\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (b^{2/3} (a f+2 b c)-a^{2/3} (2 a h+b e)\right )}{3 \sqrt [3]{a} \sqrt [3]{b}}\right )}{3 a}+\frac {x \left (2 b x (a f+2 b c)+3 b x^2 (a g+b d)+a (b e-7 a h)\right )}{3 a \left (a+b x^3\right )}}{6 a b^2}-\frac {x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 a b^2 \left (a+b x^3\right )^2}\) |
Input:
Int[(x*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3)^3,x]
Output:
-1/6*(x*(a*(b*e - a*h) - b*(b*c - a*f)*x - b*(b*d - a*g)*x^2))/(a*b^2*(a + b*x^3)^2) + ((x*(a*(b*e - 7*a*h) + 2*b*(2*b*c + a*f)*x + 3*b*(b*d + a*g)* x^2))/(3*a*(a + b*x^3)) + (2*(-1/3*((b^(2/3)*(2*b*c + a*f) - a^(2/3)*(b*e + 2*a*h))*Log[a^(1/3) + b^(1/3)*x])/(a^(1/3)*b^(1/3)) + (-((Sqrt[3]*(2*b^( 5/3)*c + a^(2/3)*b*e + a*b^(2/3)*f + 2*a^(5/3)*h)*ArcTan[(1 - (2*b^(1/3)*x )/a^(1/3))/Sqrt[3]])/b^(1/3)) + ((b^(2/3)*(2*b*c + a*f) - a^(2/3)*(b*e + 2 *a*h))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(2*b^(1/3)))/(3*a^( 1/3))))/(3*a))/(6*a*b^2)
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, 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[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numer ator[Rt[a/b, 3]], s = Denominator[Rt[a/b, 3]]}, Simp[(-r)*((B*r - A*s)/(3*a *s)) Int[1/(r + s*x), x], x] + Simp[r/(3*a*s) Int[(r*(B*r + 2*A*s) + s* (B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] & & NeQ[a*B^3 - b*A^3, 0] && PosQ[a/b]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.12 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.49
| method | result | size |
| risch | \(\frac {\frac {\left (a f +2 c b \right ) x^{5}}{9 a^{2}}-\frac {\left (7 a h -b e \right ) x^{4}}{18 a b}-\frac {g \,x^{3}}{3 b}-\frac {\left (a f -7 c b \right ) x^{2}}{18 a b}-\frac {\left (2 a h +b e \right ) x}{9 b^{2}}-\frac {a g +b d}{6 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\left (\frac {\left (a f +2 c b \right ) \textit {\_R}}{a}+\frac {2 a h +b e}{b}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{27 b^{2} a}\) | \(158\) |
| default | \(\frac {\frac {\left (a f +2 c b \right ) x^{5}}{9 a^{2}}-\frac {\left (7 a h -b e \right ) x^{4}}{18 a b}-\frac {g \,x^{3}}{3 b}-\frac {\left (a f -7 c b \right ) x^{2}}{18 a b}-\frac {\left (2 a h +b e \right ) x}{9 b^{2}}-\frac {a g +b d}{6 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (2 a^{2} h +a b e \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\left (a b f +2 b^{2} c \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 a^{2} b^{2}}\) | \(316\) |
Input:
int(x*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^3,x,method=_RETURNVERBOSE)
Output:
(1/9*(a*f+2*b*c)/a^2*x^5-1/18*(7*a*h-b*e)/a/b*x^4-1/3*g*x^3/b-1/18*(a*f-7* b*c)/a/b*x^2-1/9*(2*a*h+b*e)/b^2*x-1/6*(a*g+b*d)/b^2)/(b*x^3+a)^2+1/27/b^2 /a*sum((1/a*(a*f+2*b*c)*_R+1/b*(2*a*h+b*e))/_R^2*ln(x-_R),_R=RootOf(_Z^3*b +a))
Result contains complex when optimal does not.
Time = 2.01 (sec) , antiderivative size = 7190, normalized size of antiderivative = 22.26 \[ \int \frac {x \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(x*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^3,x, algorithm="fric as")
Output:
Too large to include
Timed out. \[ \int \frac {x \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=\text {Timed out} \] Input:
integrate(x*(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 = 0.12 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.07 \[ \int \frac {x \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {6 \, a^{2} b g x^{3} - 2 \, {\left (2 \, b^{3} c + a b^{2} f\right )} x^{5} - {\left (a b^{2} e - 7 \, a^{2} b h\right )} x^{4} + 3 \, a^{2} b d + 3 \, a^{3} g - {\left (7 \, a b^{2} c - a^{2} b f\right )} x^{2} + 2 \, {\left (a^{2} b e + 2 \, a^{3} h\right )} x}{18 \, {\left (a^{2} b^{4} x^{6} + 2 \, a^{3} b^{3} x^{3} + a^{4} b^{2}\right )}} + \frac {\sqrt {3} {\left (2 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} + a b f \left (\frac {a}{b}\right )^{\frac {1}{3}} + a b e + 2 \, a^{2} h\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{2} b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (2 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} + a b f \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b e - 2 \, a^{2} h\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{2} b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (2 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} + a b f \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b e - 2 \, a^{2} h\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{2} b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate(x*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^3,x, algorithm="maxi ma")
Output:
-1/18*(6*a^2*b*g*x^3 - 2*(2*b^3*c + a*b^2*f)*x^5 - (a*b^2*e - 7*a^2*b*h)*x ^4 + 3*a^2*b*d + 3*a^3*g - (7*a*b^2*c - a^2*b*f)*x^2 + 2*(a^2*b*e + 2*a^3* h)*x)/(a^2*b^4*x^6 + 2*a^3*b^3*x^3 + a^4*b^2) + 1/27*sqrt(3)*(2*b^2*c*(a/b )^(1/3) + a*b*f*(a/b)^(1/3) + a*b*e + 2*a^2*h)*arctan(1/3*sqrt(3)*(2*x - ( a/b)^(1/3))/(a/b)^(1/3))/(a^2*b^3*(a/b)^(2/3)) + 1/54*(2*b^2*c*(a/b)^(1/3) + a*b*f*(a/b)^(1/3) - a*b*e - 2*a^2*h)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2 /3))/(a^2*b^3*(a/b)^(2/3)) - 1/27*(2*b^2*c*(a/b)^(1/3) + a*b*f*(a/b)^(1/3) - a*b*e - 2*a^2*h)*log(x + (a/b)^(1/3))/(a^2*b^3*(a/b)^(2/3))
Time = 0.14 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.04 \[ \int \frac {x \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {\sqrt {3} {\left (a b e + 2 \, a^{2} h - 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b c - \left (-a b^{2}\right )^{\frac {1}{3}} a f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b} - \frac {{\left (a b e + 2 \, a^{2} h + 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b c + \left (-a b^{2}\right )^{\frac {1}{3}} a f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b} - \frac {{\left (2 \, b^{2} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a b f \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a b e + 2 \, a^{2} h\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{3} b^{2}} + \frac {4 \, b^{3} c x^{5} + 2 \, a b^{2} f x^{5} + a b^{2} e x^{4} - 7 \, a^{2} b h x^{4} - 6 \, a^{2} b g x^{3} + 7 \, a b^{2} c x^{2} - a^{2} b f x^{2} - 2 \, a^{2} b e x - 4 \, a^{3} h x - 3 \, a^{2} b d - 3 \, a^{3} g}{18 \, {\left (b x^{3} + a\right )}^{2} a^{2} b^{2}} \] Input:
integrate(x*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^3,x, algorithm="giac ")
Output:
-1/27*sqrt(3)*(a*b*e + 2*a^2*h - 2*(-a*b^2)^(1/3)*b*c - (-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^ 2*b) - 1/54*(a*b*e + 2*a^2*h + 2*(-a*b^2)^(1/3)*b*c + (-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^2*b) - 1/27*(2* b^2*c*(-a/b)^(1/3) + a*b*f*(-a/b)^(1/3) + a*b*e + 2*a^2*h)*(-a/b)^(1/3)*lo g(abs(x - (-a/b)^(1/3)))/(a^3*b^2) + 1/18*(4*b^3*c*x^5 + 2*a*b^2*f*x^5 + a *b^2*e*x^4 - 7*a^2*b*h*x^4 - 6*a^2*b*g*x^3 + 7*a*b^2*c*x^2 - a^2*b*f*x^2 - 2*a^2*b*e*x - 4*a^3*h*x - 3*a^2*b*d - 3*a^3*g)/((b*x^3 + a)^2*a^2*b^2)
Time = 6.38 (sec) , antiderivative size = 640, normalized size of antiderivative = 1.98 \[ \int \frac {x \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=\left (\sum _{k=1}^3\ln \left (\mathrm {root}\left (19683\,a^7\,b^7\,z^3+162\,a^5\,b^3\,f\,h\,z+324\,a^4\,b^4\,c\,h\,z+81\,a^4\,b^4\,e\,f\,z+162\,a^3\,b^5\,c\,e\,z-12\,a^4\,b\,e\,h^2+12\,a\,b^4\,c^2\,f-6\,a^3\,b^2\,e^2\,h+6\,a^2\,b^3\,c\,f^2+a^3\,b^2\,f^3-8\,a^5\,h^3+8\,b^5\,c^3-a^2\,b^3\,e^3,z,k\right )\,\left (\mathrm {root}\left (19683\,a^7\,b^7\,z^3+162\,a^5\,b^3\,f\,h\,z+324\,a^4\,b^4\,c\,h\,z+81\,a^4\,b^4\,e\,f\,z+162\,a^3\,b^5\,c\,e\,z-12\,a^4\,b\,e\,h^2+12\,a\,b^4\,c^2\,f-6\,a^3\,b^2\,e^2\,h+6\,a^2\,b^3\,c\,f^2+a^3\,b^2\,f^3-8\,a^5\,h^3+8\,b^5\,c^3-a^2\,b^3\,e^3,z,k\right )\,a\,b^2\,9+\frac {x\,\left (54\,h\,a^4\,b+27\,e\,a^3\,b^2\right )}{81\,a^4\,b}\right )+\frac {2\,b^2\,c\,e+2\,a^2\,f\,h+4\,a\,b\,c\,h+a\,b\,e\,f}{81\,a^3\,b^2}+\frac {x\,\left (a^2\,f^2+4\,a\,b\,c\,f+4\,b^2\,c^2\right )}{81\,a^4\,b}\right )\,\mathrm {root}\left (19683\,a^7\,b^7\,z^3+162\,a^5\,b^3\,f\,h\,z+324\,a^4\,b^4\,c\,h\,z+81\,a^4\,b^4\,e\,f\,z+162\,a^3\,b^5\,c\,e\,z-12\,a^4\,b\,e\,h^2+12\,a\,b^4\,c^2\,f-6\,a^3\,b^2\,e^2\,h+6\,a^2\,b^3\,c\,f^2+a^3\,b^2\,f^3-8\,a^5\,h^3+8\,b^5\,c^3-a^2\,b^3\,e^3,z,k\right )\right )-\frac {\frac {b\,d+a\,g}{6\,b^2}+\frac {x\,\left (b\,e+2\,a\,h\right )}{9\,b^2}+\frac {g\,x^3}{3\,b}-\frac {x^5\,\left (2\,b\,c+a\,f\right )}{9\,a^2}-\frac {x^2\,\left (7\,b\,c-a\,f\right )}{18\,a\,b}-\frac {x^4\,\left (b\,e-7\,a\,h\right )}{18\,a\,b}}{a^2+2\,a\,b\,x^3+b^2\,x^6} \] Input:
int((x*(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^7*b^7*z^3 + 162*a^5*b^3*f*h*z + 324*a^4*b^4*c*h*z + 81*a^4*b^4*e*f*z + 162*a^3*b^5*c*e*z - 12*a^4*b*e*h^2 + 12*a*b^4*c^2*f - 6*a^3*b^2*e^2*h + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 - 8*a^5*h^3 + 8*b^5*c^3 - a^2*b^3*e^3, z, k)*(9*root(19683*a^7*b^7*z^3 + 162*a^5*b^3*f*h*z + 324*a^ 4*b^4*c*h*z + 81*a^4*b^4*e*f*z + 162*a^3*b^5*c*e*z - 12*a^4*b*e*h^2 + 12*a *b^4*c^2*f - 6*a^3*b^2*e^2*h + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 - 8*a^5*h^3 + 8*b^5*c^3 - a^2*b^3*e^3, z, k)*a*b^2 + (x*(27*a^3*b^2*e + 54*a^4*b*h))/(8 1*a^4*b)) + (2*b^2*c*e + 2*a^2*f*h + 4*a*b*c*h + a*b*e*f)/(81*a^3*b^2) + ( x*(4*b^2*c^2 + a^2*f^2 + 4*a*b*c*f))/(81*a^4*b))*root(19683*a^7*b^7*z^3 + 162*a^5*b^3*f*h*z + 324*a^4*b^4*c*h*z + 81*a^4*b^4*e*f*z + 162*a^3*b^5*c*e *z - 12*a^4*b*e*h^2 + 12*a*b^4*c^2*f - 6*a^3*b^2*e^2*h + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 - 8*a^5*h^3 + 8*b^5*c^3 - a^2*b^3*e^3, z, k), k, 1, 3) - ((b* d + a*g)/(6*b^2) + (x*(b*e + 2*a*h))/(9*b^2) + (g*x^3)/(3*b) - (x^5*(2*b*c + a*f))/(9*a^2) - (x^2*(7*b*c - a*f))/(18*a*b) - (x^4*(b*e - 7*a*h))/(18* a*b))/(a^2 + b^2*x^6 + 2*a*b*x^3)
Time = 0.19 (sec) , antiderivative size = 1133, normalized size of antiderivative = 3.51 \[ \int \frac {x \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx =\text {Too large to display} \] Input:
int(x*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^3,x)
Output:
( - 4*b**(1/3)*a**(2/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*s qrt(3)))*a**3*h - 2*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 - 8*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 - 4*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 - 4*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 - 2*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 - 2*sqrt(3)*atan((a**(1/3) - 2*b** (1/3)*x)/(a**(1/3)*sqrt(3)))*a**3*b*f - 4*sqrt(3)*atan((a**(1/3) - 2*b**(1 /3)*x)/(a**(1/3)*sqrt(3)))*a**2*b**2*c - 4*sqrt(3)*atan((a**(1/3) - 2*b**( 1/3)*x)/(a**(1/3)*sqrt(3)))*a**2*b**2*f*x**3 - 8*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a*b**3*c*x**3 - 2*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a*b**3*f*x**6 - 4*sqrt(3)*atan((a**(1 /3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*b**4*c*x**6 - 2*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 - 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 - 4*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 - 2*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 - 2*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 - b**(1/3)*a**(2/3)*log(a**(2/3) - b*...