Integrand size = 30, antiderivative size = 307 \[ \int \frac {x^3 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {(b e-3 a f) x}{b^4}+\frac {f x^4}{4 b^3}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^4 \left (a+b x^3\right )^2}+\frac {\left (b^3 c-7 a b^2 d+13 a^2 b e-19 a^3 f\right ) x}{18 a b^4 \left (a+b x^3\right )}-\frac {\left (b^3 c+2 a b^2 d-14 a^2 b e+35 a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{5/3} b^{13/3}}+\frac {\left (b^3 c+2 a b^2 d-14 a^2 b e+35 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{13/3}}-\frac {\left (b^3 c+2 a b^2 d-14 a^2 b e+35 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{13/3}} \]
(-3*a*f+b*e)*x/b^4+1/4*f*x^4/b^3-1/6*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x/b^4/ (b*x^3+a)^2+1/18*(-19*a^3*f+13*a^2*b*e-7*a*b^2*d+b^3*c)*x/a/b^4/(b*x^3+a)+ 1/27*(35*a^3*f-14*a^2*b*e+2*a*b^2*d+b^3*c)*ln(a^(1/3)+b^(1/3)*x)/a^(5/3)/b ^(13/3)-1/54*(35*a^3*f-14*a^2*b*e+2*a*b^2*d+b^3*c)*ln(a^(2/3)-a^(1/3)*b^(1 /3)*x+b^(2/3)*x^2)/a^(5/3)/b^(13/3)-1/27*(35*a^3*f-14*a^2*b*e+2*a*b^2*d+b^ 3*c)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(5/3)/b^(13/3)*3^ (1/2)
Time = 0.22 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.96 \[ \int \frac {x^3 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {108 \sqrt [3]{b} (b e-3 a f) x+27 b^{4/3} f x^4-\frac {18 \sqrt [3]{b} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{\left (a+b x^3\right )^2}+\frac {6 \sqrt [3]{b} \left (b^3 c-7 a b^2 d+13 a^2 b e-19 a^3 f\right ) x}{a \left (a+b x^3\right )}-\frac {4 \sqrt {3} \left (b^3 c+2 a b^2 d-14 a^2 b e+35 a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{5/3}}+\frac {4 \left (b^3 c+2 a b^2 d-14 a^2 b e+35 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3}}-\frac {2 \left (b^3 c+2 a b^2 d-14 a^2 b e+35 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3}}}{108 b^{13/3}} \]
(108*b^(1/3)*(b*e - 3*a*f)*x + 27*b^(4/3)*f*x^4 - (18*b^(1/3)*(b^3*c - a*b ^2*d + a^2*b*e - a^3*f)*x)/(a + b*x^3)^2 + (6*b^(1/3)*(b^3*c - 7*a*b^2*d + 13*a^2*b*e - 19*a^3*f)*x)/(a*(a + b*x^3)) - (4*Sqrt[3]*(b^3*c + 2*a*b^2*d - 14*a^2*b*e + 35*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^( 5/3) + (4*(b^3*c + 2*a*b^2*d - 14*a^2*b*e + 35*a^3*f)*Log[a^(1/3) + b^(1/3 )*x])/a^(5/3) - (2*(b^3*c + 2*a*b^2*d - 14*a^2*b*e + 35*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(5/3))/(108*b^(13/3))
Time = 0.76 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.90, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2367, 25, 2397, 27, 1741, 27, 913, 750, 16, 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.
| \(\displaystyle \int \frac {x^3 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 2367 |
| \(\displaystyle -\frac {\int -\frac {6 a b^3 f x^9+6 a b^2 (b e-a f) x^6+6 a b \left (f a^2-b e a+b^2 d\right ) x^3+a \left (-f a^3+b e a^2-b^2 d a+b^3 c\right )}{\left (b x^3+a\right )^2}dx}{6 a b^4}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {6 a b^3 f x^9+6 a b^2 (b e-a f) x^6+6 a b \left (f a^2-b e a+b^2 d\right ) x^3+a \left (-f a^3+b e a^2-b^2 d a+b^3 c\right )}{\left (b x^3+a\right )^2}dx}{6 a b^4}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2397 |
| \(\displaystyle \frac {\frac {x \left (-19 a^3 f+13 a^2 b e-7 a b^2 d+b^3 c\right )}{3 \left (a+b x^3\right )}-\frac {\int -\frac {2 \left (9 a^2 b^5 f x^6+9 a^2 b^4 (b e-2 a f) x^3+a b^3 \left (8 f a^3-5 b e a^2+2 b^2 d a+b^3 c\right )\right )}{b x^3+a}dx}{3 a b^3}}{6 a b^4}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \int \frac {9 a^2 b^5 f x^6+9 a^2 b^4 (b e-2 a f) x^3+a b^3 \left (8 f a^3-5 b e a^2+2 b^2 d a+b^3 c\right )}{b x^3+a}dx}{3 a b^3}+\frac {x \left (-19 a^3 f+13 a^2 b e-7 a b^2 d+b^3 c\right )}{3 \left (a+b x^3\right )}}{6 a b^4}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1741 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\int \frac {4 a b^4 \left (8 f a^3-5 b e a^2+9 b (b e-3 a f) x^3 a+2 b^2 d a+b^3 c\right )}{b x^3+a}dx}{4 b}+\frac {9}{4} a^2 b^4 f x^4\right )}{3 a b^3}+\frac {x \left (-19 a^3 f+13 a^2 b e-7 a b^2 d+b^3 c\right )}{3 \left (a+b x^3\right )}}{6 a b^4}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \left (a b^3 \int \frac {8 f a^3-5 b e a^2+9 b (b e-3 a f) x^3 a+2 b^2 d a+b^3 c}{b x^3+a}dx+\frac {9}{4} a^2 b^4 f x^4\right )}{3 a b^3}+\frac {x \left (-19 a^3 f+13 a^2 b e-7 a b^2 d+b^3 c\right )}{3 \left (a+b x^3\right )}}{6 a b^4}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 913 |
| \(\displaystyle \frac {\frac {2 \left (a b^3 \left (\left (35 a^3 f-14 a^2 b e+2 a b^2 d+b^3 c\right ) \int \frac {1}{b x^3+a}dx+9 a x (b e-3 a f)\right )+\frac {9}{4} a^2 b^4 f x^4\right )}{3 a b^3}+\frac {x \left (-19 a^3 f+13 a^2 b e-7 a b^2 d+b^3 c\right )}{3 \left (a+b x^3\right )}}{6 a b^4}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {\frac {2 \left (a b^3 \left (\left (35 a^3 f-14 a^2 b e+2 a b^2 d+b^3 c\right ) \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 a^{2/3}}\right )+9 a x (b e-3 a f)\right )+\frac {9}{4} a^2 b^4 f x^4\right )}{3 a b^3}+\frac {x \left (-19 a^3 f+13 a^2 b e-7 a b^2 d+b^3 c\right )}{3 \left (a+b x^3\right )}}{6 a b^4}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {2 \left (a b^3 \left (\left (35 a^3 f-14 a^2 b e+2 a b^2 d+b^3 c\right ) \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )+9 a x (b e-3 a f)\right )+\frac {9}{4} a^2 b^4 f x^4\right )}{3 a b^3}+\frac {x \left (-19 a^3 f+13 a^2 b e-7 a b^2 d+b^3 c\right )}{3 \left (a+b x^3\right )}}{6 a b^4}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {2 \left (a b^3 \left (\left (35 a^3 f-14 a^2 b e+2 a b^2 d+b^3 c\right ) \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )+9 a x (b e-3 a f)\right )+\frac {9}{4} a^2 b^4 f x^4\right )}{3 a b^3}+\frac {x \left (-19 a^3 f+13 a^2 b e-7 a b^2 d+b^3 c\right )}{3 \left (a+b x^3\right )}}{6 a b^4}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 \left (a b^3 \left (\left (35 a^3 f-14 a^2 b e+2 a b^2 d+b^3 c\right ) \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )+9 a x (b e-3 a f)\right )+\frac {9}{4} a^2 b^4 f x^4\right )}{3 a b^3}+\frac {x \left (-19 a^3 f+13 a^2 b e-7 a b^2 d+b^3 c\right )}{3 \left (a+b x^3\right )}}{6 a b^4}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \left (a b^3 \left (\left (35 a^3 f-14 a^2 b e+2 a b^2 d+b^3 c\right ) \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )+9 a x (b e-3 a f)\right )+\frac {9}{4} a^2 b^4 f x^4\right )}{3 a b^3}+\frac {x \left (-19 a^3 f+13 a^2 b e-7 a b^2 d+b^3 c\right )}{3 \left (a+b x^3\right )}}{6 a b^4}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {2 \left (a b^3 \left (\left (35 a^3 f-14 a^2 b e+2 a b^2 d+b^3 c\right ) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )+9 a x (b e-3 a f)\right )+\frac {9}{4} a^2 b^4 f x^4\right )}{3 a b^3}+\frac {x \left (-19 a^3 f+13 a^2 b e-7 a b^2 d+b^3 c\right )}{3 \left (a+b x^3\right )}}{6 a b^4}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {2 \left (a b^3 \left (\left (35 a^3 f-14 a^2 b e+2 a b^2 d+b^3 c\right ) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )+9 a x (b e-3 a f)\right )+\frac {9}{4} a^2 b^4 f x^4\right )}{3 a b^3}+\frac {x \left (-19 a^3 f+13 a^2 b e-7 a b^2 d+b^3 c\right )}{3 \left (a+b x^3\right )}}{6 a b^4}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {x \left (-19 a^3 f+13 a^2 b e-7 a b^2 d+b^3 c\right )}{3 \left (a+b x^3\right )}+\frac {2 \left (\frac {9}{4} a^2 b^4 f x^4+a b^3 \left (\left (\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right ) \left (35 a^3 f-14 a^2 b e+2 a b^2 d+b^3 c\right )+9 a x (b e-3 a f)\right )\right )}{3 a b^3}}{6 a b^4}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^4 \left (a+b x^3\right )^2}\) |
-1/6*((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(b^4*(a + b*x^3)^2) + (((b^3* c - 7*a*b^2*d + 13*a^2*b*e - 19*a^3*f)*x)/(3*(a + b*x^3)) + (2*((9*a^2*b^4 *f*x^4)/4 + a*b^3*(9*a*(b*e - 3*a*f)*x + (b^3*c + 2*a*b^2*d - 14*a^2*b*e + 35*a^3*f)*(Log[a^(1/3) + b^(1/3)*x]/(3*a^(2/3)*b^(1/3)) + (-((Sqrt[3]*Arc Tan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3)) - Log[a^(2/3) - a^(1/3) *b^(1/3)*x + b^(2/3)*x^2]/(2*b^(1/3)))/(3*a^(2/3))))))/(3*a*b^3))/(6*a*b^4 )
3.3.92.3.1 Defintions of rubi rules used
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_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si mp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(p + 1) + 1))), x] - Simp[(a*d - b*c*(n*( p + 1) + 1))/(b*(n*(p + 1) + 1)) Int[(a + b*x^n)^p, x], x] /; FreeQ[{a, b , c, d, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 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[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_ )), x_Symbol] :> Simp[c*x^(n + 1)*((d + e*x^n)^(q + 1)/(e*(n*(q + 2) + 1))) , x] + Simp[1/(e*(n*(q + 2) + 1)) Int[(d + e*x^n)^q*(a*e*(n*(q + 2) + 1) - (c*d*(n + 1) - b*e*(n*(q + 2) + 1))*x^n), x], x] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a* e^2, 0]
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.53 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.52
| method | result | size |
| risch | \(\frac {f \,x^{4}}{4 b^{3}}-\frac {3 x a f}{b^{4}}+\frac {e x}{b^{3}}+\frac {-\frac {b \left (19 f \,a^{3}-13 a^{2} b e +7 a \,b^{2} d -b^{3} c \right ) x^{4}}{18 a}+\left (-\frac {8}{9} f \,a^{3}+\frac {5}{9} a^{2} b e -\frac {2}{9} a \,b^{2} d -\frac {1}{9} b^{3} c \right ) x}{b^{4} \left (b \,x^{3}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (35 f \,a^{3}-14 a^{2} b e +2 a \,b^{2} d +b^{3} c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{27 b^{5} a}\) | \(159\) |
| default | \(-\frac {-\frac {1}{4} b f \,x^{4}+3 a f x -b e x}{b^{4}}+\frac {\frac {-\frac {b \left (19 f \,a^{3}-13 a^{2} b e +7 a \,b^{2} d -b^{3} c \right ) x^{4}}{18 a}+\left (-\frac {8}{9} f \,a^{3}+\frac {5}{9} a^{2} b e -\frac {2}{9} a \,b^{2} d -\frac {1}{9} b^{3} c \right ) x}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (35 f \,a^{3}-14 a^{2} b e +2 a \,b^{2} d +b^{3} c \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{9 a}}{b^{4}}\) | \(227\) |
1/4*f*x^4/b^3-3/b^4*x*a*f+1/b^3*e*x+(-1/18*b*(19*a^3*f-13*a^2*b*e+7*a*b^2* d-b^3*c)/a*x^4+(-8/9*f*a^3+5/9*a^2*b*e-2/9*a*b^2*d-1/9*b^3*c)*x)/b^4/(b*x^ 3+a)^2+1/27/b^5/a*sum((35*a^3*f-14*a^2*b*e+2*a*b^2*d+b^3*c)/_R^2*ln(x-_R), _R=RootOf(_Z^3*b+a))
Leaf count of result is larger than twice the leaf count of optimal. 586 vs. \(2 (264) = 528\).
Time = 0.31 (sec) , antiderivative size = 1213, normalized size of antiderivative = 3.95 \[ \int \frac {x^3 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\text {Too large to display} \]
[1/108*(27*a^3*b^4*f*x^10 + 54*(2*a^3*b^4*e - 5*a^4*b^3*f)*x^7 + 3*(2*a^2* b^5*c - 14*a^3*b^4*d + 98*a^4*b^3*e - 245*a^5*b^2*f)*x^4 + 6*sqrt(1/3)*(a^ 3*b^4*c + 2*a^4*b^3*d - 14*a^5*b^2*e + 35*a^6*b*f + (a*b^6*c + 2*a^2*b^5*d - 14*a^3*b^4*e + 35*a^4*b^3*f)*x^6 + 2*(a^2*b^5*c + 2*a^3*b^4*d - 14*a^4* b^3*e + 35*a^5*b^2*f)*x^3)*sqrt(-(a^2*b)^(1/3)/b)*log((2*a*b*x^3 - 3*(a^2* b)^(1/3)*a*x - a^2 + 3*sqrt(1/3)*(2*a*b*x^2 + (a^2*b)^(2/3)*x - (a^2*b)^(1 /3)*a)*sqrt(-(a^2*b)^(1/3)/b))/(b*x^3 + a)) - 2*((b^5*c + 2*a*b^4*d - 14*a ^2*b^3*e + 35*a^3*b^2*f)*x^6 + a^2*b^3*c + 2*a^3*b^2*d - 14*a^4*b*e + 35*a ^5*f + 2*(a*b^4*c + 2*a^2*b^3*d - 14*a^3*b^2*e + 35*a^4*b*f)*x^3)*(a^2*b)^ (2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 4*((b^5*c + 2*a*b ^4*d - 14*a^2*b^3*e + 35*a^3*b^2*f)*x^6 + a^2*b^3*c + 2*a^3*b^2*d - 14*a^4 *b*e + 35*a^5*f + 2*(a*b^4*c + 2*a^2*b^3*d - 14*a^3*b^2*e + 35*a^4*b*f)*x^ 3)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)) - 12*(a^3*b^4*c + 2*a^4*b^3*d - 14*a^5*b^2*e + 35*a^6*b*f)*x)/(a^3*b^7*x^6 + 2*a^4*b^6*x^3 + a^5*b^5), 1 /108*(27*a^3*b^4*f*x^10 + 54*(2*a^3*b^4*e - 5*a^4*b^3*f)*x^7 + 3*(2*a^2*b^ 5*c - 14*a^3*b^4*d + 98*a^4*b^3*e - 245*a^5*b^2*f)*x^4 + 12*sqrt(1/3)*(a^3 *b^4*c + 2*a^4*b^3*d - 14*a^5*b^2*e + 35*a^6*b*f + (a*b^6*c + 2*a^2*b^5*d - 14*a^3*b^4*e + 35*a^4*b^3*f)*x^6 + 2*(a^2*b^5*c + 2*a^3*b^4*d - 14*a^4*b ^3*e + 35*a^5*b^2*f)*x^3)*sqrt((a^2*b)^(1/3)/b)*arctan(sqrt(1/3)*(2*(a^2*b )^(2/3)*x - (a^2*b)^(1/3)*a)*sqrt((a^2*b)^(1/3)/b)/a^2) - 2*((b^5*c + 2...
Timed out. \[ \int \frac {x^3 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.99 \[ \int \frac {x^3 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {{\left (b^{4} c - 7 \, a b^{3} d + 13 \, a^{2} b^{2} e - 19 \, a^{3} b f\right )} x^{4} - 2 \, {\left (a b^{3} c + 2 \, a^{2} b^{2} d - 5 \, a^{3} b e + 8 \, a^{4} f\right )} x}{18 \, {\left (a b^{6} x^{6} + 2 \, a^{2} b^{5} x^{3} + a^{3} b^{4}\right )}} + \frac {b f x^{4} + 4 \, {\left (b e - 3 \, a f\right )} x}{4 \, b^{4}} + \frac {\sqrt {3} {\left (b^{3} c + 2 \, a b^{2} d - 14 \, a^{2} b e + 35 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a b^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (b^{3} c + 2 \, a b^{2} d - 14 \, a^{2} b e + 35 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a b^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (b^{3} c + 2 \, a b^{2} d - 14 \, a^{2} b e + 35 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a b^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
1/18*((b^4*c - 7*a*b^3*d + 13*a^2*b^2*e - 19*a^3*b*f)*x^4 - 2*(a*b^3*c + 2 *a^2*b^2*d - 5*a^3*b*e + 8*a^4*f)*x)/(a*b^6*x^6 + 2*a^2*b^5*x^3 + a^3*b^4) + 1/4*(b*f*x^4 + 4*(b*e - 3*a*f)*x)/b^4 + 1/27*sqrt(3)*(b^3*c + 2*a*b^2*d - 14*a^2*b*e + 35*a^3*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/ 3))/(a*b^5*(a/b)^(2/3)) - 1/54*(b^3*c + 2*a*b^2*d - 14*a^2*b*e + 35*a^3*f) *log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a*b^5*(a/b)^(2/3)) + 1/27*(b^3*c + 2*a*b^2*d - 14*a^2*b*e + 35*a^3*f)*log(x + (a/b)^(1/3))/(a*b^5*(a/b)^(2/ 3))
Time = 0.28 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.02 \[ \int \frac {x^3 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {\sqrt {3} {\left (b^{3} c + 2 \, a b^{2} d - 14 \, a^{2} b e + 35 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{3}} - \frac {{\left (b^{3} c + 2 \, a b^{2} d - 14 \, a^{2} b e + 35 \, a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{3}} - \frac {{\left (b^{3} c + 2 \, a b^{2} d - 14 \, a^{2} b e + 35 \, a^{3} f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{2} b^{4}} + \frac {b^{4} c x^{4} - 7 \, a b^{3} d x^{4} + 13 \, a^{2} b^{2} e x^{4} - 19 \, a^{3} b f x^{4} - 2 \, a b^{3} c x - 4 \, a^{2} b^{2} d x + 10 \, a^{3} b e x - 16 \, a^{4} f x}{18 \, {\left (b x^{3} + a\right )}^{2} a b^{4}} + \frac {b^{9} f x^{4} + 4 \, b^{9} e x - 12 \, a b^{8} f x}{4 \, b^{12}} \]
-1/27*sqrt(3)*(b^3*c + 2*a*b^2*d - 14*a^2*b*e + 35*a^3*f)*arctan(1/3*sqrt( 3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/((-a*b^2)^(2/3)*a*b^3) - 1/54*(b^3*c + 2*a*b^2*d - 14*a^2*b*e + 35*a^3*f)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2 /3))/((-a*b^2)^(2/3)*a*b^3) - 1/27*(b^3*c + 2*a*b^2*d - 14*a^2*b*e + 35*a^ 3*f)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^2*b^4) + 1/18*(b^4*c*x^4 - 7*a*b^3*d*x^4 + 13*a^2*b^2*e*x^4 - 19*a^3*b*f*x^4 - 2*a*b^3*c*x - 4*a^2*b ^2*d*x + 10*a^3*b*e*x - 16*a^4*f*x)/((b*x^3 + a)^2*a*b^4) + 1/4*(b^9*f*x^4 + 4*b^9*e*x - 12*a*b^8*f*x)/b^12
Time = 9.30 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.94 \[ \int \frac {x^3 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=x\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )-\frac {x\,\left (\frac {8\,f\,a^3}{9}-\frac {5\,e\,a^2\,b}{9}+\frac {2\,d\,a\,b^2}{9}+\frac {c\,b^3}{9}\right )-\frac {x^4\,\left (-19\,f\,a^3\,b+13\,e\,a^2\,b^2-7\,d\,a\,b^3+c\,b^4\right )}{18\,a}}{a^2\,b^4+2\,a\,b^5\,x^3+b^6\,x^6}+\frac {f\,x^4}{4\,b^3}+\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (35\,f\,a^3-14\,e\,a^2\,b+2\,d\,a\,b^2+c\,b^3\right )}{27\,a^{5/3}\,b^{13/3}}+\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (35\,f\,a^3-14\,e\,a^2\,b+2\,d\,a\,b^2+c\,b^3\right )}{27\,a^{5/3}\,b^{13/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (35\,f\,a^3-14\,e\,a^2\,b+2\,d\,a\,b^2+c\,b^3\right )}{27\,a^{5/3}\,b^{13/3}} \]
x*(e/b^3 - (3*a*f)/b^4) - (x*((b^3*c)/9 + (8*a^3*f)/9 + (2*a*b^2*d)/9 - (5 *a^2*b*e)/9) - (x^4*(b^4*c + 13*a^2*b^2*e - 7*a*b^3*d - 19*a^3*b*f))/(18*a ))/(a^2*b^4 + b^6*x^6 + 2*a*b^5*x^3) + (f*x^4)/(4*b^3) + (log(b^(1/3)*x + a^(1/3))*(b^3*c + 35*a^3*f + 2*a*b^2*d - 14*a^2*b*e))/(27*a^(5/3)*b^(13/3) ) + (log(3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*x - a^(1/3))*((3^(1/2)*1i)/2 - 1/2 )*(b^3*c + 35*a^3*f + 2*a*b^2*d - 14*a^2*b*e))/(27*a^(5/3)*b^(13/3)) - (lo g(3^(1/2)*a^(1/3)*1i - 2*b^(1/3)*x + a^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(b^3* c + 35*a^3*f + 2*a*b^2*d - 14*a^2*b*e))/(27*a^(5/3)*b^(13/3))