Integrand size = 24, antiderivative size = 184 \[ \int \frac {-d+c x}{x^7 \sqrt [3]{-b+a x^3}} \, dx=\frac {\left (-b+a x^3\right )^{2/3} \left (-15 b d+18 b c x-20 a d x^3+27 a c x^4\right )}{90 b^2 x^6}+\frac {2 a^2 d \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt {3} \sqrt [3]{b}}\right )}{9 \sqrt {3} b^{7/3}}+\frac {2 a^2 d \log \left (\sqrt [3]{b}+\sqrt [3]{-b+a x^3}\right )}{27 b^{7/3}}-\frac {a^2 d \log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{-b+a x^3}+\left (-b+a x^3\right )^{2/3}\right )}{27 b^{7/3}} \]
1/90*(a*x^3-b)^(2/3)*(27*a*c*x^4-20*a*d*x^3+18*b*c*x-15*b*d)/b^2/x^6-2/27* a^2*d*arctan(-1/3*3^(1/2)+2/3*(a*x^3-b)^(1/3)*3^(1/2)/b^(1/3))*3^(1/2)/b^( 7/3)+2/27*a^2*d*ln(b^(1/3)+(a*x^3-b)^(1/3))/b^(7/3)-1/27*a^2*d*ln(b^(2/3)- b^(1/3)*(a*x^3-b)^(1/3)+(a*x^3-b)^(2/3))/b^(7/3)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.05 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.35 \[ \int \frac {-d+c x}{x^7 \sqrt [3]{-b+a x^3}} \, dx=\frac {\left (-b+a x^3\right )^{2/3} \left (b c \left (2 b+3 a x^3\right )-5 a^2 d x^5 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},3,\frac {5}{3},1-\frac {a x^3}{b}\right )\right )}{10 b^3 x^5} \]
((-b + a*x^3)^(2/3)*(b*c*(2*b + 3*a*x^3) - 5*a^2*d*x^5*Hypergeometric2F1[2 /3, 3, 5/3, 1 - (a*x^3)/b]))/(10*b^3*x^5)
Time = 0.44 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.08, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2383, 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.
| \(\displaystyle \int \frac {c x-d}{x^7 \sqrt [3]{a x^3-b}} \, dx\) |
| \(\Big \downarrow \) 2383 |
| \(\displaystyle \int \left (\frac {c}{x^6 \sqrt [3]{a x^3-b}}-\frac {d}{x^7 \sqrt [3]{a x^3-b}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 a^2 d \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a x^3-b}}{\sqrt {3} \sqrt [3]{b}}\right )}{9 \sqrt {3} b^{7/3}}+\frac {a^2 d \log \left (\sqrt [3]{a x^3-b}+\sqrt [3]{b}\right )}{9 b^{7/3}}-\frac {a^2 d \log (x)}{9 b^{7/3}}+\frac {3 a c \left (a x^3-b\right )^{2/3}}{10 b^2 x^2}-\frac {2 a d \left (a x^3-b\right )^{2/3}}{9 b^2 x^3}+\frac {c \left (a x^3-b\right )^{2/3}}{5 b x^5}-\frac {d \left (a x^3-b\right )^{2/3}}{6 b x^6}\) |
-1/6*(d*(-b + a*x^3)^(2/3))/(b*x^6) + (c*(-b + a*x^3)^(2/3))/(5*b*x^5) - ( 2*a*d*(-b + a*x^3)^(2/3))/(9*b^2*x^3) + (3*a*c*(-b + a*x^3)^(2/3))/(10*b^2 *x^2) + (2*a^2*d*ArcTan[(b^(1/3) - 2*(-b + a*x^3)^(1/3))/(Sqrt[3]*b^(1/3)) ])/(9*Sqrt[3]*b^(7/3)) - (a^2*d*Log[x])/(9*b^(7/3)) + (a^2*d*Log[b^(1/3) + (-b + a*x^3)^(1/3)])/(9*b^(7/3))
3.24.33.3.1 Defintions of rubi rules used
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> I nt[ExpandIntegrand[(c*x)^m*Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n , p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) && !IGtQ[m, 0]
\[\int \frac {c x -d}{x^{7} \left (a \,x^{3}-b \right )^{\frac {1}{3}}}d x\]
Time = 118.79 (sec) , antiderivative size = 405, normalized size of antiderivative = 2.20 \[ \int \frac {-d+c x}{x^7 \sqrt [3]{-b+a x^3}} \, dx=\left [\frac {30 \, \sqrt {\frac {1}{3}} a^{2} b d x^{6} \sqrt {-\frac {1}{b^{\frac {2}{3}}}} \log \left (\frac {2 \, a x^{3} - 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (a x^{3} - b\right )}^{\frac {2}{3}} b^{\frac {2}{3}} + {\left (a x^{3} - b\right )}^{\frac {1}{3}} b - b^{\frac {4}{3}}\right )} \sqrt {-\frac {1}{b^{\frac {2}{3}}}} - 3 \, {\left (a x^{3} - b\right )}^{\frac {1}{3}} b^{\frac {2}{3}} - 3 \, b}{x^{3}}\right ) + 20 \, a^{2} b^{\frac {2}{3}} d x^{6} \log \left (\frac {{\left (a x^{3} - b\right )}^{\frac {1}{3}} + b^{\frac {1}{3}}}{x}\right ) - 10 \, a^{2} b^{\frac {2}{3}} d x^{6} \log \left (\frac {{\left (a x^{3} - b\right )}^{\frac {2}{3}} - {\left (a x^{3} - b\right )}^{\frac {1}{3}} b^{\frac {1}{3}} + b^{\frac {2}{3}}}{x^{2}}\right ) + 3 \, {\left (27 \, a b c x^{4} - 20 \, a b d x^{3} + 18 \, b^{2} c x - 15 \, b^{2} d\right )} {\left (a x^{3} - b\right )}^{\frac {2}{3}}}{270 \, b^{3} x^{6}}, -\frac {60 \, \sqrt {\frac {1}{3}} a^{2} b^{\frac {2}{3}} d x^{6} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, {\left (a x^{3} - b\right )}^{\frac {1}{3}} - b^{\frac {1}{3}}\right )}}{b^{\frac {1}{3}}}\right ) - 20 \, a^{2} b^{\frac {2}{3}} d x^{6} \log \left (\frac {{\left (a x^{3} - b\right )}^{\frac {1}{3}} + b^{\frac {1}{3}}}{x}\right ) + 10 \, a^{2} b^{\frac {2}{3}} d x^{6} \log \left (\frac {{\left (a x^{3} - b\right )}^{\frac {2}{3}} - {\left (a x^{3} - b\right )}^{\frac {1}{3}} b^{\frac {1}{3}} + b^{\frac {2}{3}}}{x^{2}}\right ) - 3 \, {\left (27 \, a b c x^{4} - 20 \, a b d x^{3} + 18 \, b^{2} c x - 15 \, b^{2} d\right )} {\left (a x^{3} - b\right )}^{\frac {2}{3}}}{270 \, b^{3} x^{6}}\right ] \]
[1/270*(30*sqrt(1/3)*a^2*b*d*x^6*sqrt(-1/b^(2/3))*log((2*a*x^3 - 3*sqrt(1/ 3)*(2*(a*x^3 - b)^(2/3)*b^(2/3) + (a*x^3 - b)^(1/3)*b - b^(4/3))*sqrt(-1/b ^(2/3)) - 3*(a*x^3 - b)^(1/3)*b^(2/3) - 3*b)/x^3) + 20*a^2*b^(2/3)*d*x^6*l og(((a*x^3 - b)^(1/3) + b^(1/3))/x) - 10*a^2*b^(2/3)*d*x^6*log(((a*x^3 - b )^(2/3) - (a*x^3 - b)^(1/3)*b^(1/3) + b^(2/3))/x^2) + 3*(27*a*b*c*x^4 - 20 *a*b*d*x^3 + 18*b^2*c*x - 15*b^2*d)*(a*x^3 - b)^(2/3))/(b^3*x^6), -1/270*( 60*sqrt(1/3)*a^2*b^(2/3)*d*x^6*arctan(sqrt(1/3)*(2*(a*x^3 - b)^(1/3) - b^( 1/3))/b^(1/3)) - 20*a^2*b^(2/3)*d*x^6*log(((a*x^3 - b)^(1/3) + b^(1/3))/x) + 10*a^2*b^(2/3)*d*x^6*log(((a*x^3 - b)^(2/3) - (a*x^3 - b)^(1/3)*b^(1/3) + b^(2/3))/x^2) - 3*(27*a*b*c*x^4 - 20*a*b*d*x^3 + 18*b^2*c*x - 15*b^2*d) *(a*x^3 - b)^(2/3))/(b^3*x^6)]
Result contains complex when optimal does not.
Time = 2.71 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.88 \[ \int \frac {-d+c x}{x^7 \sqrt [3]{-b+a x^3}} \, dx=c \left (\begin {cases} - \frac {3 a^{\frac {11}{3}} x^{6} \left (-1 + \frac {b}{a x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{9 a^{2} b^{2} x^{6} e^{\frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right ) - 9 a b^{3} x^{3} e^{\frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right )} + \frac {a^{\frac {8}{3}} b x^{3} \left (-1 + \frac {b}{a x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{9 a^{2} b^{2} x^{6} e^{\frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right ) - 9 a b^{3} x^{3} e^{\frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right )} + \frac {2 a^{\frac {5}{3}} b^{2} \left (-1 + \frac {b}{a x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{9 a^{2} b^{2} x^{6} e^{\frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right ) - 9 a b^{3} x^{3} e^{\frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right )} & \text {for}\: \left |{\frac {b}{a x^{3}}}\right | > 1 \\\frac {a^{\frac {5}{3}} \left (1 - \frac {b}{a x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 b^{2} \Gamma \left (\frac {1}{3}\right )} + \frac {2 a^{\frac {2}{3}} \left (1 - \frac {b}{a x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{9 b x^{3} \Gamma \left (\frac {1}{3}\right )} & \text {otherwise} \end {cases}\right ) + \frac {d \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b e^{2 i \pi }}{a x^{3}}} \right )}}{3 \sqrt [3]{a} x^{7} \Gamma \left (\frac {10}{3}\right )} \]
c*Piecewise((-3*a**(11/3)*x**6*(-1 + b/(a*x**3))**(2/3)*gamma(-5/3)/(9*a** 2*b**2*x**6*exp(I*pi/3)*gamma(1/3) - 9*a*b**3*x**3*exp(I*pi/3)*gamma(1/3)) + a**(8/3)*b*x**3*(-1 + b/(a*x**3))**(2/3)*gamma(-5/3)/(9*a**2*b**2*x**6* exp(I*pi/3)*gamma(1/3) - 9*a*b**3*x**3*exp(I*pi/3)*gamma(1/3)) + 2*a**(5/3 )*b**2*(-1 + b/(a*x**3))**(2/3)*gamma(-5/3)/(9*a**2*b**2*x**6*exp(I*pi/3)* gamma(1/3) - 9*a*b**3*x**3*exp(I*pi/3)*gamma(1/3)), Abs(b/(a*x**3)) > 1), (a**(5/3)*(1 - b/(a*x**3))**(2/3)*gamma(-5/3)/(3*b**2*gamma(1/3)) + 2*a**( 2/3)*(1 - b/(a*x**3))**(2/3)*gamma(-5/3)/(9*b*x**3*gamma(1/3)), True)) + d *gamma(7/3)*hyper((1/3, 7/3), (10/3,), b*exp_polar(2*I*pi)/(a*x**3))/(3*a* *(1/3)*x**7*gamma(10/3))
Time = 0.27 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.19 \[ \int \frac {-d+c x}{x^7 \sqrt [3]{-b+a x^3}} \, dx=-\frac {1}{54} \, {\left (\frac {4 \, \sqrt {3} a^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (a x^{3} - b\right )}^{\frac {1}{3}} - b^{\frac {1}{3}}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {7}{3}}} + \frac {2 \, a^{2} \log \left ({\left (a x^{3} - b\right )}^{\frac {2}{3}} - {\left (a x^{3} - b\right )}^{\frac {1}{3}} b^{\frac {1}{3}} + b^{\frac {2}{3}}\right )}{b^{\frac {7}{3}}} - \frac {4 \, a^{2} \log \left ({\left (a x^{3} - b\right )}^{\frac {1}{3}} + b^{\frac {1}{3}}\right )}{b^{\frac {7}{3}}} + \frac {3 \, {\left (4 \, {\left (a x^{3} - b\right )}^{\frac {5}{3}} a^{2} + 7 \, {\left (a x^{3} - b\right )}^{\frac {2}{3}} a^{2} b\right )}}{{\left (a x^{3} - b\right )}^{2} b^{2} + 2 \, {\left (a x^{3} - b\right )} b^{3} + b^{4}}\right )} d + \frac {c {\left (\frac {5 \, {\left (a x^{3} - b\right )}^{\frac {2}{3}} a}{x^{2}} - \frac {2 \, {\left (a x^{3} - b\right )}^{\frac {5}{3}}}{x^{5}}\right )}}{10 \, b^{2}} \]
-1/54*(4*sqrt(3)*a^2*arctan(1/3*sqrt(3)*(2*(a*x^3 - b)^(1/3) - b^(1/3))/b^ (1/3))/b^(7/3) + 2*a^2*log((a*x^3 - b)^(2/3) - (a*x^3 - b)^(1/3)*b^(1/3) + b^(2/3))/b^(7/3) - 4*a^2*log((a*x^3 - b)^(1/3) + b^(1/3))/b^(7/3) + 3*(4* (a*x^3 - b)^(5/3)*a^2 + 7*(a*x^3 - b)^(2/3)*a^2*b)/((a*x^3 - b)^2*b^2 + 2* (a*x^3 - b)*b^3 + b^4))*d + 1/10*c*(5*(a*x^3 - b)^(2/3)*a/x^2 - 2*(a*x^3 - b)^(5/3)/x^5)/b^2
\[ \int \frac {-d+c x}{x^7 \sqrt [3]{-b+a x^3}} \, dx=\int { \frac {c x - d}{{\left (a x^{3} - b\right )}^{\frac {1}{3}} x^{7}} \,d x } \]
Time = 7.05 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.36 \[ \int \frac {-d+c x}{x^7 \sqrt [3]{-b+a x^3}} \, dx=\frac {2\,a^2\,d\,\ln \left ({\left (a\,x^3-b\right )}^{1/3}+b^{1/3}\right )}{27\,b^{7/3}}-\frac {\frac {7\,a^2\,d\,{\left (a\,x^3-b\right )}^{2/3}}{18\,b}+\frac {2\,a^2\,d\,{\left (a\,x^3-b\right )}^{5/3}}{9\,b^2}}{{\left (b-a\,x^3\right )}^2-2\,b\,\left (b-a\,x^3\right )+b^2}+\frac {2\,a^2\,d\,\ln \left (\frac {4\,a^4\,d^2\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{81\,b^{11/3}}+\frac {4\,a^4\,d^2\,{\left (a\,x^3-b\right )}^{1/3}}{81\,b^4}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{27\,b^{7/3}}-\frac {2\,a^2\,d\,\ln \left (\frac {4\,a^4\,d^2\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{81\,b^{11/3}}+\frac {4\,a^4\,d^2\,{\left (a\,x^3-b\right )}^{1/3}}{81\,b^4}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{27\,b^{7/3}}+\frac {c\,{\left (a\,x^3-b\right )}^{2/3}\,\left (3\,a\,x^3+2\,b\right )}{10\,b^2\,x^5} \]
(2*a^2*d*log((a*x^3 - b)^(1/3) + b^(1/3)))/(27*b^(7/3)) - ((7*a^2*d*(a*x^3 - b)^(2/3))/(18*b) + (2*a^2*d*(a*x^3 - b)^(5/3))/(9*b^2))/((b - a*x^3)^2 - 2*b*(b - a*x^3) + b^2) + (2*a^2*d*log((4*a^4*d^2*((3^(1/2)*1i)/2 - 1/2)^ 2)/(81*b^(11/3)) + (4*a^4*d^2*(a*x^3 - b)^(1/3))/(81*b^4))*((3^(1/2)*1i)/2 - 1/2))/(27*b^(7/3)) - (2*a^2*d*log((4*a^4*d^2*((3^(1/2)*1i)/2 + 1/2)^2)/ (81*b^(11/3)) + (4*a^4*d^2*(a*x^3 - b)^(1/3))/(81*b^4))*((3^(1/2)*1i)/2 + 1/2))/(27*b^(7/3)) + (c*(a*x^3 - b)^(2/3)*(2*b + 3*a*x^3))/(10*b^2*x^5)