Integrand size = 26, antiderivative size = 306 \[ \int \frac {-d+c x^7}{x \sqrt [3]{-b+a x^3}} \, dx=\frac {c \left (-b+a x^3\right )^{2/3} \left (4 b x+3 a x^4\right )}{18 a^2}-\frac {2 b^2 c \arctan \left (\frac {\frac {x}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt {3} \sqrt [3]{a}}}{x}\right )}{9 \sqrt {3} a^{7/3}}+\frac {d \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} \sqrt [3]{b}}+\frac {d \log \left (\sqrt [3]{b}+\sqrt [3]{-b+a x^3}\right )}{3 \sqrt [3]{b}}-\frac {2 b^2 c \log \left (-\sqrt [3]{a} x+\sqrt [3]{-b+a x^3}\right )}{27 a^{7/3}}-\frac {d \log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{-b+a x^3}+\left (-b+a x^3\right )^{2/3}\right )}{6 \sqrt [3]{b}}+\frac {b^2 c \log \left (a^{2/3} x^2+\sqrt [3]{a} x \sqrt [3]{-b+a x^3}+\left (-b+a x^3\right )^{2/3}\right )}{27 a^{7/3}} \]
1/18*c*(a*x^3-b)^(2/3)*(3*a*x^4+4*b*x)/a^2-2/27*b^2*c*arctan((1/3*x*3^(1/2 )+2/3*(a*x^3-b)^(1/3)*3^(1/2)/a^(1/3))/x)*3^(1/2)/a^(7/3)-1/3*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^(1/3)+1/3*d*ln(b^ (1/3)+(a*x^3-b)^(1/3))/b^(1/3)-2/27*b^2*c*ln(-a^(1/3)*x+(a*x^3-b)^(1/3))/a ^(7/3)-1/6*d*ln(b^(2/3)-b^(1/3)*(a*x^3-b)^(1/3)+(a*x^3-b)^(2/3))/b^(1/3)+1 /27*b^2*c*ln(a^(2/3)*x^2+a^(1/3)*x*(a*x^3-b)^(1/3)+(a*x^3-b)^(2/3))/a^(7/3 )
Time = 5.54 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.95 \[ \int \frac {-d+c x^7}{x \sqrt [3]{-b+a x^3}} \, dx=\frac {1}{54} \left (\frac {3 c \left (-b+a x^3\right )^{2/3} \left (4 b x+3 a x^4\right )}{a^2}+\frac {18 \sqrt {3} d \arctan \left (\frac {1-\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}-\frac {4 \sqrt {3} b^2 c \arctan \left (\frac {1+\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt [3]{a} x}}{\sqrt {3}}\right )}{a^{7/3}}+\frac {18 d \log \left (\sqrt [3]{b}+\sqrt [3]{-b+a x^3}\right )}{\sqrt [3]{b}}-\frac {4 b^2 c \log \left (-\sqrt [3]{a} x+\sqrt [3]{-b+a x^3}\right )}{a^{7/3}}-\frac {9 d \log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{-b+a x^3}+\left (-b+a x^3\right )^{2/3}\right )}{\sqrt [3]{b}}+\frac {2 b^2 c \log \left (a^{2/3} x^2+\sqrt [3]{a} x \sqrt [3]{-b+a x^3}+\left (-b+a x^3\right )^{2/3}\right )}{a^{7/3}}\right ) \]
((3*c*(-b + a*x^3)^(2/3)*(4*b*x + 3*a*x^4))/a^2 + (18*Sqrt[3]*d*ArcTan[(1 - (2*(-b + a*x^3)^(1/3))/b^(1/3))/Sqrt[3]])/b^(1/3) - (4*Sqrt[3]*b^2*c*Arc Tan[(1 + (2*(-b + a*x^3)^(1/3))/(a^(1/3)*x))/Sqrt[3]])/a^(7/3) + (18*d*Log [b^(1/3) + (-b + a*x^3)^(1/3)])/b^(1/3) - (4*b^2*c*Log[-(a^(1/3)*x) + (-b + a*x^3)^(1/3)])/a^(7/3) - (9*d*Log[b^(2/3) - b^(1/3)*(-b + a*x^3)^(1/3) + (-b + a*x^3)^(2/3)])/b^(1/3) + (2*b^2*c*Log[a^(2/3)*x^2 + a^(1/3)*x*(-b + a*x^3)^(1/3) + (-b + a*x^3)^(2/3)])/a^(7/3))/54
Time = 0.52 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.75, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2375, 27, 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.
| \(\displaystyle \int \frac {c x^7-d}{x \sqrt [3]{a x^3-b}} \, dx\) |
| \(\Big \downarrow \) 2375 |
| \(\displaystyle \frac {\int -\frac {2 \left (3 a d-2 b c x^4\right )}{x \sqrt [3]{a x^3-b}}dx}{6 a}+\frac {c x^4 \left (a x^3-b\right )^{2/3}}{6 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c x^4 \left (a x^3-b\right )^{2/3}}{6 a}-\frac {\int \frac {3 a d-2 b c x^4}{x \sqrt [3]{a x^3-b}}dx}{3 a}\) |
| \(\Big \downarrow \) 2383 |
| \(\displaystyle \frac {c x^4 \left (a x^3-b\right )^{2/3}}{6 a}-\frac {\int \left (\frac {3 a d}{x \sqrt [3]{a x^3-b}}-\frac {2 b c x^3}{\sqrt [3]{a x^3-b}}\right )dx}{3 a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {c x^4 \left (a x^3-b\right )^{2/3}}{6 a}-\frac {-\frac {2 b^2 c \arctan \left (\frac {\frac {2 \sqrt [3]{a} x}{\sqrt [3]{a x^3-b}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} a^{4/3}}+\frac {b^2 c \log \left (\sqrt [3]{a x^3-b}-\sqrt [3]{a} x\right )}{3 a^{4/3}}-\frac {\sqrt {3} a d \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a x^3-b}}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt [3]{b}}-\frac {2 b c x \left (a x^3-b\right )^{2/3}}{3 a}-\frac {3 a d \log \left (\sqrt [3]{a x^3-b}+\sqrt [3]{b}\right )}{2 \sqrt [3]{b}}+\frac {3 a d \log (x)}{2 \sqrt [3]{b}}}{3 a}\) |
(c*x^4*(-b + a*x^3)^(2/3))/(6*a) - ((-2*b*c*x*(-b + a*x^3)^(2/3))/(3*a) - (2*b^2*c*ArcTan[(1 + (2*a^(1/3)*x)/(-b + a*x^3)^(1/3))/Sqrt[3]])/(3*Sqrt[3 ]*a^(4/3)) - (Sqrt[3]*a*d*ArcTan[(b^(1/3) - 2*(-b + a*x^3)^(1/3))/(Sqrt[3] *b^(1/3))])/b^(1/3) + (3*a*d*Log[x])/(2*b^(1/3)) - (3*a*d*Log[b^(1/3) + (- b + a*x^3)^(1/3)])/(2*b^(1/3)) + (b^2*c*Log[-(a^(1/3)*x) + (-b + a*x^3)^(1 /3)])/(3*a^(4/3)))/(3*a)
3.29.73.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Wi th[{q = Expon[Pq, x]}, With[{Pqq = Coeff[Pq, x, q]}, Simp[Pqq*(c*x)^(m + q - n + 1)*((a + b*x^n)^(p + 1)/(b*c^(q - n + 1)*(m + q + n*p + 1))), x] + Si mp[1/(b*(m + q + n*p + 1)) Int[(c*x)^m*ExpandToSum[b*(m + q + n*p + 1)*(P q - Pqq*x^q) - a*Pqq*(m + q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x]] / ; NeQ[m + q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || IntegerQ[p + ( q + 1)/(2*n)])] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]
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]
\[\int \frac {c \,x^{7}-d}{x \left (a \,x^{3}-b \right )^{\frac {1}{3}}}d x\]
Timed out. \[ \int \frac {-d+c x^7}{x \sqrt [3]{-b+a x^3}} \, dx=\text {Timed out} \]
Result contains complex when optimal does not.
Time = 2.70 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.27 \[ \int \frac {-d+c x^7}{x \sqrt [3]{-b+a x^3}} \, dx=\frac {c x^{7} e^{- \frac {i \pi }{3}} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {a x^{3}}{b}} \right )}}{3 \sqrt [3]{b} \Gamma \left (\frac {10}{3}\right )} + \frac {d \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b e^{2 i \pi }}{a x^{3}}} \right )}}{3 \sqrt [3]{a} x \Gamma \left (\frac {4}{3}\right )} \]
c*x**7*exp(-I*pi/3)*gamma(7/3)*hyper((1/3, 7/3), (10/3,), a*x**3/b)/(3*b** (1/3)*gamma(10/3)) + d*gamma(1/3)*hyper((1/3, 1/3), (4/3,), b*exp_polar(2* I*pi)/(a*x**3))/(3*a**(1/3)*x*gamma(4/3))
Time = 0.26 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.98 \[ \int \frac {-d+c x^7}{x \sqrt [3]{-b+a x^3}} \, dx=-\frac {1}{54} \, {\left (\frac {4 \, \sqrt {3} b^{2} \arctan \left (\frac {\sqrt {3} {\left (a^{\frac {1}{3}} + \frac {2 \, {\left (a x^{3} - b\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {7}{3}}} - \frac {2 \, b^{2} \log \left (a^{\frac {2}{3}} + \frac {{\left (a x^{3} - b\right )}^{\frac {1}{3}} a^{\frac {1}{3}}}{x} + \frac {{\left (a x^{3} - b\right )}^{\frac {2}{3}}}{x^{2}}\right )}{a^{\frac {7}{3}}} + \frac {4 \, b^{2} \log \left (-a^{\frac {1}{3}} + \frac {{\left (a x^{3} - b\right )}^{\frac {1}{3}}}{x}\right )}{a^{\frac {7}{3}}} - \frac {3 \, {\left (\frac {7 \, {\left (a x^{3} - b\right )}^{\frac {2}{3}} a b^{2}}{x^{2}} - \frac {4 \, {\left (a x^{3} - b\right )}^{\frac {5}{3}} b^{2}}{x^{5}}\right )}}{a^{4} - \frac {2 \, {\left (a x^{3} - b\right )} a^{3}}{x^{3}} + \frac {{\left (a x^{3} - b\right )}^{2} a^{2}}{x^{6}}}\right )} c - \frac {1}{6} \, {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (a x^{3} - b\right )}^{\frac {1}{3}} - b^{\frac {1}{3}}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {1}{3}}} + \frac {\log \left ({\left (a x^{3} - b\right )}^{\frac {2}{3}} - {\left (a x^{3} - b\right )}^{\frac {1}{3}} b^{\frac {1}{3}} + b^{\frac {2}{3}}\right )}{b^{\frac {1}{3}}} - \frac {2 \, \log \left ({\left (a x^{3} - b\right )}^{\frac {1}{3}} + b^{\frac {1}{3}}\right )}{b^{\frac {1}{3}}}\right )} d \]
-1/54*(4*sqrt(3)*b^2*arctan(1/3*sqrt(3)*(a^(1/3) + 2*(a*x^3 - b)^(1/3)/x)/ a^(1/3))/a^(7/3) - 2*b^2*log(a^(2/3) + (a*x^3 - b)^(1/3)*a^(1/3)/x + (a*x^ 3 - b)^(2/3)/x^2)/a^(7/3) + 4*b^2*log(-a^(1/3) + (a*x^3 - b)^(1/3)/x)/a^(7 /3) - 3*(7*(a*x^3 - b)^(2/3)*a*b^2/x^2 - 4*(a*x^3 - b)^(5/3)*b^2/x^5)/(a^4 - 2*(a*x^3 - b)*a^3/x^3 + (a*x^3 - b)^2*a^2/x^6))*c - 1/6*(2*sqrt(3)*arct an(1/3*sqrt(3)*(2*(a*x^3 - b)^(1/3) - b^(1/3))/b^(1/3))/b^(1/3) + log((a*x ^3 - b)^(2/3) - (a*x^3 - b)^(1/3)*b^(1/3) + b^(2/3))/b^(1/3) - 2*log((a*x^ 3 - b)^(1/3) + b^(1/3))/b^(1/3))*d
\[ \int \frac {-d+c x^7}{x \sqrt [3]{-b+a x^3}} \, dx=\int { \frac {c x^{7} - d}{{\left (a x^{3} - b\right )}^{\frac {1}{3}} x} \,d x } \]
Timed out. \[ \int \frac {-d+c x^7}{x \sqrt [3]{-b+a x^3}} \, dx=-\int \frac {d-c\,x^7}{x\,{\left (a\,x^3-b\right )}^{1/3}} \,d x \]