Integrand size = 24, antiderivative size = 272 \[ \int \frac {x^3 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx=\frac {x \left (a+b x^3\right )^{2/3}}{3 d}-\frac {(3 b c-2 a d) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{b} d^2}+\frac {\sqrt [3]{c} (b c-a d)^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} d^2}+\frac {\sqrt [3]{c} (b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 d^2}-\frac {\sqrt [3]{c} (b c-a d)^{2/3} \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 d^2}+\frac {(3 b c-2 a d) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{6 \sqrt [3]{b} d^2} \]
1/3*x*(b*x^3+a)^(2/3)/d+1/6*c^(1/3)*(-a*d+b*c)^(2/3)*ln(d*x^3+c)/d^2-1/2*c ^(1/3)*(-a*d+b*c)^(2/3)*ln((-a*d+b*c)^(1/3)*x/c^(1/3)-(b*x^3+a)^(1/3))/d^2 +1/6*(-2*a*d+3*b*c)*ln(-b^(1/3)*x+(b*x^3+a)^(1/3))/b^(1/3)/d^2-1/9*(-2*a*d +3*b*c)*arctan(1/3*(1+2*b^(1/3)*x/(b*x^3+a)^(1/3))*3^(1/2))/b^(1/3)/d^2*3^ (1/2)+1/3*c^(1/3)*(-a*d+b*c)^(2/3)*arctan(1/3*(1+2*(-a*d+b*c)^(1/3)*x/c^(1 /3)/(b*x^3+a)^(1/3))*3^(1/2))/d^2*3^(1/2)
Result contains complex when optimal does not.
Time = 5.05 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.71 \[ \int \frac {x^3 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx=\frac {12 d x \left (a+b x^3\right )^{2/3}-\frac {4 \sqrt {3} (3 b c-2 a d) \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )}{\sqrt [3]{b}}-6 \sqrt {-6+6 i \sqrt {3}} \sqrt [3]{c} (b c-a d)^{2/3} \arctan \left (\frac {3 \sqrt [3]{b c-a d} x}{\sqrt {3} \sqrt [3]{b c-a d} x-\left (3 i+\sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )+\frac {4 (3 b c-2 a d) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{b}}+6 \left (1+i \sqrt {3}\right ) \sqrt [3]{c} (b c-a d)^{2/3} \log \left (2 \sqrt [3]{b c-a d} x+\left (1+i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}\right )+\frac {2 (-3 b c+2 a d) \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{\sqrt [3]{b}}-3 i \left (-i+\sqrt {3}\right ) \sqrt [3]{c} (b c-a d)^{2/3} \log \left (2 (b c-a d)^{2/3} x^2+\left (-1-i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{b c-a d} x \sqrt [3]{a+b x^3}+i \left (i+\sqrt {3}\right ) c^{2/3} \left (a+b x^3\right )^{2/3}\right )}{36 d^2} \]
(12*d*x*(a + b*x^3)^(2/3) - (4*Sqrt[3]*(3*b*c - 2*a*d)*ArcTan[(Sqrt[3]*b^( 1/3)*x)/(b^(1/3)*x + 2*(a + b*x^3)^(1/3))])/b^(1/3) - 6*Sqrt[-6 + (6*I)*Sq rt[3]]*c^(1/3)*(b*c - a*d)^(2/3)*ArcTan[(3*(b*c - a*d)^(1/3)*x)/(Sqrt[3]*( b*c - a*d)^(1/3)*x - (3*I + Sqrt[3])*c^(1/3)*(a + b*x^3)^(1/3))] + (4*(3*b *c - 2*a*d)*Log[-(b^(1/3)*x) + (a + b*x^3)^(1/3)])/b^(1/3) + 6*(1 + I*Sqrt [3])*c^(1/3)*(b*c - a*d)^(2/3)*Log[2*(b*c - a*d)^(1/3)*x + (1 + I*Sqrt[3]) *c^(1/3)*(a + b*x^3)^(1/3)] + (2*(-3*b*c + 2*a*d)*Log[b^(2/3)*x^2 + b^(1/3 )*x*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)])/b^(1/3) - (3*I)*(-I + Sqrt[3]) *c^(1/3)*(b*c - a*d)^(2/3)*Log[2*(b*c - a*d)^(2/3)*x^2 + (-1 - I*Sqrt[3])* c^(1/3)*(b*c - a*d)^(1/3)*x*(a + b*x^3)^(1/3) + I*(I + Sqrt[3])*c^(2/3)*(a + b*x^3)^(2/3)])/(36*d^2)
Time = 0.37 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {978, 1026, 769, 901}
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.
\(\displaystyle \int \frac {x^3 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx\) |
\(\Big \downarrow \) 978 |
\(\displaystyle \frac {x \left (a+b x^3\right )^{2/3}}{3 d}-\frac {\int \frac {(3 b c-2 a d) x^3+a c}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{3 d}\) |
\(\Big \downarrow \) 1026 |
\(\displaystyle \frac {x \left (a+b x^3\right )^{2/3}}{3 d}-\frac {\frac {(3 b c-2 a d) \int \frac {1}{\sqrt [3]{b x^3+a}}dx}{d}-\frac {3 c (b c-a d) \int \frac {1}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{d}}{3 d}\) |
\(\Big \downarrow \) 769 |
\(\displaystyle \frac {x \left (a+b x^3\right )^{2/3}}{3 d}-\frac {\frac {(3 b c-2 a d) \left (\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}}\right )}{d}-\frac {3 c (b c-a d) \int \frac {1}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{d}}{3 d}\) |
\(\Big \downarrow \) 901 |
\(\displaystyle \frac {x \left (a+b x^3\right )^{2/3}}{3 d}-\frac {\frac {(3 b c-2 a d) \left (\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}}\right )}{d}-\frac {3 c (b c-a d) \left (\frac {\arctan \left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} c^{2/3} \sqrt [3]{b c-a d}}+\frac {\log \left (c+d x^3\right )}{6 c^{2/3} \sqrt [3]{b c-a d}}-\frac {\log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{2/3} \sqrt [3]{b c-a d}}\right )}{d}}{3 d}\) |
(x*(a + b*x^3)^(2/3))/(3*d) - ((-3*c*(b*c - a*d)*(ArcTan[(1 + (2*(b*c - a* d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*c^(2/3)*(b*c - a*d)^(1/3)) + Log[c + d*x^3]/(6*c^(2/3)*(b*c - a*d)^(1/3)) - Log[((b*c - a *d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)]/(2*c^(2/3)*(b*c - a*d)^(1/3))))/ d + ((3*b*c - 2*a*d)*(ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3] ]/(Sqrt[3]*b^(1/3)) - Log[-(b^(1/3)*x) + (a + b*x^3)^(1/3)]/(2*b^(1/3))))/ d)/(3*d)
3.7.85.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]* (x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^ 3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]
Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> Wit h[{q = Rt[(b*c - a*d)/c, 3]}, Simp[ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/S qrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[e^(n - 1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)* ((c + d*x^n)^q/(b*(m + n*(p + q) + 1))), x] - Simp[e^n/(b*(m + n*(p + q) + 1)) Int[(e*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[a*c*(m - n + 1) + (a*d*(m - n + 1) - n*q*(b*c - a*d))*x^n, x], x], x] /; FreeQ[{a, b, c , d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 0] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Int[(((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)* (x_)^(n_)), x_Symbol] :> Simp[f/d Int[(a + b*x^n)^p, x], x] + Simp[(d*e - c*f)/d Int[(a + b*x^n)^p/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, p, n}, x]
Time = 4.82 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.44
method | result | size |
pseudoelliptic | \(-\frac {-\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (a d -\frac {3 b c}{2}\right ) \ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{3}+\left (a d \,b^{\frac {1}{3}}-b^{\frac {4}{3}} c \right ) \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )+\frac {2 \sqrt {3}\, \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (a d -\frac {3 b c}{2}\right ) \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right )}{3}+\frac {2 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (a d -\frac {3 b c}{2}\right ) \ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )}{3}-\left (b \,x^{3}+a \right )^{\frac {2}{3}} x \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} d \,b^{\frac {1}{3}}+\left (a d \,b^{\frac {1}{3}}-b^{\frac {4}{3}} c \right ) \left (\arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x -2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x}\right ) \sqrt {3}-\frac {\ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}\right )}{3 b^{\frac {1}{3}} \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} d^{2}}\) | \(391\) |
-1/3*(-1/3*((a*d-b*c)/c)^(1/3)*(a*d-3/2*b*c)*ln((b^(2/3)*x^2+b^(1/3)*(b*x^ 3+a)^(1/3)*x+(b*x^3+a)^(2/3))/x^2)+(a*d*b^(1/3)-b^(4/3)*c)*ln((((a*d-b*c)/ c)^(1/3)*x+(b*x^3+a)^(1/3))/x)+2/3*3^(1/2)*((a*d-b*c)/c)^(1/3)*(a*d-3/2*b* c)*arctan(1/3*3^(1/2)*(b^(1/3)*x+2*(b*x^3+a)^(1/3))/b^(1/3)/x)+2/3*((a*d-b *c)/c)^(1/3)*(a*d-3/2*b*c)*ln((-b^(1/3)*x+(b*x^3+a)^(1/3))/x)-(b*x^3+a)^(2 /3)*x*((a*d-b*c)/c)^(1/3)*d*b^(1/3)+(a*d*b^(1/3)-b^(4/3)*c)*(arctan(1/3*3^ (1/2)*(((a*d-b*c)/c)^(1/3)*x-2*(b*x^3+a)^(1/3))/((a*d-b*c)/c)^(1/3)/x)*3^( 1/2)-1/2*ln((((a*d-b*c)/c)^(2/3)*x^2-((a*d-b*c)/c)^(1/3)*(b*x^3+a)^(1/3)*x +(b*x^3+a)^(2/3))/x^2)))/b^(1/3)/((a*d-b*c)/c)^(1/3)/d^2
Leaf count of result is larger than twice the leaf count of optimal. 521 vs. \(2 (219) = 438\).
Time = 0.60 (sec) , antiderivative size = 1091, normalized size of antiderivative = 4.01 \[ \int \frac {x^3 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx=\text {Too large to display} \]
[1/18*(6*(b*x^3 + a)^(2/3)*b*d*x - 3*sqrt(1/3)*(3*b^2*c - 2*a*b*d)*sqrt((- b)^(1/3)/b)*log(3*b*x^3 - 3*(b*x^3 + a)^(1/3)*(-b)^(2/3)*x^2 - 3*sqrt(1/3) *((-b)^(1/3)*b*x^3 - (b*x^3 + a)^(1/3)*b*x^2 + 2*(b*x^3 + a)^(2/3)*(-b)^(2 /3)*x)*sqrt((-b)^(1/3)/b) + 2*a) + 6*sqrt(3)*(-b^2*c^3 + 2*a*b*c^2*d - a^2 *c*d^2)^(1/3)*b*arctan(1/3*(sqrt(3)*(b*c - a*d)*x - 2*sqrt(3)*(-b^2*c^3 + 2*a*b*c^2*d - a^2*c*d^2)^(1/3)*(b*x^3 + a)^(1/3))/((b*c - a*d)*x)) + 2*(3* b*c - 2*a*d)*(-b)^(2/3)*log(((-b)^(1/3)*x + (b*x^3 + a)^(1/3))/x) - (3*b*c - 2*a*d)*(-b)^(2/3)*log(((-b)^(2/3)*x^2 - (b*x^3 + a)^(1/3)*(-b)^(1/3)*x + (b*x^3 + a)^(2/3))/x^2) + 6*(-b^2*c^3 + 2*a*b*c^2*d - a^2*c*d^2)^(1/3)*b *log(((-b^2*c^3 + 2*a*b*c^2*d - a^2*c*d^2)^(2/3)*x - (b*x^3 + a)^(1/3)*(b* c^2 - a*c*d))/x) - 3*(-b^2*c^3 + 2*a*b*c^2*d - a^2*c*d^2)^(1/3)*b*log(((-b ^2*c^3 + 2*a*b*c^2*d - a^2*c*d^2)^(1/3)*(b*c - a*d)*x^2 - (-b^2*c^3 + 2*a* b*c^2*d - a^2*c*d^2)^(2/3)*(b*x^3 + a)^(1/3)*x - (b*x^3 + a)^(2/3)*(b*c^2 - a*c*d))/x^2))/(b*d^2), 1/18*(6*(b*x^3 + a)^(2/3)*b*d*x + 6*sqrt(1/3)*(3* b^2*c - 2*a*b*d)*sqrt(-(-b)^(1/3)/b)*arctan(-sqrt(1/3)*((-b)^(1/3)*x - 2*( b*x^3 + a)^(1/3))*sqrt(-(-b)^(1/3)/b)/x) + 6*sqrt(3)*(-b^2*c^3 + 2*a*b*c^2 *d - a^2*c*d^2)^(1/3)*b*arctan(1/3*(sqrt(3)*(b*c - a*d)*x - 2*sqrt(3)*(-b^ 2*c^3 + 2*a*b*c^2*d - a^2*c*d^2)^(1/3)*(b*x^3 + a)^(1/3))/((b*c - a*d)*x)) + 2*(3*b*c - 2*a*d)*(-b)^(2/3)*log(((-b)^(1/3)*x + (b*x^3 + a)^(1/3))/x) - (3*b*c - 2*a*d)*(-b)^(2/3)*log(((-b)^(2/3)*x^2 - (b*x^3 + a)^(1/3)*(-...
\[ \int \frac {x^3 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx=\int \frac {x^{3} \left (a + b x^{3}\right )^{\frac {2}{3}}}{c + d x^{3}}\, dx \]
\[ \int \frac {x^3 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} x^{3}}{d x^{3} + c} \,d x } \]
\[ \int \frac {x^3 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} x^{3}}{d x^{3} + c} \,d x } \]
Timed out. \[ \int \frac {x^3 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx=\int \frac {x^3\,{\left (b\,x^3+a\right )}^{2/3}}{d\,x^3+c} \,d x \]