Integrand size = 17, antiderivative size = 121 \[ \int \frac {c+d x}{\left (a+b x^3\right )^{2/3}} \, dx=-\frac {d \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^{2/3}}+\frac {c x \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}-\frac {d \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2 b^{2/3}} \] Output:
-1/3*d*arctan(1/3*(1+2*b^(1/3)*x/(b*x^3+a)^(1/3))*3^(1/2))*3^(1/2)/b^(2/3) +c*x*(1+b*x^3/a)^(2/3)*hypergeom([1/3, 2/3],[4/3],-b*x^3/a)/(b*x^3+a)^(2/3 )-1/2*d*ln(b^(1/3)*x-(b*x^3+a)^(1/3))/b^(2/3)
Time = 10.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.64 \[ \int \frac {c+d x}{\left (a+b x^3\right )^{2/3}} \, dx=\frac {x \left (2 c \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )+d x \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},\frac {b x^3}{a+b x^3}\right )\right )}{2 \left (a+b x^3\right )^{2/3}} \] Input:
Integrate[(c + d*x)/(a + b*x^3)^(2/3),x]
Output:
(x*(2*c*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -((b*x^3)/a )] + d*x*Hypergeometric2F1[2/3, 1, 5/3, (b*x^3)/(a + b*x^3)]))/(2*(a + b*x ^3)^(2/3))
Time = 0.39 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2432, 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.
\(\displaystyle \int \frac {c+d x}{\left (a+b x^3\right )^{2/3}} \, dx\) |
\(\Big \downarrow \) 2432 |
\(\displaystyle \int \left (\frac {c}{\left (a+b x^3\right )^{2/3}}+\frac {d x}{\left (a+b x^3\right )^{2/3}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {d \arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} b^{2/3}}-\frac {d \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2 b^{2/3}}+\frac {c x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}\) |
Input:
Int[(c + d*x)/(a + b*x^3)^(2/3),x]
Output:
-((d*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]])/(Sqrt[3]*b^(2/ 3))) + (c*x*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -((b*x^ 3)/a)])/(a + b*x^3)^(2/3) - (d*Log[b^(1/3)*x - (a + b*x^3)^(1/3)])/(2*b^(2 /3))
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[ Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n, p}, x] && (PolyQ[Pq, x] || Poly Q[Pq, x^n])
\[\int \frac {d x +c}{\left (b \,x^{3}+a \right )^{\frac {2}{3}}}d x\]
Input:
int((d*x+c)/(b*x^3+a)^(2/3),x)
Output:
int((d*x+c)/(b*x^3+a)^(2/3),x)
\[ \int \frac {c+d x}{\left (a+b x^3\right )^{2/3}} \, dx=\int { \frac {d x + c}{{\left (b x^{3} + a\right )}^{\frac {2}{3}}} \,d x } \] Input:
integrate((d*x+c)/(b*x^3+a)^(2/3),x, algorithm="fricas")
Output:
integral((d*x + c)/(b*x^3 + a)^(2/3), x)
Result contains complex when optimal does not.
Time = 1.14 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.64 \[ \int \frac {c+d x}{\left (a+b x^3\right )^{2/3}} \, dx=\frac {c x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {2}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {2}{3}} \Gamma \left (\frac {4}{3}\right )} + \frac {d x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {2}{3}} \Gamma \left (\frac {5}{3}\right )} \] Input:
integrate((d*x+c)/(b*x**3+a)**(2/3),x)
Output:
c*x*gamma(1/3)*hyper((1/3, 2/3), (4/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**( 2/3)*gamma(4/3)) + d*x**2*gamma(2/3)*hyper((2/3, 2/3), (5/3,), b*x**3*exp_ polar(I*pi)/a)/(3*a**(2/3)*gamma(5/3))
\[ \int \frac {c+d x}{\left (a+b x^3\right )^{2/3}} \, dx=\int { \frac {d x + c}{{\left (b x^{3} + a\right )}^{\frac {2}{3}}} \,d x } \] Input:
integrate((d*x+c)/(b*x^3+a)^(2/3),x, algorithm="maxima")
Output:
integrate((d*x + c)/(b*x^3 + a)^(2/3), x)
\[ \int \frac {c+d x}{\left (a+b x^3\right )^{2/3}} \, dx=\int { \frac {d x + c}{{\left (b x^{3} + a\right )}^{\frac {2}{3}}} \,d x } \] Input:
integrate((d*x+c)/(b*x^3+a)^(2/3),x, algorithm="giac")
Output:
integrate((d*x + c)/(b*x^3 + a)^(2/3), x)
Timed out. \[ \int \frac {c+d x}{\left (a+b x^3\right )^{2/3}} \, dx=\int \frac {c+d\,x}{{\left (b\,x^3+a\right )}^{2/3}} \,d x \] Input:
int((c + d*x)/(a + b*x^3)^(2/3),x)
Output:
int((c + d*x)/(a + b*x^3)^(2/3), x)
\[ \int \frac {c+d x}{\left (a+b x^3\right )^{2/3}} \, dx=\left (\int \frac {x}{\left (b \,x^{3}+a \right )^{\frac {2}{3}}}d x \right ) d +\left (\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {2}{3}}}d x \right ) c \] Input:
int((d*x+c)/(b*x^3+a)^(2/3),x)
Output:
int(x/(a + b*x**3)**(2/3),x)*d + int(1/(a + b*x**3)**(2/3),x)*c