Integrand size = 21, antiderivative size = 132 \[ \int \frac {\left (c+d x^3\right )^2}{\left (a+b x^3\right )^{2/3}} \, dx=\frac {2 d (2 b c-a d) x \sqrt [3]{a+b x^3}}{5 b^2}+\frac {d x \sqrt [3]{a+b x^3} \left (c+d x^3\right )}{5 b}+\frac {\left (5 b^2 c^2-5 a b c d+2 a^2 d^2\right ) x \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{5 b^2 \left (a+b x^3\right )^{2/3}} \]
2/5*d*(-a*d+2*b*c)*x*(b*x^3+a)^(1/3)/b^2+1/5*d*x*(b*x^3+a)^(1/3)*(d*x^3+c) /b+1/5*(2*a^2*d^2-5*a*b*c*d+5*b^2*c^2)*x*(1+b*x^3/a)^(2/3)*hypergeom([1/3, 2/3],[4/3],-b*x^3/a)/b^2/(b*x^3+a)^(2/3)
Time = 15.10 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.79 \[ \int \frac {\left (c+d x^3\right )^2}{\left (a+b x^3\right )^{2/3}} \, dx=\frac {x \left (-d \left (a+b x^3\right ) \left (2 a d-b \left (5 c+d x^3\right )\right )+\left (5 b^2 c^2-5 a b c d+2 a^2 d^2\right ) \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )\right )}{5 b^2 \left (a+b x^3\right )^{2/3}} \]
(x*(-(d*(a + b*x^3)*(2*a*d - b*(5*c + d*x^3))) + (5*b^2*c^2 - 5*a*b*c*d + 2*a^2*d^2)*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -((b*x^3 )/a)]))/(5*b^2*(a + b*x^3)^(2/3))
Time = 0.28 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {933, 913, 779, 778}
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.
\(\displaystyle \int \frac {\left (c+d x^3\right )^2}{\left (a+b x^3\right )^{2/3}} \, dx\) |
\(\Big \downarrow \) 933 |
\(\displaystyle \frac {\int \frac {4 d (2 b c-a d) x^3+c (5 b c-a d)}{\left (b x^3+a\right )^{2/3}}dx}{5 b}+\frac {d x \sqrt [3]{a+b x^3} \left (c+d x^3\right )}{5 b}\) |
\(\Big \downarrow \) 913 |
\(\displaystyle \frac {\frac {\left (2 a^2 d^2-5 a b c d+5 b^2 c^2\right ) \int \frac {1}{\left (b x^3+a\right )^{2/3}}dx}{b}+\frac {2 d x \sqrt [3]{a+b x^3} (2 b c-a d)}{b}}{5 b}+\frac {d x \sqrt [3]{a+b x^3} \left (c+d x^3\right )}{5 b}\) |
\(\Big \downarrow \) 779 |
\(\displaystyle \frac {\frac {\left (\frac {b x^3}{a}+1\right )^{2/3} \left (2 a^2 d^2-5 a b c d+5 b^2 c^2\right ) \int \frac {1}{\left (\frac {b x^3}{a}+1\right )^{2/3}}dx}{b \left (a+b x^3\right )^{2/3}}+\frac {2 d x \sqrt [3]{a+b x^3} (2 b c-a d)}{b}}{5 b}+\frac {d x \sqrt [3]{a+b x^3} \left (c+d x^3\right )}{5 b}\) |
\(\Big \downarrow \) 778 |
\(\displaystyle \frac {\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \left (2 a^2 d^2-5 a b c d+5 b^2 c^2\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{b \left (a+b x^3\right )^{2/3}}+\frac {2 d x \sqrt [3]{a+b x^3} (2 b c-a d)}{b}}{5 b}+\frac {d x \sqrt [3]{a+b x^3} \left (c+d x^3\right )}{5 b}\) |
(d*x*(a + b*x^3)^(1/3)*(c + d*x^3))/(5*b) + ((2*d*(2*b*c - a*d)*x*(a + b*x ^3)^(1/3))/b + ((5*b^2*c^2 - 5*a*b*c*d + 2*a^2*d^2)*x*(1 + (b*x^3)/a)^(2/3 )*Hypergeometric2F1[1/3, 2/3, 4/3, -((b*x^3)/a)])/(b*(a + b*x^3)^(2/3)))/( 5*b)
3.1.82.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, n, p}, x] && !IGtQ[p , 0] && !IntegerQ[1/n] && !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p] || GtQ[a, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x ^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]) Int[(1 + b*(x^n/a))^p, x], x ] /; FreeQ[{a, b, n, p}, x] && !IGtQ[p, 0] && !IntegerQ[1/n] && !ILtQ[Si mplify[1/n + p], 0] && !(IntegerQ[p] || GtQ[a, 0])
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_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(b*(n*(p + q) + 1))), x] + Simp[1/(b*(n*(p + q) + 1)) Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Sim p[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d , 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] && !IGtQ[p, 1] && IntBinomialQ[ a, b, c, d, n, p, q, x]
\[\int \frac {\left (d \,x^{3}+c \right )^{2}}{\left (b \,x^{3}+a \right )^{\frac {2}{3}}}d x\]
\[ \int \frac {\left (c+d x^3\right )^2}{\left (a+b x^3\right )^{2/3}} \, dx=\int { \frac {{\left (d x^{3} + c\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {2}{3}}} \,d x } \]
Result contains complex when optimal does not.
Time = 1.52 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.95 \[ \int \frac {\left (c+d x^3\right )^2}{\left (a+b x^3\right )^{2/3}} \, dx=\frac {c^{2} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {2}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {2}{3}} \Gamma \left (\frac {4}{3}\right )} + \frac {2 c d x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {2}{3}} \Gamma \left (\frac {7}{3}\right )} + \frac {d^{2} x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {2}{3}} \Gamma \left (\frac {10}{3}\right )} \]
c**2*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)) + 2*c*d*x**4*gamma(4/3)*hyper((2/3, 4/3), (7/3,), b*x* *3*exp_polar(I*pi)/a)/(3*a**(2/3)*gamma(7/3)) + d**2*x**7*gamma(7/3)*hyper ((2/3, 7/3), (10/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**(2/3)*gamma(10/3))
\[ \int \frac {\left (c+d x^3\right )^2}{\left (a+b x^3\right )^{2/3}} \, dx=\int { \frac {{\left (d x^{3} + c\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {2}{3}}} \,d x } \]
\[ \int \frac {\left (c+d x^3\right )^2}{\left (a+b x^3\right )^{2/3}} \, dx=\int { \frac {{\left (d x^{3} + c\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {2}{3}}} \,d x } \]
Timed out. \[ \int \frac {\left (c+d x^3\right )^2}{\left (a+b x^3\right )^{2/3}} \, dx=\int \frac {{\left (d\,x^3+c\right )}^2}{{\left (b\,x^3+a\right )}^{2/3}} \,d x \]