Integrand size = 28, antiderivative size = 485 \[ \int \frac {x^4 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=-\frac {x^2 \left (a+b x^3\right )^{2/3}}{4 b d}+\frac {2^{2/3} a^{4/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^{5/3} d}+\frac {a^{4/3} \arctan \left (\frac {1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} b^{5/3} d}-\frac {3 a x^2 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )}{4 b d \sqrt [3]{a+b x^3}}+\frac {a^{4/3} \log \left (\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a}\right )}{6 \sqrt [3]{2} b^{5/3} d}+\frac {a^{4/3} \log \left (1+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{2} b^{5/3} d}-\frac {2^{2/3} a^{4/3} \log \left (1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{3 b^{5/3} d}-\frac {a^{4/3} \log \left (\frac {\sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}-\frac {2^{2/3} \sqrt [3]{b} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{2 \sqrt [3]{2} b^{5/3} d} \] Output:
-1/4*x^2*(b*x^3+a)^(2/3)/b/d+1/3*2^(2/3)*a^(4/3)*arctan(1/3*(1-2*2^(1/3)*( a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))*3^(1/2))*3^(1/2)/b^(5/3)/d+1/6*a^(4/3) *arctan(1/3*(1+2^(1/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))*3^(1/2))*2^(2/ 3)*3^(1/2)/b^(5/3)/d-3/4*a*x^2*(1+b*x^3/a)^(1/3)*hypergeom([1/3, 2/3],[5/3 ],-b*x^3/a)/b/d/(b*x^3+a)^(1/3)+1/12*a^(4/3)*ln((a^(1/3)-b^(1/3)*x)^2*(a^( 1/3)+b^(1/3)*x)/a)*2^(2/3)/b^(5/3)/d+1/6*a^(4/3)*ln(1+2^(2/3)*(a^(1/3)+b^( 1/3)*x)^2/(b*x^3+a)^(2/3)-2^(1/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))*2^( 2/3)/b^(5/3)/d-1/3*2^(2/3)*a^(4/3)*ln(1+2^(1/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3 +a)^(1/3))/b^(5/3)/d-1/4*a^(4/3)*ln(b^(1/3)*(a^(1/3)+b^(1/3)*x)/a^(1/3)-2^ (2/3)*b^(1/3)*(b*x^3+a)^(1/3)/a^(1/3))*2^(2/3)/b^(5/3)/d
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 6.95 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.26 \[ \int \frac {x^4 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=\frac {x^2 \left (-5 \left (a+b x^3\right )+5 a \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )+6 b x^3 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{3},1,\frac {8}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )\right )}{20 b d \sqrt [3]{a+b x^3}} \] Input:
Integrate[(x^4*(a + b*x^3)^(2/3))/(a*d - b*d*x^3),x]
Output:
(x^2*(-5*(a + b*x^3) + 5*a*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3 , -((b*x^3)/a), (b*x^3)/a] + 6*b*x^3*(1 + (b*x^3)/a)^(1/3)*AppellF1[5/3, 1 /3, 1, 8/3, -((b*x^3)/a), (b*x^3)/a]))/(20*b*d*(a + b*x^3)^(1/3))
Time = 1.10 (sec) , antiderivative size = 471, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {978, 27, 1054, 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.
| \(\displaystyle \int \frac {x^4 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx\) |
| \(\Big \downarrow \) 978 |
| \(\displaystyle \frac {\int \frac {2 a x \left (3 b x^3+a\right )}{\left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx}{4 b d}-\frac {x^2 \left (a+b x^3\right )^{2/3}}{4 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \int \frac {x \left (3 b x^3+a\right )}{\left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx}{2 b d}-\frac {x^2 \left (a+b x^3\right )^{2/3}}{4 b d}\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle \frac {a \int \left (\frac {4 a x}{\left (a-b x^3\right ) \sqrt [3]{b x^3+a}}-\frac {3 x}{\sqrt [3]{b x^3+a}}\right )dx}{2 b d}-\frac {x^2 \left (a+b x^3\right )^{2/3}}{4 b d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a \left (\frac {2\ 2^{2/3} \sqrt [3]{a} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^{2/3}}+\frac {2^{2/3} \sqrt [3]{a} \arctan \left (\frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} b^{2/3}}+\frac {2^{2/3} \sqrt [3]{a} \log \left (\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3 b^{2/3}}-\frac {2\ 2^{2/3} \sqrt [3]{a} \log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3 b^{2/3}}-\frac {\sqrt [3]{a} \log \left (\frac {\sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}-\frac {2^{2/3} \sqrt [3]{b} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{\sqrt [3]{2} b^{2/3}}+\frac {\sqrt [3]{a} \log \left (\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a}\right )}{3 \sqrt [3]{2} b^{2/3}}-\frac {3 x^2 \sqrt [3]{\frac {b x^3}{a}+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )}{2 \sqrt [3]{a+b x^3}}\right )}{2 b d}-\frac {x^2 \left (a+b x^3\right )^{2/3}}{4 b d}\) |
Input:
Int[(x^4*(a + b*x^3)^(2/3))/(a*d - b*d*x^3),x]
Output:
-1/4*(x^2*(a + b*x^3)^(2/3))/(b*d) + (a*((2*2^(2/3)*a^(1/3)*ArcTan[(1 - (2 *2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/Sqrt[3]])/(Sqrt[3]*b^(2 /3)) + (2^(2/3)*a^(1/3)*ArcTan[(1 + (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b *x^3)^(1/3))/Sqrt[3]])/(Sqrt[3]*b^(2/3)) - (3*x^2*(1 + (b*x^3)/a)^(1/3)*Hy pergeometric2F1[1/3, 2/3, 5/3, -((b*x^3)/a)])/(2*(a + b*x^3)^(1/3)) + (a^( 1/3)*Log[((a^(1/3) - b^(1/3)*x)^2*(a^(1/3) + b^(1/3)*x))/a])/(3*2^(1/3)*b^ (2/3)) + (2^(2/3)*a^(1/3)*Log[1 + (2^(2/3)*(a^(1/3) + b^(1/3)*x)^2)/(a + b *x^3)^(2/3) - (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3)])/(3*b^(2/ 3)) - (2*2^(2/3)*a^(1/3)*Log[1 + (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^ 3)^(1/3)])/(3*b^(2/3)) - (a^(1/3)*Log[(b^(1/3)*(a^(1/3) + b^(1/3)*x))/a^(1 /3) - (2^(2/3)*b^(1/3)*(a + b*x^3)^(1/3))/a^(1/3)])/(2^(1/3)*b^(2/3))))/(2 *b*d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n _)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0]
\[\int \frac {x^{4} \left (b \,x^{3}+a \right )^{\frac {2}{3}}}{-b d \,x^{3}+a d}d x\]
Input:
int(x^4*(b*x^3+a)^(2/3)/(-b*d*x^3+a*d),x)
Output:
int(x^4*(b*x^3+a)^(2/3)/(-b*d*x^3+a*d),x)
Timed out. \[ \int \frac {x^4 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=\text {Timed out} \] Input:
integrate(x^4*(b*x^3+a)^(2/3)/(-b*d*x^3+a*d),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {x^4 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=- \frac {\int \frac {x^{4} \left (a + b x^{3}\right )^{\frac {2}{3}}}{- a + b x^{3}}\, dx}{d} \] Input:
integrate(x**4*(b*x**3+a)**(2/3)/(-b*d*x**3+a*d),x)
Output:
-Integral(x**4*(a + b*x**3)**(2/3)/(-a + b*x**3), x)/d
\[ \int \frac {x^4 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=\int { -\frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} x^{4}}{b d x^{3} - a d} \,d x } \] Input:
integrate(x^4*(b*x^3+a)^(2/3)/(-b*d*x^3+a*d),x, algorithm="maxima")
Output:
-integrate((b*x^3 + a)^(2/3)*x^4/(b*d*x^3 - a*d), x)
\[ \int \frac {x^4 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=\int { -\frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} x^{4}}{b d x^{3} - a d} \,d x } \] Input:
integrate(x^4*(b*x^3+a)^(2/3)/(-b*d*x^3+a*d),x, algorithm="giac")
Output:
integrate(-(b*x^3 + a)^(2/3)*x^4/(b*d*x^3 - a*d), x)
Timed out. \[ \int \frac {x^4 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=\int \frac {x^4\,{\left (b\,x^3+a\right )}^{2/3}}{a\,d-b\,d\,x^3} \,d x \] Input:
int((x^4*(a + b*x^3)^(2/3))/(a*d - b*d*x^3),x)
Output:
int((x^4*(a + b*x^3)^(2/3))/(a*d - b*d*x^3), x)
\[ \int \frac {x^4 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=\frac {-\left (b \,x^{3}+a \right )^{\frac {2}{3}} x^{2}+6 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} x^{4}}{-b^{2} x^{6}+a^{2}}d x \right ) a b +2 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} x}{-b^{2} x^{6}+a^{2}}d x \right ) a^{2}}{4 b d} \] Input:
int(x^4*(b*x^3+a)^(2/3)/(-b*d*x^3+a*d),x)
Output:
( - (a + b*x**3)**(2/3)*x**2 + 6*int(((a + b*x**3)**(2/3)*x**4)/(a**2 - b* *2*x**6),x)*a*b + 2*int(((a + b*x**3)**(2/3)*x)/(a**2 - b**2*x**6),x)*a**2 )/(4*b*d)