Integrand size = 22, antiderivative size = 595 \[ \int \frac {x^4 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {(b c-a d) x^5}{9 a b \left (a+b x^6\right )^{3/2}}+\frac {(4 b c+5 a d) x^5}{27 a^2 b \sqrt {a+b x^6}}-\frac {\left (1+\sqrt {3}\right ) (4 b c+5 a d) x \sqrt {a+b x^6}}{27 a^2 b^{5/3} \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )}+\frac {(4 b c+5 a d) x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{9\ 3^{3/4} a^{5/3} b^{5/3} \sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}+\frac {\left (1-\sqrt {3}\right ) (4 b c+5 a d) x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{54 \sqrt [4]{3} a^{5/3} b^{5/3} \sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}} \] Output:
1/9*(-a*d+b*c)*x^5/a/b/(b*x^6+a)^(3/2)+1/27*(5*a*d+4*b*c)*x^5/a^2/b/(b*x^6 +a)^(1/2)-1/27*(1+3^(1/2))*(5*a*d+4*b*c)*x*(b*x^6+a)^(1/2)/a^2/b^(5/3)/(a^ (1/3)+(1+3^(1/2))*b^(1/3)*x^2)+1/27*(5*a*d+4*b*c)*x*(a^(1/3)+b^(1/3)*x^2)* ((a^(2/3)-a^(1/3)*b^(1/3)*x^2+b^(2/3)*x^4)/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x^ 2)^2)^(1/2)*EllipticE((1-(a^(1/3)+(1-3^(1/2))*b^(1/3)*x^2)^2/(a^(1/3)+(1+3 ^(1/2))*b^(1/3)*x^2)^2)^(1/2),1/4*6^(1/2)+1/4*2^(1/2))*3^(1/4)/a^(5/3)/b^( 5/3)/(b^(1/3)*x^2*(a^(1/3)+b^(1/3)*x^2)/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x^2)^ 2)^(1/2)/(b*x^6+a)^(1/2)+1/162*(1-3^(1/2))*(5*a*d+4*b*c)*x*(a^(1/3)+b^(1/3 )*x^2)*((a^(2/3)-a^(1/3)*b^(1/3)*x^2+b^(2/3)*x^4)/(a^(1/3)+(1+3^(1/2))*b^( 1/3)*x^2)^2)^(1/2)*InverseJacobiAM(arccos((a^(1/3)+(1-3^(1/2))*b^(1/3)*x^2 )/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x^2)),1/4*6^(1/2)+1/4*2^(1/2))*3^(3/4)/a^(5 /3)/b^(5/3)/(b^(1/3)*x^2*(a^(1/3)+b^(1/3)*x^2)/(a^(1/3)+(1+3^(1/2))*b^(1/3 )*x^2)^2)^(1/2)/(b*x^6+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.09 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.14 \[ \int \frac {x^4 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {x^5 \left (-5 a^2 d+(4 b c+5 a d) \left (a+b x^6\right ) \sqrt {1+\frac {b x^6}{a}} \operatorname {Hypergeometric2F1}\left (\frac {5}{6},\frac {5}{2},\frac {11}{6},-\frac {b x^6}{a}\right )\right )}{20 a^2 b \left (a+b x^6\right )^{3/2}} \] Input:
Integrate[(x^4*(c + d*x^6))/(a + b*x^6)^(5/2),x]
Output:
(x^5*(-5*a^2*d + (4*b*c + 5*a*d)*(a + b*x^6)*Sqrt[1 + (b*x^6)/a]*Hypergeom etric2F1[5/6, 5/2, 11/6, -((b*x^6)/a)]))/(20*a^2*b*(a + b*x^6)^(3/2))
Time = 0.95 (sec) , antiderivative size = 576, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {957, 819, 837, 25, 766, 2420}
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.
| \(\displaystyle \int \frac {x^4 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 957 |
| \(\displaystyle \frac {(5 a d+4 b c) \int \frac {x^4}{\left (b x^6+a\right )^{3/2}}dx}{9 a b}+\frac {x^5 (b c-a d)}{9 a b \left (a+b x^6\right )^{3/2}}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {(5 a d+4 b c) \left (\frac {x^5}{3 a \sqrt {a+b x^6}}-\frac {2 \int \frac {x^4}{\sqrt {b x^6+a}}dx}{3 a}\right )}{9 a b}+\frac {x^5 (b c-a d)}{9 a b \left (a+b x^6\right )^{3/2}}\) |
| \(\Big \downarrow \) 837 |
| \(\displaystyle \frac {(5 a d+4 b c) \left (\frac {x^5}{3 a \sqrt {a+b x^6}}-\frac {2 \left (-\frac {\left (1-\sqrt {3}\right ) a^{2/3} \int \frac {1}{\sqrt {b x^6+a}}dx}{2 b^{2/3}}-\frac {\int -\frac {2 b^{2/3} x^4+\left (1-\sqrt {3}\right ) a^{2/3}}{\sqrt {b x^6+a}}dx}{2 b^{2/3}}\right )}{3 a}\right )}{9 a b}+\frac {x^5 (b c-a d)}{9 a b \left (a+b x^6\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(5 a d+4 b c) \left (\frac {x^5}{3 a \sqrt {a+b x^6}}-\frac {2 \left (\frac {\int \frac {2 b^{2/3} x^4+\left (1-\sqrt {3}\right ) a^{2/3}}{\sqrt {b x^6+a}}dx}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) a^{2/3} \int \frac {1}{\sqrt {b x^6+a}}dx}{2 b^{2/3}}\right )}{3 a}\right )}{9 a b}+\frac {x^5 (b c-a d)}{9 a b \left (a+b x^6\right )^{3/2}}\) |
| \(\Big \downarrow \) 766 |
| \(\displaystyle \frac {(5 a d+4 b c) \left (\frac {x^5}{3 a \sqrt {a+b x^6}}-\frac {2 \left (\frac {\int \frac {2 b^{2/3} x^4+\left (1-\sqrt {3}\right ) a^{2/3}}{\sqrt {b x^6+a}}dx}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2+\sqrt [3]{a}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}\right )}{3 a}\right )}{9 a b}+\frac {x^5 (b c-a d)}{9 a b \left (a+b x^6\right )^{3/2}}\) |
| \(\Big \downarrow \) 2420 |
| \(\displaystyle \frac {(5 a d+4 b c) \left (\frac {x^5}{3 a \sqrt {a+b x^6}}-\frac {2 \left (\frac {\frac {\left (1+\sqrt {3}\right ) x \sqrt {a+b x^6}}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2}-\frac {\sqrt [4]{3} \sqrt [3]{a} x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} E\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2+\sqrt [3]{a}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2+\sqrt [3]{a}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}\right )}{3 a}\right )}{9 a b}+\frac {x^5 (b c-a d)}{9 a b \left (a+b x^6\right )^{3/2}}\) |
Input:
Int[(x^4*(c + d*x^6))/(a + b*x^6)^(5/2),x]
Output:
((b*c - a*d)*x^5)/(9*a*b*(a + b*x^6)^(3/2)) + ((4*b*c + 5*a*d)*(x^5/(3*a*S qrt[a + b*x^6]) - (2*((((1 + Sqrt[3])*x*Sqrt[a + b*x^6])/(a^(1/3) + (1 + S qrt[3])*b^(1/3)*x^2) - (3^(1/4)*a^(1/3)*x*(a^(1/3) + b^(1/3)*x^2)*Sqrt[(a^ (2/3) - a^(1/3)*b^(1/3)*x^2 + b^(2/3)*x^4)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3 )*x^2)^2]*EllipticE[ArcCos[(a^(1/3) + (1 - Sqrt[3])*b^(1/3)*x^2)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x^2)], (2 + Sqrt[3])/4])/(Sqrt[(b^(1/3)*x^2*(a^(1/ 3) + b^(1/3)*x^2))/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x^2)^2]*Sqrt[a + b*x^6 ]))/(2*b^(2/3)) - ((1 - Sqrt[3])*a^(1/3)*x*(a^(1/3) + b^(1/3)*x^2)*Sqrt[(a ^(2/3) - a^(1/3)*b^(1/3)*x^2 + b^(2/3)*x^4)/(a^(1/3) + (1 + Sqrt[3])*b^(1/ 3)*x^2)^2]*EllipticF[ArcCos[(a^(1/3) + (1 - Sqrt[3])*b^(1/3)*x^2)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x^2)], (2 + Sqrt[3])/4])/(4*3^(1/4)*b^(2/3)*Sqrt[ (b^(1/3)*x^2*(a^(1/3) + b^(1/3)*x^2))/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x^2 )^2]*Sqrt[a + b*x^6])))/(3*a)))/(9*a*b)
Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[x*(s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/ (s + (1 + Sqrt[3])*r*x^2)^2]/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[r*x^2*((s + r*x^2)/(s + (1 + Sqrt[3])*r*x^2)^2)]))*EllipticF[ArcCos[(s + (1 - Sqrt[3])* r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b}, x ]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-( c*x)^(m + 1))*((a + b*x^n)^(p + 1)/(a*c*n*(p + 1))), x] + Simp[(m + n*(p + 1) + 1)/(a*n*(p + 1)) Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a , b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[(x_)^4/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(Sqrt[3] - 1)*(s^2/(2*r^2)) Int[1/Sqrt[ a + b*x^6], x], x] - Simp[1/(2*r^2) Int[((Sqrt[3] - 1)*s^2 - 2*r^2*x^4)/S qrt[a + b*x^6], x], x]] /; FreeQ[{a, b}, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a *b*e*n*(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b*n* (p + 1)) Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && N eQ[p, -5/4]) || !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] && LeQ[-1 , m, (-n)*(p + 1)]))
Int[((c_) + (d_.)*(x_)^4)/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(1 + Sqrt[3])*d*s^3*x*(Sqr t[a + b*x^6]/(2*a*r^2*(s + (1 + Sqrt[3])*r*x^2))), x] - Simp[3^(1/4)*d*s*x* (s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]/(2 *r^2*Sqrt[(r*x^2*(s + r*x^2))/(s + (1 + Sqrt[3])*r*x^2)^2]*Sqrt[a + b*x^6]) )*EllipticE[ArcCos[(s + (1 - Sqrt[3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[2*Rt[b/a, 3]^2*c - (1 - Sqrt[3])*d, 0]
\[\int \frac {x^{4} \left (d \,x^{6}+c \right )}{\left (b \,x^{6}+a \right )^{\frac {5}{2}}}d x\]
Input:
int(x^4*(d*x^6+c)/(b*x^6+a)^(5/2),x)
Output:
int(x^4*(d*x^6+c)/(b*x^6+a)^(5/2),x)
\[ \int \frac {x^4 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\int { \frac {{\left (d x^{6} + c\right )} x^{4}}{{\left (b x^{6} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(x^4*(d*x^6+c)/(b*x^6+a)^(5/2),x, algorithm="fricas")
Output:
integral((d*x^10 + c*x^4)*sqrt(b*x^6 + a)/(b^3*x^18 + 3*a*b^2*x^12 + 3*a^2 *b*x^6 + a^3), x)
Result contains complex when optimal does not.
Time = 137.27 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.13 \[ \int \frac {x^4 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {c x^{5} \Gamma \left (\frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{6}, \frac {5}{2} \\ \frac {11}{6} \end {matrix}\middle | {\frac {b x^{6} e^{i \pi }}{a}} \right )}}{6 a^{\frac {5}{2}} \Gamma \left (\frac {11}{6}\right )} + \frac {d x^{11} \Gamma \left (\frac {11}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {11}{6}, \frac {5}{2} \\ \frac {17}{6} \end {matrix}\middle | {\frac {b x^{6} e^{i \pi }}{a}} \right )}}{6 a^{\frac {5}{2}} \Gamma \left (\frac {17}{6}\right )} \] Input:
integrate(x**4*(d*x**6+c)/(b*x**6+a)**(5/2),x)
Output:
c*x**5*gamma(5/6)*hyper((5/6, 5/2), (11/6,), b*x**6*exp_polar(I*pi)/a)/(6* a**(5/2)*gamma(11/6)) + d*x**11*gamma(11/6)*hyper((11/6, 5/2), (17/6,), b* x**6*exp_polar(I*pi)/a)/(6*a**(5/2)*gamma(17/6))
\[ \int \frac {x^4 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\int { \frac {{\left (d x^{6} + c\right )} x^{4}}{{\left (b x^{6} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(x^4*(d*x^6+c)/(b*x^6+a)^(5/2),x, algorithm="maxima")
Output:
integrate((d*x^6 + c)*x^4/(b*x^6 + a)^(5/2), x)
\[ \int \frac {x^4 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\int { \frac {{\left (d x^{6} + c\right )} x^{4}}{{\left (b x^{6} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(x^4*(d*x^6+c)/(b*x^6+a)^(5/2),x, algorithm="giac")
Output:
integrate((d*x^6 + c)*x^4/(b*x^6 + a)^(5/2), x)
Timed out. \[ \int \frac {x^4 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\int \frac {x^4\,\left (d\,x^6+c\right )}{{\left (b\,x^6+a\right )}^{5/2}} \,d x \] Input:
int((x^4*(c + d*x^6))/(a + b*x^6)^(5/2),x)
Output:
int((x^4*(c + d*x^6))/(a + b*x^6)^(5/2), x)
\[ \int \frac {x^4 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {-\sqrt {b \,x^{6}+a}\, d \,x^{5}+5 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{4}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{3} d +4 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{4}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{2} b c +10 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{4}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{2} b d \,x^{6}+8 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{4}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a \,b^{2} c \,x^{6}+5 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{4}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a \,b^{2} d \,x^{12}+4 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{4}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) b^{3} c \,x^{12}}{4 b \left (b^{2} x^{12}+2 a b \,x^{6}+a^{2}\right )} \] Input:
int(x^4*(d*x^6+c)/(b*x^6+a)^(5/2),x)
Output:
( - sqrt(a + b*x**6)*d*x**5 + 5*int((sqrt(a + b*x**6)*x**4)/(a**3 + 3*a**2 *b*x**6 + 3*a*b**2*x**12 + b**3*x**18),x)*a**3*d + 4*int((sqrt(a + b*x**6) *x**4)/(a**3 + 3*a**2*b*x**6 + 3*a*b**2*x**12 + b**3*x**18),x)*a**2*b*c + 10*int((sqrt(a + b*x**6)*x**4)/(a**3 + 3*a**2*b*x**6 + 3*a*b**2*x**12 + b* *3*x**18),x)*a**2*b*d*x**6 + 8*int((sqrt(a + b*x**6)*x**4)/(a**3 + 3*a**2* b*x**6 + 3*a*b**2*x**12 + b**3*x**18),x)*a*b**2*c*x**6 + 5*int((sqrt(a + b *x**6)*x**4)/(a**3 + 3*a**2*b*x**6 + 3*a*b**2*x**12 + b**3*x**18),x)*a*b** 2*d*x**12 + 4*int((sqrt(a + b*x**6)*x**4)/(a**3 + 3*a**2*b*x**6 + 3*a*b**2 *x**12 + b**3*x**18),x)*b**3*c*x**12)/(4*b*(a**2 + 2*a*b*x**6 + b**2*x**12 ))