\(\int \frac {x^3 (c+d x^6)}{(a+b x^6)^{5/2}} \, dx\) [54]

Optimal result

Integrand size = 22, antiderivative size = 593 \[ \int \frac {x^3 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {(b c-a d) x^4}{9 a b \left (a+b x^6\right )^{3/2}}+\frac {(5 b c+4 a d) x^4}{27 a^2 b \sqrt {a+b x^6}}-\frac {(5 b c+4 a d) \sqrt {a+b x^6}}{27 a^2 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )}+\frac {\sqrt {2-\sqrt {3}} (5 b c+4 a d) \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2}\right )|-7-4 \sqrt {3}\right )}{18\ 3^{3/4} a^{5/3} b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}-\frac {\sqrt {2} (5 b c+4 a d) \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2}\right ),-7-4 \sqrt {3}\right )}{27 \sqrt [4]{3} a^{5/3} b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}} \] Output:

1/9*(-a*d+b*c)*x^4/a/b/(b*x^6+a)^(3/2)+1/27*(4*a*d+5*b*c)*x^4/a^2/b/(b*x^6 
+a)^(1/2)-1/27*(4*a*d+5*b*c)*(b*x^6+a)^(1/2)/a^2/b^(5/3)/((1+3^(1/2))*a^(1 
/3)+b^(1/3)*x^2)+1/54*(1/2*6^(1/2)-1/2*2^(1/2))*(4*a*d+5*b*c)*(a^(1/3)+b^( 
1/3)*x^2)*((a^(2/3)-a^(1/3)*b^(1/3)*x^2+b^(2/3)*x^4)/((1+3^(1/2))*a^(1/3)+ 
b^(1/3)*x^2)^2)^(1/2)*EllipticE(((1-3^(1/2))*a^(1/3)+b^(1/3)*x^2)/((1+3^(1 
/2))*a^(1/3)+b^(1/3)*x^2),I*3^(1/2)+2*I)*3^(1/4)/a^(5/3)/b^(5/3)/(a^(1/3)* 
(a^(1/3)+b^(1/3)*x^2)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x^2)^2)^(1/2)/(b*x^6+a) 
^(1/2)-1/81*2^(1/2)*(4*a*d+5*b*c)*(a^(1/3)+b^(1/3)*x^2)*((a^(2/3)-a^(1/3)* 
b^(1/3)*x^2+b^(2/3)*x^4)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x^2)^2)^(1/2)*Ellipt 
icF(((1-3^(1/2))*a^(1/3)+b^(1/3)*x^2)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x^2),I* 
3^(1/2)+2*I)*3^(3/4)/a^(5/3)/b^(5/3)/(a^(1/3)*(a^(1/3)+b^(1/3)*x^2)/((1+3^ 
(1/2))*a^(1/3)+b^(1/3)*x^2)^2)^(1/2)/(b*x^6+a)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.06 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.14 \[ \int \frac {x^3 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {x^4 \left (-4 a^2 d+(5 b c+4 a d) \left (a+b x^6\right ) \sqrt {1+\frac {b x^6}{a}} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {5}{2},\frac {5}{3},-\frac {b x^6}{a}\right )\right )}{20 a^2 b \left (a+b x^6\right )^{3/2}} \] Input:

Integrate[(x^3*(c + d*x^6))/(a + b*x^6)^(5/2),x]
 

Output:

(x^4*(-4*a^2*d + (5*b*c + 4*a*d)*(a + b*x^6)*Sqrt[1 + (b*x^6)/a]*Hypergeom 
etric2F1[2/3, 5/2, 5/3, -((b*x^6)/a)]))/(20*a^2*b*(a + b*x^6)^(3/2))
 

Rubi [A] (warning: unable to verify)

Time = 0.97 (sec) , antiderivative size = 595, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {957, 807, 819, 832, 759, 2416}

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.

\(\displaystyle \int \frac {x^3 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 957

\(\displaystyle \frac {(4 a d+5 b c) \int \frac {x^3}{\left (b x^6+a\right )^{3/2}}dx}{9 a b}+\frac {x^4 (b c-a d)}{9 a b \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 807

\(\displaystyle \frac {(4 a d+5 b c) \int \frac {x^2}{\left (b x^6+a\right )^{3/2}}dx^2}{18 a b}+\frac {x^4 (b c-a d)}{9 a b \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 819

\(\displaystyle \frac {(4 a d+5 b c) \left (\frac {2 x^4}{3 a \sqrt {a+b x^6}}-\frac {\int \frac {x^2}{\sqrt {b x^6+a}}dx^2}{3 a}\right )}{18 a b}+\frac {x^4 (b c-a d)}{9 a b \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 832

\(\displaystyle \frac {(4 a d+5 b c) \left (\frac {2 x^4}{3 a \sqrt {a+b x^6}}-\frac {\frac {\int \frac {\sqrt [3]{b} x^2+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^6+a}}dx^2}{\sqrt [3]{b}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {b x^6+a}}dx^2}{\sqrt [3]{b}}}{3 a}\right )}{18 a b}+\frac {x^4 (b c-a d)}{9 a b \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 759

\(\displaystyle \frac {(4 a d+5 b c) \left (\frac {2 x^4}{3 a \sqrt {a+b x^6}}-\frac {\frac {\int \frac {\sqrt [3]{b} x^2+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^6+a}}dx^2}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x^2+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x^2+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}}{3 a}\right )}{18 a b}+\frac {x^4 (b c-a d)}{9 a b \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 2416

\(\displaystyle \frac {(4 a d+5 b c) \left (\frac {2 x^4}{3 a \sqrt {a+b x^6}}-\frac {\frac {\frac {2 \sqrt {a+b x^6}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} x^2+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x^2+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x^2+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x^2+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}}{3 a}\right )}{18 a b}+\frac {x^4 (b c-a d)}{9 a b \left (a+b x^6\right )^{3/2}}\)

Input:

Int[(x^3*(c + d*x^6))/(a + b*x^6)^(5/2),x]
 

Output:

((b*c - a*d)*x^4)/(9*a*b*(a + b*x^6)^(3/2)) + ((5*b*c + 4*a*d)*((2*x^4)/(3 
*a*Sqrt[a + b*x^6]) - (((2*Sqrt[a + b*x^6])/(b^(1/3)*((1 + Sqrt[3])*a^(1/3 
) + b^(1/3)*x^2)) - (3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(1/3)*(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)/((1 + Sqrt[3])*a^( 
1/3) + b^(1/3)*x^2)^2]*EllipticE[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x 
^2)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x^2)], -7 - 4*Sqrt[3]])/(b^(1/3)*Sqrt 
[(a^(1/3)*(a^(1/3) + b^(1/3)*x^2))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x^2)^2 
]*Sqrt[a + b*x^6]))/b^(1/3) - (2*(1 - Sqrt[3])*Sqrt[2 + Sqrt[3]]*a^(1/3)*( 
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)/ 
((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x^2)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a 
^(1/3) + b^(1/3)*x^2)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x^2)], -7 - 4*Sqrt[ 
3]])/(3^(1/4)*b^(2/3)*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*x^2))/((1 + Sqrt[3] 
)*a^(1/3) + b^(1/3)*x^2)^2]*Sqrt[a + b*x^6]))/(3*a)))/(18*a*b)
 

Defintions of rubi rules used

rule 759

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* 
((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s 
+ r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & 
& PosQ[a]
 

rule 807

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m 
+ 1, n]}, Simp[1/k   Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, 
x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
 

rule 819

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]
 

rule 832

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] 
], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 - Sqrt[3]))*(s/r)   Int[1/Sqrt[a + b*x 
^3], x], x] + Simp[1/r   Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x 
]] /; FreeQ[{a, b}, x] && PosQ[a]
 

rule 957

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)]))
 

rule 2416

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Simplify[(1 - Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 - Sqrt[3])*(d/c) 
]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x] - S 
imp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( 
(1 + Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt 
[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3]) 
*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && Eq 
Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
 

Maple [F]

\[\int \frac {x^{3} \left (d \,x^{6}+c \right )}{\left (b \,x^{6}+a \right )^{\frac {5}{2}}}d x\]

Input:

int(x^3*(d*x^6+c)/(b*x^6+a)^(5/2),x)
 

Output:

int(x^3*(d*x^6+c)/(b*x^6+a)^(5/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.26 \[ \int \frac {x^3 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {{\left ({\left (5 \, b^{3} c + 4 \, a b^{2} d\right )} x^{12} + 2 \, {\left (5 \, a b^{2} c + 4 \, a^{2} b d\right )} x^{6} + 5 \, a^{2} b c + 4 \, a^{3} d\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x^{2}\right )\right ) + {\left ({\left (5 \, b^{3} c + 4 \, a b^{2} d\right )} x^{10} + {\left (8 \, a b^{2} c + a^{2} b d\right )} x^{4}\right )} \sqrt {b x^{6} + a}}{27 \, {\left (a^{2} b^{4} x^{12} + 2 \, a^{3} b^{3} x^{6} + a^{4} b^{2}\right )}} \] Input:

integrate(x^3*(d*x^6+c)/(b*x^6+a)^(5/2),x, algorithm="fricas")
 

Output:

1/27*(((5*b^3*c + 4*a*b^2*d)*x^12 + 2*(5*a*b^2*c + 4*a^2*b*d)*x^6 + 5*a^2* 
b*c + 4*a^3*d)*sqrt(b)*weierstrassZeta(0, -4*a/b, weierstrassPInverse(0, - 
4*a/b, x^2)) + ((5*b^3*c + 4*a*b^2*d)*x^10 + (8*a*b^2*c + a^2*b*d)*x^4)*sq 
rt(b*x^6 + a))/(a^2*b^4*x^12 + 2*a^3*b^3*x^6 + a^4*b^2)
 

Sympy [A] (verification not implemented)

Time = 100.83 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.13 \[ \int \frac {x^3 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {c x^{4} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {5}{2} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{6} e^{i \pi }}{a}} \right )}}{6 a^{\frac {5}{2}} \Gamma \left (\frac {5}{3}\right )} + \frac {d x^{10} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{3}, \frac {5}{2} \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{6} e^{i \pi }}{a}} \right )}}{6 a^{\frac {5}{2}} \Gamma \left (\frac {8}{3}\right )} \] Input:

integrate(x**3*(d*x**6+c)/(b*x**6+a)**(5/2),x)
 

Output:

c*x**4*gamma(2/3)*hyper((2/3, 5/2), (5/3,), b*x**6*exp_polar(I*pi)/a)/(6*a 
**(5/2)*gamma(5/3)) + d*x**10*gamma(5/3)*hyper((5/3, 5/2), (8/3,), b*x**6* 
exp_polar(I*pi)/a)/(6*a**(5/2)*gamma(8/3))
 

Maxima [F]

\[ \int \frac {x^3 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\int { \frac {{\left (d x^{6} + c\right )} x^{3}}{{\left (b x^{6} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^3*(d*x^6+c)/(b*x^6+a)^(5/2),x, algorithm="maxima")
 

Output:

integrate((d*x^6 + c)*x^3/(b*x^6 + a)^(5/2), x)
 

Giac [F]

\[ \int \frac {x^3 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\int { \frac {{\left (d x^{6} + c\right )} x^{3}}{{\left (b x^{6} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^3*(d*x^6+c)/(b*x^6+a)^(5/2),x, algorithm="giac")
 

Output:

integrate((d*x^6 + c)*x^3/(b*x^6 + a)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\int \frac {x^3\,\left (d\,x^6+c\right )}{{\left (b\,x^6+a\right )}^{5/2}} \,d x \] Input:

int((x^3*(c + d*x^6))/(a + b*x^6)^(5/2),x)
 

Output:

int((x^3*(c + d*x^6))/(a + b*x^6)^(5/2), x)
 

Reduce [F]

\[ \int \frac {x^3 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {-\sqrt {b \,x^{6}+a}\, d \,x^{4}+4 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{3}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{3} d +5 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{3}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{2} b c +8 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{3}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{2} b d \,x^{6}+10 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{3}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a \,b^{2} c \,x^{6}+4 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{3}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a \,b^{2} d \,x^{12}+5 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{3}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) b^{3} c \,x^{12}}{5 b \left (b^{2} x^{12}+2 a b \,x^{6}+a^{2}\right )} \] Input:

int(x^3*(d*x^6+c)/(b*x^6+a)^(5/2),x)
                                                                                    
                                                                                    
 

Output:

( - sqrt(a + b*x**6)*d*x**4 + 4*int((sqrt(a + b*x**6)*x**3)/(a**3 + 3*a**2 
*b*x**6 + 3*a*b**2*x**12 + b**3*x**18),x)*a**3*d + 5*int((sqrt(a + b*x**6) 
*x**3)/(a**3 + 3*a**2*b*x**6 + 3*a*b**2*x**12 + b**3*x**18),x)*a**2*b*c + 
8*int((sqrt(a + b*x**6)*x**3)/(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 + 10*int((sqrt(a + b*x**6)*x**3)/(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 + 4*int((sqrt(a + b 
*x**6)*x**3)/(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 + 5*int((sqrt(a + b*x**6)*x**3)/(a**3 + 3*a**2*b*x**6 + 3*a*b**2 
*x**12 + b**3*x**18),x)*b**3*c*x**12)/(5*b*(a**2 + 2*a*b*x**6 + b**2*x**12 
))