\(\int \frac {c+d x^6}{x^5 (a+b x^6)^{3/2}} \, dx\) [13]

Optimal result

Integrand size = 22, antiderivative size = 288 \[ \int \frac {c+d x^6}{x^5 \left (a+b x^6\right )^{3/2}} \, dx=-\frac {c}{4 a x^4 \sqrt {a+b x^6}}-\frac {(7 b c-4 a d) x^2}{12 a^2 \sqrt {a+b x^6}}-\frac {\sqrt {2+\sqrt {3}} (7 b c-4 a d) \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2}\right ),-7-4 \sqrt {3}\right )}{12 \sqrt [4]{3} a^2 \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}} \] Output:

-1/4*c/a/x^4/(b*x^6+a)^(1/2)-1/12*(-4*a*d+7*b*c)*x^2/a^2/(b*x^6+a)^(1/2)-1 
/36*(1/2*6^(1/2)+1/2*2^(1/2))*(-4*a*d+7*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)*EllipticF(((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^2/b^(1/3)/(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 = 10.04 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.30 \[ \int \frac {c+d x^6}{x^5 \left (a+b x^6\right )^{3/2}} \, dx=\frac {-6 a c-14 b c x^6+8 a d x^6+(-7 b c+4 a d) x^6 \sqrt {1+\frac {b x^6}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-\frac {b x^6}{a}\right )}{24 a^2 x^4 \sqrt {a+b x^6}} \] Input:

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

Output:

(-6*a*c - 14*b*c*x^6 + 8*a*d*x^6 + (-7*b*c + 4*a*d)*x^6*Sqrt[1 + (b*x^6)/a 
]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^6)/a)])/(24*a^2*x^4*Sqrt[a + b*x 
^6])
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {955, 807, 749, 759}

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.

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

\(\Big \downarrow \) 955

\(\displaystyle -\frac {(7 b c-4 a d) \int \frac {x}{\left (b x^6+a\right )^{3/2}}dx}{4 a}-\frac {c}{4 a x^4 \sqrt {a+b x^6}}\)

\(\Big \downarrow \) 807

\(\displaystyle -\frac {(7 b c-4 a d) \int \frac {1}{\left (b x^6+a\right )^{3/2}}dx^2}{8 a}-\frac {c}{4 a x^4 \sqrt {a+b x^6}}\)

\(\Big \downarrow \) 749

\(\displaystyle -\frac {(7 b c-4 a d) \left (\frac {\int \frac {1}{\sqrt {b x^6+a}}dx^2}{3 a}+\frac {2 x^2}{3 a \sqrt {a+b x^6}}\right )}{8 a}-\frac {c}{4 a x^4 \sqrt {a+b x^6}}\)

\(\Big \downarrow \) 759

\(\displaystyle -\frac {(7 b c-4 a d) \left (\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x^2+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x^2+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} a \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}+\frac {2 x^2}{3 a \sqrt {a+b x^6}}\right )}{8 a}-\frac {c}{4 a x^4 \sqrt {a+b x^6}}\)

Input:

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

Output:

-1/4*c/(a*x^4*Sqrt[a + b*x^6]) - ((7*b*c - 4*a*d)*((2*x^2)/(3*a*Sqrt[a + b 
*x^6]) + (2*Sqrt[2 + Sqrt[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]*El 
lipticF[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*3^(1/4)*a*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])))/(8*a)
 

Defintions of rubi rules used

rule 749

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 
 1)/(a*n*(p + 1))), x] + Simp[(n*(p + 1) + 1)/(a*n*(p + 1))   Int[(a + b*x^ 
n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (Inte 
gerQ[2*p] || Denominator[p + 1/n] < Denominator[p])
 

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 955

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_)), x_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), 
 x] + Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1))   Int[(e 
*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b* 
c - a*d, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || 
(LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.36 \[ \int \frac {c+d x^6}{x^5 \left (a+b x^6\right )^{3/2}} \, dx=-\frac {{\left ({\left (7 \, b^{2} c - 4 \, a b d\right )} x^{10} + {\left (7 \, a b c - 4 \, a^{2} d\right )} x^{4}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x^{2}\right ) + {\left ({\left (7 \, b^{2} c - 4 \, a b d\right )} x^{6} + 3 \, a b c\right )} \sqrt {b x^{6} + a}}{12 \, {\left (a^{2} b^{2} x^{10} + a^{3} b x^{4}\right )}} \] Input:

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

Output:

-1/12*(((7*b^2*c - 4*a*b*d)*x^10 + (7*a*b*c - 4*a^2*d)*x^4)*sqrt(b)*weiers 
trassPInverse(0, -4*a/b, x^2) + ((7*b^2*c - 4*a*b*d)*x^6 + 3*a*b*c)*sqrt(b 
*x^6 + a))/(a^2*b^2*x^10 + a^3*b*x^4)
 

Sympy [A] (verification not implemented)

Time = 28.38 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.29 \[ \int \frac {c+d x^6}{x^5 \left (a+b x^6\right )^{3/2}} \, dx=\frac {c \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {3}{2} \\ \frac {1}{3} \end {matrix}\middle | {\frac {b x^{6} e^{i \pi }}{a}} \right )}}{6 a^{\frac {3}{2}} x^{4} \Gamma \left (\frac {1}{3}\right )} + \frac {d x^{2} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {3}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{6} e^{i \pi }}{a}} \right )}}{6 a^{\frac {3}{2}} \Gamma \left (\frac {4}{3}\right )} \] Input:

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

Output:

c*gamma(-2/3)*hyper((-2/3, 3/2), (1/3,), b*x**6*exp_polar(I*pi)/a)/(6*a**( 
3/2)*x**4*gamma(1/3)) + d*x**2*gamma(1/3)*hyper((1/3, 3/2), (4/3,), b*x**6 
*exp_polar(I*pi)/a)/(6*a**(3/2)*gamma(4/3))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {c+d x^6}{x^5 \left (a+b x^6\right )^{3/2}} \, dx=\frac {-\sqrt {b \,x^{6}+a}\, d -4 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{2} x^{17}+2 a b \,x^{11}+a^{2} x^{5}}d x \right ) a^{2} d \,x^{4}+7 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{2} x^{17}+2 a b \,x^{11}+a^{2} x^{5}}d x \right ) a b c \,x^{4}-4 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{2} x^{17}+2 a b \,x^{11}+a^{2} x^{5}}d x \right ) a b d \,x^{10}+7 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{2} x^{17}+2 a b \,x^{11}+a^{2} x^{5}}d x \right ) b^{2} c \,x^{10}}{7 b \,x^{4} \left (b \,x^{6}+a \right )} \] Input:

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

Output:

( - sqrt(a + b*x**6)*d - 4*int(sqrt(a + b*x**6)/(a**2*x**5 + 2*a*b*x**11 + 
 b**2*x**17),x)*a**2*d*x**4 + 7*int(sqrt(a + b*x**6)/(a**2*x**5 + 2*a*b*x* 
*11 + b**2*x**17),x)*a*b*c*x**4 - 4*int(sqrt(a + b*x**6)/(a**2*x**5 + 2*a* 
b*x**11 + b**2*x**17),x)*a*b*d*x**10 + 7*int(sqrt(a + b*x**6)/(a**2*x**5 + 
 2*a*b*x**11 + b**2*x**17),x)*b**2*c*x**10)/(7*b*x**4*(a + b*x**6))