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

Optimal result

Integrand size = 21, antiderivative size = 142 \[ \int \frac {\left (d+e x^3\right )^{3/2}}{a+c x^6} \, dx=\frac {d x \sqrt {d+e x^3} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {3}{2},1,\frac {4}{3},-\frac {e x^3}{d},-\frac {\sqrt {c} x^3}{\sqrt {-a}}\right )}{2 a \sqrt {1+\frac {e x^3}{d}}}+\frac {d x \sqrt {d+e x^3} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {3}{2},1,\frac {4}{3},-\frac {e x^3}{d},\frac {\sqrt {c} x^3}{\sqrt {-a}}\right )}{2 a \sqrt {1+\frac {e x^3}{d}}} \] Output:

1/2*d*x*(e*x^3+d)^(1/2)*AppellF1(1/3,1,-3/2,4/3,-c^(1/2)*x^3/(-a)^(1/2),-e 
*x^3/d)/a/(1+e*x^3/d)^(1/2)+1/2*d*x*(e*x^3+d)^(1/2)*AppellF1(1/3,1,-3/2,4/ 
3,c^(1/2)*x^3/(-a)^(1/2),-e*x^3/d)/a/(1+e*x^3/d)^(1/2)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {1759, 27, 937, 936}

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.

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

\(\Big \downarrow \) 1759

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 937

\(\displaystyle -\frac {d \sqrt {d+e x^3} \int \frac {\left (\frac {e x^3}{d}+1\right )^{3/2}}{\sqrt {-a}-\sqrt {c} x^3}dx}{2 \sqrt {-a} \sqrt {\frac {e x^3}{d}+1}}-\frac {d \sqrt {d+e x^3} \int \frac {\left (\frac {e x^3}{d}+1\right )^{3/2}}{\sqrt {c} x^3+\sqrt {-a}}dx}{2 \sqrt {-a} \sqrt {\frac {e x^3}{d}+1}}\)

\(\Big \downarrow \) 936

\(\displaystyle \frac {d x \sqrt {d+e x^3} \operatorname {AppellF1}\left (\frac {1}{3},1,-\frac {3}{2},\frac {4}{3},-\frac {\sqrt {c} x^3}{\sqrt {-a}},-\frac {e x^3}{d}\right )}{2 a \sqrt {\frac {e x^3}{d}+1}}+\frac {d x \sqrt {d+e x^3} \operatorname {AppellF1}\left (\frac {1}{3},1,-\frac {3}{2},\frac {4}{3},\frac {\sqrt {c} x^3}{\sqrt {-a}},-\frac {e x^3}{d}\right )}{2 a \sqrt {\frac {e x^3}{d}+1}}\)

Input:

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

Output:

(d*x*Sqrt[d + e*x^3]*AppellF1[1/3, 1, -3/2, 4/3, -((Sqrt[c]*x^3)/Sqrt[-a]) 
, -((e*x^3)/d)])/(2*a*Sqrt[1 + (e*x^3)/d]) + (d*x*Sqrt[d + e*x^3]*AppellF1 
[1/3, 1, -3/2, 4/3, (Sqrt[c]*x^3)/Sqrt[-a], -((e*x^3)/d)])/(2*a*Sqrt[1 + ( 
e*x^3)/d])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 936

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[a^p*c^q*x*AppellF1[1/n, -p, -q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c) 
], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] 
 && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 937

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), 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*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, p, q 
}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])
 

rule 1759

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> W 
ith[{r = Rt[(-a)*c, 2]}, Simp[-c/(2*r)   Int[(d + e*x^n)^q/(r - c*x^n), x], 
 x] - Simp[c/(2*r)   Int[(d + e*x^n)^q/(r + c*x^n), x], x]] /; FreeQ[{a, c, 
 d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[q]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 6.

Time = 0.72 (sec) , antiderivative size = 902, normalized size of antiderivative = 6.35

method	result	size
default	\(\text {Expression too large to display}\)	\(902\)
elliptic	\(\text {Expression too large to display}\)	\(902\)

Input:

int((e*x^3+d)^(3/2)/(c*x^6+a),x,method=_RETURNVERBOSE)
 

Output:

-2/3*I*e/c*3^(1/2)*(-d*e^2)^(1/3)*(I*(x+1/2/e*(-d*e^2)^(1/3)-1/2*I*3^(1/2) 
/e*(-d*e^2)^(1/3))*3^(1/2)*e/(-d*e^2)^(1/3))^(1/2)*((x-1/e*(-d*e^2)^(1/3)) 
/(-3/2/e*(-d*e^2)^(1/3)+1/2*I*3^(1/2)/e*(-d*e^2)^(1/3)))^(1/2)*(-I*(x+1/2/ 
e*(-d*e^2)^(1/3)+1/2*I*3^(1/2)/e*(-d*e^2)^(1/3))*3^(1/2)*e/(-d*e^2)^(1/3)) 
^(1/2)/(e*x^3+d)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/e*(-d*e^2)^(1/3)-1/ 
2*I*3^(1/2)/e*(-d*e^2)^(1/3))*3^(1/2)*e/(-d*e^2)^(1/3))^(1/2),(I*3^(1/2)/e 
*(-d*e^2)^(1/3)/(-3/2/e*(-d*e^2)^(1/3)+1/2*I*3^(1/2)/e*(-d*e^2)^(1/3)))^(1 
/2))-1/6*I/c/e^2*2^(1/2)*sum((-2*_alpha^3*c*d*e+a*e^2-c*d^2)/_alpha^5/(a*e 
^2+c*d^2)*(-d*e^2)^(1/3)*(1/2*I*e*(2*x+1/e*(-I*3^(1/2)*(-d*e^2)^(1/3)+(-d* 
e^2)^(1/3)))/(-d*e^2)^(1/3))^(1/2)*(e*(x-1/e*(-d*e^2)^(1/3))/(-3*(-d*e^2)^ 
(1/3)+I*3^(1/2)*(-d*e^2)^(1/3)))^(1/2)*(-1/2*I*e*(2*x+1/e*(I*3^(1/2)*(-d*e 
^2)^(1/3)+(-d*e^2)^(1/3)))/(-d*e^2)^(1/3))^(1/2)/(e*x^3+d)^(1/2)*(2*e^2*(- 
_alpha^5*e+_alpha^2*d)-I*(-d*e^2)^(1/3)*_alpha^4*3^(1/2)*e^2+I*_alpha^3*3^ 
(1/2)*(-d*e^2)^(2/3)*e+(-d*e^2)^(1/3)*_alpha^4*e^2+I*(-d*e^2)^(1/3)*d*_alp 
ha*3^(1/2)*e+_alpha^3*(-d*e^2)^(2/3)*e-I*3^(1/2)*(-d*e^2)^(2/3)*d-(-d*e^2) 
^(1/3)*d*_alpha*e-(-d*e^2)^(2/3)*d)*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/e*(-d 
*e^2)^(1/3)-1/2*I*3^(1/2)/e*(-d*e^2)^(1/3))*3^(1/2)*e/(-d*e^2)^(1/3))^(1/2 
),-1/2*c/e*(-2*I*3^(1/2)*(-d*e^2)^(1/3)*_alpha^5*e^2+I*3^(1/2)*(-d*e^2)^(2 
/3)*_alpha^4*e-I*3^(1/2)*_alpha^3*d*e^2+2*I*3^(1/2)*(-d*e^2)^(1/3)*_alpha^ 
2*d*e+3*(-d*e^2)^(2/3)*_alpha^4*e-I*3^(1/2)*(-d*e^2)^(2/3)*_alpha*d+3*_...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

Integral((d + e*x**3)**(3/2)/(a + c*x**6), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(d + e*x**3)/(a + c*x**6),x)*d + int((sqrt(d + e*x**3)*x**3)/(a + 
c*x**6),x)*e