Integrand size = 26, antiderivative size = 200 \[ \int \frac {1}{\left (d+e x^3\right )^{3/2} \left (a+b x^3+c x^6\right )} \, dx=-\frac {2 c x \sqrt {1+\frac {e x^3}{d}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {3}{2},1,\frac {4}{3},-\frac {e x^3}{d},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) d \sqrt {d+e x^3}}-\frac {2 c x \sqrt {1+\frac {e x^3}{d}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {3}{2},1,\frac {4}{3},-\frac {e x^3}{d},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) d \sqrt {d+e x^3}} \] Output:
-2*c*x*(1+e*x^3/d)^(1/2)*AppellF1(1/3,1,3/2,4/3,-2*c*x^3/(b-(-4*a*c+b^2)^( 1/2)),-e*x^3/d)/(b^2-4*a*c-b*(-4*a*c+b^2)^(1/2))/d/(e*x^3+d)^(1/2)-2*c*x*( 1+e*x^3/d)^(1/2)*AppellF1(1/3,1,3/2,4/3,-2*c*x^3/(b+(-4*a*c+b^2)^(1/2)),-e *x^3/d)/(b*(-4*a*c+b^2)^(1/2)-4*a*c+b^2)/d/(e*x^3+d)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(13575\) vs. \(2(200)=400\).
Time = 16.53 (sec) , antiderivative size = 13575, normalized size of antiderivative = 67.88 \[ \int \frac {1}{\left (d+e x^3\right )^{3/2} \left (a+b x^3+c x^6\right )} \, dx=\text {Result too large to show} \] Input:
Integrate[1/((d + e*x^3)^(3/2)*(a + b*x^3 + c*x^6)),x]
Output:
Result too large to show
Leaf count is larger than twice the leaf count of optimal. \(502\) vs. \(2(200)=400\).
Time = 1.20 (sec) , antiderivative size = 502, normalized size of antiderivative = 2.51, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1756, 749, 759, 7279, 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.
\(\displaystyle \int \frac {1}{\left (d+e x^3\right )^{3/2} \left (a+b x^3+c x^6\right )} \, dx\) |
\(\Big \downarrow \) 1756 |
\(\displaystyle \frac {e^2 \int \frac {1}{\left (e x^3+d\right )^{3/2}}dx}{a e^2-b d e+c d^2}+\frac {\int \frac {-c e x^3+c d-b e}{\sqrt {e x^3+d} \left (c x^6+b x^3+a\right )}dx}{a e^2-b d e+c d^2}\) |
\(\Big \downarrow \) 749 |
\(\displaystyle \frac {e^2 \left (\frac {\int \frac {1}{\sqrt {e x^3+d}}dx}{3 d}+\frac {2 x}{3 d \sqrt {d+e x^3}}\right )}{a e^2-b d e+c d^2}+\frac {\int \frac {-c e x^3+c d-b e}{\sqrt {e x^3+d} \left (c x^6+b x^3+a\right )}dx}{a e^2-b d e+c d^2}\) |
\(\Big \downarrow \) 759 |
\(\displaystyle \frac {\int \frac {-c e x^3+c d-b e}{\sqrt {e x^3+d} \left (c x^6+b x^3+a\right )}dx}{a e^2-b d e+c d^2}+\frac {e^2 \left (\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{e} x+\left (1-\sqrt {3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt {3}\right ) \sqrt [3]{d}}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} d \sqrt [3]{e} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}}+\frac {2 x}{3 d \sqrt {d+e x^3}}\right )}{a e^2-b d e+c d^2}\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \frac {\int \left (\frac {-c e-\frac {c (b e-2 c d)}{\sqrt {b^2-4 a c}}}{\left (2 c x^3+b-\sqrt {b^2-4 a c}\right ) \sqrt {e x^3+d}}+\frac {\frac {c (b e-2 c d)}{\sqrt {b^2-4 a c}}-c e}{\left (2 c x^3+b+\sqrt {b^2-4 a c}\right ) \sqrt {e x^3+d}}\right )dx}{a e^2-b d e+c d^2}+\frac {e^2 \left (\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{e} x+\left (1-\sqrt {3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt {3}\right ) \sqrt [3]{d}}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} d \sqrt [3]{e} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}}+\frac {2 x}{3 d \sqrt {d+e x^3}}\right )}{a e^2-b d e+c d^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {c x \sqrt {\frac {e x^3}{d}+1} \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \operatorname {AppellF1}\left (\frac {1}{3},1,\frac {1}{2},\frac {4}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {e x^3}{d}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {d+e x^3}}-\frac {c x \sqrt {\frac {e x^3}{d}+1} \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \operatorname {AppellF1}\left (\frac {1}{3},1,\frac {1}{2},\frac {4}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},-\frac {e x^3}{d}\right )}{\left (\sqrt {b^2-4 a c}+b\right ) \sqrt {d+e x^3}}}{a e^2-b d e+c d^2}+\frac {e^2 \left (\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{e} x+\left (1-\sqrt {3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt {3}\right ) \sqrt [3]{d}}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} d \sqrt [3]{e} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}}+\frac {2 x}{3 d \sqrt {d+e x^3}}\right )}{a e^2-b d e+c d^2}\) |
Input:
Int[1/((d + e*x^3)^(3/2)*(a + b*x^3 + c*x^6)),x]
Output:
(-((c*(e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*x*Sqrt[1 + (e*x^3)/d]*AppellF1 [1/3, 1, 1/2, 4/3, (-2*c*x^3)/(b - Sqrt[b^2 - 4*a*c]), -((e*x^3)/d)])/((b - Sqrt[b^2 - 4*a*c])*Sqrt[d + e*x^3])) - (c*(e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*x*Sqrt[1 + (e*x^3)/d]*AppellF1[1/3, 1, 1/2, 4/3, (-2*c*x^3)/(b + S qrt[b^2 - 4*a*c]), -((e*x^3)/d)])/((b + Sqrt[b^2 - 4*a*c])*Sqrt[d + e*x^3] ))/(c*d^2 - b*d*e + a*e^2) + (e^2*((2*x)/(3*d*Sqrt[d + e*x^3]) + (2*Sqrt[2 + Sqrt[3]]*(d^(1/3) + e^(1/3)*x)*Sqrt[(d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2 /3)*x^2)/((1 + Sqrt[3])*d^(1/3) + e^(1/3)*x)^2]*EllipticF[ArcSin[((1 - Sqr t[3])*d^(1/3) + e^(1/3)*x)/((1 + Sqrt[3])*d^(1/3) + e^(1/3)*x)], -7 - 4*Sq rt[3]])/(3*3^(1/4)*d*e^(1/3)*Sqrt[(d^(1/3)*(d^(1/3) + e^(1/3)*x))/((1 + Sq rt[3])*d^(1/3) + e^(1/3)*x)^2]*Sqrt[d + e*x^3])))/(c*d^2 - b*d*e + a*e^2)
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])
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]
Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_ )), x_Symbol] :> Simp[e^2/(c*d^2 - b*d*e + a*e^2) Int[(d + e*x^n)^q, x], x] + Simp[1/(c*d^2 - b*d*e + a*e^2) Int[(d + e*x^n)^(q + 1)*((c*d - b*e - c*e*x^n)/(a + b*x^n + c*x^(2*n))), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[q] && LtQ[q, -1]
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 6.
Time = 0.40 (sec) , antiderivative size = 1132, normalized size of antiderivative = 5.66
method | result | size |
default | \(\text {Expression too large to display}\) | \(1132\) |
elliptic | \(\text {Expression too large to display}\) | \(1132\) |
Input:
int(1/(e*x^3+d)^(3/2)/(c*x^6+b*x^3+a),x,method=_RETURNVERBOSE)
Output:
2/3*e^2*x/d/(a*e^2-b*d*e+c*d^2)/((x^3+d/e)*e)^(1/2)-2/9*I/d*e/(a*e^2-b*d*e +c*d^2)*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) )+1/3*I/e^2*2^(1/2)*sum((_alpha^3*c*e+b*e-c*d)/(a*e^2-b*d*e+c*d^2)^2/_alph a^2/(2*_alpha^3*c+b)*(-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*c*e+_alpha^2*b*e-_alpha^2*c*d)+I*(-d*e^2)^(1/3)*3^(1/2) *_alpha^4*c*e^2-I*(-d*e^2)^(2/3)*3^(1/2)*_alpha^3*c*e-(-d*e^2)^(1/3)*_alph a^4*c*e^2-(-d*e^2)^(2/3)*_alpha^3*c*e+I*(-d*e^2)^(1/3)*3^(1/2)*_alpha*b*e^ 2-I*(-d*e^2)^(1/3)*3^(1/2)*_alpha*c*d*e-I*(-d*e^2)^(2/3)*3^(1/2)*b*e+I*(-d *e^2)^(2/3)*3^(1/2)*c*d-(-d*e^2)^(1/3)*_alpha*b*e^2+(-d*e^2)^(1/3)*_alpha* c*d*e-(-d*e^2)^(2/3)*b*e+(-d*e^2)^(2/3)*c*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)...
Timed out. \[ \int \frac {1}{\left (d+e x^3\right )^{3/2} \left (a+b x^3+c x^6\right )} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x^3+d)^(3/2)/(c*x^6+b*x^3+a),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{\left (d+e x^3\right )^{3/2} \left (a+b x^3+c x^6\right )} \, dx=\int \frac {1}{\left (d + e x^{3}\right )^{\frac {3}{2}} \left (a + b x^{3} + c x^{6}\right )}\, dx \] Input:
integrate(1/(e*x**3+d)**(3/2)/(c*x**6+b*x**3+a),x)
Output:
Integral(1/((d + e*x**3)**(3/2)*(a + b*x**3 + c*x**6)), x)
\[ \int \frac {1}{\left (d+e x^3\right )^{3/2} \left (a+b x^3+c x^6\right )} \, dx=\int { \frac {1}{{\left (c x^{6} + b x^{3} + a\right )} {\left (e x^{3} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(e*x^3+d)^(3/2)/(c*x^6+b*x^3+a),x, algorithm="maxima")
Output:
integrate(1/((c*x^6 + b*x^3 + a)*(e*x^3 + d)^(3/2)), x)
\[ \int \frac {1}{\left (d+e x^3\right )^{3/2} \left (a+b x^3+c x^6\right )} \, dx=\int { \frac {1}{{\left (c x^{6} + b x^{3} + a\right )} {\left (e x^{3} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(e*x^3+d)^(3/2)/(c*x^6+b*x^3+a),x, algorithm="giac")
Output:
integrate(1/((c*x^6 + b*x^3 + a)*(e*x^3 + d)^(3/2)), x)
Timed out. \[ \int \frac {1}{\left (d+e x^3\right )^{3/2} \left (a+b x^3+c x^6\right )} \, dx=\text {Hanged} \] Input:
int(1/((d + e*x^3)^(3/2)*(a + b*x^3 + c*x^6)),x)
Output:
\text{Hanged}
\[ \int \frac {1}{\left (d+e x^3\right )^{3/2} \left (a+b x^3+c x^6\right )} \, dx=\int \frac {\sqrt {e \,x^{3}+d}}{c \,e^{2} x^{12}+b \,e^{2} x^{9}+2 c d e \,x^{9}+a \,e^{2} x^{6}+2 b d e \,x^{6}+c \,d^{2} x^{6}+2 a d e \,x^{3}+b \,d^{2} x^{3}+a \,d^{2}}d x \] Input:
int(1/(e*x^3+d)^(3/2)/(c*x^6+b*x^3+a),x)
Output:
int(sqrt(d + e*x**3)/(a*d**2 + 2*a*d*e*x**3 + a*e**2*x**6 + b*d**2*x**3 + 2*b*d*e*x**6 + b*e**2*x**9 + c*d**2*x**6 + 2*c*d*e*x**9 + c*e**2*x**12),x)