Integrand size = 23, antiderivative size = 64 \[ \int \frac {1}{\sqrt {c+d x^3} \left (4 c+d x^3\right )} \, dx=\frac {x \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {1}{3},1,\frac {1}{2},\frac {4}{3},-\frac {d x^3}{4 c},-\frac {d x^3}{c}\right )}{4 c \sqrt {c+d x^3}} \]
Leaf count is larger than twice the leaf count of optimal. \(165\) vs. \(2(64)=128\).
Time = 10.05 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.58 \[ \int \frac {1}{\sqrt {c+d x^3} \left (4 c+d x^3\right )} \, dx=\frac {16 c x \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {d x^3}{c},-\frac {d x^3}{4 c}\right )}{\sqrt {c+d x^3} \left (4 c+d x^3\right ) \left (16 c \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {d x^3}{c},-\frac {d x^3}{4 c}\right )-3 d x^3 \left (\operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},2,\frac {7}{3},-\frac {d x^3}{c},-\frac {d x^3}{4 c}\right )+2 \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},1,\frac {7}{3},-\frac {d x^3}{c},-\frac {d x^3}{4 c}\right )\right )\right )} \]
(16*c*x*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -1/4*(d*x^3)/c])/(Sqrt[c + d*x^3]*(4*c + d*x^3)*(16*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -1/4 *(d*x^3)/c] - 3*d*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -1/4*(d*x^ 3)/c] + 2*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -1/4*(d*x^3)/c])))
Time = 0.19 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {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.
\(\displaystyle \int \frac {1}{\sqrt {c+d x^3} \left (4 c+d x^3\right )} \, dx\) |
\(\Big \downarrow \) 937 |
\(\displaystyle \frac {\sqrt {\frac {d x^3}{c}+1} \int \frac {1}{\left (d x^3+4 c\right ) \sqrt {\frac {d x^3}{c}+1}}dx}{\sqrt {c+d x^3}}\) |
\(\Big \downarrow \) 936 |
\(\displaystyle \frac {x \sqrt {\frac {d x^3}{c}+1} \operatorname {AppellF1}\left (\frac {1}{3},1,\frac {1}{2},\frac {4}{3},-\frac {d x^3}{4 c},-\frac {d x^3}{c}\right )}{4 c \sqrt {c+d x^3}}\) |
(x*Sqrt[1 + (d*x^3)/c]*AppellF1[1/3, 1, 1/2, 4/3, -1/4*(d*x^3)/c, -((d*x^3 )/c)])/(4*c*Sqrt[c + d*x^3])
3.3.79.3.1 Defintions of rubi rules used
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])
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])
Result contains higher order function than in optimal. Order 9 vs. order 6.
Time = 4.33 (sec) , antiderivative size = 416, normalized size of antiderivative = 6.50
method | result | size |
default | \(-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}+4 c \right )}{\sum }\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i d \left (2 x +\frac {-i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {d \left (x -\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{-3 \left (-c \,d^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i d \left (2 x +\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{2 \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \left (i \left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, d -i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} d^{2}-\left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha d -\left (-c \,d^{2}\right )^{\frac {2}{3}}\right ) \Pi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}}{3}, \frac {2 i \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d -i \left (-c \,d^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, c d -3 \left (-c \,d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 c d}{6 d c}, \sqrt {\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{d \left (-\frac {3 \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {d \,x^{3}+c}}\right )}{9 d^{3} c}\) | \(416\) |
elliptic | \(-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}+4 c \right )}{\sum }\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i d \left (2 x +\frac {-i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {d \left (x -\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{-3 \left (-c \,d^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i d \left (2 x +\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{2 \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \left (i \left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, d -i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} d^{2}-\left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha d -\left (-c \,d^{2}\right )^{\frac {2}{3}}\right ) \Pi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}}{3}, \frac {2 i \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d -i \left (-c \,d^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, c d -3 \left (-c \,d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 c d}{6 d c}, \sqrt {\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{d \left (-\frac {3 \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {d \,x^{3}+c}}\right )}{9 d^{3} c}\) | \(416\) |
-1/9*I/d^3/c*2^(1/2)*sum(1/_alpha^2*(-c*d^2)^(1/3)*(1/2*I*d*(2*x+1/d*(-I*3 ^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)*(d*(x-1/d*(-c *d^2)^(1/3))/(-3*(-c*d^2)^(1/3)+I*3^(1/2)*(-c*d^2)^(1/3)))^(1/2)*(-1/2*I*d *(2*x+1/d*(I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2) /(d*x^3+c)^(1/2)*(I*(-c*d^2)^(1/3)*_alpha*3^(1/2)*d-I*3^(1/2)*(-c*d^2)^(2/ 3)+2*_alpha^2*d^2-(-c*d^2)^(1/3)*_alpha*d-(-c*d^2)^(2/3))*EllipticPi(1/3*3 ^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)* d/(-c*d^2)^(1/3))^(1/2),1/6/d*(2*I*(-c*d^2)^(1/3)*3^(1/2)*_alpha^2*d-I*(-c *d^2)^(2/3)*3^(1/2)*_alpha+I*3^(1/2)*c*d-3*(-c*d^2)^(2/3)*_alpha-3*c*d)/c, (I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2 )^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*d+4*c))
Leaf count of result is larger than twice the leaf count of optimal. 2347 vs. \(2 (50) = 100\).
Time = 0.72 (sec) , antiderivative size = 2347, normalized size of antiderivative = 36.67 \[ \int \frac {1}{\sqrt {c+d x^3} \left (4 c+d x^3\right )} \, dx=\text {Too large to display} \]
-1/72*(2*(1/108)^(1/6)*c*d*(-1/(c^7*d^2))^(1/6)*log((d^3*x^9 - 66*c*d^2*x^ 6 - 72*c^2*d*x^3 - 32*c^3 - 24*(1/4)^(2/3)*(c^5*d^4*x^8 - 7*c^6*d^3*x^5 - 8*c^7*d^2*x^2)*(-1/(c^7*d^2))^(2/3) - 48*(1/4)^(1/3)*(c^3*d^3*x^7 - c^4*d^ 2*x^4 - 2*c^5*d*x)*(-1/(c^7*d^2))^(1/3) + 6*(18*(1/108)^(1/6)*c^2*d^2*x^5* (-1/(c^7*d^2))^(1/6) + 36*(1/108)^(5/6)*(c^6*d^4*x^7 - 16*c^7*d^3*x^4 - 8* c^8*d^2*x)*(-1/(c^7*d^2))^(5/6) + sqrt(1/3)*(5*c^4*d^3*x^6 - 20*c^5*d^2*x^ 3 - 16*c^6*d)*sqrt(-1/(c^7*d^2)))*sqrt(d*x^3 + c))/(d^3*x^9 + 12*c*d^2*x^6 + 48*c^2*d*x^3 + 64*c^3)) - 2*(1/108)^(1/6)*c*d*(-1/(c^7*d^2))^(1/6)*log( (d^3*x^9 - 66*c*d^2*x^6 - 72*c^2*d*x^3 - 32*c^3 - 24*(1/4)^(2/3)*(c^5*d^4* x^8 - 7*c^6*d^3*x^5 - 8*c^7*d^2*x^2)*(-1/(c^7*d^2))^(2/3) - 48*(1/4)^(1/3) *(c^3*d^3*x^7 - c^4*d^2*x^4 - 2*c^5*d*x)*(-1/(c^7*d^2))^(1/3) - 6*(18*(1/1 08)^(1/6)*c^2*d^2*x^5*(-1/(c^7*d^2))^(1/6) + 36*(1/108)^(5/6)*(c^6*d^4*x^7 - 16*c^7*d^3*x^4 - 8*c^8*d^2*x)*(-1/(c^7*d^2))^(5/6) + sqrt(1/3)*(5*c^4*d ^3*x^6 - 20*c^5*d^2*x^3 - 16*c^6*d)*sqrt(-1/(c^7*d^2)))*sqrt(d*x^3 + c))/( d^3*x^9 + 12*c*d^2*x^6 + 48*c^2*d*x^3 + 64*c^3)) + (1/108)^(1/6)*(sqrt(-3) *c*d + c*d)*(-1/(c^7*d^2))^(1/6)*log((d^3*x^9 - 66*c*d^2*x^6 - 72*c^2*d*x^ 3 - 32*c^3 + 12*(1/4)^(2/3)*(c^5*d^4*x^8 - 7*c^6*d^3*x^5 - 8*c^7*d^2*x^2 + sqrt(-3)*(c^5*d^4*x^8 - 7*c^6*d^3*x^5 - 8*c^7*d^2*x^2))*(-1/(c^7*d^2))^(2 /3) + 24*(1/4)^(1/3)*(c^3*d^3*x^7 - c^4*d^2*x^4 - 2*c^5*d*x - sqrt(-3)*(c^ 3*d^3*x^7 - c^4*d^2*x^4 - 2*c^5*d*x))*(-1/(c^7*d^2))^(1/3) + 6*sqrt(d*x...
\[ \int \frac {1}{\sqrt {c+d x^3} \left (4 c+d x^3\right )} \, dx=\int \frac {1}{\sqrt {c + d x^{3}} \cdot \left (4 c + d x^{3}\right )}\, dx \]
\[ \int \frac {1}{\sqrt {c+d x^3} \left (4 c+d x^3\right )} \, dx=\int { \frac {1}{{\left (d x^{3} + 4 \, c\right )} \sqrt {d x^{3} + c}} \,d x } \]
\[ \int \frac {1}{\sqrt {c+d x^3} \left (4 c+d x^3\right )} \, dx=\int { \frac {1}{{\left (d x^{3} + 4 \, c\right )} \sqrt {d x^{3} + c}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {c+d x^3} \left (4 c+d x^3\right )} \, dx=\int \frac {1}{\sqrt {d\,x^3+c}\,\left (d\,x^3+4\,c\right )} \,d x \]