Integrand size = 21, antiderivative size = 773 \[ \int \frac {1}{\sqrt {a+b x^3} \left (c+d x^3\right )} \, dx=\frac {2 (-1)^{2/3} \left (-\sqrt [3]{a} b^{2/3} c^{2/3}-a^{2/3} \sqrt [3]{b} \sqrt [3]{c} \sqrt [3]{d}-a d^{2/3}\right ) \sqrt {\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}+\frac {b^{2/3} x^2}{a^{2/3}}} \operatorname {EllipticPi}\left (\frac {\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a} \sqrt [3]{d}}{-\sqrt [3]{b} \sqrt [3]{c}+\sqrt [3]{a} \sqrt [3]{d}},\arcsin \left (\sqrt {\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right ),\sqrt [3]{-1}\right )}{\sqrt {3} \left (1+\sqrt [3]{-1}\right ) c^{2/3} (b c-a d) \sqrt {a+b x^3}}+\frac {2 (-1)^{2/3} \sqrt [3]{a} \left (-\sqrt [3]{b} \sqrt [3]{c}+\sqrt [3]{a} \sqrt [3]{d}\right ) \left (\sqrt [3]{b} \sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{d}\right ) \sqrt {\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}+\frac {b^{2/3} x^2}{a^{2/3}}} \operatorname {EllipticPi}\left (\frac {\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a} \sqrt [3]{d}}{\sqrt [3]{-1} \sqrt [3]{b} \sqrt [3]{c}+\sqrt [3]{a} \sqrt [3]{d}},\arcsin \left (\sqrt {\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right ),\sqrt [3]{-1}\right )}{\sqrt {3} \left (1+\sqrt [3]{-1}\right ) c^{2/3} (b c-a d) \sqrt {a+b x^3}}+\frac {2 (-1)^{2/3} \sqrt [3]{a} \left (-\sqrt [3]{b} \sqrt [3]{c}+\sqrt [3]{a} \sqrt [3]{d}\right ) \left (\sqrt [3]{b} \sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{a} \sqrt [3]{d}\right ) \sqrt {\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}+\frac {b^{2/3} x^2}{a^{2/3}}} \operatorname {EllipticPi}\left (\frac {i \sqrt {3} \sqrt [3]{a} \sqrt [3]{d}}{\sqrt [3]{b} \sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{d}},\arcsin \left (\sqrt {\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right ),\sqrt [3]{-1}\right )}{\sqrt {3} \left (1+\sqrt [3]{-1}\right ) c^{2/3} (b c-a d) \sqrt {a+b x^3}} \] Output:
2/3*(-1)^(2/3)*(-a^(1/3)*b^(2/3)*c^(2/3)-a^(2/3)*b^(1/3)*c^(1/3)*d^(1/3)-a *d^(2/3))*((a^(1/3)+b^(1/3)*x)/(1+(-1)^(1/3))/a^(1/3))^(1/2)*(1-b^(1/3)*x/ a^(1/3)+b^(2/3)*x^2/a^(2/3))^(1/2)*EllipticPi(((a^(1/3)+(-1)^(2/3)*b^(1/3) *x)/(1+(-1)^(1/3))/a^(1/3))^(1/2),(1+(-1)^(1/3))*a^(1/3)*d^(1/3)/(-b^(1/3) *c^(1/3)+a^(1/3)*d^(1/3)),(-1)^(1/6))*3^(1/2)/(1+(-1)^(1/3))/c^(2/3)/(-a*d +b*c)/(b*x^3+a)^(1/2)+2/3*(-1)^(2/3)*a^(1/3)*(-b^(1/3)*c^(1/3)+a^(1/3)*d^( 1/3))*(b^(1/3)*c^(1/3)+(-1)^(1/3)*a^(1/3)*d^(1/3))*((a^(1/3)+b^(1/3)*x)/(1 +(-1)^(1/3))/a^(1/3))^(1/2)*(1-b^(1/3)*x/a^(1/3)+b^(2/3)*x^2/a^(2/3))^(1/2 )*EllipticPi(((a^(1/3)+(-1)^(2/3)*b^(1/3)*x)/(1+(-1)^(1/3))/a^(1/3))^(1/2) ,(1+(-1)^(1/3))*a^(1/3)*d^(1/3)/((-1)^(1/3)*b^(1/3)*c^(1/3)+a^(1/3)*d^(1/3 )),(-1)^(1/6))*3^(1/2)/(1+(-1)^(1/3))/c^(2/3)/(-a*d+b*c)/(b*x^3+a)^(1/2)+2 /3*(-1)^(2/3)*a^(1/3)*(-b^(1/3)*c^(1/3)+a^(1/3)*d^(1/3))*(b^(1/3)*c^(1/3)- (-1)^(2/3)*a^(1/3)*d^(1/3))*((a^(1/3)+b^(1/3)*x)/(1+(-1)^(1/3))/a^(1/3))^( 1/2)*(1-b^(1/3)*x/a^(1/3)+b^(2/3)*x^2/a^(2/3))^(1/2)*EllipticPi(((a^(1/3)+ (-1)^(2/3)*b^(1/3)*x)/(1+(-1)^(1/3))/a^(1/3))^(1/2),I*3^(1/2)*a^(1/3)*d^(1 /3)/(b^(1/3)*c^(1/3)+(-1)^(1/3)*a^(1/3)*d^(1/3)),(-1)^(1/6))*3^(1/2)/(1+(- 1)^(1/3))/c^(2/3)/(-a*d+b*c)/(b*x^3+a)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.06 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.21 \[ \int \frac {1}{\sqrt {a+b x^3} \left (c+d x^3\right )} \, dx=-\frac {8 a c x \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{\sqrt {a+b x^3} \left (c+d x^3\right ) \left (-8 a c \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+3 x^3 \left (2 a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},2,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )} \] Input:
Integrate[1/(Sqrt[a + b*x^3]*(c + d*x^3)),x]
Output:
(-8*a*c*x*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/(Sqrt[a + b*x^3]*(c + d*x^3)*(-8*a*c*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), -((d *x^3)/c)] + 3*x^3*(2*a*d*AppellF1[4/3, 1/2, 2, 7/3, -((b*x^3)/a), -((d*x^3 )/c)] + b*c*AppellF1[4/3, 3/2, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)])))
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 0.31 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.08, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, 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.
| \(\displaystyle \int \frac {1}{\sqrt {a+b x^3} \left (c+d x^3\right )} \, dx\) |
| \(\Big \downarrow \) 937 |
| \(\displaystyle \frac {\sqrt {\frac {b x^3}{a}+1} \int \frac {1}{\sqrt {\frac {b x^3}{a}+1} \left (d x^3+c\right )}dx}{\sqrt {a+b x^3}}\) |
| \(\Big \downarrow \) 936 |
| \(\displaystyle \frac {x \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c \sqrt {a+b x^3}}\) |
Input:
Int[1/(Sqrt[a + b*x^3]*(c + d*x^3)),x]
Output:
(x*Sqrt[1 + (b*x^3)/a]*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), -((d*x^3)/ c)])/(c*Sqrt[a + b*x^3])
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 4.
Time = 1.04 (sec) , antiderivative size = 429, normalized size of antiderivative = 0.55
| method | result | size |
| default | \(\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}+c \right )}{\sum }\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i b \left (2 x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}-i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {b \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{-3 \left (-a \,b^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i b \left (2 x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{2 \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (i \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, b -i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} b^{2}-\left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b -\left (-a \,b^{2}\right )^{\frac {2}{3}}\right ) \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {d \left (2 i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b -i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, a b -3 \left (-a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 a b \right )}{2 b \left (a d -b c \right )}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha ^{2} \left (a d -b c \right ) \sqrt {b \,x^{3}+a}}\right )}{3 b^{2}}\) | \(429\) |
| elliptic | \(\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}+c \right )}{\sum }\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i b \left (2 x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}-i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {b \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{-3 \left (-a \,b^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i b \left (2 x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{2 \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (i \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, b -i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} b^{2}-\left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b -\left (-a \,b^{2}\right )^{\frac {2}{3}}\right ) \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {d \left (2 i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b -i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, a b -3 \left (-a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 a b \right )}{2 b \left (a d -b c \right )}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha ^{2} \left (a d -b c \right ) \sqrt {b \,x^{3}+a}}\right )}{3 b^{2}}\) | \(429\) |
Input:
int(1/(b*x^3+a)^(1/2)/(d*x^3+c),x,method=_RETURNVERBOSE)
Output:
1/3*I/b^2*2^(1/2)*sum(1/_alpha^2/(a*d-b*c)*(-a*b^2)^(1/3)*(1/2*I*b*(2*x+1/ b*((-a*b^2)^(1/3)-I*3^(1/2)*(-a*b^2)^(1/3)))/(-a*b^2)^(1/3))^(1/2)*(b*(x-1 /b*(-a*b^2)^(1/3))/(-3*(-a*b^2)^(1/3)+I*3^(1/2)*(-a*b^2)^(1/3)))^(1/2)*(-1 /2*I*b*(2*x+1/b*((-a*b^2)^(1/3)+I*3^(1/2)*(-a*b^2)^(1/3)))/(-a*b^2)^(1/3)) ^(1/2)/(b*x^3+a)^(1/2)*(I*(-a*b^2)^(1/3)*_alpha*3^(1/2)*b-I*3^(1/2)*(-a*b^ 2)^(2/3)+2*_alpha^2*b^2-(-a*b^2)^(1/3)*_alpha*b-(-a*b^2)^(2/3))*EllipticPi (1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^ (1/2)*b/(-a*b^2)^(1/3))^(1/2),-1/2/b*d*(2*I*(-a*b^2)^(1/3)*3^(1/2)*_alpha^ 2*b-I*(-a*b^2)^(2/3)*3^(1/2)*_alpha+I*3^(1/2)*a*b-3*(-a*b^2)^(2/3)*_alpha- 3*a*b)/(a*d-b*c),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I* 3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*d+c))
Timed out. \[ \int \frac {1}{\sqrt {a+b x^3} \left (c+d x^3\right )} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x^3+a)^(1/2)/(d*x^3+c),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{\sqrt {a+b x^3} \left (c+d x^3\right )} \, dx=\int \frac {1}{\sqrt {a + b x^{3}} \left (c + d x^{3}\right )}\, dx \] Input:
integrate(1/(b*x**3+a)**(1/2)/(d*x**3+c),x)
Output:
Integral(1/(sqrt(a + b*x**3)*(c + d*x**3)), x)
\[ \int \frac {1}{\sqrt {a+b x^3} \left (c+d x^3\right )} \, dx=\int { \frac {1}{\sqrt {b x^{3} + a} {\left (d x^{3} + c\right )}} \,d x } \] Input:
integrate(1/(b*x^3+a)^(1/2)/(d*x^3+c),x, algorithm="maxima")
Output:
integrate(1/(sqrt(b*x^3 + a)*(d*x^3 + c)), x)
\[ \int \frac {1}{\sqrt {a+b x^3} \left (c+d x^3\right )} \, dx=\int { \frac {1}{\sqrt {b x^{3} + a} {\left (d x^{3} + c\right )}} \,d x } \] Input:
integrate(1/(b*x^3+a)^(1/2)/(d*x^3+c),x, algorithm="giac")
Output:
integrate(1/(sqrt(b*x^3 + a)*(d*x^3 + c)), x)
Timed out. \[ \int \frac {1}{\sqrt {a+b x^3} \left (c+d x^3\right )} \, dx=\int \frac {1}{\sqrt {b\,x^3+a}\,\left (d\,x^3+c\right )} \,d x \] Input:
int(1/((a + b*x^3)^(1/2)*(c + d*x^3)),x)
Output:
int(1/((a + b*x^3)^(1/2)*(c + d*x^3)), x)
\[ \int \frac {1}{\sqrt {a+b x^3} \left (c+d x^3\right )} \, dx=\int \frac {\sqrt {b \,x^{3}+a}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \] Input:
int(1/(b*x^3+a)^(1/2)/(d*x^3+c),x)
Output:
int(sqrt(a + b*x**3)/(a*c + a*d*x**3 + b*c*x**3 + b*d*x**6),x)