\(\int \frac {1}{\sqrt [3]{-a+b x^2} (-\frac {9 a d}{b}+d x^2)} \, dx\) [362]

Optimal result

Integrand size = 29, antiderivative size = 151 \[ \int \frac {1}{\sqrt [3]{-a+b x^2} \left (-\frac {9 a d}{b}+d x^2\right )} \, dx=\frac {\sqrt {b} \arctan \left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{-a+b x^2}\right )}{\sqrt {b} x}\right )}{4 \sqrt {3} a^{5/6} d}+\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{3 \sqrt {a}}\right )}{12 a^{5/6} d}-\frac {\sqrt {b} \text {arctanh}\left (\frac {\left (\sqrt [3]{a}+\sqrt [3]{-a+b x^2}\right )^2}{3 \sqrt [6]{a} \sqrt {b} x}\right )}{12 a^{5/6} d} \] Output:

1/12*b^(1/2)*arctan(3^(1/2)*a^(1/6)*(a^(1/3)+(b*x^2-a)^(1/3))/b^(1/2)/x)*3 
^(1/2)/a^(5/6)/d+1/12*b^(1/2)*arctanh(1/3*b^(1/2)*x/a^(1/2))/a^(5/6)/d-1/1 
2*b^(1/2)*arctanh(1/3*(a^(1/3)+(b*x^2-a)^(1/3))^2/a^(1/6)/b^(1/2)/x)/a^(5/ 
6)/d
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 6.32 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\sqrt [3]{-a+b x^2} \left (-\frac {9 a d}{b}+d x^2\right )} \, dx=-\frac {27 a b x \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},\frac {b x^2}{a},\frac {b x^2}{9 a}\right )}{d \left (9 a-b x^2\right ) \sqrt [3]{-a+b x^2} \left (27 a \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},\frac {b x^2}{a},\frac {b x^2}{9 a}\right )+2 b x^2 \left (\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},\frac {b x^2}{a},\frac {b x^2}{9 a}\right )+3 \operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},\frac {b x^2}{a},\frac {b x^2}{9 a}\right )\right )\right )} \] Input:

Integrate[1/((-a + b*x^2)^(1/3)*((-9*a*d)/b + d*x^2)),x]
 

Output:

(-27*a*b*x*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, (b*x^2)/(9*a)])/(d*(9*a - 
 b*x^2)*(-a + b*x^2)^(1/3)*(27*a*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, (b* 
x^2)/(9*a)] + 2*b*x^2*(AppellF1[3/2, 1/3, 2, 5/2, (b*x^2)/a, (b*x^2)/(9*a) 
] + 3*AppellF1[3/2, 4/3, 1, 5/2, (b*x^2)/a, (b*x^2)/(9*a)])))
 

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {307}

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.

\(\displaystyle \int \frac {1}{\sqrt [3]{b x^2-a} \left (d x^2-\frac {9 a d}{b}\right )} \, dx\)

\(\Big \downarrow \) 307

\(\displaystyle \frac {\sqrt {b} \arctan \left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{b x^2-a}+\sqrt [3]{a}\right )}{\sqrt {b} x}\right )}{4 \sqrt {3} a^{5/6} d}-\frac {\sqrt {b} \text {arctanh}\left (\frac {\left (\sqrt [3]{b x^2-a}+\sqrt [3]{a}\right )^2}{3 \sqrt [6]{a} \sqrt {b} x}\right )}{12 a^{5/6} d}+\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{3 \sqrt {a}}\right )}{12 a^{5/6} d}\)

Input:

Int[1/((-a + b*x^2)^(1/3)*((-9*a*d)/b + d*x^2)),x]
 

Output:

(Sqrt[b]*ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) + (-a + b*x^2)^(1/3)))/(Sqrt[b]* 
x)])/(4*Sqrt[3]*a^(5/6)*d) + (Sqrt[b]*ArcTanh[(Sqrt[b]*x)/(3*Sqrt[a])])/(1 
2*a^(5/6)*d) - (Sqrt[b]*ArcTanh[(a^(1/3) + (-a + b*x^2)^(1/3))^2/(3*a^(1/6 
)*Sqrt[b]*x)])/(12*a^(5/6)*d)
 

Defintions of rubi rules used

rule 307

Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Wit 
h[{q = Rt[-b/a, 2]}, Simp[(-q)*(ArcTanh[q*(x/3)]/(12*Rt[a, 3]*d)), x] + (Si 
mp[q*(ArcTanh[(Rt[a, 3] - (a + b*x^2)^(1/3))^2/(3*Rt[a, 3]^2*q*x)]/(12*Rt[a 
, 3]*d)), x] - Simp[q*(ArcTan[(Sqrt[3]*(Rt[a, 3] - (a + b*x^2)^(1/3)))/(Rt[ 
a, 3]*q*x)]/(4*Sqrt[3]*Rt[a, 3]*d)), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[ 
b*c - a*d, 0] && EqQ[b*c - 9*a*d, 0] && NegQ[b/a]
 

Maple [F]

\[\int \frac {1}{\left (b \,x^{2}-a \right )^{\frac {1}{3}} \left (-\frac {9 a d}{b}+x^{2} d \right )}d x\]

Input:

int(1/(b*x^2-a)^(1/3)/(-9*a*d/b+x^2*d),x)
 

Output:

int(1/(b*x^2-a)^(1/3)/(-9*a*d/b+x^2*d),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [3]{-a+b x^2} \left (-\frac {9 a d}{b}+d x^2\right )} \, dx=\text {Timed out} \] Input:

integrate(1/(b*x^2-a)^(1/3)/(-9*a*d/b+d*x^2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{\sqrt [3]{-a+b x^2} \left (-\frac {9 a d}{b}+d x^2\right )} \, dx=\frac {b \int \frac {1}{- 9 a \sqrt [3]{- a + b x^{2}} + b x^{2} \sqrt [3]{- a + b x^{2}}}\, dx}{d} \] Input:

integrate(1/(b*x**2-a)**(1/3)/(-9*a*d/b+d*x**2),x)
 

Output:

b*Integral(1/(-9*a*(-a + b*x**2)**(1/3) + b*x**2*(-a + b*x**2)**(1/3)), x) 
/d
 

Maxima [F]

\[ \int \frac {1}{\sqrt [3]{-a+b x^2} \left (-\frac {9 a d}{b}+d x^2\right )} \, dx=\int { \frac {1}{{\left (b x^{2} - a\right )}^{\frac {1}{3}} {\left (d x^{2} - \frac {9 \, a d}{b}\right )}} \,d x } \] Input:

integrate(1/(b*x^2-a)^(1/3)/(-9*a*d/b+d*x^2),x, algorithm="maxima")
 

Output:

integrate(1/((b*x^2 - a)^(1/3)*(d*x^2 - 9*a*d/b)), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt [3]{-a+b x^2} \left (-\frac {9 a d}{b}+d x^2\right )} \, dx=\int { \frac {1}{{\left (b x^{2} - a\right )}^{\frac {1}{3}} {\left (d x^{2} - \frac {9 \, a d}{b}\right )}} \,d x } \] Input:

integrate(1/(b*x^2-a)^(1/3)/(-9*a*d/b+d*x^2),x, algorithm="giac")
 

Output:

integrate(1/((b*x^2 - a)^(1/3)*(d*x^2 - 9*a*d/b)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [3]{-a+b x^2} \left (-\frac {9 a d}{b}+d x^2\right )} \, dx=\int \frac {1}{{\left (b\,x^2-a\right )}^{1/3}\,\left (d\,x^2-\frac {9\,a\,d}{b}\right )} \,d x \] Input:

int(1/((b*x^2 - a)^(1/3)*(d*x^2 - (9*a*d)/b)),x)
 

Output:

int(1/((b*x^2 - a)^(1/3)*(d*x^2 - (9*a*d)/b)), x)
 

Reduce [F]

\[ \int \frac {1}{\sqrt [3]{-a+b x^2} \left (-\frac {9 a d}{b}+d x^2\right )} \, dx=-\frac {\left (\int \frac {1}{9 \left (b \,x^{2}-a \right )^{\frac {1}{3}} a -\left (b \,x^{2}-a \right )^{\frac {1}{3}} b \,x^{2}}d x \right ) b}{d} \] Input:

int(1/(b*x^2-a)^(1/3)/(-9*a*d/b+d*x^2),x)
 

Output:

( - int(1/(9*( - a + b*x**2)**(1/3)*a - ( - a + b*x**2)**(1/3)*b*x**2),x)* 
b)/d