\(\int \frac {1}{x (c+d x) \sqrt [3]{a+b x^2}} \, dx\) [1624]

Optimal result

Integrand size = 22, antiderivative size = 333 \[ \int \frac {1}{x (c+d x) \sqrt [3]{a+b x^2}} \, dx=-\frac {d x \sqrt [3]{1+\frac {b x^2}{a}} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^2 \sqrt [3]{a+b x^2}}+\frac {\sqrt {3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^2}}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \sqrt [3]{a} c}-\frac {\sqrt {3} d^{2/3} \arctan \left (\frac {1+\frac {2 d^{2/3} \sqrt [3]{a+b x^2}}{\sqrt [3]{b c^2+a d^2}}}{\sqrt {3}}\right )}{2 c \sqrt [3]{b c^2+a d^2}}-\frac {\log (x)}{2 \sqrt [3]{a} c}+\frac {d^{2/3} \log \left (c^2-d^2 x^2\right )}{4 c \sqrt [3]{b c^2+a d^2}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{4 \sqrt [3]{a} c}-\frac {3 d^{2/3} \log \left (\sqrt [3]{b c^2+a d^2}-d^{2/3} \sqrt [3]{a+b x^2}\right )}{4 c \sqrt [3]{b c^2+a d^2}} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.98, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {621, 334, 333, 354, 97, 67, 16, 1082, 217}

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.

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

\(\Big \downarrow \) 621

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

\(\Big \downarrow \) 334

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

\(\Big \downarrow \) 333

\(\displaystyle c \int \frac {1}{x \sqrt [3]{b x^2+a} \left (c^2-d^2 x^2\right )}dx-\frac {d x \sqrt [3]{\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^2 \sqrt [3]{a+b x^2}}\)

\(\Big \downarrow \) 354

\(\displaystyle \frac {1}{2} c \int \frac {1}{x^2 \sqrt [3]{b x^2+a} \left (c^2-d^2 x^2\right )}dx^2-\frac {d x \sqrt [3]{\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^2 \sqrt [3]{a+b x^2}}\)

\(\Big \downarrow \) 97

\(\displaystyle \frac {1}{2} c \left (\frac {d^2 \int \frac {1}{\sqrt [3]{b x^2+a} \left (c^2-d^2 x^2\right )}dx^2}{c^2}+\frac {\int \frac {1}{x^2 \sqrt [3]{b x^2+a}}dx^2}{c^2}\right )-\frac {d x \sqrt [3]{\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^2 \sqrt [3]{a+b x^2}}\)

\(\Big \downarrow \) 67

\(\displaystyle \frac {1}{2} c \left (\frac {\frac {3}{2} \int \frac {1}{x^4+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^2+a}}d\sqrt [3]{b x^2+a}-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{b x^2+a}}d\sqrt [3]{b x^2+a}}{2 \sqrt [3]{a}}-\frac {\log \left (x^2\right )}{2 \sqrt [3]{a}}}{c^2}+\frac {d^2 \left (\frac {3 \int \frac {1}{\frac {\sqrt [3]{b c^2+a d^2}}{d^{2/3}}-\sqrt [3]{b x^2+a}}d\sqrt [3]{b x^2+a}}{2 d^{4/3} \sqrt [3]{a d^2+b c^2}}-\frac {3 \int \frac {1}{x^4+\frac {\left (b c^2+a d^2\right )^{2/3}}{d^{4/3}}+\frac {\sqrt [3]{b c^2+a d^2} \sqrt [3]{b x^2+a}}{d^{2/3}}}d\sqrt [3]{b x^2+a}}{2 d^2}+\frac {\log \left (c^2-d^2 x^2\right )}{2 d^{4/3} \sqrt [3]{a d^2+b c^2}}\right )}{c^2}\right )-\frac {d x \sqrt [3]{\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^2 \sqrt [3]{a+b x^2}}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {1}{2} c \left (\frac {\frac {3}{2} \int \frac {1}{x^4+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^2+a}}d\sqrt [3]{b x^2+a}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^2\right )}{2 \sqrt [3]{a}}}{c^2}+\frac {d^2 \left (-\frac {3 \int \frac {1}{x^4+\frac {\left (b c^2+a d^2\right )^{2/3}}{d^{4/3}}+\frac {\sqrt [3]{b c^2+a d^2} \sqrt [3]{b x^2+a}}{d^{2/3}}}d\sqrt [3]{b x^2+a}}{2 d^2}+\frac {\log \left (c^2-d^2 x^2\right )}{2 d^{4/3} \sqrt [3]{a d^2+b c^2}}-\frac {3 \log \left (\sqrt [3]{a d^2+b c^2}-d^{2/3} \sqrt [3]{a+b x^2}\right )}{2 d^{4/3} \sqrt [3]{a d^2+b c^2}}\right )}{c^2}\right )-\frac {d x \sqrt [3]{\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^2 \sqrt [3]{a+b x^2}}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {1}{2} c \left (\frac {d^2 \left (\frac {3 \int \frac {1}{-x^4-3}d\left (\frac {2 \sqrt [3]{b x^2+a} d^{2/3}}{\sqrt [3]{b c^2+a d^2}}+1\right )}{d^{4/3} \sqrt [3]{a d^2+b c^2}}+\frac {\log \left (c^2-d^2 x^2\right )}{2 d^{4/3} \sqrt [3]{a d^2+b c^2}}-\frac {3 \log \left (\sqrt [3]{a d^2+b c^2}-d^{2/3} \sqrt [3]{a+b x^2}\right )}{2 d^{4/3} \sqrt [3]{a d^2+b c^2}}\right )}{c^2}+\frac {-\frac {3 \int \frac {1}{-x^4-3}d\left (\frac {2 \sqrt [3]{b x^2+a}}{\sqrt [3]{a}}+1\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^2\right )}{2 \sqrt [3]{a}}}{c^2}\right )-\frac {d x \sqrt [3]{\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^2 \sqrt [3]{a+b x^2}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{2} c \left (\frac {d^2 \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {2 d^{2/3} \sqrt [3]{a+b x^2}}{\sqrt [3]{a d^2+b c^2}}+1}{\sqrt {3}}\right )}{d^{4/3} \sqrt [3]{a d^2+b c^2}}+\frac {\log \left (c^2-d^2 x^2\right )}{2 d^{4/3} \sqrt [3]{a d^2+b c^2}}-\frac {3 \log \left (\sqrt [3]{a d^2+b c^2}-d^{2/3} \sqrt [3]{a+b x^2}\right )}{2 d^{4/3} \sqrt [3]{a d^2+b c^2}}\right )}{c^2}+\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^2\right )}{2 \sqrt [3]{a}}}{c^2}\right )-\frac {d x \sqrt [3]{\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^2 \sqrt [3]{a+b x^2}}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 67

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ 
{q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x 
] + (Simp[3/(2*b)   Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], 
 x] - Simp[3/(2*b*q)   Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] / 
; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
 

rule 97

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), 
x_] :> Simp[b/(b*c - a*d)   Int[(e + f*x)^p/(a + b*x), x], x] - Simp[d/(b*c 
 - a*d)   Int[(e + f*x)^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p}, 
 x] &&  !IntegerQ[p]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 333

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[a^p*c^q*x*AppellF1[1/2, -p, -q, 3/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; F 
reeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[p] || GtQ[a, 
0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 334

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[a^IntPart[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[ 
(1 + b*(x^2/a))^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, p, q}, x] && 
NeQ[b*c - a*d, 0] &&  !(IntegerQ[p] || GtQ[a, 0])
 

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

rule 621

Int[((x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_))/((c_) + (d_.)*(x_)), x_Symbol] 
:> Simp[c   Int[x^m*((a + b*x^2)^p/(c^2 - d^2*x^2)), x], x] - Simp[d   Int[ 
x^(m + 1)*((a + b*x^2)^p/(c^2 - d^2*x^2)), x], x] /; FreeQ[{a, b, c, d, m, 
p}, x]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

integrate(1/((b*x^2 + a)^(1/3)*(d*x + c)*x), x)
 

Giac [F]

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

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

Output:

integrate(1/((b*x^2 + a)^(1/3)*(d*x + c)*x), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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