\(\int \frac {1}{x^4 \sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx\) [875]

Optimal result

Integrand size = 26, antiderivative size = 91 \[ \int \frac {1}{x^4 \sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx=-\frac {\sqrt {a+b x^3} \sqrt {c+d x^3}}{3 a c x^3}+\frac {(b c+a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x^3}}{\sqrt {a} \sqrt {c+d x^3}}\right )}{3 a^{3/2} c^{3/2}} \] Output:

-1/3*(b*x^3+a)^(1/2)*(d*x^3+c)^(1/2)/a/c/x^3+1/3*(a*d+b*c)*arctanh(c^(1/2) 
*(b*x^3+a)^(1/2)/a^(1/2)/(d*x^3+c)^(1/2))/a^(3/2)/c^(3/2)
 

Mathematica [A] (verified)

Time = 1.41 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^4 \sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx=-\frac {\sqrt {a+b x^3} \sqrt {c+d x^3}}{3 a c x^3}+\frac {(b c+a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x^3}}{\sqrt {a} \sqrt {c+d x^3}}\right )}{3 a^{3/2} c^{3/2}} \] Input:

Integrate[1/(x^4*Sqrt[a + b*x^3]*Sqrt[c + d*x^3]),x]
 

Output:

-1/3*(Sqrt[a + b*x^3]*Sqrt[c + d*x^3])/(a*c*x^3) + ((b*c + a*d)*ArcTanh[(S 
qrt[c]*Sqrt[a + b*x^3])/(Sqrt[a]*Sqrt[c + d*x^3])])/(3*a^(3/2)*c^(3/2))
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {948, 107, 104, 221}

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.

\(\displaystyle \int \frac {1}{x^4 \sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx\)

\(\Big \downarrow \) 948

\(\displaystyle \frac {1}{3} \int \frac {1}{x^6 \sqrt {b x^3+a} \sqrt {d x^3+c}}dx^3\)

\(\Big \downarrow \) 107

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

\(\Big \downarrow \) 104

\(\displaystyle \frac {1}{3} \left (-\frac {(a d+b c) \int \frac {1}{c x^6-a}d\frac {\sqrt {b x^3+a}}{\sqrt {d x^3+c}}}{a c}-\frac {\sqrt {a+b x^3} \sqrt {c+d x^3}}{a c x^3}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{3} \left (\frac {(a d+b c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x^3}}{\sqrt {a} \sqrt {c+d x^3}}\right )}{a^{3/2} c^{3/2}}-\frac {\sqrt {a+b x^3} \sqrt {c+d x^3}}{a c x^3}\right )\)

Input:

Int[1/(x^4*Sqrt[a + b*x^3]*Sqrt[c + d*x^3]),x]
 

Output:

(-((Sqrt[a + b*x^3]*Sqrt[c + d*x^3])/(a*c*x^3)) + ((b*c + a*d)*ArcTanh[(Sq 
rt[c]*Sqrt[a + b*x^3])/(Sqrt[a]*Sqrt[c + d*x^3])])/(a^(3/2)*c^(3/2)))/3
 

Defintions of rubi rules used

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 107

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 
 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 
 1)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x 
] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
 

rule 221

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

rule 948

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

Maple [F]

\[\int \frac {1}{x^{4} \sqrt {b \,x^{3}+a}\, \sqrt {d \,x^{3}+c}}d x\]

Input:

int(1/x^4/(b*x^3+a)^(1/2)/(d*x^3+c)^(1/2),x)
 

Output:

int(1/x^4/(b*x^3+a)^(1/2)/(d*x^3+c)^(1/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 278, normalized size of antiderivative = 3.05 \[ \int \frac {1}{x^4 \sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx=\left [\frac {\sqrt {a c} {\left (b c + a d\right )} x^{3} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{6} + 8 \, a^{2} c^{2} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x^{3} + 4 \, {\left ({\left (b c + a d\right )} x^{3} + 2 \, a c\right )} \sqrt {b x^{3} + a} \sqrt {d x^{3} + c} \sqrt {a c}}{x^{6}}\right ) - 4 \, \sqrt {b x^{3} + a} \sqrt {d x^{3} + c} a c}{12 \, a^{2} c^{2} x^{3}}, -\frac {\sqrt {-a c} {\left (b c + a d\right )} x^{3} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{3} + 2 \, a c\right )} \sqrt {b x^{3} + a} \sqrt {d x^{3} + c} \sqrt {-a c}}{2 \, {\left (a b c d x^{6} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x^{3}\right )}}\right ) + 2 \, \sqrt {b x^{3} + a} \sqrt {d x^{3} + c} a c}{6 \, a^{2} c^{2} x^{3}}\right ] \] Input:

integrate(1/x^4/(b*x^3+a)^(1/2)/(d*x^3+c)^(1/2),x, algorithm="fricas")
 

Output:

[1/12*(sqrt(a*c)*(b*c + a*d)*x^3*log(((b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^6 
+ 8*a^2*c^2 + 8*(a*b*c^2 + a^2*c*d)*x^3 + 4*((b*c + a*d)*x^3 + 2*a*c)*sqrt 
(b*x^3 + a)*sqrt(d*x^3 + c)*sqrt(a*c))/x^6) - 4*sqrt(b*x^3 + a)*sqrt(d*x^3 
 + c)*a*c)/(a^2*c^2*x^3), -1/6*(sqrt(-a*c)*(b*c + a*d)*x^3*arctan(1/2*((b* 
c + a*d)*x^3 + 2*a*c)*sqrt(b*x^3 + a)*sqrt(d*x^3 + c)*sqrt(-a*c)/(a*b*c*d* 
x^6 + a^2*c^2 + (a*b*c^2 + a^2*c*d)*x^3)) + 2*sqrt(b*x^3 + a)*sqrt(d*x^3 + 
 c)*a*c)/(a^2*c^2*x^3)]
 

Sympy [F]

\[ \int \frac {1}{x^4 \sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx=\int \frac {1}{x^{4} \sqrt {a + b x^{3}} \sqrt {c + d x^{3}}}\, dx \] Input:

integrate(1/x**4/(b*x**3+a)**(1/2)/(d*x**3+c)**(1/2),x)
 

Output:

Integral(1/(x**4*sqrt(a + b*x**3)*sqrt(c + d*x**3)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{x^4 \sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/x^4/(b*x^3+a)^(1/2)/(d*x^3+c)^(1/2),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m 
ore detail
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 413 vs. \(2 (71) = 142\).

Time = 0.14 (sec) , antiderivative size = 413, normalized size of antiderivative = 4.54 \[ \int \frac {1}{x^4 \sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx=\frac {\sqrt {b d} b^{4} d {\left (\frac {{\left (b c + a d\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b x^{3} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{3} + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} a b^{3} c d} - \frac {2 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2} - {\left (\sqrt {b x^{3} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{3} + a\right )} b d - a b d}\right )}^{2} b c - {\left (\sqrt {b x^{3} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{3} + a\right )} b d - a b d}\right )}^{2} a d\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b x^{3} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{3} + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b x^{3} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{3} + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b x^{3} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{3} + a\right )} b d - a b d}\right )}^{4}\right )} a b^{2} c d}\right )}}{3 \, {\left | b \right |}} \] Input:

integrate(1/x^4/(b*x^3+a)^(1/2)/(d*x^3+c)^(1/2),x, algorithm="giac")
 

Output:

1/3*sqrt(b*d)*b^4*d*((b*c + a*d)*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*x^3 
+ a)*sqrt(b*d) - sqrt(b^2*c + (b*x^3 + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d) 
*b))/(sqrt(-a*b*c*d)*a*b^3*c*d) - 2*(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2 - ( 
sqrt(b*x^3 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^3 + a)*b*d - a*b*d))^2*b*c - 
 (sqrt(b*x^3 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^3 + a)*b*d - a*b*d))^2*a*d 
)/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2 - 2*(sqrt(b*x^3 + a)*sqrt(b*d) - s 
qrt(b^2*c + (b*x^3 + a)*b*d - a*b*d))^2*b^2*c - 2*(sqrt(b*x^3 + a)*sqrt(b* 
d) - sqrt(b^2*c + (b*x^3 + a)*b*d - a*b*d))^2*a*b*d + (sqrt(b*x^3 + a)*sqr 
t(b*d) - sqrt(b^2*c + (b*x^3 + a)*b*d - a*b*d))^4)*a*b^2*c*d))/abs(b)
 

Mupad [B] (verification not implemented)

Time = 10.00 (sec) , antiderivative size = 481, normalized size of antiderivative = 5.29 \[ \int \frac {1}{x^4 \sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx=\frac {\frac {\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )\,\left (\frac {c\,b^2}{12}+\frac {a\,d\,b}{12}\right )}{a^{3/2}\,c^{3/2}\,d\,\left (\sqrt {d\,x^3+c}-\sqrt {c}\right )}-\frac {b^2}{12\,a\,c\,d}+\frac {{\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}^2\,\left (\frac {a^2\,d^2}{12}-\frac {a\,b\,c\,d}{4}+\frac {b^2\,c^2}{12}\right )}{a^2\,c^2\,d\,{\left (\sqrt {d\,x^3+c}-\sqrt {c}\right )}^2}}{\frac {{\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}^3}{{\left (\sqrt {d\,x^3+c}-\sqrt {c}\right )}^3}+\frac {b\,\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}{d\,\left (\sqrt {d\,x^3+c}-\sqrt {c}\right )}-\frac {{\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}^2\,\left (a\,d+b\,c\right )}{\sqrt {a}\,\sqrt {c}\,d\,{\left (\sqrt {d\,x^3+c}-\sqrt {c}\right )}^2}}+\frac {\ln \left (\frac {\sqrt {b\,x^3+a}-\sqrt {a}}{\sqrt {d\,x^3+c}-\sqrt {c}}\right )\,\left (\sqrt {a}\,b\,c^{3/2}+a^{3/2}\,\sqrt {c}\,d\right )}{6\,a^2\,c^2}-\frac {\ln \left (\frac {\left (\sqrt {c}\,\sqrt {b\,x^3+a}-\sqrt {a}\,\sqrt {d\,x^3+c}\right )\,\left (b\,\sqrt {c}-\frac {\sqrt {a}\,d\,\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}{\sqrt {d\,x^3+c}-\sqrt {c}}\right )}{\sqrt {d\,x^3+c}-\sqrt {c}}\right )\,\left (\sqrt {a}\,b\,c^{3/2}+a^{3/2}\,\sqrt {c}\,d\right )}{6\,a^2\,c^2}-\frac {d\,\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}{12\,a\,c\,\left (\sqrt {d\,x^3+c}-\sqrt {c}\right )} \] Input:

int(1/(x^4*(a + b*x^3)^(1/2)*(c + d*x^3)^(1/2)),x)
 

Output:

((((a + b*x^3)^(1/2) - a^(1/2))*((b^2*c)/12 + (a*b*d)/12))/(a^(3/2)*c^(3/2 
)*d*((c + d*x^3)^(1/2) - c^(1/2))) - b^2/(12*a*c*d) + (((a + b*x^3)^(1/2) 
- a^(1/2))^2*((a^2*d^2)/12 + (b^2*c^2)/12 - (a*b*c*d)/4))/(a^2*c^2*d*((c + 
 d*x^3)^(1/2) - c^(1/2))^2))/(((a + b*x^3)^(1/2) - a^(1/2))^3/((c + d*x^3) 
^(1/2) - c^(1/2))^3 + (b*((a + b*x^3)^(1/2) - a^(1/2)))/(d*((c + d*x^3)^(1 
/2) - c^(1/2))) - (((a + b*x^3)^(1/2) - a^(1/2))^2*(a*d + b*c))/(a^(1/2)*c 
^(1/2)*d*((c + d*x^3)^(1/2) - c^(1/2))^2)) + (log(((a + b*x^3)^(1/2) - a^( 
1/2))/((c + d*x^3)^(1/2) - c^(1/2)))*(a^(1/2)*b*c^(3/2) + a^(3/2)*c^(1/2)* 
d))/(6*a^2*c^2) - (log(((c^(1/2)*(a + b*x^3)^(1/2) - a^(1/2)*(c + d*x^3)^( 
1/2))*(b*c^(1/2) - (a^(1/2)*d*((a + b*x^3)^(1/2) - a^(1/2)))/((c + d*x^3)^ 
(1/2) - c^(1/2))))/((c + d*x^3)^(1/2) - c^(1/2)))*(a^(1/2)*b*c^(3/2) + a^( 
3/2)*c^(1/2)*d))/(6*a^2*c^2) - (d*((a + b*x^3)^(1/2) - a^(1/2)))/(12*a*c*( 
(c + d*x^3)^(1/2) - c^(1/2)))
 

Reduce [F]

\[ \int \frac {1}{x^4 \sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx=\frac {-2 \sqrt {d \,x^{3}+c}\, \sqrt {b \,x^{3}+a}-3 \left (\int \frac {\sqrt {d \,x^{3}+c}\, \sqrt {b \,x^{3}+a}}{b d \,x^{7}+a d \,x^{4}+b c \,x^{4}+a c x}d x \right ) a d \,x^{3}-3 \left (\int \frac {\sqrt {d \,x^{3}+c}\, \sqrt {b \,x^{3}+a}}{b d \,x^{7}+a d \,x^{4}+b c \,x^{4}+a c x}d x \right ) b c \,x^{3}}{6 a c \,x^{3}} \] Input:

int(1/x^4/(b*x^3+a)^(1/2)/(d*x^3+c)^(1/2),x)
 

Output:

( - 2*sqrt(c + d*x**3)*sqrt(a + b*x**3) - 3*int((sqrt(c + d*x**3)*sqrt(a + 
 b*x**3))/(a*c*x + a*d*x**4 + b*c*x**4 + b*d*x**7),x)*a*d*x**3 - 3*int((sq 
rt(c + d*x**3)*sqrt(a + b*x**3))/(a*c*x + a*d*x**4 + b*c*x**4 + b*d*x**7), 
x)*b*c*x**3)/(6*a*c*x**3)