\(\int \frac {1}{x^3 (c+d x) \sqrt [4]{a+b x^2}} \, dx\) [1650]

Optimal result

Integrand size = 22, antiderivative size = 474 \[ \int \frac {1}{x^3 (c+d x) \sqrt [4]{a+b x^2}} \, dx=\frac {d}{c^2 x \sqrt [4]{a+b x^2}}-\frac {\left (a+b x^2\right )^{3/4}}{2 a c x^2}-\frac {\left (b c^2-4 a d^2\right ) \arctan \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )}{4 a^{5/4} c^3}-\frac {d^{5/2} \arctan \left (\frac {\sqrt {d} \sqrt [4]{a+b x^2}}{\sqrt [4]{b c^2+a d^2}}\right )}{c^3 \sqrt [4]{b c^2+a d^2}}+\frac {\left (b c^2-4 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )}{4 a^{5/4} c^3}+\frac {d^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt [4]{a+b x^2}}{\sqrt [4]{b c^2+a d^2}}\right )}{c^3 \sqrt [4]{b c^2+a d^2}}+\frac {\sqrt {b} d \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a} c^2 \sqrt [4]{a+b x^2}}+\frac {\sqrt [4]{a} d^2 \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b c^2+a d^2}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{c^2 \sqrt {b c^2+a d^2} x}-\frac {\sqrt [4]{a} d^2 \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} d}{\sqrt {b c^2+a d^2}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{c^2 \sqrt {b c^2+a d^2} x} \] Output:

d/c^2/x/(b*x^2+a)^(1/4)-1/2*(b*x^2+a)^(3/4)/a/c/x^2-1/4*(-4*a*d^2+b*c^2)*a 
rctan((b*x^2+a)^(1/4)/a^(1/4))/a^(5/4)/c^3-d^(5/2)*arctan(d^(1/2)*(b*x^2+a 
)^(1/4)/(a*d^2+b*c^2)^(1/4))/c^3/(a*d^2+b*c^2)^(1/4)+1/4*(-4*a*d^2+b*c^2)* 
arctanh((b*x^2+a)^(1/4)/a^(1/4))/a^(5/4)/c^3+d^(5/2)*arctanh(d^(1/2)*(b*x^ 
2+a)^(1/4)/(a*d^2+b*c^2)^(1/4))/c^3/(a*d^2+b*c^2)^(1/4)+b^(1/2)*d*(1+b*x^2 
/a)^(1/4)*EllipticE(sin(1/2*arctan(b^(1/2)*x/a^(1/2))),2^(1/2))/a^(1/2)/c^ 
2/(b*x^2+a)^(1/4)+a^(1/4)*d^2*(-b*x^2/a)^(1/2)*EllipticPi((b*x^2+a)^(1/4)/ 
a^(1/4),-a^(1/2)/(a*d^2+b*c^2)^(1/2)*d,I)/c^2/(a*d^2+b*c^2)^(1/2)/x-a^(1/4 
)*d^2*(-b*x^2/a)^(1/2)*EllipticPi((b*x^2+a)^(1/4)/a^(1/4),a^(1/2)/(a*d^2+b 
*c^2)^(1/2)*d,I)/c^2/(a*d^2+b*c^2)^(1/2)/x
                                                                                    
                                                                                    
 

Mathematica [F]

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

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

Output:

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

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 0.54 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.62, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {621, 354, 114, 27, 174, 73, 25, 27, 395, 394, 827, 216, 218, 219, 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.

\(\displaystyle \int \frac {1}{x^3 \sqrt [4]{a+b x^2} (c+d x)} \, dx\)

\(\Big \downarrow \) 621

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

\(\Big \downarrow \) 354

\(\displaystyle \frac {1}{2} c \int \frac {1}{x^4 \sqrt [4]{b x^2+a} \left (c^2-d^2 x^2\right )}dx^2-d \int \frac {1}{x^2 \sqrt [4]{b x^2+a} \left (c^2-d^2 x^2\right )}dx\)

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{2} c \left (-\frac {\frac {4 \left (b-\frac {4 a d^2}{c^2}\right ) \int -\frac {b x^4}{a-x^8}d\sqrt [4]{b x^2+a}}{b}-\frac {16 a d^4 \int \frac {b x^4}{b \left (c^2+\frac {a d^2}{b}\right )-d^2 x^8}d\sqrt [4]{b x^2+a}}{b c^2}}{4 a c^2}-\frac {\left (a+b x^2\right )^{3/4}}{a c^2 x^2}\right )-d \int \frac {1}{x^2 \sqrt [4]{b x^2+a} \left (c^2-d^2 x^2\right )}dx\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} c \left (-\frac {-\frac {4 \left (b-\frac {4 a d^2}{c^2}\right ) \int \frac {b x^4}{a-x^8}d\sqrt [4]{b x^2+a}}{b}-\frac {16 a d^4 \int \frac {b x^4}{b \left (c^2+\frac {a d^2}{b}\right )-d^2 x^8}d\sqrt [4]{b x^2+a}}{b c^2}}{4 a c^2}-\frac {\left (a+b x^2\right )^{3/4}}{a c^2 x^2}\right )-d \int \frac {1}{x^2 \sqrt [4]{b x^2+a} \left (c^2-d^2 x^2\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} c \left (-\frac {-4 \left (b-\frac {4 a d^2}{c^2}\right ) \int \frac {x^4}{a-x^8}d\sqrt [4]{b x^2+a}-\frac {16 a d^4 \int \frac {x^4}{-d^2 x^8+b c^2+a d^2}d\sqrt [4]{b x^2+a}}{c^2}}{4 a c^2}-\frac {\left (a+b x^2\right )^{3/4}}{a c^2 x^2}\right )-d \int \frac {1}{x^2 \sqrt [4]{b x^2+a} \left (c^2-d^2 x^2\right )}dx\)

\(\Big \downarrow \) 395

\(\displaystyle \frac {1}{2} c \left (-\frac {-4 \left (b-\frac {4 a d^2}{c^2}\right ) \int \frac {x^4}{a-x^8}d\sqrt [4]{b x^2+a}-\frac {16 a d^4 \int \frac {x^4}{-d^2 x^8+b c^2+a d^2}d\sqrt [4]{b x^2+a}}{c^2}}{4 a c^2}-\frac {\left (a+b x^2\right )^{3/4}}{a c^2 x^2}\right )-\frac {d \sqrt [4]{\frac {b x^2}{a}+1} \int \frac {1}{x^2 \sqrt [4]{\frac {b x^2}{a}+1} \left (c^2-d^2 x^2\right )}dx}{\sqrt [4]{a+b x^2}}\)

\(\Big \downarrow \) 394

\(\displaystyle \frac {1}{2} c \left (-\frac {-4 \left (b-\frac {4 a d^2}{c^2}\right ) \int \frac {x^4}{a-x^8}d\sqrt [4]{b x^2+a}-\frac {16 a d^4 \int \frac {x^4}{-d^2 x^8+b c^2+a d^2}d\sqrt [4]{b x^2+a}}{c^2}}{4 a c^2}-\frac {\left (a+b x^2\right )^{3/4}}{a c^2 x^2}\right )+\frac {d \sqrt [4]{\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (-\frac {1}{2},\frac {1}{4},1,\frac {1}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^2 x \sqrt [4]{a+b x^2}}\)

\(\Big \downarrow \) 827

\(\displaystyle \frac {1}{2} c \left (-\frac {-4 \left (b-\frac {4 a d^2}{c^2}\right ) \left (\frac {1}{2} \int \frac {1}{\sqrt {a}-x^4}d\sqrt [4]{b x^2+a}-\frac {1}{2} \int \frac {1}{x^4+\sqrt {a}}d\sqrt [4]{b x^2+a}\right )-\frac {16 a d^4 \left (\frac {\int \frac {1}{\sqrt {b c^2+a d^2}-d x^4}d\sqrt [4]{b x^2+a}}{2 d}-\frac {\int \frac {1}{d x^4+\sqrt {b c^2+a d^2}}d\sqrt [4]{b x^2+a}}{2 d}\right )}{c^2}}{4 a c^2}-\frac {\left (a+b x^2\right )^{3/4}}{a c^2 x^2}\right )+\frac {d \sqrt [4]{\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (-\frac {1}{2},\frac {1}{4},1,\frac {1}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^2 x \sqrt [4]{a+b x^2}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {1}{2} c \left (-\frac {-4 \left (b-\frac {4 a d^2}{c^2}\right ) \left (\frac {1}{2} \int \frac {1}{\sqrt {a}-x^4}d\sqrt [4]{b x^2+a}-\frac {\arctan \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}}\right )-\frac {16 a d^4 \left (\frac {\int \frac {1}{\sqrt {b c^2+a d^2}-d x^4}d\sqrt [4]{b x^2+a}}{2 d}-\frac {\int \frac {1}{d x^4+\sqrt {b c^2+a d^2}}d\sqrt [4]{b x^2+a}}{2 d}\right )}{c^2}}{4 a c^2}-\frac {\left (a+b x^2\right )^{3/4}}{a c^2 x^2}\right )+\frac {d \sqrt [4]{\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (-\frac {1}{2},\frac {1}{4},1,\frac {1}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^2 x \sqrt [4]{a+b x^2}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {1}{2} c \left (-\frac {-4 \left (b-\frac {4 a d^2}{c^2}\right ) \left (\frac {1}{2} \int \frac {1}{\sqrt {a}-x^4}d\sqrt [4]{b x^2+a}-\frac {\arctan \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}}\right )-\frac {16 a d^4 \left (\frac {\int \frac {1}{\sqrt {b c^2+a d^2}-d x^4}d\sqrt [4]{b x^2+a}}{2 d}-\frac {\arctan \left (\frac {\sqrt {d} \sqrt [4]{a+b x^2}}{\sqrt [4]{a d^2+b c^2}}\right )}{2 d^{3/2} \sqrt [4]{a d^2+b c^2}}\right )}{c^2}}{4 a c^2}-\frac {\left (a+b x^2\right )^{3/4}}{a c^2 x^2}\right )+\frac {d \sqrt [4]{\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (-\frac {1}{2},\frac {1}{4},1,\frac {1}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^2 x \sqrt [4]{a+b x^2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{2} c \left (-\frac {-\frac {16 a d^4 \left (\frac {\int \frac {1}{\sqrt {b c^2+a d^2}-d x^4}d\sqrt [4]{b x^2+a}}{2 d}-\frac {\arctan \left (\frac {\sqrt {d} \sqrt [4]{a+b x^2}}{\sqrt [4]{a d^2+b c^2}}\right )}{2 d^{3/2} \sqrt [4]{a d^2+b c^2}}\right )}{c^2}-4 \left (b-\frac {4 a d^2}{c^2}\right ) \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}}-\frac {\arctan \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}}\right )}{4 a c^2}-\frac {\left (a+b x^2\right )^{3/4}}{a c^2 x^2}\right )+\frac {d \sqrt [4]{\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (-\frac {1}{2},\frac {1}{4},1,\frac {1}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^2 x \sqrt [4]{a+b x^2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {d \sqrt [4]{\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (-\frac {1}{2},\frac {1}{4},1,\frac {1}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^2 x \sqrt [4]{a+b x^2}}+\frac {1}{2} c \left (-\frac {-4 \left (b-\frac {4 a d^2}{c^2}\right ) \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}}-\frac {\arctan \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}}\right )-\frac {16 a d^4 \left (\frac {\text {arctanh}\left (\frac {\sqrt {d} \sqrt [4]{a+b x^2}}{\sqrt [4]{a d^2+b c^2}}\right )}{2 d^{3/2} \sqrt [4]{a d^2+b c^2}}-\frac {\arctan \left (\frac {\sqrt {d} \sqrt [4]{a+b x^2}}{\sqrt [4]{a d^2+b c^2}}\right )}{2 d^{3/2} \sqrt [4]{a d^2+b c^2}}\right )}{c^2}}{4 a c^2}-\frac {\left (a+b x^2\right )^{3/4}}{a c^2 x^2}\right )\)

Input:

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

Output:

(d*(1 + (b*x^2)/a)^(1/4)*AppellF1[-1/2, 1/4, 1, 1/2, -((b*x^2)/a), (d^2*x^ 
2)/c^2])/(c^2*x*(a + b*x^2)^(1/4)) + (c*(-((a + b*x^2)^(3/4)/(a*c^2*x^2)) 
- (-4*(b - (4*a*d^2)/c^2)*(-1/2*ArcTan[(a + b*x^2)^(1/4)/a^(1/4)]/a^(1/4) 
+ ArcTanh[(a + b*x^2)^(1/4)/a^(1/4)]/(2*a^(1/4))) - (16*a*d^4*(-1/2*ArcTan 
[(Sqrt[d]*(a + b*x^2)^(1/4))/(b*c^2 + a*d^2)^(1/4)]/(d^(3/2)*(b*c^2 + a*d^ 
2)^(1/4)) + ArcTanh[(Sqrt[d]*(a + b*x^2)^(1/4))/(b*c^2 + a*d^2)^(1/4)]/(2* 
d^(3/2)*(b*c^2 + a*d^2)^(1/4))))/c^2)/(4*a*c^2)))/2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 114

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[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*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

rule 174

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

rule 216

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

rule 218

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

rule 219

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

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 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 394

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/2 
, -p, -q, 1 + (m + 1)/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; FreeQ[{a, b, c, 
 d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 1] && (Int 
egerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 395

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

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 827

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 
 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b)   Int[1/(r + s*x^2), x], 
x] - Simp[s/(2*b)   Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ 
[a/b, 0]
 

Maple [F]

\[\int \frac {1}{x^{3} \left (d x +c \right ) \left (b \,x^{2}+a \right )^{\frac {1}{4}}}d x\]

Input:

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

Output:

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

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{x^3 (c+d x) \sqrt [4]{a+b x^2}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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