\(\int \frac {\sqrt {q+p x^5} (-2 q+3 p x^5)}{b x^4+a (q+p x^5)^2} \, dx\) [2248]

Optimal result

Integrand size = 41, antiderivative size = 167 \[ \int \frac {\sqrt {q+p x^5} \left (-2 q+3 p x^5\right )}{b x^4+a \left (q+p x^5\right )^2} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x \sqrt {q+p x^5}}{\sqrt {a} q-\sqrt {b} x^2+\sqrt {a} p x^5}\right )}{\sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\text {arctanh}\left (\frac {\frac {\sqrt [4]{a} q}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{b} x^2}{\sqrt {2} \sqrt [4]{a}}+\frac {\sqrt [4]{a} p x^5}{\sqrt {2} \sqrt [4]{b}}}{x \sqrt {q+p x^5}}\right )}{\sqrt {2} a^{3/4} \sqrt [4]{b}} \] Output:

-1/2*arctan(2^(1/2)*a^(1/4)*b^(1/4)*x*(p*x^5+q)^(1/2)/(a^(1/2)*q-b^(1/2)*x 
^2+a^(1/2)*p*x^5))*2^(1/2)/a^(3/4)/b^(1/4)-1/2*arctanh((1/2*a^(1/4)*q*2^(1 
/2)/b^(1/4)+1/2*2^(1/2)*b^(1/4)*x^2/a^(1/4)+1/2*a^(1/4)*p*x^5*2^(1/2)/b^(1 
/4))/x/(p*x^5+q)^(1/2))*2^(1/2)/a^(3/4)/b^(1/4)
 

Mathematica [A] (verified)

Time = 1.75 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {q+p x^5} \left (-2 q+3 p x^5\right )}{b x^4+a \left (q+p x^5\right )^2} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x \sqrt {q+p x^5}}{-\sqrt {b} x^2+\sqrt {a} \left (q+p x^5\right )}\right )+\text {arctanh}\left (\frac {\sqrt {b} x^2+\sqrt {a} \left (q+p x^5\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x \sqrt {q+p x^5}}\right )}{\sqrt {2} a^{3/4} \sqrt [4]{b}} \] Input:

Integrate[(Sqrt[q + p*x^5]*(-2*q + 3*p*x^5))/(b*x^4 + a*(q + p*x^5)^2),x]
 

Output:

-((ArcTan[(Sqrt[2]*a^(1/4)*b^(1/4)*x*Sqrt[q + p*x^5])/(-(Sqrt[b]*x^2) + Sq 
rt[a]*(q + p*x^5))] + ArcTanh[(Sqrt[b]*x^2 + Sqrt[a]*(q + p*x^5))/(Sqrt[2] 
*a^(1/4)*b^(1/4)*x*Sqrt[q + p*x^5])])/(Sqrt[2]*a^(3/4)*b^(1/4)))
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.58, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.220, Rules used = {7263, 755, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

\(\displaystyle \int \frac {\sqrt {p x^5+q} \left (3 p x^5-2 q\right )}{a \left (p x^5+q\right )^2+b x^4} \, dx\)

\(\Big \downarrow \) 7263

\(\displaystyle -2 \int \frac {1}{\frac {b x^4}{\left (p x^5+q\right )^2}+a}d\frac {x}{\sqrt {p x^5+q}}\)

\(\Big \downarrow \) 755

\(\displaystyle -2 \left (\frac {\int \frac {\sqrt {a}-\frac {\sqrt {b} x^2}{p x^5+q}}{\frac {b x^4}{\left (p x^5+q\right )^2}+a}d\frac {x}{\sqrt {p x^5+q}}}{2 \sqrt {a}}+\frac {\int \frac {\frac {\sqrt {b} x^2}{p x^5+q}+\sqrt {a}}{\frac {b x^4}{\left (p x^5+q\right )^2}+a}d\frac {x}{\sqrt {p x^5+q}}}{2 \sqrt {a}}\right )\)

\(\Big \downarrow \) 1476

\(\displaystyle -2 \left (\frac {\frac {\int \frac {1}{\frac {x^2}{p x^5+q}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b} \sqrt {p x^5+q}}+\frac {\sqrt {a}}{\sqrt {b}}}d\frac {x}{\sqrt {p x^5+q}}}{2 \sqrt {b}}+\frac {\int \frac {1}{\frac {x^2}{p x^5+q}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b} \sqrt {p x^5+q}}+\frac {\sqrt {a}}{\sqrt {b}}}d\frac {x}{\sqrt {p x^5+q}}}{2 \sqrt {b}}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {a}-\frac {\sqrt {b} x^2}{p x^5+q}}{\frac {b x^4}{\left (p x^5+q\right )^2}+a}d\frac {x}{\sqrt {p x^5+q}}}{2 \sqrt {a}}\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle -2 \left (\frac {\frac {\int \frac {1}{-\frac {x^2}{p x^5+q}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {p x^5+q}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-\frac {x^2}{p x^5+q}-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {p x^5+q}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {a}-\frac {\sqrt {b} x^2}{p x^5+q}}{\frac {b x^4}{\left (p x^5+q\right )^2}+a}d\frac {x}{\sqrt {p x^5+q}}}{2 \sqrt {a}}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle -2 \left (\frac {\int \frac {\sqrt {a}-\frac {\sqrt {b} x^2}{p x^5+q}}{\frac {b x^4}{\left (p x^5+q\right )^2}+a}d\frac {x}{\sqrt {p x^5+q}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {p x^5+q}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {p x^5+q}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )\)

\(\Big \downarrow \) 1479

\(\displaystyle -2 \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-\frac {2 \sqrt [4]{b} x}{\sqrt {p x^5+q}}}{\sqrt [4]{b} \left (\frac {x^2}{p x^5+q}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b} \sqrt {p x^5+q}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\frac {x}{\sqrt {p x^5+q}}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt {p x^5+q}}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (\frac {x^2}{p x^5+q}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b} \sqrt {p x^5+q}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\frac {x}{\sqrt {p x^5+q}}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {p x^5+q}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {p x^5+q}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -2 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-\frac {2 \sqrt [4]{b} x}{\sqrt {p x^5+q}}}{\sqrt [4]{b} \left (\frac {x^2}{p x^5+q}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b} \sqrt {p x^5+q}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\frac {x}{\sqrt {p x^5+q}}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt {p x^5+q}}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (\frac {x^2}{p x^5+q}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b} \sqrt {p x^5+q}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\frac {x}{\sqrt {p x^5+q}}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {p x^5+q}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {p x^5+q}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -2 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-\frac {2 \sqrt [4]{b} x}{\sqrt {p x^5+q}}}{\frac {x^2}{p x^5+q}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b} \sqrt {p x^5+q}}+\frac {\sqrt {a}}{\sqrt {b}}}d\frac {x}{\sqrt {p x^5+q}}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt {p x^5+q}}+\sqrt [4]{a}}{\frac {x^2}{p x^5+q}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b} \sqrt {p x^5+q}}+\frac {\sqrt {a}}{\sqrt {b}}}d\frac {x}{\sqrt {p x^5+q}}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {p x^5+q}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {p x^5+q}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )\)

\(\Big \downarrow \) 1103

\(\displaystyle -2 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {p x^5+q}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {p x^5+q}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {p x^5+q}}+\sqrt {a}+\frac {\sqrt {b} x^2}{p x^5+q}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {p x^5+q}}+\sqrt {a}+\frac {\sqrt {b} x^2}{p x^5+q}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )\)

Input:

Int[(Sqrt[q + p*x^5]*(-2*q + 3*p*x^5))/(b*x^4 + a*(q + p*x^5)^2),x]
 

Output:

-2*((-(ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/(a^(1/4)*Sqrt[q + p*x^5])]/(Sqrt[2]* 
a^(1/4)*b^(1/4))) + ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/(a^(1/4)*Sqrt[q + p*x^5 
])]/(Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[a]) + (-1/2*Log[Sqrt[a] + (Sqrt[b]* 
x^2)/(q + p*x^5) - (Sqrt[2]*a^(1/4)*b^(1/4)*x)/Sqrt[q + p*x^5]]/(Sqrt[2]*a 
^(1/4)*b^(1/4)) + Log[Sqrt[a] + (Sqrt[b]*x^2)/(q + p*x^5) + (Sqrt[2]*a^(1/ 
4)*b^(1/4)*x)/Sqrt[q + p*x^5]]/(2*Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[a]))
 

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 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 755

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

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]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

rule 7263

Int[(u_)*(v_)^(r_.)*((a_.)*(v_)^(p_.) + (b_.)*(w_)^(q_.))^(m_.), x_Symbol] 
:> With[{c = Simplify[u/(p*w*D[v, x] - q*v*D[w, x])]}, Simp[(-c)*q   Subst[ 
Int[(a + b*x^q)^m, x], x, v^(m*p + r + 1)*w], x] /; FreeQ[c, x]] /; FreeQ[{ 
a, b, m, p, q, r}, x] && EqQ[p + q*(m*p + r + 1), 0] && IntegerQ[q] && Inte 
gerQ[m]
 

Maple [A] (verified)

Time = 1.82 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.94

method	result	size
pseudoelliptic	\(\frac {\sqrt {2}\, \left (\ln \left (\frac {p \,x^{5}-\left (\frac {b}{a}\right )^{\frac {1}{4}} \sqrt {p \,x^{5}+q}\, \sqrt {2}\, x +\sqrt {\frac {b}{a}}\, x^{2}+q}{p \,x^{5}+\left (\frac {b}{a}\right )^{\frac {1}{4}} \sqrt {p \,x^{5}+q}\, \sqrt {2}\, x +\sqrt {\frac {b}{a}}\, x^{2}+q}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {p \,x^{5}+q}}{\left (\frac {b}{a}\right )^{\frac {1}{4}} x}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {p \,x^{5}+q}}{\left (\frac {b}{a}\right )^{\frac {1}{4}} x}+1\right )\right )}{4 \left (\frac {b}{a}\right )^{\frac {1}{4}} a}\)	\(157\)

Input:

int((p*x^5+q)^(1/2)*(3*p*x^5-2*q)/(b*x^4+a*(p*x^5+q)^2),x,method=_RETURNVE 
RBOSE)
 

Output:

1/4/(b/a)^(1/4)*2^(1/2)*(ln((p*x^5-(b/a)^(1/4)*(p*x^5+q)^(1/2)*2^(1/2)*x+( 
b/a)^(1/2)*x^2+q)/(p*x^5+(b/a)^(1/4)*(p*x^5+q)^(1/2)*2^(1/2)*x+(b/a)^(1/2) 
*x^2+q))+2*arctan(2^(1/2)/(b/a)^(1/4)*(p*x^5+q)^(1/2)/x+1)-2*arctan(-2^(1/ 
2)/(b/a)^(1/4)*(p*x^5+q)^(1/2)/x+1))/a
                                                                                    
                                                                                    
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.91 (sec) , antiderivative size = 639, normalized size of antiderivative = 3.83 \[ \int \frac {\sqrt {q+p x^5} \left (-2 q+3 p x^5\right )}{b x^4+a \left (q+p x^5\right )^2} \, dx=\frac {1}{4} \, \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} \log \left (\frac {a p^{2} x^{10} + 2 \, a p q x^{5} - b x^{4} + a q^{2} + 2 \, \sqrt {p x^{5} + q} {\left (a b x^{3} \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} + {\left (a^{3} b p x^{6} + a^{3} b q x\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}}\right )} - 2 \, {\left (a^{2} b p x^{7} + a^{2} b q x^{2}\right )} \sqrt {-\frac {1}{a^{3} b}}}{a p^{2} x^{10} + 2 \, a p q x^{5} + b x^{4} + a q^{2}}\right ) - \frac {1}{4} \, \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} \log \left (\frac {a p^{2} x^{10} + 2 \, a p q x^{5} - b x^{4} + a q^{2} - 2 \, \sqrt {p x^{5} + q} {\left (a b x^{3} \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} + {\left (a^{3} b p x^{6} + a^{3} b q x\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}}\right )} - 2 \, {\left (a^{2} b p x^{7} + a^{2} b q x^{2}\right )} \sqrt {-\frac {1}{a^{3} b}}}{a p^{2} x^{10} + 2 \, a p q x^{5} + b x^{4} + a q^{2}}\right ) - \frac {1}{4} i \, \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} \log \left (\frac {a p^{2} x^{10} + 2 \, a p q x^{5} - b x^{4} + a q^{2} - 2 \, \sqrt {p x^{5} + q} {\left (i \, a b x^{3} \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} + {\left (-i \, a^{3} b p x^{6} - i \, a^{3} b q x\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}}\right )} + 2 \, {\left (a^{2} b p x^{7} + a^{2} b q x^{2}\right )} \sqrt {-\frac {1}{a^{3} b}}}{a p^{2} x^{10} + 2 \, a p q x^{5} + b x^{4} + a q^{2}}\right ) + \frac {1}{4} i \, \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} \log \left (\frac {a p^{2} x^{10} + 2 \, a p q x^{5} - b x^{4} + a q^{2} - 2 \, \sqrt {p x^{5} + q} {\left (-i \, a b x^{3} \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} + {\left (i \, a^{3} b p x^{6} + i \, a^{3} b q x\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}}\right )} + 2 \, {\left (a^{2} b p x^{7} + a^{2} b q x^{2}\right )} \sqrt {-\frac {1}{a^{3} b}}}{a p^{2} x^{10} + 2 \, a p q x^{5} + b x^{4} + a q^{2}}\right ) \] Input:

integrate((p*x^5+q)^(1/2)*(3*p*x^5-2*q)/(b*x^4+a*(p*x^5+q)^2),x, algorithm 
="fricas")
 

Output:

1/4*(-1/(a^3*b))^(1/4)*log((a*p^2*x^10 + 2*a*p*q*x^5 - b*x^4 + a*q^2 + 2*s 
qrt(p*x^5 + q)*(a*b*x^3*(-1/(a^3*b))^(1/4) + (a^3*b*p*x^6 + a^3*b*q*x)*(-1 
/(a^3*b))^(3/4)) - 2*(a^2*b*p*x^7 + a^2*b*q*x^2)*sqrt(-1/(a^3*b)))/(a*p^2* 
x^10 + 2*a*p*q*x^5 + b*x^4 + a*q^2)) - 1/4*(-1/(a^3*b))^(1/4)*log((a*p^2*x 
^10 + 2*a*p*q*x^5 - b*x^4 + a*q^2 - 2*sqrt(p*x^5 + q)*(a*b*x^3*(-1/(a^3*b) 
)^(1/4) + (a^3*b*p*x^6 + a^3*b*q*x)*(-1/(a^3*b))^(3/4)) - 2*(a^2*b*p*x^7 + 
 a^2*b*q*x^2)*sqrt(-1/(a^3*b)))/(a*p^2*x^10 + 2*a*p*q*x^5 + b*x^4 + a*q^2) 
) - 1/4*I*(-1/(a^3*b))^(1/4)*log((a*p^2*x^10 + 2*a*p*q*x^5 - b*x^4 + a*q^2 
 - 2*sqrt(p*x^5 + q)*(I*a*b*x^3*(-1/(a^3*b))^(1/4) + (-I*a^3*b*p*x^6 - I*a 
^3*b*q*x)*(-1/(a^3*b))^(3/4)) + 2*(a^2*b*p*x^7 + a^2*b*q*x^2)*sqrt(-1/(a^3 
*b)))/(a*p^2*x^10 + 2*a*p*q*x^5 + b*x^4 + a*q^2)) + 1/4*I*(-1/(a^3*b))^(1/ 
4)*log((a*p^2*x^10 + 2*a*p*q*x^5 - b*x^4 + a*q^2 - 2*sqrt(p*x^5 + q)*(-I*a 
*b*x^3*(-1/(a^3*b))^(1/4) + (I*a^3*b*p*x^6 + I*a^3*b*q*x)*(-1/(a^3*b))^(3/ 
4)) + 2*(a^2*b*p*x^7 + a^2*b*q*x^2)*sqrt(-1/(a^3*b)))/(a*p^2*x^10 + 2*a*p* 
q*x^5 + b*x^4 + a*q^2))
 

Sympy [F]

\[ \int \frac {\sqrt {q+p x^5} \left (-2 q+3 p x^5\right )}{b x^4+a \left (q+p x^5\right )^2} \, dx=\int \frac {\sqrt {p x^{5} + q} \left (3 p x^{5} - 2 q\right )}{a p^{2} x^{10} + 2 a p q x^{5} + a q^{2} + b x^{4}}\, dx \] Input:

integrate((p*x**5+q)**(1/2)*(3*p*x**5-2*q)/(b*x**4+a*(p*x**5+q)**2),x)
 

Output:

Integral(sqrt(p*x**5 + q)*(3*p*x**5 - 2*q)/(a*p**2*x**10 + 2*a*p*q*x**5 + 
a*q**2 + b*x**4), x)
 

Maxima [F]

\[ \int \frac {\sqrt {q+p x^5} \left (-2 q+3 p x^5\right )}{b x^4+a \left (q+p x^5\right )^2} \, dx=\int { \frac {{\left (3 \, p x^{5} - 2 \, q\right )} \sqrt {p x^{5} + q}}{b x^{4} + {\left (p x^{5} + q\right )}^{2} a} \,d x } \] Input:

integrate((p*x^5+q)^(1/2)*(3*p*x^5-2*q)/(b*x^4+a*(p*x^5+q)^2),x, algorithm 
="maxima")
 

Output:

integrate((3*p*x^5 - 2*q)*sqrt(p*x^5 + q)/(b*x^4 + (p*x^5 + q)^2*a), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {\sqrt {q+p x^5} \left (-2 q+3 p x^5\right )}{b x^4+a \left (q+p x^5\right )^2} \, dx=\text {Timed out} \] Input:

integrate((p*x^5+q)^(1/2)*(3*p*x^5-2*q)/(b*x^4+a*(p*x^5+q)^2),x, algorithm 
="giac")
 

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 162.72 (sec) , antiderivative size = 498, normalized size of antiderivative = 2.98 \[ \int \frac {\sqrt {q+p x^5} \left (-2 q+3 p x^5\right )}{b x^4+a \left (q+p x^5\right )^2} \, dx=\frac {\ln \left (\frac {\left (2\,\sqrt {p\,x^5+q}\,{\left (-a^3\,b\right )}^{15/4}-a^{23/2}\,b^{7/2}\,p\,x^4+a^{19/2}\,b^{7/2}\,x\,\sqrt {-a^3\,b}\right )\,\left (2\,a^{27/2}\,b^{7/2}\,q\,\sqrt {p\,x^5+q}\,{\left (-a^3\,b\right )}^{15/4}-a^{24}\,b^8\,x^3+a^{25}\,b^7\,p\,x^4\,\left (p\,x^5+q\right )+a^{23}\,b^7\,q\,x\,\sqrt {-a^3\,b}+2\,a^{23}\,b^7\,x\,\left (p\,x^5+q\right )\,\sqrt {-a^3\,b}\right )}{\left (x^2\,\sqrt {-a^3\,b}+a^2\,q+a^2\,p\,x^5\right )\,\left (4\,q\,{\left (-a^3\,b\right )}^{15/2}+2\,p\,x^5\,{\left (-a^3\,b\right )}^{15/2}+a^{22}\,b^8\,x^2-a^{23}\,b^7\,p^2\,x^8\right )}\right )\,{\left (-a^3\,b\right )}^{1/4}}{2\,a^{3/2}\,\sqrt {b}}+\frac {\ln \left (\frac {\left (-2\,\sqrt {p\,x^5+q}\,{\left (-a^3\,b\right )}^{15/4}+a^{23/2}\,b^{7/2}\,p\,x^4\,1{}\mathrm {i}+a^{19/2}\,b^{7/2}\,x\,\sqrt {-a^3\,b}\,1{}\mathrm {i}\right )\,\left (a^{24}\,b^8\,x^3\,1{}\mathrm {i}-2\,a^{27/2}\,b^{7/2}\,q\,\sqrt {p\,x^5+q}\,{\left (-a^3\,b\right )}^{15/4}-a^{25}\,b^7\,p\,x^4\,\left (p\,x^5+q\right )\,1{}\mathrm {i}+a^{23}\,b^7\,q\,x\,\sqrt {-a^3\,b}\,1{}\mathrm {i}+a^{23}\,b^7\,x\,\left (p\,x^5+q\right )\,\sqrt {-a^3\,b}\,2{}\mathrm {i}\right )}{\left (a^2\,q-x^2\,\sqrt {-a^3\,b}+a^2\,p\,x^5\right )\,\left (4\,q\,{\left (-a^3\,b\right )}^{15/2}+2\,p\,x^5\,{\left (-a^3\,b\right )}^{15/2}-a^{22}\,b^8\,x^2+a^{23}\,b^7\,p^2\,x^8\right )}\right )\,{\left (-a^3\,b\right )}^{1/4}\,1{}\mathrm {i}}{2\,a^{3/2}\,\sqrt {b}} \] Input:

int(-((q + p*x^5)^(1/2)*(2*q - 3*p*x^5))/(a*(q + p*x^5)^2 + b*x^4),x)
 

Output:

(log(((2*(q + p*x^5)^(1/2)*(-a^3*b)^(15/4) - a^(23/2)*b^(7/2)*p*x^4 + a^(1 
9/2)*b^(7/2)*x*(-a^3*b)^(1/2))*(2*a^(27/2)*b^(7/2)*q*(q + p*x^5)^(1/2)*(-a 
^3*b)^(15/4) - a^24*b^8*x^3 + a^25*b^7*p*x^4*(q + p*x^5) + a^23*b^7*q*x*(- 
a^3*b)^(1/2) + 2*a^23*b^7*x*(q + p*x^5)*(-a^3*b)^(1/2)))/((x^2*(-a^3*b)^(1 
/2) + a^2*q + a^2*p*x^5)*(4*q*(-a^3*b)^(15/2) + 2*p*x^5*(-a^3*b)^(15/2) + 
a^22*b^8*x^2 - a^23*b^7*p^2*x^8)))*(-a^3*b)^(1/4))/(2*a^(3/2)*b^(1/2)) + ( 
log(((a^(23/2)*b^(7/2)*p*x^4*1i - 2*(q + p*x^5)^(1/2)*(-a^3*b)^(15/4) + a^ 
(19/2)*b^(7/2)*x*(-a^3*b)^(1/2)*1i)*(a^24*b^8*x^3*1i - 2*a^(27/2)*b^(7/2)* 
q*(q + p*x^5)^(1/2)*(-a^3*b)^(15/4) - a^25*b^7*p*x^4*(q + p*x^5)*1i + a^23 
*b^7*q*x*(-a^3*b)^(1/2)*1i + a^23*b^7*x*(q + p*x^5)*(-a^3*b)^(1/2)*2i))/(( 
a^2*q - x^2*(-a^3*b)^(1/2) + a^2*p*x^5)*(4*q*(-a^3*b)^(15/2) + 2*p*x^5*(-a 
^3*b)^(15/2) - a^22*b^8*x^2 + a^23*b^7*p^2*x^8)))*(-a^3*b)^(1/4)*1i)/(2*a^ 
(3/2)*b^(1/2))
 

Reduce [F]

\[ \int \frac {\sqrt {q+p x^5} \left (-2 q+3 p x^5\right )}{b x^4+a \left (q+p x^5\right )^2} \, dx=-2 \left (\int \frac {\sqrt {p \,x^{5}+q}}{a \,p^{2} x^{10}+2 a p q \,x^{5}+b \,x^{4}+a \,q^{2}}d x \right ) q +3 \left (\int \frac {\sqrt {p \,x^{5}+q}\, x^{5}}{a \,p^{2} x^{10}+2 a p q \,x^{5}+b \,x^{4}+a \,q^{2}}d x \right ) p \] Input:

int((p*x^5+q)^(1/2)*(3*p*x^5-2*q)/(b*x^4+a*(p*x^5+q)^2),x)
 

Output:

 - 2*int(sqrt(p*x**5 + q)/(a*p**2*x**10 + 2*a*p*q*x**5 + a*q**2 + b*x**4), 
x)*q + 3*int((sqrt(p*x**5 + q)*x**5)/(a*p**2*x**10 + 2*a*p*q*x**5 + a*q**2 
 + b*x**4),x)*p