\(\int \frac {(b+a x^2) \sqrt {b x+a x^3}}{x^2 (-b+a x^2)} \, dx\) [1879]

Optimal result

Integrand size = 35, antiderivative size = 130 \[ \int \frac {\left (b+a x^2\right ) \sqrt {b x+a x^3}}{x^2 \left (-b+a x^2\right )} \, dx=\frac {2 \sqrt {b x+a x^3}}{x}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}{b+a x^2}\right )-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}{b+a x^2}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.35, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2081, 477, 485, 12, 504, 1225, 226, 1713, 211, 214} \[ \int \frac {\left (b+a x^2\right ) \sqrt {b x+a x^3}}{x^2 \left (-b+a x^2\right )} \, dx=\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^3+b x} \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a x^2+b}}\right )}{\sqrt {x} \sqrt {a x^2+b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^3+b x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a x^2+b}}\right )}{\sqrt {x} \sqrt {a x^2+b}}+\frac {2 \sqrt {a x^3+b x}}{x} \]

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 211

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

Rule 214

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 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 477

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 485

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

Rule 504

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s
= Denominator[Rt[-a/b, 2]]}, Dist[s/(2*b), Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Dist[s/(2*b), Int[1/
((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 1225

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[1/(2*d), Int[1/Sqrt[a + c*x^4], x],
x] + Dist[1/(2*d), Int[(d - e*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d
^2 + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0]

Rule 1713

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[A, Subst[Int[1/
(d + 2*a*e*x^2), x], x, x/Sqrt[a + c*x^4]], x] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ
[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {b x+a x^3} \int \frac {\left (b+a x^2\right )^{3/2}}{x^{3/2} \left (-b+a x^2\right )} \, dx}{\sqrt {x} \sqrt {b+a x^2}} \\ & = \frac {\left (2 \sqrt {b x+a x^3}\right ) \text {Subst}\left (\int \frac {\left (b+a x^4\right )^{3/2}}{x^2 \left (-b+a x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {b+a x^2}} \\ & = \frac {2 \sqrt {b x+a x^3}}{x}-\frac {\left (2 \sqrt {b x+a x^3}\right ) \text {Subst}\left (\int -\frac {4 a b^2 x^2}{\left (-b+a x^4\right ) \sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{b \sqrt {x} \sqrt {b+a x^2}} \\ & = \frac {2 \sqrt {b x+a x^3}}{x}+\frac {\left (8 a b \sqrt {b x+a x^3}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-b+a x^4\right ) \sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {b+a x^2}} \\ & = \frac {2 \sqrt {b x+a x^3}}{x}-\frac {\left (4 \sqrt {a} b \sqrt {b x+a x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {b}-\sqrt {a} x^2\right ) \sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {b+a x^2}}+\frac {\left (4 \sqrt {a} b \sqrt {b x+a x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {b}+\sqrt {a} x^2\right ) \sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {b+a x^2}} \\ & = \frac {2 \sqrt {b x+a x^3}}{x}+\frac {\left (2 \sqrt {a} \sqrt {b} \sqrt {b x+a x^3}\right ) \text {Subst}\left (\int \frac {\sqrt {b}-\sqrt {a} x^2}{\left (\sqrt {b}+\sqrt {a} x^2\right ) \sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {b+a x^2}}-\frac {\left (2 \sqrt {a} \sqrt {b} \sqrt {b x+a x^3}\right ) \text {Subst}\left (\int \frac {\sqrt {b}+\sqrt {a} x^2}{\left (\sqrt {b}-\sqrt {a} x^2\right ) \sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {b+a x^2}} \\ & = \frac {2 \sqrt {b x+a x^3}}{x}-\frac {\left (2 \sqrt {a} b \sqrt {b x+a x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}-2 \sqrt {a} b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {b+a x^2}}\right )}{\sqrt {x} \sqrt {b+a x^2}}+\frac {\left (2 \sqrt {a} b \sqrt {b x+a x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}+2 \sqrt {a} b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {b+a x^2}}\right )}{\sqrt {x} \sqrt {b+a x^2}} \\ & = \frac {2 \sqrt {b x+a x^3}}{x}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3} \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {b+a x^2}}\right )}{\sqrt {x} \sqrt {b+a x^2}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {b+a x^2}}\right )}{\sqrt {x} \sqrt {b+a x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.14 \[ \int \frac {\left (b+a x^2\right ) \sqrt {b x+a x^3}}{x^2 \left (-b+a x^2\right )} \, dx=\frac {\sqrt {b+a x^2} \left (2 \sqrt {b+a x^2}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x} \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {b+a x^2}}\right )-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {b+a x^2}}\right )\right )}{\sqrt {x \left (b+a x^2\right )}} \]

[In]

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

[Out]

(Sqrt[b + a*x^2]*(2*Sqrt[b + a*x^2] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x]*ArcTan[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])
/Sqrt[b + a*x^2]] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x]*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/Sqrt[b + a*x^2]]
))/Sqrt[x*(b + a*x^2)]

Maple [A] (verified)

Time = 0.86 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.84

method	result	size
risch	\(\frac {2 a \,x^{2}+2 b}{\sqrt {\left (a \,x^{2}+b \right ) x}}-\frac {\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} \left (\ln \left (\frac {\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x +\sqrt {\left (a \,x^{2}+b \right ) x}}{-\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x +\sqrt {\left (a \,x^{2}+b \right ) x}}\right )+2 \arctan \left (\frac {\sqrt {\left (a \,x^{2}+b \right ) x}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right )\right )}{2}\)	\(109\)
default	\(\frac {-\ln \left (\frac {-\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}{\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}\right ) \sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -2 \arctan \left (\frac {\sqrt {\left (a \,x^{2}+b \right ) x}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right ) \sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x +4 \sqrt {\left (a \,x^{2}+b \right ) x}}{2 x}\)	\(120\)
pseudoelliptic	\(\frac {-\ln \left (\frac {-\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}{\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}\right ) \sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -2 \arctan \left (\frac {\sqrt {\left (a \,x^{2}+b \right ) x}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right ) \sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x +4 \sqrt {\left (a \,x^{2}+b \right ) x}}{2 x}\)	\(120\)
elliptic	\(\frac {2 a \,x^{2}+2 b}{\sqrt {\left (a \,x^{2}+b \right ) x}}+\frac {2 b \sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}+\frac {2 b \sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}\)	\(318\)

[In]

int((a*x^2+b)*(a*x^3+b*x)^(1/2)/x^2/(a*x^2-b),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 40.91 (sec) , antiderivative size = 1301, normalized size of antiderivative = 10.01 \[ \int \frac {\left (b+a x^2\right ) \sqrt {b x+a x^3}}{x^2 \left (-b+a x^2\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/4*(4^(1/4)*(a*b)^(1/4)*x*log(-(4^(3/4)*((5*a^3 + a^2*b)*x^4 - 8*(a^3 + 2*a^2*b)*x^3 + 5*a*b^2 + b^3 + 6*(5*
a^2*b + a*b^2)*x^2 - 8*(a^2*b + 2*a*b^2)*x)*(a*b)^(3/4) + 8*(5*a^2*b^2 + a*b^3 + (5*a^3*b + a^2*b^2)*x^2 - 4*(
a^3*b + 2*a^2*b^2)*x - 2*(a^2*b + 2*a*b^2 + (a^3 + 2*a^2*b)*x^2 - (5*a^2*b + a*b^2)*x)*sqrt(a*b))*sqrt(a*x^3 +
b*x) - 4*4^(1/4)*((a^4 + 2*a^3*b)*x^4 + a^2*b^2 + 2*a*b^3 - 2*(5*a^3*b + a^2*b^2)*x^3 + 6*(a^3*b + 2*a^2*b^2)
*x^2 - 2*(5*a^2*b^2 + a*b^3)*x)*(a*b)^(1/4))/(a^2*x^4 - 2*a*b*x^2 + b^2)) - 4^(1/4)*(a*b)^(1/4)*x*log((4^(3/4)
*((5*a^3 + a^2*b)*x^4 - 8*(a^3 + 2*a^2*b)*x^3 + 5*a*b^2 + b^3 + 6*(5*a^2*b + a*b^2)*x^2 - 8*(a^2*b + 2*a*b^2)*
x)*(a*b)^(3/4) - 8*(5*a^2*b^2 + a*b^3 + (5*a^3*b + a^2*b^2)*x^2 - 4*(a^3*b + 2*a^2*b^2)*x - 2*(a^2*b + 2*a*b^2
+ (a^3 + 2*a^2*b)*x^2 - (5*a^2*b + a*b^2)*x)*sqrt(a*b))*sqrt(a*x^3 + b*x) - 4*4^(1/4)*((a^4 + 2*a^3*b)*x^4 +
a^2*b^2 + 2*a*b^3 - 2*(5*a^3*b + a^2*b^2)*x^3 + 6*(a^3*b + 2*a^2*b^2)*x^2 - 2*(5*a^2*b^2 + a*b^3)*x)*(a*b)^(1/
4))/(a^2*x^4 - 2*a*b*x^2 + b^2)) + I*4^(1/4)*(a*b)^(1/4)*x*log((4^(3/4)*(I*(5*a^3 + a^2*b)*x^4 - 8*I*(a^3 + 2*
a^2*b)*x^3 + 5*I*a*b^2 + I*b^3 + 6*I*(5*a^2*b + a*b^2)*x^2 - 8*I*(a^2*b + 2*a*b^2)*x)*(a*b)^(3/4) - 8*(5*a^2*b
^2 + a*b^3 + (5*a^3*b + a^2*b^2)*x^2 - 4*(a^3*b + 2*a^2*b^2)*x + 2*(a^2*b + 2*a*b^2 + (a^3 + 2*a^2*b)*x^2 - (5
*a^2*b + a*b^2)*x)*sqrt(a*b))*sqrt(a*x^3 + b*x) - 4*4^(1/4)*(-I*(a^4 + 2*a^3*b)*x^4 - I*a^2*b^2 - 2*I*a*b^3 +
2*I*(5*a^3*b + a^2*b^2)*x^3 - 6*I*(a^3*b + 2*a^2*b^2)*x^2 + 2*I*(5*a^2*b^2 + a*b^3)*x)*(a*b)^(1/4))/(a^2*x^4 -
2*a*b*x^2 + b^2)) - I*4^(1/4)*(a*b)^(1/4)*x*log((4^(3/4)*(-I*(5*a^3 + a^2*b)*x^4 + 8*I*(a^3 + 2*a^2*b)*x^3 -
5*I*a*b^2 - I*b^3 - 6*I*(5*a^2*b + a*b^2)*x^2 + 8*I*(a^2*b + 2*a*b^2)*x)*(a*b)^(3/4) - 8*(5*a^2*b^2 + a*b^3 +
(5*a^3*b + a^2*b^2)*x^2 - 4*(a^3*b + 2*a^2*b^2)*x + 2*(a^2*b + 2*a*b^2 + (a^3 + 2*a^2*b)*x^2 - (5*a^2*b + a*b^
2)*x)*sqrt(a*b))*sqrt(a*x^3 + b*x) - 4*4^(1/4)*(I*(a^4 + 2*a^3*b)*x^4 + I*a^2*b^2 + 2*I*a*b^3 - 2*I*(5*a^3*b +
a^2*b^2)*x^3 + 6*I*(a^3*b + 2*a^2*b^2)*x^2 - 2*I*(5*a^2*b^2 + a*b^3)*x)*(a*b)^(1/4))/(a^2*x^4 - 2*a*b*x^2 + b
^2)) - 8*sqrt(a*x^3 + b*x))/x

Sympy [F]

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

[In]

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

[Out]

Integral(sqrt(x*(a*x**2 + b))*(a*x**2 + b)/(x**2*(a*x**2 - b)), x)

Maxima [F]

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

[In]

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

[Out]

integrate(sqrt(a*x^3 + b*x)*(a*x^2 + b)/((a*x^2 - b)*x^2), x)

Giac [F]

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

[In]

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

[Out]

integrate(sqrt(a*x^3 + b*x)*(a*x^2 + b)/((a*x^2 - b)*x^2), x)

Mupad [F(-1)]

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

[In]

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

[Out]

\text{Hanged}