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

Optimal. Leaf size=130 \[ \frac {2 \sqrt {a x^3+b x}}{x}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^3+b x}}{a x^2+b}\right )-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^3+b x}}{a x^2+b}\right ) \]

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Rubi [A]  time = 0.60, antiderivative size = 176, normalized size of antiderivative = 1.35, number of steps used = 13, number of rules used = 10, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2056, 466, 474, 12, 490, 1211, 220, 1699, 205, 208} \begin {gather*} \frac {2 \sqrt {a x^3+b x}}{x}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^3+b x} \tan ^{-1}\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} \tanh ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[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 205

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

Rule 208

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

Rule 220

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)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 466

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 474

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 490

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), In
t[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 1211

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 1699

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 2056

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*} \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 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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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} \tan ^{-1}\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} \tanh ^{-1}\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*}

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Mathematica [C]  time = 0.07, size = 75, normalized size = 0.58 \begin {gather*} \frac {6 \left (a x^2+b\right )-8 a x^2 \sqrt {\frac {a x^2}{b}+1} F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};-\frac {a x^2}{b},\frac {a x^2}{b}\right )}{3 \sqrt {x \left (a x^2+b\right )}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(6*(b + a*x^2) - 8*a*x^2*Sqrt[1 + (a*x^2)/b]*AppellF1[3/4, 1/2, 1, 7/4, -((a*x^2)/b), (a*x^2)/b])/(3*Sqrt[x*(b
+ a*x^2)])

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IntegrateAlgebraic [A]  time = 0.37, size = 130, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {b x+a x^3}}{x}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan ^{-1}\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} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}{b+a x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

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fricas [B]  time = 74.41, size = 1139, normalized size = 8.76

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[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*4^(1/4)*(a*b)^(1/4)*x*arctan(-1/2*(sqrt(2)*sqrt(20*a^4*b + 44*a^3*b^2 + 8*a^2*b^3 + (4*a^4 + 41*a^3*b
+ 26*a^2*b^2 + a*b^3)*sqrt(a*b))*(4^(3/4)*((a^3 + 2*a^2*b)*x^4 - 2*(5*a^2*b + a*b^2)*x^3 + a*b^2 + 2*b^3 + 6*(
a^2*b + 2*a*b^2)*x^2 - 2*(5*a*b^2 + b^3)*x)*(a*b)^(3/4) - 4^(1/4)*((5*a^3*b + a^2*b^2)*x^4 + 5*a*b^3 + b^4 - 8
*(a^3*b + 2*a^2*b^2)*x^3 + 6*(5*a^2*b^2 + a*b^3)*x^2 - 8*(a^2*b^2 + 2*a*b^3)*x)*(a*b)^(1/4)) + 4*sqrt(a*x^3 +
b*x)*(4^(3/4)*(4*a^4*b - 9*a^3*b^2 + 6*a^2*b^3 - a*b^4)*(a*b)^(3/4)*x + 4^(1/4)*(4*a^4*b^2 - 9*a^3*b^3 + 6*a^2
*b^4 - a*b^5 + (4*a^5*b - 9*a^4*b^2 + 6*a^3*b^3 - a^2*b^4)*x^2)*(a*b)^(1/4)))/(4*a^4*b^3 - 9*a^3*b^4 + 6*a^2*b
^5 - a*b^6 + (4*a^6*b - 9*a^5*b^2 + 6*a^4*b^3 - a^3*b^4)*x^4 - 2*(4*a^5*b^2 - 9*a^4*b^3 + 6*a^3*b^4 - a^2*b^5)
*x^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)) - 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)) - 8*sqrt(a*x^3 + b*x))/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x^{3} + b x} {\left (a x^{2} + b\right )}}{{\left (a x^{2} - b\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [C]  time = 0.19, size = 318, normalized size = 2.45

method	result	size
elliptic	\(\frac {2 a \,x^{2}+2 b}{\sqrt {x \left (a \,x^{2}+b \right )}}+\frac {2 b \sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}+\frac {2 b \sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}\)	\(318\)
risch	\(\frac {2 a \,x^{2}+2 b}{\sqrt {x \left (a \,x^{2}+b \right )}}+4 b a \left (\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 a^{2} \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}+\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 a^{2} \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}\right )\)	\(321\)
default	\(2 a \left (\frac {2 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}}}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}+b x}}-\frac {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}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}+b x}}+\frac {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}}}\, \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^{2} \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}+\frac {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}}}\, \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^{2} \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}\right )+\frac {2 a \,x^{2}+2 b}{\sqrt {x \left (a \,x^{2}+b \right )}}-\frac {2 \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}\right )}{\sqrt {a \,x^{3}+b x}}\)	\(643\)

Verification of antiderivative is not currently implemented for this CAS.

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x^{3} + b x} {\left (a x^{2} + b\right )}}{{\left (a x^{2} - b\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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