3.8.47 \(\int \frac {-7+x+7 x^2}{(1+x^2) \sqrt {-x+x^3}} \, dx\)

Optimal. Leaf size=58 \[ -\frac {15}{4} \tan ^{-1}\left (\frac {2 \sqrt {x^3-x}}{x^2-2 x-1}\right )-\frac {13}{4} \tanh ^{-1}\left (\frac {\frac {x^2}{2}+x-\frac {1}{2}}{\sqrt {x^3-x}}\right ) \]

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Rubi [C]  time = 0.69, antiderivative size = 183, normalized size of antiderivative = 3.16, number of steps used = 13, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2056, 6725, 329, 222, 933, 168, 537} \begin {gather*} \frac {7 \sqrt {2} \sqrt {x-1} \sqrt {x} \sqrt {x+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}\right )|\frac {1}{2}\right )}{\sqrt {x^3-x}}+\frac {\left (\frac {13}{2}+\frac {15 i}{2}\right ) \sqrt {x} \sqrt {1-x^2} \Pi \left (\frac {1}{2}-\frac {i}{2};\sin ^{-1}\left (\sqrt {1-x}\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^3-x}}+\frac {\left (\frac {13}{2}-\frac {15 i}{2}\right ) \sqrt {x} \sqrt {1-x^2} \Pi \left (\frac {1}{2}+\frac {i}{2};\sin ^{-1}\left (\sqrt {1-x}\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^3-x}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[(-7 + x + 7*x^2)/((1 + x^2)*Sqrt[-x + x^3]),x]

[Out]

(7*Sqrt[2]*Sqrt[-1 + x]*Sqrt[x]*Sqrt[1 + x]*EllipticF[ArcSin[(Sqrt[2]*Sqrt[x])/Sqrt[-1 + x]], 1/2])/Sqrt[-x +
x^3] + ((13/2 + (15*I)/2)*Sqrt[x]*Sqrt[1 - x^2]*EllipticPi[1/2 - I/2, ArcSin[Sqrt[1 - x]], 1/2])/(Sqrt[2]*Sqrt
[-x + x^3]) + ((13/2 - (15*I)/2)*Sqrt[x]*Sqrt[1 - x^2]*EllipticPi[1/2 + I/2, ArcSin[Sqrt[1 - x]], 1/2])/(Sqrt[
2]*Sqrt[-x + x^3])

Rule 168

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + (f*x^2)/d, x]]*Sqrt[Simp[(d
*g - c*h)/d + (h*x^2)/d, x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c
*f)/d, 0]

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-(a*b), 2]}, Simp[(Sqrt[-a + q*x^2]*Sqrt[(a + q*x^2
)/q]*EllipticF[ArcSin[x/Sqrt[(a + q*x^2)/(2*q)]], 1/2])/(Sqrt[2]*Sqrt[-a]*Sqrt[a + b*x^4]), x] /; IntegerQ[q]]
/; FreeQ[{a, b}, x] && LtQ[a, 0] && GtQ[b, 0]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])

Rule 933

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[-(c
/a), 2]}, Dist[Sqrt[1 + (c*x^2)/a]/Sqrt[a + c*x^2], Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]
), x], x]] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] &&  !GtQ[a, 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]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {-7+x+7 x^2}{\left (1+x^2\right ) \sqrt {-x+x^3}} \, dx &=\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \int \frac {-7+x+7 x^2}{\sqrt {x} \sqrt {-1+x^2} \left (1+x^2\right )} \, dx}{\sqrt {-x+x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \int \left (\frac {7}{\sqrt {x} \sqrt {-1+x^2}}-\frac {14-x}{\sqrt {x} \sqrt {-1+x^2} \left (1+x^2\right )}\right ) \, dx}{\sqrt {-x+x^3}}\\ &=-\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \int \frac {14-x}{\sqrt {x} \sqrt {-1+x^2} \left (1+x^2\right )} \, dx}{\sqrt {-x+x^3}}+\frac {\left (7 \sqrt {x} \sqrt {-1+x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {-1+x^2}} \, dx}{\sqrt {-x+x^3}}\\ &=-\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \int \left (\frac {\frac {1}{2}+7 i}{(i-x) \sqrt {x} \sqrt {-1+x^2}}-\frac {\frac {1}{2}-7 i}{\sqrt {x} (i+x) \sqrt {-1+x^2}}\right ) \, dx}{\sqrt {-x+x^3}}+\frac {\left (14 \sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}}\\ &=\frac {7 \sqrt {2} \sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{\sqrt {-x+x^3}}--\frac {\left (\left (\frac {1}{2}-7 i\right ) \sqrt {x} \sqrt {-1+x^2}\right ) \int \frac {1}{\sqrt {x} (i+x) \sqrt {-1+x^2}} \, dx}{\sqrt {-x+x^3}}-\frac {\left (\left (\frac {1}{2}+7 i\right ) \sqrt {x} \sqrt {-1+x^2}\right ) \int \frac {1}{(i-x) \sqrt {x} \sqrt {-1+x^2}} \, dx}{\sqrt {-x+x^3}}\\ &=\frac {7 \sqrt {2} \sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{\sqrt {-x+x^3}}--\frac {\left (\left (\frac {1}{2}-7 i\right ) \sqrt {x} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {1-x} \sqrt {x} (i+x) \sqrt {1+x}} \, dx}{\sqrt {-x+x^3}}-\frac {\left (\left (\frac {1}{2}+7 i\right ) \sqrt {x} \sqrt {1-x^2}\right ) \int \frac {1}{(i-x) \sqrt {1-x} \sqrt {x} \sqrt {1+x}} \, dx}{\sqrt {-x+x^3}}\\ &=\frac {7 \sqrt {2} \sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{\sqrt {-x+x^3}}--\frac {\left ((1+14 i) \sqrt {x} \sqrt {1-x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {2-x^2} \left ((-1+i)+x^2\right )} \, dx,x,\sqrt {1-x}\right )}{\sqrt {-x+x^3}}-\frac {\left ((1-14 i) \sqrt {x} \sqrt {1-x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \left ((1+i)-x^2\right ) \sqrt {2-x^2}} \, dx,x,\sqrt {1-x}\right )}{\sqrt {-x+x^3}}\\ &=\frac {7 \sqrt {2} \sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{\sqrt {-x+x^3}}+\frac {\left (\frac {13}{2}+\frac {15 i}{2}\right ) \sqrt {x} \sqrt {1-x^2} \Pi \left (\frac {1}{2}-\frac {i}{2};\sin ^{-1}\left (\sqrt {1-x}\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt {-x+x^3}}+\frac {\left (\frac {13}{2}-\frac {15 i}{2}\right ) \sqrt {x} \sqrt {1-x^2} \Pi \left (\frac {1}{2}+\frac {i}{2};\sin ^{-1}\left (\sqrt {1-x}\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt {-x+x^3}}\\ \end {align*}

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Mathematica [C]  time = 0.53, size = 95, normalized size = 1.64 \begin {gather*} \frac {2 x \sqrt {1-x^2} \left (x \left (5 F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};x^2,-x^2\right )+21 x F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};x^2,-x^2\right )\right )-105 F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};x^2,-x^2\right )\right )}{15 \sqrt {x \left (x^2-1\right )}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(-7 + x + 7*x^2)/((1 + x^2)*Sqrt[-x + x^3]),x]

[Out]

(2*x*Sqrt[1 - x^2]*(-105*AppellF1[1/4, 1/2, 1, 5/4, x^2, -x^2] + x*(5*AppellF1[3/4, 1/2, 1, 7/4, x^2, -x^2] +
21*x*AppellF1[5/4, 1/2, 1, 9/4, x^2, -x^2])))/(15*Sqrt[x*(-1 + x^2)])

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IntegrateAlgebraic [A]  time = 0.28, size = 58, normalized size = 1.00 \begin {gather*} -\frac {15}{4} \tan ^{-1}\left (\frac {2 \sqrt {-x+x^3}}{-1-2 x+x^2}\right )-\frac {13}{4} \tanh ^{-1}\left (\frac {-\frac {1}{2}+x+\frac {x^2}{2}}{\sqrt {-x+x^3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-7 + x + 7*x^2)/((1 + x^2)*Sqrt[-x + x^3]),x]

[Out]

(-15*ArcTan[(2*Sqrt[-x + x^3])/(-1 - 2*x + x^2)])/4 - (13*ArcTanh[(-1/2 + x + x^2/2)/Sqrt[-x + x^3]])/4

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fricas [A]  time = 0.74, size = 76, normalized size = 1.31 \begin {gather*} \frac {15}{4} \, \arctan \left (\frac {x^{2} - 2 \, x - 1}{2 \, \sqrt {x^{3} - x}}\right ) + \frac {13}{8} \, \log \left (\frac {x^{4} + 8 \, x^{3} + 2 \, x^{2} - 4 \, \sqrt {x^{3} - x} {\left (x^{2} + 2 \, x - 1\right )} - 8 \, x + 1}{x^{4} + 2 \, x^{2} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((7*x^2+x-7)/(x^2+1)/(x^3-x)^(1/2),x, algorithm="fricas")

[Out]

15/4*arctan(1/2*(x^2 - 2*x - 1)/sqrt(x^3 - x)) + 13/8*log((x^4 + 8*x^3 + 2*x^2 - 4*sqrt(x^3 - x)*(x^2 + 2*x -
1) - 8*x + 1)/(x^4 + 2*x^2 + 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {7 \, x^{2} + x - 7}{\sqrt {x^{3} - x} {\left (x^{2} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((7*x^2+x-7)/(x^2+1)/(x^3-x)^(1/2),x, algorithm="giac")

[Out]

integrate((7*x^2 + x - 7)/(sqrt(x^3 - x)*(x^2 + 1)), x)

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maple [C]  time = 1.60, size = 211, normalized size = 3.64

method	result	size
default	\(\frac {7 \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticF \left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}-x}}-\frac {15 \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}-\frac {13 i \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}-\frac {15 \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}+\frac {13 i \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}\)	\(211\)
elliptic	\(\frac {7 \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticF \left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}-x}}-\frac {15 \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}-\frac {13 i \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}-\frac {15 \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}+\frac {13 i \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}\)	\(211\)
trager	\(-\frac {13 \ln \left (\frac {464 \RootOf \left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )^{2} x^{2}-696 \RootOf \left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )^{2} x +4141 \RootOf \left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right ) x^{2}+6000 \sqrt {x^{3}-x}\, \RootOf \left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )-464 \RootOf \left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )^{2}-774 \RootOf \left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right ) x +8051 x^{2}+13875 \sqrt {x^{3}-x}-4141 \RootOf \left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )+3486 x -8051}{\left (4 \RootOf \left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right ) x -8 \RootOf \left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )-17 x -41\right )^{2}}\right )}{4}-\frac {\ln \left (\frac {464 \RootOf \left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )^{2} x^{2}-696 \RootOf \left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )^{2} x +4141 \RootOf \left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right ) x^{2}+6000 \sqrt {x^{3}-x}\, \RootOf \left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )-464 \RootOf \left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )^{2}-774 \RootOf \left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right ) x +8051 x^{2}+13875 \sqrt {x^{3}-x}-4141 \RootOf \left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )+3486 x -8051}{\left (4 \RootOf \left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right ) x -8 \RootOf \left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )-17 x -41\right )^{2}}\right ) \RootOf \left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )}{2}+\frac {\RootOf \left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right ) \ln \left (-\frac {-928 \RootOf \left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )^{2} x^{2}+1392 \RootOf \left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )^{2} x -3782 \RootOf \left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right ) x^{2}+12000 \sqrt {x^{3}-x}\, \RootOf \left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )+928 \RootOf \left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )^{2}+16548 \RootOf \left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right ) x -1477 x^{2}+50250 \sqrt {x^{3}-x}+3782 \RootOf \left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )+41778 x +1477}{\left (4 \RootOf \left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right ) x -8 \RootOf \left (8 \textit {\_Z}^{2}+52 \textit {\_Z} +197\right )+43 x -11\right )^{2}}\right )}{2}\)	\(544\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((7*x^2+x-7)/(x^2+1)/(x^3-x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

7*(1+x)^(1/2)*(2-2*x)^(1/2)*(-x)^(1/2)/(x^3-x)^(1/2)*EllipticF((1+x)^(1/2),1/2*2^(1/2))-15/4*(1+x)^(1/2)*(2-2*
x)^(1/2)*(-x)^(1/2)/(x^3-x)^(1/2)*EllipticPi((1+x)^(1/2),1/2-1/2*I,1/2*2^(1/2))-13/4*I*(1+x)^(1/2)*(2-2*x)^(1/
2)*(-x)^(1/2)/(x^3-x)^(1/2)*EllipticPi((1+x)^(1/2),1/2-1/2*I,1/2*2^(1/2))-15/4*(1+x)^(1/2)*(2-2*x)^(1/2)*(-x)^
(1/2)/(x^3-x)^(1/2)*EllipticPi((1+x)^(1/2),1/2+1/2*I,1/2*2^(1/2))+13/4*I*(1+x)^(1/2)*(2-2*x)^(1/2)*(-x)^(1/2)/
(x^3-x)^(1/2)*EllipticPi((1+x)^(1/2),1/2+1/2*I,1/2*2^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {7 \, x^{2} + x - 7}{\sqrt {x^{3} - x} {\left (x^{2} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((7*x^2+x-7)/(x^2+1)/(x^3-x)^(1/2),x, algorithm="maxima")

[Out]

integrate((7*x^2 + x - 7)/(sqrt(x^3 - x)*(x^2 + 1)), x)

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mupad [B]  time = 0.56, size = 100, normalized size = 1.72 \begin {gather*} \frac {-14\,\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )+\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (-\mathrm {i};\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )\,\left (14+1{}\mathrm {i}\right )+\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (1{}\mathrm {i};\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )\,\left (14-\mathrm {i}\right )}{\sqrt {x^3-x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 7*x^2 - 7)/((x^3 - x)^(1/2)*(x^2 + 1)),x)

[Out]

((-x)^(1/2)*(1 - x)^(1/2)*(x + 1)^(1/2)*ellipticPi(-1i, asin((-x)^(1/2)), -1)*(14 + 1i) - 14*(-x)^(1/2)*(1 - x
)^(1/2)*(x + 1)^(1/2)*ellipticF(asin((-x)^(1/2)), -1) + (-x)^(1/2)*(1 - x)^(1/2)*(x + 1)^(1/2)*ellipticPi(1i,
asin((-x)^(1/2)), -1)*(14 - 1i))/(x^3 - x)^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {7 x^{2} + x - 7}{\sqrt {x \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((7*x**2+x-7)/(x**2+1)/(x**3-x)**(1/2),x)

[Out]

Integral((7*x**2 + x - 7)/(sqrt(x*(x - 1)*(x + 1))*(x**2 + 1)), x)

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