3.23.86 \(\int \frac {(-b+a x^2) \sqrt [4]{-b x^2+a x^4}}{b+a x^2} \, dx\)

Optimal. Leaf size=174 \[ \frac {9 b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b x^2}}\right )}{4 a^{3/4}}-\frac {2 \sqrt [4]{2} b \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b x^2}}\right )}{a^{3/4}}-\frac {9 b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b x^2}}\right )}{4 a^{3/4}}+\frac {2 \sqrt [4]{2} b \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b x^2}}\right )}{a^{3/4}}+\frac {1}{2} x \sqrt [4]{a x^4-b x^2} \]

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Rubi [C]  time = 0.26, antiderivative size = 64, normalized size of antiderivative = 0.37, number of steps used = 4, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {2056, 466, 511, 510} \begin {gather*} -\frac {2 x \sqrt [4]{a x^4-b x^2} F_1\left (\frac {3}{4};1,-\frac {5}{4};\frac {7}{4};-\frac {a x^2}{b},\frac {a x^2}{b}\right )}{3 \sqrt [4]{1-\frac {a x^2}{b}}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*x*(-(b*x^2) + a*x^4)^(1/4)*AppellF1[3/4, 1, -5/4, 7/4, -((a*x^2)/b), (a*x^2)/b])/(3*(1 - (a*x^2)/b)^(1/4))

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 510

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

Rule 511

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

Rubi steps

\begin {align*} \int \frac {\left (-b+a x^2\right ) \sqrt [4]{-b x^2+a x^4}}{b+a x^2} \, dx &=\frac {\sqrt [4]{-b x^2+a x^4} \int \frac {\sqrt {x} \left (-b+a x^2\right )^{5/4}}{b+a x^2} \, dx}{\sqrt {x} \sqrt [4]{-b+a x^2}}\\ &=\frac {\left (2 \sqrt [4]{-b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-b+a x^4\right )^{5/4}}{b+a x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{-b+a x^2}}\\ &=-\frac {\left (2 b \sqrt [4]{-b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (1-\frac {a x^4}{b}\right )^{5/4}}{b+a x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1-\frac {a x^2}{b}}}\\ &=-\frac {2 x \sqrt [4]{-b x^2+a x^4} F_1\left (\frac {3}{4};1,-\frac {5}{4};\frac {7}{4};-\frac {a x^2}{b},\frac {a x^2}{b}\right )}{3 \sqrt [4]{1-\frac {a x^2}{b}}}\\ \end {align*}

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Mathematica [C]  time = 0.17, size = 164, normalized size = 0.94 \begin {gather*} \frac {x \sqrt [4]{a x^4-b x^2} \left (27 a x^2 \left (1-\frac {a^2 x^4}{b^2}\right )^{3/4} F_1\left (\frac {7}{4};\frac {3}{4},1;\frac {11}{4};\frac {a x^2}{b},-\frac {a x^2}{b}\right )-49 b \left (1-\frac {a x^2}{b}\right )^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};\frac {2 a x^2}{a x^2+b}\right )+21 \left (b-a x^2\right ) \left (\frac {a x^2}{b}+1\right )^{3/4}\right )}{42 \left (b-a x^2\right ) \left (\frac {a x^2}{b}+1\right )^{3/4}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x*(-(b*x^2) + a*x^4)^(1/4)*(21*(b - a*x^2)*(1 + (a*x^2)/b)^(3/4) + 27*a*x^2*(1 - (a^2*x^4)/b^2)^(3/4)*AppellF
1[7/4, 3/4, 1, 11/4, (a*x^2)/b, -((a*x^2)/b)] - 49*b*(1 - (a*x^2)/b)^(3/4)*Hypergeometric2F1[3/4, 3/4, 7/4, (2
*a*x^2)/(b + a*x^2)]))/(42*(b - a*x^2)*(1 + (a*x^2)/b)^(3/4))

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IntegrateAlgebraic [A]  time = 0.64, size = 174, normalized size = 1.00 \begin {gather*} \frac {1}{2} x \sqrt [4]{-b x^2+a x^4}+\frac {9 b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )}{4 a^{3/4}}-\frac {2 \sqrt [4]{2} b \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )}{a^{3/4}}-\frac {9 b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )}{4 a^{3/4}}+\frac {2 \sqrt [4]{2} b \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )}{a^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-b + a*x^2)*(-(b*x^2) + a*x^4)^(1/4))/(b + a*x^2),x]

[Out]

(x*(-(b*x^2) + a*x^4)^(1/4))/2 + (9*b*ArcTan[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/(4*a^(3/4)) - (2*2^(1/4)*b
*ArcTan[(2^(1/4)*a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/a^(3/4) - (9*b*ArcTanh[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(
1/4)])/(4*a^(3/4)) + (2*2^(1/4)*b*ArcTanh[(2^(1/4)*a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/a^(3/4)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [B]  time = 0.38, size = 403, normalized size = 2.32 \begin {gather*} \frac {1}{2} \, {\left (a - \frac {b}{x^{2}}\right )}^{\frac {1}{4}} x^{2} + \frac {2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} b \arctan \left (\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a - \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a} + \frac {2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} b \arctan \left (-\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a - \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a} + \frac {2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} b \log \left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a - \frac {b}{x^{2}}}\right )}{2 \, a} - \frac {2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} b \log \left (-2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a - \frac {b}{x^{2}}}\right )}{2 \, a} + \frac {9 \, \sqrt {2} b \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a - \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{8 \, \left (-a\right )^{\frac {3}{4}}} + \frac {9 \, \sqrt {2} b \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a - \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{8 \, \left (-a\right )^{\frac {3}{4}}} + \frac {9 \, \sqrt {2} b \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a - \frac {b}{x^{2}}}\right )}{16 \, \left (-a\right )^{\frac {3}{4}}} + \frac {9 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a - \frac {b}{x^{2}}}\right )}{16 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a - b/x^2)^(1/4)*x^2 + 2^(3/4)*(-a)^(1/4)*b*arctan(1/2*2^(1/4)*(2^(3/4)*(-a)^(1/4) + 2*(a - b/x^2)^(1/4))
/(-a)^(1/4))/a + 2^(3/4)*(-a)^(1/4)*b*arctan(-1/2*2^(1/4)*(2^(3/4)*(-a)^(1/4) - 2*(a - b/x^2)^(1/4))/(-a)^(1/4
))/a + 1/2*2^(3/4)*(-a)^(1/4)*b*log(2^(3/4)*(-a)^(1/4)*(a - b/x^2)^(1/4) + sqrt(2)*sqrt(-a) + sqrt(a - b/x^2))
/a - 1/2*2^(3/4)*(-a)^(1/4)*b*log(-2^(3/4)*(-a)^(1/4)*(a - b/x^2)^(1/4) + sqrt(2)*sqrt(-a) + sqrt(a - b/x^2))/
a + 9/8*sqrt(2)*b*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(a - b/x^2)^(1/4))/(-a)^(1/4))/(-a)^(3/4) + 9/8*s
qrt(2)*b*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) - 2*(a - b/x^2)^(1/4))/(-a)^(1/4))/(-a)^(3/4) + 9/16*sqrt(2)*
b*log(sqrt(2)*(-a)^(1/4)*(a - b/x^2)^(1/4) + sqrt(-a) + sqrt(a - b/x^2))/(-a)^(3/4) + 9/16*sqrt(2)*(-a)^(1/4)*
b*log(-sqrt(2)*(-a)^(1/4)*(a - b/x^2)^(1/4) + sqrt(-a) + sqrt(a - b/x^2))/a

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{2}-b \right ) \left (a \,x^{4}-b \,x^{2}\right )^{\frac {1}{4}}}{a \,x^{2}+b}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^2-b)*(a*x^4-b*x^2)^(1/4)/(a*x^2+b),x)

[Out]

int((a*x^2-b)*(a*x^4-b*x^2)^(1/4)/(a*x^2+b),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{2} - b\right )}}{a x^{2} + b}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*x^4 - b*x^2)^(1/4)*(a*x^2 - b)/(a*x^2 + b), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {\left (b-a\,x^2\right )\,{\left (a\,x^4-b\,x^2\right )}^{1/4}}{a\,x^2+b} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((b - a*x^2)*(a*x^4 - b*x^2)^(1/4))/(b + a*x^2),x)

[Out]

int(-((b - a*x^2)*(a*x^4 - b*x^2)^(1/4))/(b + a*x^2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{x^{2} \left (a x^{2} - b\right )} \left (a x^{2} - b\right )}{a x^{2} + b}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**2-b)*(a*x**4-b*x**2)**(1/4)/(a*x**2+b),x)

[Out]

Integral((x**2*(a*x**2 - b))**(1/4)*(a*x**2 - b)/(a*x**2 + b), x)

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