3.9.72 \(\int \frac {1}{x^6 (a+b x^4)^{3/2}} \, dx\) [872]

Optimal. Leaf size=282 \[ \frac {1}{2 a x^5 \sqrt {a+b x^4}}-\frac {7 \sqrt {a+b x^4}}{10 a^2 x^5}+\frac {21 b \sqrt {a+b x^4}}{10 a^3 x}-\frac {21 b^{3/2} x \sqrt {a+b x^4}}{10 a^3 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {21 b^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{10 a^{11/4} \sqrt {a+b x^4}}-\frac {21 b^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{20 a^{11/4} \sqrt {a+b x^4}} \]

[Out]

1/2/a/x^5/(b*x^4+a)^(1/2)-7/10*(b*x^4+a)^(1/2)/a^2/x^5+21/10*b*(b*x^4+a)^(1/2)/a^3/x-21/10*b^(3/2)*x*(b*x^4+a)
^(1/2)/a^3/(a^(1/2)+x^2*b^(1/2))+21/10*b^(5/4)*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)
*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x^2*b^(1/2))*((b*x^4+a)/(a^(1/2)
+x^2*b^(1/2))^2)^(1/2)/a^(11/4)/(b*x^4+a)^(1/2)-21/20*b^(5/4)*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2
*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x^2*b^(1/2))*((b*
x^4+a)/(a^(1/2)+x^2*b^(1/2))^2)^(1/2)/a^(11/4)/(b*x^4+a)^(1/2)

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Rubi [A]
time = 0.08, antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {296, 331, 311, 226, 1210} \begin {gather*} -\frac {21 b^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{20 a^{11/4} \sqrt {a+b x^4}}+\frac {21 b^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{10 a^{11/4} \sqrt {a+b x^4}}-\frac {21 b^{3/2} x \sqrt {a+b x^4}}{10 a^3 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {21 b \sqrt {a+b x^4}}{10 a^3 x}-\frac {7 \sqrt {a+b x^4}}{10 a^2 x^5}+\frac {1}{2 a x^5 \sqrt {a+b x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^6*(a + b*x^4)^(3/2)),x]

[Out]

1/(2*a*x^5*Sqrt[a + b*x^4]) - (7*Sqrt[a + b*x^4])/(10*a^2*x^5) + (21*b*Sqrt[a + b*x^4])/(10*a^3*x) - (21*b^(3/
2)*x*Sqrt[a + b*x^4])/(10*a^3*(Sqrt[a] + Sqrt[b]*x^2)) + (21*b^(5/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/
(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticE[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(10*a^(11/4)*Sqrt[a + b*x^4]) - (21*
b^(5/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(
1/4)], 1/2])/(20*a^(11/4)*Sqrt[a + b*x^4])

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 296

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-(c*x)^(m + 1))*((a + b*x^n)^(p + 1)/
(a*c*n*(p + 1))), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 311

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

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 1210

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

Rubi steps

\begin {align*} \int \frac {1}{x^6 \left (a+b x^4\right )^{3/2}} \, dx &=\frac {1}{2 a x^5 \sqrt {a+b x^4}}+\frac {7 \int \frac {1}{x^6 \sqrt {a+b x^4}} \, dx}{2 a}\\ &=\frac {1}{2 a x^5 \sqrt {a+b x^4}}-\frac {7 \sqrt {a+b x^4}}{10 a^2 x^5}-\frac {(21 b) \int \frac {1}{x^2 \sqrt {a+b x^4}} \, dx}{10 a^2}\\ &=\frac {1}{2 a x^5 \sqrt {a+b x^4}}-\frac {7 \sqrt {a+b x^4}}{10 a^2 x^5}+\frac {21 b \sqrt {a+b x^4}}{10 a^3 x}-\frac {\left (21 b^2\right ) \int \frac {x^2}{\sqrt {a+b x^4}} \, dx}{10 a^3}\\ &=\frac {1}{2 a x^5 \sqrt {a+b x^4}}-\frac {7 \sqrt {a+b x^4}}{10 a^2 x^5}+\frac {21 b \sqrt {a+b x^4}}{10 a^3 x}-\frac {\left (21 b^{3/2}\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{10 a^{5/2}}+\frac {\left (21 b^{3/2}\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx}{10 a^{5/2}}\\ &=\frac {1}{2 a x^5 \sqrt {a+b x^4}}-\frac {7 \sqrt {a+b x^4}}{10 a^2 x^5}+\frac {21 b \sqrt {a+b x^4}}{10 a^3 x}-\frac {21 b^{3/2} x \sqrt {a+b x^4}}{10 a^3 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {21 b^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{10 a^{11/4} \sqrt {a+b x^4}}-\frac {21 b^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{20 a^{11/4} \sqrt {a+b x^4}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.01, size = 54, normalized size = 0.19 \begin {gather*} -\frac {\sqrt {1+\frac {b x^4}{a}} \, _2F_1\left (-\frac {5}{4},\frac {3}{2};-\frac {1}{4};-\frac {b x^4}{a}\right )}{5 a x^5 \sqrt {a+b x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^6*(a + b*x^4)^(3/2)),x]

[Out]

-1/5*(Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[-5/4, 3/2, -1/4, -((b*x^4)/a)])/(a*x^5*Sqrt[a + b*x^4])

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Maple [C] Result contains complex when optimal does not.
time = 0.16, size = 157, normalized size = 0.56

method	result	size
default	\(-\frac {\sqrt {b \,x^{4}+a}}{5 a^{2} x^{5}}+\frac {8 b \sqrt {b \,x^{4}+a}}{5 a^{3} x}+\frac {b^{2} x^{3}}{2 a^{3} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {21 i b^{\frac {3}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{10 a^{\frac {5}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\)	\(157\)
elliptic	\(-\frac {\sqrt {b \,x^{4}+a}}{5 a^{2} x^{5}}+\frac {8 b \sqrt {b \,x^{4}+a}}{5 a^{3} x}+\frac {b^{2} x^{3}}{2 a^{3} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {21 i b^{\frac {3}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{10 a^{\frac {5}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\)	\(157\)
risch	\(-\frac {\sqrt {b \,x^{4}+a}\, \left (-8 b \,x^{4}+a \right )}{5 a^{3} x^{5}}-\frac {b^{2} \left (8 b \left (-\frac {x^{3}}{2 b \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {3 i \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 b^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+3 a \left (\frac {x^{3}}{2 a \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {i \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\right )\right )}{5 a^{3}}\)	\(278\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^6/(b*x^4+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/5*(b*x^4+a)^(1/2)/a^2/x^5+8/5*b*(b*x^4+a)^(1/2)/a^3/x+1/2*b^2*x^3/a^3/((x^4+a/b)*b)^(1/2)-21/10*I*b^(3/2)/a
^(5/2)/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4+a)^(1/
2)*(EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/2),I)-EllipticE(x*(I/a^(1/2)*b^(1/2))^(1/2),I))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(b*x^4+a)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(3/2)*x^6), x)

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Fricas [A]
time = 0.08, size = 132, normalized size = 0.47 \begin {gather*} \frac {21 \, {\left (b^{2} x^{9} + a b x^{5}\right )} \sqrt {a} \left (-\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - 21 \, {\left (b^{2} x^{9} + a b x^{5}\right )} \sqrt {a} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + {\left (21 \, b^{2} x^{8} + 14 \, a b x^{4} - 2 \, a^{2}\right )} \sqrt {b x^{4} + a}}{10 \, {\left (a^{3} b x^{9} + a^{4} x^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(b*x^4+a)^(3/2),x, algorithm="fricas")

[Out]

1/10*(21*(b^2*x^9 + a*b*x^5)*sqrt(a)*(-b/a)^(3/4)*elliptic_e(arcsin(x*(-b/a)^(1/4)), -1) - 21*(b^2*x^9 + a*b*x
^5)*sqrt(a)*(-b/a)^(3/4)*elliptic_f(arcsin(x*(-b/a)^(1/4)), -1) + (21*b^2*x^8 + 14*a*b*x^4 - 2*a^2)*sqrt(b*x^4
+ a))/(a^3*b*x^9 + a^4*x^5)

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Sympy [C] Result contains complex when optimal does not.
time = 0.58, size = 44, normalized size = 0.16 \begin {gather*} \frac {\Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {3}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} x^{5} \Gamma \left (- \frac {1}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**6/(b*x**4+a)**(3/2),x)

[Out]

gamma(-5/4)*hyper((-5/4, 3/2), (-1/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**(3/2)*x**5*gamma(-1/4))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(b*x^4+a)^(3/2),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(3/2)*x^6), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x^6\,{\left (b\,x^4+a\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^6*(a + b*x^4)^(3/2)),x)

[Out]

int(1/(x^6*(a + b*x^4)^(3/2)), x)

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