3.9.36 \(\int \frac {x^6}{(a-b x^2)^{3/4}} \, dx\) [836]

Optimal. Leaf size=129 \[ -\frac {40 a^2 x \sqrt [4]{a-b x^2}}{77 b^3}-\frac {20 a x^3 \sqrt [4]{a-b x^2}}{77 b^2}-\frac {2 x^5 \sqrt [4]{a-b x^2}}{11 b}+\frac {80 a^{7/2} \left (1-\frac {b x^2}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{77 b^{7/2} \left (a-b x^2\right )^{3/4}} \]

[Out]

-40/77*a^2*x*(-b*x^2+a)^(1/4)/b^3-20/77*a*x^3*(-b*x^2+a)^(1/4)/b^2-2/11*x^5*(-b*x^2+a)^(1/4)/b+80/77*a^(7/2)*(
1-b*x^2/a)^(3/4)*(cos(1/2*arcsin(x*b^(1/2)/a^(1/2)))^2)^(1/2)/cos(1/2*arcsin(x*b^(1/2)/a^(1/2)))*EllipticF(sin
(1/2*arcsin(x*b^(1/2)/a^(1/2))),2^(1/2))/b^(7/2)/(-b*x^2+a)^(3/4)

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Rubi [A]
time = 0.03, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {327, 239, 238} \begin {gather*} \frac {80 a^{7/2} \left (1-\frac {b x^2}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \text {ArcSin}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{77 b^{7/2} \left (a-b x^2\right )^{3/4}}-\frac {40 a^2 x \sqrt [4]{a-b x^2}}{77 b^3}-\frac {20 a x^3 \sqrt [4]{a-b x^2}}{77 b^2}-\frac {2 x^5 \sqrt [4]{a-b x^2}}{11 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^6/(a - b*x^2)^(3/4),x]

[Out]

(-40*a^2*x*(a - b*x^2)^(1/4))/(77*b^3) - (20*a*x^3*(a - b*x^2)^(1/4))/(77*b^2) - (2*x^5*(a - b*x^2)^(1/4))/(11
*b) + (80*a^(7/2)*(1 - (b*x^2)/a)^(3/4)*EllipticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(77*b^(7/2)*(a - b*x^2)^(
3/4))

Rule 238

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[-b/a, 2]))*EllipticF[(1/2)*ArcSin[Rt[-b/a,
2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b/a]

Rule 239

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Dist[(1 + b*(x^2/a))^(3/4)/(a + b*x^2)^(3/4), Int[1/(1 + b*(x^2
/a))^(3/4), x], x] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 327

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

Rubi steps

\begin {align*} \int \frac {x^6}{\left (a-b x^2\right )^{3/4}} \, dx &=-\frac {2 x^5 \sqrt [4]{a-b x^2}}{11 b}+\frac {(10 a) \int \frac {x^4}{\left (a-b x^2\right )^{3/4}} \, dx}{11 b}\\ &=-\frac {20 a x^3 \sqrt [4]{a-b x^2}}{77 b^2}-\frac {2 x^5 \sqrt [4]{a-b x^2}}{11 b}+\frac {\left (60 a^2\right ) \int \frac {x^2}{\left (a-b x^2\right )^{3/4}} \, dx}{77 b^2}\\ &=-\frac {40 a^2 x \sqrt [4]{a-b x^2}}{77 b^3}-\frac {20 a x^3 \sqrt [4]{a-b x^2}}{77 b^2}-\frac {2 x^5 \sqrt [4]{a-b x^2}}{11 b}+\frac {\left (40 a^3\right ) \int \frac {1}{\left (a-b x^2\right )^{3/4}} \, dx}{77 b^3}\\ &=-\frac {40 a^2 x \sqrt [4]{a-b x^2}}{77 b^3}-\frac {20 a x^3 \sqrt [4]{a-b x^2}}{77 b^2}-\frac {2 x^5 \sqrt [4]{a-b x^2}}{11 b}+\frac {\left (40 a^3 \left (1-\frac {b x^2}{a}\right )^{3/4}\right ) \int \frac {1}{\left (1-\frac {b x^2}{a}\right )^{3/4}} \, dx}{77 b^3 \left (a-b x^2\right )^{3/4}}\\ &=-\frac {40 a^2 x \sqrt [4]{a-b x^2}}{77 b^3}-\frac {20 a x^3 \sqrt [4]{a-b x^2}}{77 b^2}-\frac {2 x^5 \sqrt [4]{a-b x^2}}{11 b}+\frac {80 a^{7/2} \left (1-\frac {b x^2}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{77 b^{7/2} \left (a-b x^2\right )^{3/4}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 7.13, size = 91, normalized size = 0.71 \begin {gather*} \frac {2 \left (-20 a^3 x+10 a^2 b x^3+3 a b^2 x^5+7 b^3 x^7+20 a^3 x \left (1-\frac {b x^2}{a}\right )^{3/4} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {3}{2};\frac {b x^2}{a}\right )\right )}{77 b^3 \left (a-b x^2\right )^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^6/(a - b*x^2)^(3/4),x]

[Out]

(2*(-20*a^3*x + 10*a^2*b*x^3 + 3*a*b^2*x^5 + 7*b^3*x^7 + 20*a^3*x*(1 - (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/2,
3/4, 3/2, (b*x^2)/a]))/(77*b^3*(a - b*x^2)^(3/4))

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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {x^{6}}{\left (-b \,x^{2}+a \right )^{\frac {3}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(-b*x^2+a)^(3/4),x)

[Out]

int(x^6/(-b*x^2+a)^(3/4),x)

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

[Out]

integrate(x^6/(-b*x^2 + a)^(3/4), x)

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Fricas [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(x^6/(-b*x^2+a)^(3/4),x, algorithm="fricas")

[Out]

integral(-(-b*x^2 + a)^(1/4)*x^6/(b*x^2 - a), x)

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Sympy [C] Result contains complex when optimal does not.
time = 0.60, size = 29, normalized size = 0.22 \begin {gather*} \frac {x^{7} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{7 a^{\frac {3}{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**7*hyper((3/4, 7/2), (9/2,), b*x**2*exp_polar(2*I*pi)/a)/(7*a**(3/4))

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

[Out]

integrate(x^6/(-b*x^2 + a)^(3/4), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^6}{{\left (a-b\,x^2\right )}^{3/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(a - b*x^2)^(3/4),x)

[Out]

int(x^6/(a - b*x^2)^(3/4), x)

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