3.9.43 \(\int \frac {x^6}{(a+b x^2)^{5/4}} \, dx\) [843]

Optimal. Leaf size=124 \[ \frac {8 a^2 x}{3 b^3 \sqrt [4]{a+b x^2}}-\frac {4 a x^3}{9 b^2 \sqrt [4]{a+b x^2}}+\frac {2 x^5}{9 b \sqrt [4]{a+b x^2}}-\frac {16 a^{5/2} \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{3 b^{7/2} \sqrt [4]{a+b x^2}} \]

[Out]

8/3*a^2*x/b^3/(b*x^2+a)^(1/4)-4/9*a*x^3/b^2/(b*x^2+a)^(1/4)+2/9*x^5/b/(b*x^2+a)^(1/4)-16/3*a^(5/2)*(1+b*x^2/a)
^(1/4)*(cos(1/2*arctan(x*b^(1/2)/a^(1/2)))^2)^(1/2)/cos(1/2*arctan(x*b^(1/2)/a^(1/2)))*EllipticE(sin(1/2*arcta
n(x*b^(1/2)/a^(1/2))),2^(1/2))/b^(7/2)/(b*x^2+a)^(1/4)

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Rubi [A]
time = 0.03, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {291, 203, 202} \begin {gather*} -\frac {16 a^{5/2} \sqrt [4]{\frac {b x^2}{a}+1} E\left (\left .\frac {1}{2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{3 b^{7/2} \sqrt [4]{a+b x^2}}+\frac {8 a^2 x}{3 b^3 \sqrt [4]{a+b x^2}}-\frac {4 a x^3}{9 b^2 \sqrt [4]{a+b x^2}}+\frac {2 x^5}{9 b \sqrt [4]{a+b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^6/(a + b*x^2)^(5/4),x]

[Out]

(8*a^2*x)/(3*b^3*(a + b*x^2)^(1/4)) - (4*a*x^3)/(9*b^2*(a + b*x^2)^(1/4)) + (2*x^5)/(9*b*(a + b*x^2)^(1/4)) -
(16*a^(5/2)*(1 + (b*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(3*b^(7/2)*(a + b*x^2)^(1/4))

Rule 202

Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(2/(a^(5/4)*Rt[b/a, 2]))*EllipticE[(1/2)*ArcTan[Rt[b/a, 2]
*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Dist[(1 + b*(x^2/a))^(1/4)/(a*(a + b*x^2)^(1/4)), Int[1/(1 + b*
(x^2/a))^(5/4), x], x] /; FreeQ[{a, b}, x] && PosQ[a] && PosQ[b/a]

Rule 291

Int[((c_.)*(x_))^(m_)/((a_) + (b_.)*(x_)^2)^(5/4), x_Symbol] :> Simp[2*c*((c*x)^(m - 1)/(b*(2*m - 3)*(a + b*x^
2)^(1/4))), x] - Dist[2*a*c^2*((m - 1)/(b*(2*m - 3))), Int[(c*x)^(m - 2)/(a + b*x^2)^(5/4), x], x] /; FreeQ[{a
, b, c}, x] && PosQ[b/a] && IntegerQ[2*m] && GtQ[m, 3/2]

Rubi steps

\begin {align*} \int \frac {x^6}{\left (a+b x^2\right )^{5/4}} \, dx &=\frac {2 x^5}{9 b \sqrt [4]{a+b x^2}}-\frac {(10 a) \int \frac {x^4}{\left (a+b x^2\right )^{5/4}} \, dx}{9 b}\\ &=-\frac {4 a x^3}{9 b^2 \sqrt [4]{a+b x^2}}+\frac {2 x^5}{9 b \sqrt [4]{a+b x^2}}+\frac {\left (4 a^2\right ) \int \frac {x^2}{\left (a+b x^2\right )^{5/4}} \, dx}{3 b^2}\\ &=\frac {8 a^2 x}{3 b^3 \sqrt [4]{a+b x^2}}-\frac {4 a x^3}{9 b^2 \sqrt [4]{a+b x^2}}+\frac {2 x^5}{9 b \sqrt [4]{a+b x^2}}-\frac {\left (8 a^3\right ) \int \frac {1}{\left (a+b x^2\right )^{5/4}} \, dx}{3 b^3}\\ &=\frac {8 a^2 x}{3 b^3 \sqrt [4]{a+b x^2}}-\frac {4 a x^3}{9 b^2 \sqrt [4]{a+b x^2}}+\frac {2 x^5}{9 b \sqrt [4]{a+b x^2}}-\frac {\left (8 a^2 \sqrt [4]{1+\frac {b x^2}{a}}\right ) \int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{5/4}} \, dx}{3 b^3 \sqrt [4]{a+b x^2}}\\ &=\frac {8 a^2 x}{3 b^3 \sqrt [4]{a+b x^2}}-\frac {4 a x^3}{9 b^2 \sqrt [4]{a+b x^2}}+\frac {2 x^5}{9 b \sqrt [4]{a+b x^2}}-\frac {16 a^{5/2} \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{3 b^{7/2} \sqrt [4]{a+b x^2}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 7.66, size = 78, normalized size = 0.63 \begin {gather*} \frac {2 \left (-12 a^2 x-2 a b x^3+b^2 x^5+12 a^2 x \sqrt [4]{1+\frac {b x^2}{a}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};-\frac {b x^2}{a}\right )\right )}{9 b^3 \sqrt [4]{a+b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^6/(a + b*x^2)^(5/4),x]

[Out]

(2*(-12*a^2*x - 2*a*b*x^3 + b^2*x^5 + 12*a^2*x*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, -((b*x^2
)/a)]))/(9*b^3*(a + b*x^2)^(1/4))

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{6}}{\left (b \,x^{2}+a \right )^{\frac {5}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(b*x^2+a)^(5/4),x)

[Out]

int(x^6/(b*x^2+a)^(5/4),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)^(5/4),x, algorithm="maxima")

[Out]

integrate(x^6/(b*x^2 + a)^(5/4), 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)^(5/4),x, algorithm="fricas")

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 0.53, size = 27, normalized size = 0.22 \begin {gather*} \frac {x^{7} {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{7 a^{\frac {5}{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(b*x**2+a)**(5/4),x)

[Out]

x**7*hyper((5/4, 7/2), (9/2,), b*x**2*exp_polar(I*pi)/a)/(7*a**(5/4))

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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)^(5/4),x, algorithm="giac")

[Out]

integrate(x^6/(b*x^2 + a)^(5/4), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^6}{{\left (b\,x^2+a\right )}^{5/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(a + b*x^2)^(5/4),x)

[Out]

int(x^6/(a + b*x^2)^(5/4), x)

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