3.1029 \(\int \frac{(a+b x^4)^{3/4}}{x^7} \, dx\)

Optimal. Leaf size=125 \[ \frac{b^2 x^2}{4 a \sqrt [4]{a+b x^4}}-\frac{b^{3/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 \sqrt{a} \sqrt [4]{a+b x^4}}-\frac{b \left (a+b x^4\right )^{3/4}}{4 a x^2}-\frac{\left (a+b x^4\right )^{3/4}}{6 x^6} \]

[Out]

(b^2*x^2)/(4*a*(a + b*x^4)^(1/4)) - (a + b*x^4)^(3/4)/(6*x^6) - (b*(a + b*x^4)^(3/4))/(4*a*x^2) - (b^(3/2)*(1
+ (b*x^4)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(4*Sqrt[a]*(a + b*x^4)^(1/4))

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Rubi [A]  time = 0.0725773, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {275, 277, 325, 229, 227, 196} \[ \frac{b^2 x^2}{4 a \sqrt [4]{a+b x^4}}-\frac{b^{3/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 \sqrt{a} \sqrt [4]{a+b x^4}}-\frac{b \left (a+b x^4\right )^{3/4}}{4 a x^2}-\frac{\left (a+b x^4\right )^{3/4}}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(3/4)/x^7,x]

[Out]

(b^2*x^2)/(4*a*(a + b*x^4)^(1/4)) - (a + b*x^4)^(3/4)/(6*x^6) - (b*(a + b*x^4)^(3/4))/(4*a*x^2) - (b^(3/2)*(1
+ (b*x^4)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(4*Sqrt[a]*(a + b*x^4)^(1/4))

Rule 275

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

Rule 277

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

Rule 325

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 229

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

Rule 227

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

Rule 196

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

Rubi steps

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

Mathematica [C]  time = 0.0111968, size = 51, normalized size = 0.41 \[ -\frac{\left (a+b x^4\right )^{3/4} \, _2F_1\left (-\frac{3}{2},-\frac{3}{4};-\frac{1}{2};-\frac{b x^4}{a}\right )}{6 x^6 \left (\frac{b x^4}{a}+1\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^(3/4)/x^7,x]

[Out]

-((a + b*x^4)^(3/4)*Hypergeometric2F1[-3/2, -3/4, -1/2, -((b*x^4)/a)])/(6*x^6*(1 + (b*x^4)/a)^(3/4))

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Maple [F]  time = 0.031, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{7}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)^(3/4)/x^7,x)

[Out]

int((b*x^4+a)^(3/4)/x^7,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(3/4)/x^7,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(3/4)/x^7, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{7}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(3/4)/x^7,x, algorithm="fricas")

[Out]

integral((b*x^4 + a)^(3/4)/x^7, x)

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Sympy [C]  time = 2.06082, size = 34, normalized size = 0.27 \begin{align*} - \frac{a^{\frac{3}{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, - \frac{3}{4} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{6 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)**(3/4)/x**7,x)

[Out]

-a**(3/4)*hyper((-3/2, -3/4), (-1/2,), b*x**4*exp_polar(I*pi)/a)/(6*x**6)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(3/4)/x^7,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/4)/x^7, x)