3.860 \(\int \frac{1}{x^7 (a+b x^4)^{3/2}} \, dx\)

Optimal. Leaf size=68 \[ \frac{4 b^2 x^2}{3 a^3 \sqrt{a+b x^4}}+\frac{2 b}{3 a^2 x^2 \sqrt{a+b x^4}}-\frac{1}{6 a x^6 \sqrt{a+b x^4}} \]

[Out]

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

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Rubi [A]  time = 0.0177637, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac{4 b^2 x^2}{3 a^3 \sqrt{a+b x^4}}+\frac{2 b}{3 a^2 x^2 \sqrt{a+b x^4}}-\frac{1}{6 a x^6 \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^7*(a + b*x^4)^(3/2)),x]

[Out]

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

Rule 271

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

Rule 264

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] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{x^7 \left (a+b x^4\right )^{3/2}} \, dx &=-\frac{1}{6 a x^6 \sqrt{a+b x^4}}-\frac{(4 b) \int \frac{1}{x^3 \left (a+b x^4\right )^{3/2}} \, dx}{3 a}\\ &=-\frac{1}{6 a x^6 \sqrt{a+b x^4}}+\frac{2 b}{3 a^2 x^2 \sqrt{a+b x^4}}+\frac{\left (8 b^2\right ) \int \frac{x}{\left (a+b x^4\right )^{3/2}} \, dx}{3 a^2}\\ &=-\frac{1}{6 a x^6 \sqrt{a+b x^4}}+\frac{2 b}{3 a^2 x^2 \sqrt{a+b x^4}}+\frac{4 b^2 x^2}{3 a^3 \sqrt{a+b x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0085471, size = 40, normalized size = 0.59 \[ -\frac{a^2-4 a b x^4-8 b^2 x^8}{6 a^3 x^6 \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^2 - 4*a*b*x^4 - 8*b^2*x^8)/(6*a^3*x^6*Sqrt[a + b*x^4])

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Maple [A]  time = 0.004, size = 37, normalized size = 0.5 \begin{align*} -{\frac{-8\,{b}^{2}{x}^{8}-4\,ab{x}^{4}+{a}^{2}}{6\,{a}^{3}{x}^{6}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^7/(b*x^4+a)^(3/2),x)

[Out]

-1/6*(-8*b^2*x^8-4*a*b*x^4+a^2)/x^6/(b*x^4+a)^(1/2)/a^3

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Maxima [A]  time = 0.96744, size = 76, normalized size = 1.12 \begin{align*} \frac{b^{2} x^{2}}{2 \, \sqrt{b x^{4} + a} a^{3}} + \frac{\frac{6 \, \sqrt{b x^{4} + a} b}{x^{2}} - \frac{{\left (b x^{4} + a\right )}^{\frac{3}{2}}}{x^{6}}}{6 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^4+a)^(3/2),x, algorithm="maxima")

[Out]

1/2*b^2*x^2/(sqrt(b*x^4 + a)*a^3) + 1/6*(6*sqrt(b*x^4 + a)*b/x^2 - (b*x^4 + a)^(3/2)/x^6)/a^3

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Fricas [A]  time = 1.51145, size = 100, normalized size = 1.47 \begin{align*} \frac{{\left (8 \, b^{2} x^{8} + 4 \, a b x^{4} - a^{2}\right )} \sqrt{b x^{4} + a}}{6 \,{\left (a^{3} b x^{10} + a^{4} x^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^4+a)^(3/2),x, algorithm="fricas")

[Out]

1/6*(8*b^2*x^8 + 4*a*b*x^4 - a^2)*sqrt(b*x^4 + a)/(a^3*b*x^10 + a^4*x^6)

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Sympy [B]  time = 2.18055, size = 233, normalized size = 3.43 \begin{align*} - \frac{a^{3} b^{\frac{9}{2}} \sqrt{\frac{a}{b x^{4}} + 1}}{6 a^{5} b^{4} x^{4} + 12 a^{4} b^{5} x^{8} + 6 a^{3} b^{6} x^{12}} + \frac{3 a^{2} b^{\frac{11}{2}} x^{4} \sqrt{\frac{a}{b x^{4}} + 1}}{6 a^{5} b^{4} x^{4} + 12 a^{4} b^{5} x^{8} + 6 a^{3} b^{6} x^{12}} + \frac{12 a b^{\frac{13}{2}} x^{8} \sqrt{\frac{a}{b x^{4}} + 1}}{6 a^{5} b^{4} x^{4} + 12 a^{4} b^{5} x^{8} + 6 a^{3} b^{6} x^{12}} + \frac{8 b^{\frac{15}{2}} x^{12} \sqrt{\frac{a}{b x^{4}} + 1}}{6 a^{5} b^{4} x^{4} + 12 a^{4} b^{5} x^{8} + 6 a^{3} b^{6} x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**7/(b*x**4+a)**(3/2),x)

[Out]

-a**3*b**(9/2)*sqrt(a/(b*x**4) + 1)/(6*a**5*b**4*x**4 + 12*a**4*b**5*x**8 + 6*a**3*b**6*x**12) + 3*a**2*b**(11
/2)*x**4*sqrt(a/(b*x**4) + 1)/(6*a**5*b**4*x**4 + 12*a**4*b**5*x**8 + 6*a**3*b**6*x**12) + 12*a*b**(13/2)*x**8
*sqrt(a/(b*x**4) + 1)/(6*a**5*b**4*x**4 + 12*a**4*b**5*x**8 + 6*a**3*b**6*x**12) + 8*b**(15/2)*x**12*sqrt(a/(b
*x**4) + 1)/(6*a**5*b**4*x**4 + 12*a**4*b**5*x**8 + 6*a**3*b**6*x**12)

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Giac [A]  time = 1.16699, size = 65, normalized size = 0.96 \begin{align*} -\frac{{\left (b + \frac{a}{x^{4}}\right )}^{\frac{3}{2}} - 6 \, \sqrt{b + \frac{a}{x^{4}}} b}{6 \, a^{3}} - \frac{x^{2}}{256 \, \sqrt{b x^{4} + a} a^{3} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^4+a)^(3/2),x, algorithm="giac")

[Out]

-1/6*((b + a/x^4)^(3/2) - 6*sqrt(b + a/x^4)*b)/a^3 - 1/256*x^2/(sqrt(b*x^4 + a)*a^3*b^2)