3.9.60 \(\int \frac {1}{x^7 (a+b x^4)^{3/2}} \, dx\) [860]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 68, normalized size of antiderivative = 1.00, 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 = {277, 270} \begin {gather*} \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}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^7*(a + b*x^4)^(3/2)),x]

[Out]

-1/6*1/(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 270

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]

Rule 277

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]

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*}

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Mathematica [A]
time = 0.24, size = 42, normalized size = 0.62 \begin {gather*} \frac {-a^2+4 a b x^4+8 b^2 x^8}{6 a^3 x^6 \sqrt {a+b x^4}} \end {gather*}

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 = 0.16, size = 37, normalized size = 0.54

method	result	size
gosper	\(-\frac {-8 b^{2} x^{8}-4 a b \,x^{4}+a^{2}}{6 x^{6} \sqrt {b \,x^{4}+a}\, a^{3}}\)	\(37\)
default	\(-\frac {-8 b^{2} x^{8}-4 a b \,x^{4}+a^{2}}{6 x^{6} \sqrt {b \,x^{4}+a}\, a^{3}}\)	\(37\)
trager	\(-\frac {-8 b^{2} x^{8}-4 a b \,x^{4}+a^{2}}{6 x^{6} \sqrt {b \,x^{4}+a}\, a^{3}}\)	\(37\)
elliptic	\(-\frac {-8 b^{2} x^{8}-4 a b \,x^{4}+a^{2}}{6 x^{6} \sqrt {b \,x^{4}+a}\, a^{3}}\)	\(37\)
risch	\(-\frac {\sqrt {b \,x^{4}+a}\, \left (-5 b \,x^{4}+a \right )}{6 a^{3} x^{6}}+\frac {b^{2} x^{2}}{2 a^{3} \sqrt {b \,x^{4}+a}}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.29, size = 56, normalized size = 0.82 \begin {gather*} \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 {gather*}

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 = 0.36, size = 50, normalized size = 0.74 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (63) = 126\).
time = 0.70, size = 233, normalized size = 3.43 \begin {gather*} - \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 {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (56) = 112\).
time = 3.16, size = 115, normalized size = 1.69 \begin {gather*} \frac {b^{2} x^{2}}{2 \, \sqrt {b x^{4} + a} a^{3}} - \frac {3 \, {\left (\sqrt {b} x^{2} - \sqrt {b x^{4} + a}\right )}^{4} b^{\frac {3}{2}} - 12 \, {\left (\sqrt {b} x^{2} - \sqrt {b x^{4} + a}\right )}^{2} a b^{\frac {3}{2}} + 5 \, a^{2} b^{\frac {3}{2}}}{3 \, {\left ({\left (\sqrt {b} x^{2} - \sqrt {b x^{4} + a}\right )}^{2} - a\right )}^{3} a^{2}} \end {gather*}

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]

1/2*b^2*x^2/(sqrt(b*x^4 + a)*a^3) - 1/3*(3*(sqrt(b)*x^2 - sqrt(b*x^4 + a))^4*b^(3/2) - 12*(sqrt(b)*x^2 - sqrt(
b*x^4 + a))^2*a*b^(3/2) + 5*a^2*b^(3/2))/(((sqrt(b)*x^2 - sqrt(b*x^4 + a))^2 - a)^3*a^2)

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Mupad [B]
time = 1.28, size = 70, normalized size = 1.03 \begin {gather*} -\frac {8\,{\left (b\,x^4+a\right )}^2-12\,a\,\left (b\,x^4+a\right )+3\,a^2}{\left (\frac {6\,a^4\,x^2}{b}-\frac {6\,a^3\,x^2\,\left (b\,x^4+a\right )}{b}\right )\,\sqrt {b\,x^4+a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^7*(a + b*x^4)^(3/2)),x)

[Out]

-(8*(a + b*x^4)^2 - 12*a*(a + b*x^4) + 3*a^2)/(((6*a^4*x^2)/b - (6*a^3*x^2*(a + b*x^4))/b)*(a + b*x^4)^(1/2))

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