3.13.54 \(\int \frac {1}{x^{10} (a-b x^4)^{3/4}} \, dx\) [1254]

Optimal. Leaf size=71 \[ -\frac {\sqrt [4]{a-b x^4}}{9 a x^9}-\frac {8 b \sqrt [4]{a-b x^4}}{45 a^2 x^5}-\frac {32 b^2 \sqrt [4]{a-b x^4}}{45 a^3 x} \]

[Out]

-1/9*(-b*x^4+a)^(1/4)/a/x^9-8/45*b*(-b*x^4+a)^(1/4)/a^2/x^5-32/45*b^2*(-b*x^4+a)^(1/4)/a^3/x

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Rubi [A]
time = 0.02, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {277, 270} \begin {gather*} -\frac {32 b^2 \sqrt [4]{a-b x^4}}{45 a^3 x}-\frac {8 b \sqrt [4]{a-b x^4}}{45 a^2 x^5}-\frac {\sqrt [4]{a-b x^4}}{9 a x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^10*(a - b*x^4)^(3/4)),x]

[Out]

-1/9*(a - b*x^4)^(1/4)/(a*x^9) - (8*b*(a - b*x^4)^(1/4))/(45*a^2*x^5) - (32*b^2*(a - b*x^4)^(1/4))/(45*a^3*x)

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^{10} \left (a-b x^4\right )^{3/4}} \, dx &=-\frac {\sqrt [4]{a-b x^4}}{9 a x^9}+\frac {(8 b) \int \frac {1}{x^6 \left (a-b x^4\right )^{3/4}} \, dx}{9 a}\\ &=-\frac {\sqrt [4]{a-b x^4}}{9 a x^9}-\frac {8 b \sqrt [4]{a-b x^4}}{45 a^2 x^5}+\frac {\left (32 b^2\right ) \int \frac {1}{x^2 \left (a-b x^4\right )^{3/4}} \, dx}{45 a^2}\\ &=-\frac {\sqrt [4]{a-b x^4}}{9 a x^9}-\frac {8 b \sqrt [4]{a-b x^4}}{45 a^2 x^5}-\frac {32 b^2 \sqrt [4]{a-b x^4}}{45 a^3 x}\\ \end {align*}

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Mathematica [A]
time = 0.28, size = 43, normalized size = 0.61 \begin {gather*} \frac {\sqrt [4]{a-b x^4} \left (-5 a^2-8 a b x^4-32 b^2 x^8\right )}{45 a^3 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^10*(a - b*x^4)^(3/4)),x]

[Out]

((a - b*x^4)^(1/4)*(-5*a^2 - 8*a*b*x^4 - 32*b^2*x^8))/(45*a^3*x^9)

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Maple [A]
time = 0.17, size = 40, normalized size = 0.56

method	result	size
gosper	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}} \left (32 b^{2} x^{8}+8 a b \,x^{4}+5 a^{2}\right )}{45 a^{3} x^{9}}\)	\(40\)
trager	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}} \left (32 b^{2} x^{8}+8 a b \,x^{4}+5 a^{2}\right )}{45 a^{3} x^{9}}\)	\(40\)
risch	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}} \left (\left (-b \,x^{4}+a \right )^{3}\right )^{\frac {1}{4}} \left (32 b^{2} x^{8}+8 a b \,x^{4}+5 a^{2}\right )}{45 a^{3} x^{9} \left (-\left (b \,x^{4}-a \right )^{3}\right )^{\frac {1}{4}}}\)	\(67\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^10/(-b*x^4+a)^(3/4),x,method=_RETURNVERBOSE)

[Out]

-1/45*(-b*x^4+a)^(1/4)*(32*b^2*x^8+8*a*b*x^4+5*a^2)/a^3/x^9

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Maxima [A]
time = 0.31, size = 55, normalized size = 0.77 \begin {gather*} -\frac {\frac {45 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}} b^{2}}{x} + \frac {18 \, {\left (-b x^{4} + a\right )}^{\frac {5}{4}} b}{x^{5}} + \frac {5 \, {\left (-b x^{4} + a\right )}^{\frac {9}{4}}}{x^{9}}}{45 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^10/(-b*x^4+a)^(3/4),x, algorithm="maxima")

[Out]

-1/45*(45*(-b*x^4 + a)^(1/4)*b^2/x + 18*(-b*x^4 + a)^(5/4)*b/x^5 + 5*(-b*x^4 + a)^(9/4)/x^9)/a^3

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Fricas [A]
time = 0.36, size = 39, normalized size = 0.55 \begin {gather*} -\frac {{\left (32 \, b^{2} x^{8} + 8 \, a b x^{4} + 5 \, a^{2}\right )} {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{45 \, a^{3} x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^10/(-b*x^4+a)^(3/4),x, algorithm="fricas")

[Out]

-1/45*(32*b^2*x^8 + 8*a*b*x^4 + 5*a^2)*(-b*x^4 + a)^(1/4)/(a^3*x^9)

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Sympy [C] Result contains complex when optimal does not.
time = 0.94, size = 1110, normalized size = 15.63 \begin {gather*} \begin {cases} - \frac {5 a^{4} b^{\frac {17}{4}} \sqrt [4]{\frac {a}{b x^{4}} - 1} e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {9}{4}\right )}{64 a^{5} b^{4} x^{8} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right ) - 128 a^{4} b^{5} x^{12} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right ) + 64 a^{3} b^{6} x^{16} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right )} + \frac {2 a^{3} b^{\frac {21}{4}} x^{4} \sqrt [4]{\frac {a}{b x^{4}} - 1} e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {9}{4}\right )}{64 a^{5} b^{4} x^{8} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right ) - 128 a^{4} b^{5} x^{12} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right ) + 64 a^{3} b^{6} x^{16} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right )} - \frac {21 a^{2} b^{\frac {25}{4}} x^{8} \sqrt [4]{\frac {a}{b x^{4}} - 1} e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {9}{4}\right )}{64 a^{5} b^{4} x^{8} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right ) - 128 a^{4} b^{5} x^{12} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right ) + 64 a^{3} b^{6} x^{16} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right )} + \frac {56 a b^{\frac {29}{4}} x^{12} \sqrt [4]{\frac {a}{b x^{4}} - 1} e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {9}{4}\right )}{64 a^{5} b^{4} x^{8} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right ) - 128 a^{4} b^{5} x^{12} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right ) + 64 a^{3} b^{6} x^{16} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right )} - \frac {32 b^{\frac {33}{4}} x^{16} \sqrt [4]{\frac {a}{b x^{4}} - 1} e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {9}{4}\right )}{64 a^{5} b^{4} x^{8} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right ) - 128 a^{4} b^{5} x^{12} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right ) + 64 a^{3} b^{6} x^{16} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right )} & \text {for}\: \left |{\frac {a}{b x^{4}}}\right | > 1 \\- \frac {5 a^{4} b^{\frac {17}{4}} \sqrt [4]{- \frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {9}{4}\right )}{64 a^{5} b^{4} x^{8} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right ) - 128 a^{4} b^{5} x^{12} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right ) + 64 a^{3} b^{6} x^{16} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right )} + \frac {2 a^{3} b^{\frac {21}{4}} x^{4} \sqrt [4]{- \frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {9}{4}\right )}{64 a^{5} b^{4} x^{8} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right ) - 128 a^{4} b^{5} x^{12} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right ) + 64 a^{3} b^{6} x^{16} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right )} - \frac {21 a^{2} b^{\frac {25}{4}} x^{8} \sqrt [4]{- \frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {9}{4}\right )}{64 a^{5} b^{4} x^{8} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right ) - 128 a^{4} b^{5} x^{12} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right ) + 64 a^{3} b^{6} x^{16} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right )} + \frac {56 a b^{\frac {29}{4}} x^{12} \sqrt [4]{- \frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {9}{4}\right )}{64 a^{5} b^{4} x^{8} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right ) - 128 a^{4} b^{5} x^{12} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right ) + 64 a^{3} b^{6} x^{16} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right )} - \frac {32 b^{\frac {33}{4}} x^{16} \sqrt [4]{- \frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {9}{4}\right )}{64 a^{5} b^{4} x^{8} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right ) - 128 a^{4} b^{5} x^{12} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right ) + 64 a^{3} b^{6} x^{16} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**10/(-b*x**4+a)**(3/4),x)

[Out]

Piecewise((-5*a**4*b**(17/4)*(a/(b*x**4) - 1)**(1/4)*exp(-I*pi/4)*gamma(-9/4)/(64*a**5*b**4*x**8*exp(3*I*pi/4)
*gamma(3/4) - 128*a**4*b**5*x**12*exp(3*I*pi/4)*gamma(3/4) + 64*a**3*b**6*x**16*exp(3*I*pi/4)*gamma(3/4)) + 2*
a**3*b**(21/4)*x**4*(a/(b*x**4) - 1)**(1/4)*exp(-I*pi/4)*gamma(-9/4)/(64*a**5*b**4*x**8*exp(3*I*pi/4)*gamma(3/
4) - 128*a**4*b**5*x**12*exp(3*I*pi/4)*gamma(3/4) + 64*a**3*b**6*x**16*exp(3*I*pi/4)*gamma(3/4)) - 21*a**2*b**
(25/4)*x**8*(a/(b*x**4) - 1)**(1/4)*exp(-I*pi/4)*gamma(-9/4)/(64*a**5*b**4*x**8*exp(3*I*pi/4)*gamma(3/4) - 128
*a**4*b**5*x**12*exp(3*I*pi/4)*gamma(3/4) + 64*a**3*b**6*x**16*exp(3*I*pi/4)*gamma(3/4)) + 56*a*b**(29/4)*x**1
2*(a/(b*x**4) - 1)**(1/4)*exp(-I*pi/4)*gamma(-9/4)/(64*a**5*b**4*x**8*exp(3*I*pi/4)*gamma(3/4) - 128*a**4*b**5
*x**12*exp(3*I*pi/4)*gamma(3/4) + 64*a**3*b**6*x**16*exp(3*I*pi/4)*gamma(3/4)) - 32*b**(33/4)*x**16*(a/(b*x**4
) - 1)**(1/4)*exp(-I*pi/4)*gamma(-9/4)/(64*a**5*b**4*x**8*exp(3*I*pi/4)*gamma(3/4) - 128*a**4*b**5*x**12*exp(3
*I*pi/4)*gamma(3/4) + 64*a**3*b**6*x**16*exp(3*I*pi/4)*gamma(3/4)), Abs(a/(b*x**4)) > 1), (-5*a**4*b**(17/4)*(
-a/(b*x**4) + 1)**(1/4)*gamma(-9/4)/(64*a**5*b**4*x**8*exp(3*I*pi/4)*gamma(3/4) - 128*a**4*b**5*x**12*exp(3*I*
pi/4)*gamma(3/4) + 64*a**3*b**6*x**16*exp(3*I*pi/4)*gamma(3/4)) + 2*a**3*b**(21/4)*x**4*(-a/(b*x**4) + 1)**(1/
4)*gamma(-9/4)/(64*a**5*b**4*x**8*exp(3*I*pi/4)*gamma(3/4) - 128*a**4*b**5*x**12*exp(3*I*pi/4)*gamma(3/4) + 64
*a**3*b**6*x**16*exp(3*I*pi/4)*gamma(3/4)) - 21*a**2*b**(25/4)*x**8*(-a/(b*x**4) + 1)**(1/4)*gamma(-9/4)/(64*a
**5*b**4*x**8*exp(3*I*pi/4)*gamma(3/4) - 128*a**4*b**5*x**12*exp(3*I*pi/4)*gamma(3/4) + 64*a**3*b**6*x**16*exp
(3*I*pi/4)*gamma(3/4)) + 56*a*b**(29/4)*x**12*(-a/(b*x**4) + 1)**(1/4)*gamma(-9/4)/(64*a**5*b**4*x**8*exp(3*I*
pi/4)*gamma(3/4) - 128*a**4*b**5*x**12*exp(3*I*pi/4)*gamma(3/4) + 64*a**3*b**6*x**16*exp(3*I*pi/4)*gamma(3/4))
- 32*b**(33/4)*x**16*(-a/(b*x**4) + 1)**(1/4)*gamma(-9/4)/(64*a**5*b**4*x**8*exp(3*I*pi/4)*gamma(3/4) - 128*a
**4*b**5*x**12*exp(3*I*pi/4)*gamma(3/4) + 64*a**3*b**6*x**16*exp(3*I*pi/4)*gamma(3/4)), True))

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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(1/x^10/(-b*x^4+a)^(3/4),x, algorithm="giac")

[Out]

integrate(1/((-b*x^4 + a)^(3/4)*x^10), x)

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Mupad [B]
time = 1.29, size = 39, normalized size = 0.55 \begin {gather*} -\frac {{\left (a-b\,x^4\right )}^{1/4}\,\left (5\,a^2+8\,a\,b\,x^4+32\,b^2\,x^8\right )}{45\,a^3\,x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^10*(a - b*x^4)^(3/4)),x)

[Out]

-((a - b*x^4)^(1/4)*(5*a^2 + 32*b^2*x^8 + 8*a*b*x^4))/(45*a^3*x^9)

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