\(\int \frac {(a+b x^4)^{5/4}}{x^{10}} \, dx\) [1067]

Optimal result

Integrand size = 15, antiderivative size = 21 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{10}} \, dx=-\frac {\left (a+b x^4\right )^{9/4}}{9 a x^9} \]

[Out]

-1/9*(b*x^4+a)^(9/4)/a/x^9

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {270} \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{10}} \, dx=-\frac {\left (a+b x^4\right )^{9/4}}{9 a x^9} \]

[In]

Int[(a + b*x^4)^(5/4)/x^10,x]

[Out]

-1/9*(a + b*x^4)^(9/4)/(a*x^9)

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b x^4\right )^{9/4}}{9 a x^9} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{10}} \, dx=-\frac {\left (a+b x^4\right )^{9/4}}{9 a x^9} \]

[In]

Integrate[(a + b*x^4)^(5/4)/x^10,x]

[Out]

-1/9*(a + b*x^4)^(9/4)/(a*x^9)

Maple [A] (verified)

Time = 4.47 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86

method	result	size
gosper	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {9}{4}}}{9 a \,x^{9}}\)	\(18\)
pseudoelliptic	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {9}{4}}}{9 a \,x^{9}}\)	\(18\)
trager	\(-\frac {\left (b^{2} x^{8}+2 a b \,x^{4}+a^{2}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{9 x^{9} a}\)	\(36\)
risch	\(-\frac {\left (b^{2} x^{8}+2 a b \,x^{4}+a^{2}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{9 x^{9} a}\)	\(36\)

[In]

int((b*x^4+a)^(5/4)/x^10,x,method=_RETURNVERBOSE)

[Out]

-1/9*(b*x^4+a)^(9/4)/a/x^9

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (17) = 34\).

Time = 0.31 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.67 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{10}} \, dx=-\frac {{\left (b^{2} x^{8} + 2 \, a b x^{4} + a^{2}\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{9 \, a x^{9}} \]

[In]

integrate((b*x^4+a)^(5/4)/x^10,x, algorithm="fricas")

[Out]

-1/9*(b^2*x^8 + 2*a*b*x^4 + a^2)*(b*x^4 + a)^(1/4)/(a*x^9)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (17) = 34\).

Time = 0.79 (sec) , antiderivative size = 105, normalized size of antiderivative = 5.00 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{10}} \, dx=\frac {a \sqrt [4]{b} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {9}{4}\right )}{4 x^{8} \Gamma \left (- \frac {5}{4}\right )} + \frac {b^{\frac {5}{4}} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {9}{4}\right )}{2 x^{4} \Gamma \left (- \frac {5}{4}\right )} + \frac {b^{\frac {9}{4}} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {9}{4}\right )}{4 a \Gamma \left (- \frac {5}{4}\right )} \]

[In]

integrate((b*x**4+a)**(5/4)/x**10,x)

[Out]

a*b**(1/4)*(a/(b*x**4) + 1)**(1/4)*gamma(-9/4)/(4*x**8*gamma(-5/4)) + b**(5/4)*(a/(b*x**4) + 1)**(1/4)*gamma(-
9/4)/(2*x**4*gamma(-5/4)) + b**(9/4)*(a/(b*x**4) + 1)**(1/4)*gamma(-9/4)/(4*a*gamma(-5/4))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{10}} \, dx=-\frac {{\left (b x^{4} + a\right )}^{\frac {9}{4}}}{9 \, a x^{9}} \]

[In]

integrate((b*x^4+a)^(5/4)/x^10,x, algorithm="maxima")

[Out]

-1/9*(b*x^4 + a)^(9/4)/(a*x^9)

Giac [F]

\[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{10}} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {5}{4}}}{x^{10}} \,d x } \]

[In]

integrate((b*x^4+a)^(5/4)/x^10,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(5/4)/x^10, x)

Mupad [B] (verification not implemented)

Time = 6.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{10}} \, dx=-\frac {{\left (b\,x^4+a\right )}^{9/4}}{9\,a\,x^9} \]

[In]

int((a + b*x^4)^(5/4)/x^10,x)

[Out]

-(a + b*x^4)^(9/4)/(9*a*x^9)