3.1070 \(\int \frac {(a+b x^4)^{5/4}}{x^{22}} \, dx\)

Optimal. Leaf size=92 \[ \frac {128 b^3 \left (a+b x^4\right )^{9/4}}{13923 a^4 x^9}-\frac {32 b^2 \left (a+b x^4\right )^{9/4}}{1547 a^3 x^{13}}+\frac {4 b \left (a+b x^4\right )^{9/4}}{119 a^2 x^{17}}-\frac {\left (a+b x^4\right )^{9/4}}{21 a x^{21}} \]

[Out]

-1/21*(b*x^4+a)^(9/4)/a/x^21+4/119*b*(b*x^4+a)^(9/4)/a^2/x^17-32/1547*b^2*(b*x^4+a)^(9/4)/a^3/x^13+128/13923*b
^3*(b*x^4+a)^(9/4)/a^4/x^9

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac {128 b^3 \left (a+b x^4\right )^{9/4}}{13923 a^4 x^9}-\frac {32 b^2 \left (a+b x^4\right )^{9/4}}{1547 a^3 x^{13}}+\frac {4 b \left (a+b x^4\right )^{9/4}}{119 a^2 x^{17}}-\frac {\left (a+b x^4\right )^{9/4}}{21 a x^{21}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(5/4)/x^22,x]

[Out]

-(a + b*x^4)^(9/4)/(21*a*x^21) + (4*b*(a + b*x^4)^(9/4))/(119*a^2*x^17) - (32*b^2*(a + b*x^4)^(9/4))/(1547*a^3
*x^13) + (128*b^3*(a + b*x^4)^(9/4))/(13923*a^4*x^9)

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]

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]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^4\right )^{5/4}}{x^{22}} \, dx &=-\frac {\left (a+b x^4\right )^{9/4}}{21 a x^{21}}-\frac {(4 b) \int \frac {\left (a+b x^4\right )^{5/4}}{x^{18}} \, dx}{7 a}\\ &=-\frac {\left (a+b x^4\right )^{9/4}}{21 a x^{21}}+\frac {4 b \left (a+b x^4\right )^{9/4}}{119 a^2 x^{17}}+\frac {\left (32 b^2\right ) \int \frac {\left (a+b x^4\right )^{5/4}}{x^{14}} \, dx}{119 a^2}\\ &=-\frac {\left (a+b x^4\right )^{9/4}}{21 a x^{21}}+\frac {4 b \left (a+b x^4\right )^{9/4}}{119 a^2 x^{17}}-\frac {32 b^2 \left (a+b x^4\right )^{9/4}}{1547 a^3 x^{13}}-\frac {\left (128 b^3\right ) \int \frac {\left (a+b x^4\right )^{5/4}}{x^{10}} \, dx}{1547 a^3}\\ &=-\frac {\left (a+b x^4\right )^{9/4}}{21 a x^{21}}+\frac {4 b \left (a+b x^4\right )^{9/4}}{119 a^2 x^{17}}-\frac {32 b^2 \left (a+b x^4\right )^{9/4}}{1547 a^3 x^{13}}+\frac {128 b^3 \left (a+b x^4\right )^{9/4}}{13923 a^4 x^9}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 53, normalized size = 0.58 \[ \frac {\left (a+b x^4\right )^{9/4} \left (-663 a^3+468 a^2 b x^4-288 a b^2 x^8+128 b^3 x^{12}\right )}{13923 a^4 x^{21}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^(5/4)/x^22,x]

[Out]

((a + b*x^4)^(9/4)*(-663*a^3 + 468*a^2*b*x^4 - 288*a*b^2*x^8 + 128*b^3*x^12))/(13923*a^4*x^21)

________________________________________________________________________________________

fricas [A]  time = 0.82, size = 71, normalized size = 0.77 \[ \frac {{\left (128 \, b^{5} x^{20} - 32 \, a b^{4} x^{16} + 20 \, a^{2} b^{3} x^{12} - 15 \, a^{3} b^{2} x^{8} - 858 \, a^{4} b x^{4} - 663 \, a^{5}\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{13923 \, a^{4} x^{21}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(5/4)/x^22,x, algorithm="fricas")

[Out]

1/13923*(128*b^5*x^20 - 32*a*b^4*x^16 + 20*a^2*b^3*x^12 - 15*a^3*b^2*x^8 - 858*a^4*b*x^4 - 663*a^5)*(b*x^4 + a
)^(1/4)/(a^4*x^21)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{4} + a\right )}^{\frac {5}{4}}}{x^{22}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(5/4)/x^22,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(5/4)/x^22, x)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 50, normalized size = 0.54 \[ -\frac {\left (b \,x^{4}+a \right )^{\frac {9}{4}} \left (-128 b^{3} x^{12}+288 a \,b^{2} x^{8}-468 a^{2} b \,x^{4}+663 a^{3}\right )}{13923 a^{4} x^{21}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)^(5/4)/x^22,x)

[Out]

-1/13923*(b*x^4+a)^(9/4)*(-128*b^3*x^12+288*a*b^2*x^8-468*a^2*b*x^4+663*a^3)/x^21/a^4

________________________________________________________________________________________

maxima [A]  time = 1.37, size = 69, normalized size = 0.75 \[ \frac {\frac {1547 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}} b^{3}}{x^{9}} - \frac {3213 \, {\left (b x^{4} + a\right )}^{\frac {13}{4}} b^{2}}{x^{13}} + \frac {2457 \, {\left (b x^{4} + a\right )}^{\frac {17}{4}} b}{x^{17}} - \frac {663 \, {\left (b x^{4} + a\right )}^{\frac {21}{4}}}{x^{21}}}{13923 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(5/4)/x^22,x, algorithm="maxima")

[Out]

1/13923*(1547*(b*x^4 + a)^(9/4)*b^3/x^9 - 3213*(b*x^4 + a)^(13/4)*b^2/x^13 + 2457*(b*x^4 + a)^(17/4)*b/x^17 -
663*(b*x^4 + a)^(21/4)/x^21)/a^4

________________________________________________________________________________________

mupad [B]  time = 2.52, size = 111, normalized size = 1.21 \[ \frac {128\,b^5\,{\left (b\,x^4+a\right )}^{1/4}}{13923\,a^4\,x}-\frac {22\,b\,{\left (b\,x^4+a\right )}^{1/4}}{357\,x^{17}}-\frac {a\,{\left (b\,x^4+a\right )}^{1/4}}{21\,x^{21}}-\frac {32\,b^4\,{\left (b\,x^4+a\right )}^{1/4}}{13923\,a^3\,x^5}+\frac {20\,b^3\,{\left (b\,x^4+a\right )}^{1/4}}{13923\,a^2\,x^9}-\frac {5\,b^2\,{\left (b\,x^4+a\right )}^{1/4}}{4641\,a\,x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^4)^(5/4)/x^22,x)

[Out]

(128*b^5*(a + b*x^4)^(1/4))/(13923*a^4*x) - (22*b*(a + b*x^4)^(1/4))/(357*x^17) - (a*(a + b*x^4)^(1/4))/(21*x^
21) - (32*b^4*(a + b*x^4)^(1/4))/(13923*a^3*x^5) + (20*b^3*(a + b*x^4)^(1/4))/(13923*a^2*x^9) - (5*b^2*(a + b*
x^4)^(1/4))/(4641*a*x^13)

________________________________________________________________________________________

sympy [B]  time = 15.29, size = 954, normalized size = 10.37 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)**(5/4)/x**22,x)

[Out]

-1989*a**8*b**(37/4)*(a/(b*x**4) + 1)**(1/4)*gamma(-21/4)/(256*a**7*b**9*x**20*gamma(-5/4) + 768*a**6*b**10*x*
*24*gamma(-5/4) + 768*a**5*b**11*x**28*gamma(-5/4) + 256*a**4*b**12*x**32*gamma(-5/4)) - 8541*a**7*b**(41/4)*x
**4*(a/(b*x**4) + 1)**(1/4)*gamma(-21/4)/(256*a**7*b**9*x**20*gamma(-5/4) + 768*a**6*b**10*x**24*gamma(-5/4) +
768*a**5*b**11*x**28*gamma(-5/4) + 256*a**4*b**12*x**32*gamma(-5/4)) - 13734*a**6*b**(45/4)*x**8*(a/(b*x**4)
+ 1)**(1/4)*gamma(-21/4)/(256*a**7*b**9*x**20*gamma(-5/4) + 768*a**6*b**10*x**24*gamma(-5/4) + 768*a**5*b**11*
x**28*gamma(-5/4) + 256*a**4*b**12*x**32*gamma(-5/4)) - 9786*a**5*b**(49/4)*x**12*(a/(b*x**4) + 1)**(1/4)*gamm
a(-21/4)/(256*a**7*b**9*x**20*gamma(-5/4) + 768*a**6*b**10*x**24*gamma(-5/4) + 768*a**5*b**11*x**28*gamma(-5/4
) + 256*a**4*b**12*x**32*gamma(-5/4)) - 2625*a**4*b**(53/4)*x**16*(a/(b*x**4) + 1)**(1/4)*gamma(-21/4)/(256*a*
*7*b**9*x**20*gamma(-5/4) + 768*a**6*b**10*x**24*gamma(-5/4) + 768*a**5*b**11*x**28*gamma(-5/4) + 256*a**4*b**
12*x**32*gamma(-5/4)) + 231*a**3*b**(57/4)*x**20*(a/(b*x**4) + 1)**(1/4)*gamma(-21/4)/(256*a**7*b**9*x**20*gam
ma(-5/4) + 768*a**6*b**10*x**24*gamma(-5/4) + 768*a**5*b**11*x**28*gamma(-5/4) + 256*a**4*b**12*x**32*gamma(-5
/4)) + 924*a**2*b**(61/4)*x**24*(a/(b*x**4) + 1)**(1/4)*gamma(-21/4)/(256*a**7*b**9*x**20*gamma(-5/4) + 768*a*
*6*b**10*x**24*gamma(-5/4) + 768*a**5*b**11*x**28*gamma(-5/4) + 256*a**4*b**12*x**32*gamma(-5/4)) + 1056*a*b**
(65/4)*x**28*(a/(b*x**4) + 1)**(1/4)*gamma(-21/4)/(256*a**7*b**9*x**20*gamma(-5/4) + 768*a**6*b**10*x**24*gamm
a(-5/4) + 768*a**5*b**11*x**28*gamma(-5/4) + 256*a**4*b**12*x**32*gamma(-5/4)) + 384*b**(69/4)*x**32*(a/(b*x**
4) + 1)**(1/4)*gamma(-21/4)/(256*a**7*b**9*x**20*gamma(-5/4) + 768*a**6*b**10*x**24*gamma(-5/4) + 768*a**5*b**
11*x**28*gamma(-5/4) + 256*a**4*b**12*x**32*gamma(-5/4))

________________________________________________________________________________________