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]

-(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)

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Rubi [A]  time = 0.028267, antiderivative size = 92, normalized size of antiderivative = 1., 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 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{\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 = 0.0146056, size = 53, normalized size = 0.58 \[ \frac{\left (a+b x^4\right )^{9/4} \left (468 a^2 b x^4-663 a^3-288 a b^2 x^8+128 b^3 x^{12}\right )}{13923 a^4 x^{21}} \]

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)

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Maple [A]  time = 0.005, size = 50, normalized size = 0.5 \begin{align*} -{\frac{-128\,{b}^{3}{x}^{12}+288\,a{b}^{2}{x}^{8}-468\,{a}^{2}b{x}^{4}+663\,{a}^{3}}{13923\,{x}^{21}{a}^{4}} \left ( b{x}^{4}+a \right ) ^{{\frac{9}{4}}}} \end{align*}

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

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Maxima [A]  time = 0.966351, size = 93, normalized size = 1.01 \begin{align*} \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}} \end{align*}

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

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Fricas [A]  time = 1.58285, size = 176, normalized size = 1.91 \begin{align*} \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}} \end{align*}

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)

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Sympy [B]  time = 36.2823, size = 954, normalized size = 10.37 \begin{align*} \text{result too large to display} \end{align*}

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))

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Giac [B]  time = 1.12364, size = 541, normalized size = 5.88 \begin{align*} \frac{\frac{21 \,{\left (\frac{663 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}{\left (b + \frac{a}{x^{4}}\right )} b^{3}}{x} - \frac{1105 \,{\left (b^{2} x^{8} + 2 \, a b x^{4} + a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{2}}{x^{9}} + \frac{765 \,{\left (b^{3} x^{12} + 3 \, a b^{2} x^{8} + 3 \, a^{2} b x^{4} + a^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}} b}{x^{13}} - \frac{195 \,{\left (b^{4} x^{16} + 4 \, a b^{3} x^{12} + 6 \, a^{2} b^{2} x^{8} + 4 \, a^{3} b x^{4} + a^{4}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x^{17}}\right )} b}{a^{3}} - \frac{\frac{13923 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}{\left (b + \frac{a}{x^{4}}\right )} b^{4}}{x} - \frac{30940 \,{\left (b^{2} x^{8} + 2 \, a b x^{4} + a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{3}}{x^{9}} + \frac{32130 \,{\left (b^{3} x^{12} + 3 \, a b^{2} x^{8} + 3 \, a^{2} b x^{4} + a^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{2}}{x^{13}} - \frac{16380 \,{\left (b^{4} x^{16} + 4 \, a b^{3} x^{12} + 6 \, a^{2} b^{2} x^{8} + 4 \, a^{3} b x^{4} + a^{4}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}} b}{x^{17}} + \frac{3315 \,{\left (b^{5} x^{20} + 5 \, a b^{4} x^{16} + 10 \, a^{2} b^{3} x^{12} + 10 \, a^{3} b^{2} x^{8} + 5 \, a^{4} b x^{4} + a^{5}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x^{21}}}{a^{3}}}{69615 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/69615*(21*(663*(b*x^4 + a)^(1/4)*(b + a/x^4)*b^3/x - 1105*(b^2*x^8 + 2*a*b*x^4 + a^2)*(b*x^4 + a)^(1/4)*b^2/
x^9 + 765*(b^3*x^12 + 3*a*b^2*x^8 + 3*a^2*b*x^4 + a^3)*(b*x^4 + a)^(1/4)*b/x^13 - 195*(b^4*x^16 + 4*a*b^3*x^12
+ 6*a^2*b^2*x^8 + 4*a^3*b*x^4 + a^4)*(b*x^4 + a)^(1/4)/x^17)*b/a^3 - (13923*(b*x^4 + a)^(1/4)*(b + a/x^4)*b^4
/x - 30940*(b^2*x^8 + 2*a*b*x^4 + a^2)*(b*x^4 + a)^(1/4)*b^3/x^9 + 32130*(b^3*x^12 + 3*a*b^2*x^8 + 3*a^2*b*x^4
+ a^3)*(b*x^4 + a)^(1/4)*b^2/x^13 - 16380*(b^4*x^16 + 4*a*b^3*x^12 + 6*a^2*b^2*x^8 + 4*a^3*b*x^4 + a^4)*(b*x^
4 + a)^(1/4)*b/x^17 + 3315*(b^5*x^20 + 5*a*b^4*x^16 + 10*a^2*b^3*x^12 + 10*a^3*b^2*x^8 + 5*a^4*b*x^4 + a^5)*(b
*x^4 + a)^(1/4)/x^21)/a^3)/a