3.1039 \(\int \frac {(a+b x^4)^{3/4}}{x^{20}} \, dx\)

Optimal. Leaf size=92 \[ \frac {128 b^3 \left (a+b x^4\right )^{7/4}}{7315 a^4 x^7}-\frac {32 b^2 \left (a+b x^4\right )^{7/4}}{1045 a^3 x^{11}}+\frac {4 b \left (a+b x^4\right )^{7/4}}{95 a^2 x^{15}}-\frac {\left (a+b x^4\right )^{7/4}}{19 a x^{19}} \]

[Out]

-1/19*(b*x^4+a)^(7/4)/a/x^19+4/95*b*(b*x^4+a)^(7/4)/a^2/x^15-32/1045*b^2*(b*x^4+a)^(7/4)/a^3/x^11+128/7315*b^3
*(b*x^4+a)^(7/4)/a^4/x^7

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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} \[ \frac {128 b^3 \left (a+b x^4\right )^{7/4}}{7315 a^4 x^7}-\frac {32 b^2 \left (a+b x^4\right )^{7/4}}{1045 a^3 x^{11}}+\frac {4 b \left (a+b x^4\right )^{7/4}}{95 a^2 x^{15}}-\frac {\left (a+b x^4\right )^{7/4}}{19 a x^{19}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(3/4)/x^20,x]

[Out]

-(a + b*x^4)^(7/4)/(19*a*x^19) + (4*b*(a + b*x^4)^(7/4))/(95*a^2*x^15) - (32*b^2*(a + b*x^4)^(7/4))/(1045*a^3*
x^11) + (128*b^3*(a + b*x^4)^(7/4))/(7315*a^4*x^7)

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 )^{3/4}}{x^{20}} \, dx &=-\frac {\left (a+b x^4\right )^{7/4}}{19 a x^{19}}-\frac {(12 b) \int \frac {\left (a+b x^4\right )^{3/4}}{x^{16}} \, dx}{19 a}\\ &=-\frac {\left (a+b x^4\right )^{7/4}}{19 a x^{19}}+\frac {4 b \left (a+b x^4\right )^{7/4}}{95 a^2 x^{15}}+\frac {\left (32 b^2\right ) \int \frac {\left (a+b x^4\right )^{3/4}}{x^{12}} \, dx}{95 a^2}\\ &=-\frac {\left (a+b x^4\right )^{7/4}}{19 a x^{19}}+\frac {4 b \left (a+b x^4\right )^{7/4}}{95 a^2 x^{15}}-\frac {32 b^2 \left (a+b x^4\right )^{7/4}}{1045 a^3 x^{11}}-\frac {\left (128 b^3\right ) \int \frac {\left (a+b x^4\right )^{3/4}}{x^8} \, dx}{1045 a^3}\\ &=-\frac {\left (a+b x^4\right )^{7/4}}{19 a x^{19}}+\frac {4 b \left (a+b x^4\right )^{7/4}}{95 a^2 x^{15}}-\frac {32 b^2 \left (a+b x^4\right )^{7/4}}{1045 a^3 x^{11}}+\frac {128 b^3 \left (a+b x^4\right )^{7/4}}{7315 a^4 x^7}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 53, normalized size = 0.58 \[ \frac {\left (a+b x^4\right )^{7/4} \left (-385 a^3+308 a^2 b x^4-224 a b^2 x^8+128 b^3 x^{12}\right )}{7315 a^4 x^{19}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^(3/4)/x^20,x]

[Out]

((a + b*x^4)^(7/4)*(-385*a^3 + 308*a^2*b*x^4 - 224*a*b^2*x^8 + 128*b^3*x^12))/(7315*a^4*x^19)

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fricas [A]  time = 1.06, size = 60, normalized size = 0.65 \[ \frac {{\left (128 \, b^{4} x^{16} - 96 \, a b^{3} x^{12} + 84 \, a^{2} b^{2} x^{8} - 77 \, a^{3} b x^{4} - 385 \, a^{4}\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{7315 \, a^{4} x^{19}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(3/4)/x^20,x, algorithm="fricas")

[Out]

1/7315*(128*b^4*x^16 - 96*a*b^3*x^12 + 84*a^2*b^2*x^8 - 77*a^3*b*x^4 - 385*a^4)*(b*x^4 + a)^(3/4)/(a^4*x^19)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{x^{20}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(3/4)/x^20,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/4)/x^20, x)

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maple [A]  time = 0.01, size = 50, normalized size = 0.54 \[ -\frac {\left (b \,x^{4}+a \right )^{\frac {7}{4}} \left (-128 b^{3} x^{12}+224 a \,b^{2} x^{8}-308 a^{2} b \,x^{4}+385 a^{3}\right )}{7315 a^{4} x^{19}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)^(3/4)/x^20,x)

[Out]

-1/7315*(b*x^4+a)^(7/4)*(-128*b^3*x^12+224*a*b^2*x^8-308*a^2*b*x^4+385*a^3)/x^19/a^4

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maxima [A]  time = 1.37, size = 69, normalized size = 0.75 \[ \frac {\frac {1045 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} b^{3}}{x^{7}} - \frac {1995 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}} b^{2}}{x^{11}} + \frac {1463 \, {\left (b x^{4} + a\right )}^{\frac {15}{4}} b}{x^{15}} - \frac {385 \, {\left (b x^{4} + a\right )}^{\frac {19}{4}}}{x^{19}}}{7315 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(3/4)/x^20,x, algorithm="maxima")

[Out]

1/7315*(1045*(b*x^4 + a)^(7/4)*b^3/x^7 - 1995*(b*x^4 + a)^(11/4)*b^2/x^11 + 1463*(b*x^4 + a)^(15/4)*b/x^15 - 3
85*(b*x^4 + a)^(19/4)/x^19)/a^4

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mupad [B]  time = 1.81, size = 93, normalized size = 1.01 \[ \frac {128\,b^4\,{\left (b\,x^4+a\right )}^{3/4}}{7315\,a^4\,x^3}-\frac {b\,{\left (b\,x^4+a\right )}^{3/4}}{95\,a\,x^{15}}-\frac {{\left (b\,x^4+a\right )}^{3/4}}{19\,x^{19}}-\frac {96\,b^3\,{\left (b\,x^4+a\right )}^{3/4}}{7315\,a^3\,x^7}+\frac {12\,b^2\,{\left (b\,x^4+a\right )}^{3/4}}{1045\,a^2\,x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^4)^(3/4)/x^20,x)

[Out]

(128*b^4*(a + b*x^4)^(3/4))/(7315*a^4*x^3) - (b*(a + b*x^4)^(3/4))/(95*a*x^15) - (a + b*x^4)^(3/4)/(19*x^19) -
(96*b^3*(a + b*x^4)^(3/4))/(7315*a^3*x^7) + (12*b^2*(a + b*x^4)^(3/4))/(1045*a^2*x^11)

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sympy [B]  time = 14.53, size = 847, normalized size = 9.21 \[ - \frac {1155 a^{7} b^{\frac {39}{4}} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {3}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {3}{4}\right )} - \frac {3696 a^{6} b^{\frac {43}{4}} x^{4} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {3}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {3}{4}\right )} - \frac {3906 a^{5} b^{\frac {47}{4}} x^{8} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {3}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {3}{4}\right )} - \frac {1380 a^{4} b^{\frac {51}{4}} x^{12} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {3}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {3}{4}\right )} + \frac {45 a^{3} b^{\frac {55}{4}} x^{16} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {3}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {3}{4}\right )} + \frac {540 a^{2} b^{\frac {59}{4}} x^{20} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {3}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {3}{4}\right )} + \frac {864 a b^{\frac {63}{4}} x^{24} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {3}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {3}{4}\right )} + \frac {384 b^{\frac {67}{4}} x^{28} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {3}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)**(3/4)/x**20,x)

[Out]

-1155*a**7*b**(39/4)*(a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(256*a**7*b**9*x**16*gamma(-3/4) + 768*a**6*b**10*x*
*20*gamma(-3/4) + 768*a**5*b**11*x**24*gamma(-3/4) + 256*a**4*b**12*x**28*gamma(-3/4)) - 3696*a**6*b**(43/4)*x
**4*(a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(256*a**7*b**9*x**16*gamma(-3/4) + 768*a**6*b**10*x**20*gamma(-3/4) +
768*a**5*b**11*x**24*gamma(-3/4) + 256*a**4*b**12*x**28*gamma(-3/4)) - 3906*a**5*b**(47/4)*x**8*(a/(b*x**4) +
1)**(3/4)*gamma(-19/4)/(256*a**7*b**9*x**16*gamma(-3/4) + 768*a**6*b**10*x**20*gamma(-3/4) + 768*a**5*b**11*x
**24*gamma(-3/4) + 256*a**4*b**12*x**28*gamma(-3/4)) - 1380*a**4*b**(51/4)*x**12*(a/(b*x**4) + 1)**(3/4)*gamma
(-19/4)/(256*a**7*b**9*x**16*gamma(-3/4) + 768*a**6*b**10*x**20*gamma(-3/4) + 768*a**5*b**11*x**24*gamma(-3/4)
+ 256*a**4*b**12*x**28*gamma(-3/4)) + 45*a**3*b**(55/4)*x**16*(a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(256*a**7*
b**9*x**16*gamma(-3/4) + 768*a**6*b**10*x**20*gamma(-3/4) + 768*a**5*b**11*x**24*gamma(-3/4) + 256*a**4*b**12*
x**28*gamma(-3/4)) + 540*a**2*b**(59/4)*x**20*(a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(256*a**7*b**9*x**16*gamma(
-3/4) + 768*a**6*b**10*x**20*gamma(-3/4) + 768*a**5*b**11*x**24*gamma(-3/4) + 256*a**4*b**12*x**28*gamma(-3/4)
) + 864*a*b**(63/4)*x**24*(a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(256*a**7*b**9*x**16*gamma(-3/4) + 768*a**6*b**
10*x**20*gamma(-3/4) + 768*a**5*b**11*x**24*gamma(-3/4) + 256*a**4*b**12*x**28*gamma(-3/4)) + 384*b**(67/4)*x*
*28*(a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(256*a**7*b**9*x**16*gamma(-3/4) + 768*a**6*b**10*x**20*gamma(-3/4) +
768*a**5*b**11*x**24*gamma(-3/4) + 256*a**4*b**12*x**28*gamma(-3/4))

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