3.11.38 \(\int \frac {(a+b x^4)^{3/4}}{x^{16}} \, dx\) [1038]

Optimal. Leaf size=68 \[ -\frac {\left (a+b x^4\right )^{7/4}}{15 a x^{15}}+\frac {8 b \left (a+b x^4\right )^{7/4}}{165 a^2 x^{11}}-\frac {32 b^2 \left (a+b x^4\right )^{7/4}}{1155 a^3 x^7} \]

[Out]

-1/15*(b*x^4+a)^(7/4)/a/x^15+8/165*b*(b*x^4+a)^(7/4)/a^2/x^11-32/1155*b^2*(b*x^4+a)^(7/4)/a^3/x^7

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Rubi [A]
time = 0.01, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {277, 270} \begin {gather*} -\frac {32 b^2 \left (a+b x^4\right )^{7/4}}{1155 a^3 x^7}+\frac {8 b \left (a+b x^4\right )^{7/4}}{165 a^2 x^{11}}-\frac {\left (a+b x^4\right )^{7/4}}{15 a x^{15}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(3/4)/x^16,x]

[Out]

-1/15*(a + b*x^4)^(7/4)/(a*x^15) + (8*b*(a + b*x^4)^(7/4))/(165*a^2*x^11) - (32*b^2*(a + b*x^4)^(7/4))/(1155*a
^3*x^7)

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 {\left (a+b x^4\right )^{3/4}}{x^{16}} \, dx &=-\frac {\left (a+b x^4\right )^{7/4}}{15 a x^{15}}-\frac {(8 b) \int \frac {\left (a+b x^4\right )^{3/4}}{x^{12}} \, dx}{15 a}\\ &=-\frac {\left (a+b x^4\right )^{7/4}}{15 a x^{15}}+\frac {8 b \left (a+b x^4\right )^{7/4}}{165 a^2 x^{11}}+\frac {\left (32 b^2\right ) \int \frac {\left (a+b x^4\right )^{3/4}}{x^8} \, dx}{165 a^2}\\ &=-\frac {\left (a+b x^4\right )^{7/4}}{15 a x^{15}}+\frac {8 b \left (a+b x^4\right )^{7/4}}{165 a^2 x^{11}}-\frac {32 b^2 \left (a+b x^4\right )^{7/4}}{1155 a^3 x^7}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 42, normalized size = 0.62 \begin {gather*} \frac {\left (a+b x^4\right )^{7/4} \left (-77 a^2+56 a b x^4-32 b^2 x^8\right )}{1155 a^3 x^{15}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^(3/4)/x^16,x]

[Out]

((a + b*x^4)^(7/4)*(-77*a^2 + 56*a*b*x^4 - 32*b^2*x^8))/(1155*a^3*x^15)

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Maple [A]
time = 0.18, size = 39, normalized size = 0.57

method	result	size
gosper	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {7}{4}} \left (32 b^{2} x^{8}-56 a b \,x^{4}+77 a^{2}\right )}{1155 a^{3} x^{15}}\)	\(39\)
trager	\(-\frac {\left (32 b^{3} x^{12}-24 a \,b^{2} x^{8}+21 a^{2} b \,x^{4}+77 a^{3}\right ) \left (b \,x^{4}+a \right )^{\frac {3}{4}}}{1155 a^{3} x^{15}}\)	\(50\)
risch	\(-\frac {\left (32 b^{3} x^{12}-24 a \,b^{2} x^{8}+21 a^{2} b \,x^{4}+77 a^{3}\right ) \left (b \,x^{4}+a \right )^{\frac {3}{4}}}{1155 a^{3} x^{15}}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)^(3/4)/x^16,x,method=_RETURNVERBOSE)

[Out]

-1/1155*(b*x^4+a)^(7/4)*(32*b^2*x^8-56*a*b*x^4+77*a^2)/a^3/x^15

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Maxima [A]
time = 0.29, size = 52, normalized size = 0.76 \begin {gather*} -\frac {\frac {165 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} b^{2}}{x^{7}} - \frac {210 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}} b}{x^{11}} + \frac {77 \, {\left (b x^{4} + a\right )}^{\frac {15}{4}}}{x^{15}}}{1155 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(3/4)/x^16,x, algorithm="maxima")

[Out]

-1/1155*(165*(b*x^4 + a)^(7/4)*b^2/x^7 - 210*(b*x^4 + a)^(11/4)*b/x^11 + 77*(b*x^4 + a)^(15/4)/x^15)/a^3

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Fricas [A]
time = 0.42, size = 49, normalized size = 0.72 \begin {gather*} -\frac {{\left (32 \, b^{3} x^{12} - 24 \, a b^{2} x^{8} + 21 \, a^{2} b x^{4} + 77 \, a^{3}\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{1155 \, a^{3} x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(3/4)/x^16,x, algorithm="fricas")

[Out]

-1/1155*(32*b^3*x^12 - 24*a*b^2*x^8 + 21*a^2*b*x^4 + 77*a^3)*(b*x^4 + a)^(3/4)/(a^3*x^15)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 520 vs. \(2 (61) = 122\).
time = 1.48, size = 520, normalized size = 7.65 \begin {gather*} \frac {77 a^{5} b^{\frac {19}{4}} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {3}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {3}{4}\right )} + \frac {175 a^{4} b^{\frac {23}{4}} x^{4} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {3}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {3}{4}\right )} + \frac {95 a^{3} b^{\frac {27}{4}} x^{8} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {3}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {3}{4}\right )} + \frac {5 a^{2} b^{\frac {31}{4}} x^{12} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {3}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {3}{4}\right )} + \frac {40 a b^{\frac {35}{4}} x^{16} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {3}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {3}{4}\right )} + \frac {32 b^{\frac {39}{4}} x^{20} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {3}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {3}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)**(3/4)/x**16,x)

[Out]

77*a**5*b**(19/4)*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(64*a**5*b**4*x**12*gamma(-3/4) + 128*a**4*b**5*x**16*g
amma(-3/4) + 64*a**3*b**6*x**20*gamma(-3/4)) + 175*a**4*b**(23/4)*x**4*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(6
4*a**5*b**4*x**12*gamma(-3/4) + 128*a**4*b**5*x**16*gamma(-3/4) + 64*a**3*b**6*x**20*gamma(-3/4)) + 95*a**3*b*
*(27/4)*x**8*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(64*a**5*b**4*x**12*gamma(-3/4) + 128*a**4*b**5*x**16*gamma(
-3/4) + 64*a**3*b**6*x**20*gamma(-3/4)) + 5*a**2*b**(31/4)*x**12*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(64*a**5
*b**4*x**12*gamma(-3/4) + 128*a**4*b**5*x**16*gamma(-3/4) + 64*a**3*b**6*x**20*gamma(-3/4)) + 40*a*b**(35/4)*x
**16*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(64*a**5*b**4*x**12*gamma(-3/4) + 128*a**4*b**5*x**16*gamma(-3/4) +
64*a**3*b**6*x**20*gamma(-3/4)) + 32*b**(39/4)*x**20*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(64*a**5*b**4*x**12*
gamma(-3/4) + 128*a**4*b**5*x**16*gamma(-3/4) + 64*a**3*b**6*x**20*gamma(-3/4))

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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((b*x^4+a)^(3/4)/x^16,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/4)/x^16, x)

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Mupad [B]
time = 1.55, size = 73, normalized size = 1.07 \begin {gather*} \frac {8\,b^2\,{\left (b\,x^4+a\right )}^{3/4}}{385\,a^2\,x^7}-\frac {b\,{\left (b\,x^4+a\right )}^{3/4}}{55\,a\,x^{11}}-\frac {32\,b^3\,{\left (b\,x^4+a\right )}^{3/4}}{1155\,a^3\,x^3}-\frac {{\left (b\,x^4+a\right )}^{3/4}}{15\,x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^4)^(3/4)/x^16,x)

[Out]

(8*b^2*(a + b*x^4)^(3/4))/(385*a^2*x^7) - (b*(a + b*x^4)^(3/4))/(55*a*x^11) - (32*b^3*(a + b*x^4)^(3/4))/(1155
*a^3*x^3) - (a + b*x^4)^(3/4)/(15*x^15)

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