3.12.60 \(\int \frac {1}{x^{12} (a+b x^4)^{5/4}} \, dx\) [1160]

Optimal. Leaf size=90 \[ -\frac {1}{11 a x^{11} \sqrt [4]{a+b x^4}}+\frac {12 b}{77 a^2 x^7 \sqrt [4]{a+b x^4}}-\frac {32 b^2}{77 a^3 x^3 \sqrt [4]{a+b x^4}}-\frac {128 b^3 x}{77 a^4 \sqrt [4]{a+b x^4}} \]

[Out]

-1/11/a/x^11/(b*x^4+a)^(1/4)+12/77*b/a^2/x^7/(b*x^4+a)^(1/4)-32/77*b^2/a^3/x^3/(b*x^4+a)^(1/4)-128/77*b^3*x/a^
4/(b*x^4+a)^(1/4)

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Rubi [A]
time = 0.02, antiderivative size = 90, 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 = {277, 197} \begin {gather*} -\frac {128 b^3 x}{77 a^4 \sqrt [4]{a+b x^4}}-\frac {32 b^2}{77 a^3 x^3 \sqrt [4]{a+b x^4}}+\frac {12 b}{77 a^2 x^7 \sqrt [4]{a+b x^4}}-\frac {1}{11 a x^{11} \sqrt [4]{a+b x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^12*(a + b*x^4)^(5/4)),x]

[Out]

-1/11*1/(a*x^11*(a + b*x^4)^(1/4)) + (12*b)/(77*a^2*x^7*(a + b*x^4)^(1/4)) - (32*b^2)/(77*a^3*x^3*(a + b*x^4)^
(1/4)) - (128*b^3*x)/(77*a^4*(a + b*x^4)^(1/4))

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

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 {1}{x^{12} \left (a+b x^4\right )^{5/4}} \, dx &=-\frac {1}{11 a x^{11} \sqrt [4]{a+b x^4}}-\frac {(12 b) \int \frac {1}{x^8 \left (a+b x^4\right )^{5/4}} \, dx}{11 a}\\ &=-\frac {1}{11 a x^{11} \sqrt [4]{a+b x^4}}+\frac {12 b}{77 a^2 x^7 \sqrt [4]{a+b x^4}}+\frac {\left (96 b^2\right ) \int \frac {1}{x^4 \left (a+b x^4\right )^{5/4}} \, dx}{77 a^2}\\ &=-\frac {1}{11 a x^{11} \sqrt [4]{a+b x^4}}+\frac {12 b}{77 a^2 x^7 \sqrt [4]{a+b x^4}}-\frac {32 b^2}{77 a^3 x^3 \sqrt [4]{a+b x^4}}-\frac {\left (128 b^3\right ) \int \frac {1}{\left (a+b x^4\right )^{5/4}} \, dx}{77 a^3}\\ &=-\frac {1}{11 a x^{11} \sqrt [4]{a+b x^4}}+\frac {12 b}{77 a^2 x^7 \sqrt [4]{a+b x^4}}-\frac {32 b^2}{77 a^3 x^3 \sqrt [4]{a+b x^4}}-\frac {128 b^3 x}{77 a^4 \sqrt [4]{a+b x^4}}\\ \end {align*}

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Mathematica [A]
time = 0.46, size = 53, normalized size = 0.59 \begin {gather*} \frac {-7 a^3+12 a^2 b x^4-32 a b^2 x^8-128 b^3 x^{12}}{77 a^4 x^{11} \sqrt [4]{a+b x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^12*(a + b*x^4)^(5/4)),x]

[Out]

(-7*a^3 + 12*a^2*b*x^4 - 32*a*b^2*x^8 - 128*b^3*x^12)/(77*a^4*x^11*(a + b*x^4)^(1/4))

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Maple [A]
time = 0.18, size = 50, normalized size = 0.56

method	result	size
gosper	\(-\frac {128 b^{3} x^{12}+32 a \,b^{2} x^{8}-12 a^{2} b \,x^{4}+7 a^{3}}{77 x^{11} \left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{4}}\)	\(50\)
trager	\(-\frac {128 b^{3} x^{12}+32 a \,b^{2} x^{8}-12 a^{2} b \,x^{4}+7 a^{3}}{77 x^{11} \left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{4}}\)	\(50\)
risch	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (51 b^{2} x^{8}-19 a b \,x^{4}+7 a^{2}\right )}{77 a^{4} x^{11}}-\frac {b^{3} x}{a^{4} \left (b \,x^{4}+a \right )^{\frac {1}{4}}}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^12/(b*x^4+a)^(5/4),x,method=_RETURNVERBOSE)

[Out]

-1/77*(128*b^3*x^12+32*a*b^2*x^8-12*a^2*b*x^4+7*a^3)/x^11/(b*x^4+a)^(1/4)/a^4

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Maxima [A]
time = 0.29, size = 71, normalized size = 0.79 \begin {gather*} -\frac {b^{3} x}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{4}} - \frac {\frac {77 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} b^{2}}{x^{3}} - \frac {33 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} b}{x^{7}} + \frac {7 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}}}{x^{11}}}{77 \, a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^12/(b*x^4+a)^(5/4),x, algorithm="maxima")

[Out]

-b^3*x/((b*x^4 + a)^(1/4)*a^4) - 1/77*(77*(b*x^4 + a)^(3/4)*b^2/x^3 - 33*(b*x^4 + a)^(7/4)*b/x^7 + 7*(b*x^4 +
a)^(11/4)/x^11)/a^4

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Fricas [A]
time = 0.40, size = 61, normalized size = 0.68 \begin {gather*} -\frac {{\left (128 \, b^{3} x^{12} + 32 \, a b^{2} x^{8} - 12 \, a^{2} b x^{4} + 7 \, a^{3}\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{77 \, {\left (a^{4} b x^{15} + a^{5} x^{11}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^12/(b*x^4+a)^(5/4),x, algorithm="fricas")

[Out]

-1/77*(128*b^3*x^12 + 32*a*b^2*x^8 - 12*a^2*b*x^4 + 7*a^3)*(b*x^4 + a)^(3/4)/(a^4*b*x^15 + a^5*x^11)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 592 vs. \(2 (85) = 170\).
time = 1.36, size = 592, normalized size = 6.58 \begin {gather*} \frac {21 a^{5} b^{\frac {39}{4}} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{256 a^{7} b^{9} x^{8} \Gamma \left (\frac {5}{4}\right ) + 768 a^{6} b^{10} x^{12} \Gamma \left (\frac {5}{4}\right ) + 768 a^{5} b^{11} x^{16} \Gamma \left (\frac {5}{4}\right ) + 256 a^{4} b^{12} x^{20} \Gamma \left (\frac {5}{4}\right )} + \frac {6 a^{4} b^{\frac {43}{4}} x^{4} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{256 a^{7} b^{9} x^{8} \Gamma \left (\frac {5}{4}\right ) + 768 a^{6} b^{10} x^{12} \Gamma \left (\frac {5}{4}\right ) + 768 a^{5} b^{11} x^{16} \Gamma \left (\frac {5}{4}\right ) + 256 a^{4} b^{12} x^{20} \Gamma \left (\frac {5}{4}\right )} + \frac {45 a^{3} b^{\frac {47}{4}} x^{8} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{256 a^{7} b^{9} x^{8} \Gamma \left (\frac {5}{4}\right ) + 768 a^{6} b^{10} x^{12} \Gamma \left (\frac {5}{4}\right ) + 768 a^{5} b^{11} x^{16} \Gamma \left (\frac {5}{4}\right ) + 256 a^{4} b^{12} x^{20} \Gamma \left (\frac {5}{4}\right )} + \frac {540 a^{2} b^{\frac {51}{4}} x^{12} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{256 a^{7} b^{9} x^{8} \Gamma \left (\frac {5}{4}\right ) + 768 a^{6} b^{10} x^{12} \Gamma \left (\frac {5}{4}\right ) + 768 a^{5} b^{11} x^{16} \Gamma \left (\frac {5}{4}\right ) + 256 a^{4} b^{12} x^{20} \Gamma \left (\frac {5}{4}\right )} + \frac {864 a b^{\frac {55}{4}} x^{16} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{256 a^{7} b^{9} x^{8} \Gamma \left (\frac {5}{4}\right ) + 768 a^{6} b^{10} x^{12} \Gamma \left (\frac {5}{4}\right ) + 768 a^{5} b^{11} x^{16} \Gamma \left (\frac {5}{4}\right ) + 256 a^{4} b^{12} x^{20} \Gamma \left (\frac {5}{4}\right )} + \frac {384 b^{\frac {59}{4}} x^{20} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{256 a^{7} b^{9} x^{8} \Gamma \left (\frac {5}{4}\right ) + 768 a^{6} b^{10} x^{12} \Gamma \left (\frac {5}{4}\right ) + 768 a^{5} b^{11} x^{16} \Gamma \left (\frac {5}{4}\right ) + 256 a^{4} b^{12} x^{20} \Gamma \left (\frac {5}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**12/(b*x**4+a)**(5/4),x)

[Out]

21*a**5*b**(39/4)*(a/(b*x**4) + 1)**(3/4)*gamma(-11/4)/(256*a**7*b**9*x**8*gamma(5/4) + 768*a**6*b**10*x**12*g
amma(5/4) + 768*a**5*b**11*x**16*gamma(5/4) + 256*a**4*b**12*x**20*gamma(5/4)) + 6*a**4*b**(43/4)*x**4*(a/(b*x
**4) + 1)**(3/4)*gamma(-11/4)/(256*a**7*b**9*x**8*gamma(5/4) + 768*a**6*b**10*x**12*gamma(5/4) + 768*a**5*b**1
1*x**16*gamma(5/4) + 256*a**4*b**12*x**20*gamma(5/4)) + 45*a**3*b**(47/4)*x**8*(a/(b*x**4) + 1)**(3/4)*gamma(-
11/4)/(256*a**7*b**9*x**8*gamma(5/4) + 768*a**6*b**10*x**12*gamma(5/4) + 768*a**5*b**11*x**16*gamma(5/4) + 256
*a**4*b**12*x**20*gamma(5/4)) + 540*a**2*b**(51/4)*x**12*(a/(b*x**4) + 1)**(3/4)*gamma(-11/4)/(256*a**7*b**9*x
**8*gamma(5/4) + 768*a**6*b**10*x**12*gamma(5/4) + 768*a**5*b**11*x**16*gamma(5/4) + 256*a**4*b**12*x**20*gamm
a(5/4)) + 864*a*b**(55/4)*x**16*(a/(b*x**4) + 1)**(3/4)*gamma(-11/4)/(256*a**7*b**9*x**8*gamma(5/4) + 768*a**6
*b**10*x**12*gamma(5/4) + 768*a**5*b**11*x**16*gamma(5/4) + 256*a**4*b**12*x**20*gamma(5/4)) + 384*b**(59/4)*x
**20*(a/(b*x**4) + 1)**(3/4)*gamma(-11/4)/(256*a**7*b**9*x**8*gamma(5/4) + 768*a**6*b**10*x**12*gamma(5/4) + 7
68*a**5*b**11*x**16*gamma(5/4) + 256*a**4*b**12*x**20*gamma(5/4))

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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(1/x^12/(b*x^4+a)^(5/4),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(5/4)*x^12), x)

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Mupad [B]
time = 1.46, size = 74, normalized size = 0.82 \begin {gather*} \frac {19\,b\,{\left (b\,x^4+a\right )}^{3/4}}{77\,a^3\,x^7}-\frac {b^3\,x}{a^4\,{\left (b\,x^4+a\right )}^{1/4}}-\frac {{\left (b\,x^4+a\right )}^{3/4}}{11\,a^2\,x^{11}}-\frac {51\,b^2\,{\left (b\,x^4+a\right )}^{3/4}}{77\,a^4\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^12*(a + b*x^4)^(5/4)),x)

[Out]

(19*b*(a + b*x^4)^(3/4))/(77*a^3*x^7) - (b^3*x)/(a^4*(a + b*x^4)^(1/4)) - (a + b*x^4)^(3/4)/(11*a^2*x^11) - (5
1*b^2*(a + b*x^4)^(3/4))/(77*a^4*x^3)

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