3.12.74 \(\int \frac {1}{(a+b x^4)^{17/4}} \, dx\) [1174]

Optimal. Leaf size=77 \[ \frac {x}{13 a \left (a+b x^4\right )^{13/4}}+\frac {4 x}{39 a^2 \left (a+b x^4\right )^{9/4}}+\frac {32 x}{195 a^3 \left (a+b x^4\right )^{5/4}}+\frac {128 x}{195 a^4 \sqrt [4]{a+b x^4}} \]

[Out]

1/13*x/a/(b*x^4+a)^(13/4)+4/39*x/a^2/(b*x^4+a)^(9/4)+32/195*x/a^3/(b*x^4+a)^(5/4)+128/195*x/a^4/(b*x^4+a)^(1/4
)

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Rubi [A]
time = 0.01, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {198, 197} \begin {gather*} \frac {128 x}{195 a^4 \sqrt [4]{a+b x^4}}+\frac {32 x}{195 a^3 \left (a+b x^4\right )^{5/4}}+\frac {4 x}{39 a^2 \left (a+b x^4\right )^{9/4}}+\frac {x}{13 a \left (a+b x^4\right )^{13/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(-17/4),x]

[Out]

x/(13*a*(a + b*x^4)^(13/4)) + (4*x)/(39*a^2*(a + b*x^4)^(9/4)) + (32*x)/(195*a^3*(a + b*x^4)^(5/4)) + (128*x)/
(195*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 198

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

Rubi steps

\begin {align*} \int \frac {1}{\left (a+b x^4\right )^{17/4}} \, dx &=\frac {x}{13 a \left (a+b x^4\right )^{13/4}}+\frac {12 \int \frac {1}{\left (a+b x^4\right )^{13/4}} \, dx}{13 a}\\ &=\frac {x}{13 a \left (a+b x^4\right )^{13/4}}+\frac {4 x}{39 a^2 \left (a+b x^4\right )^{9/4}}+\frac {32 \int \frac {1}{\left (a+b x^4\right )^{9/4}} \, dx}{39 a^2}\\ &=\frac {x}{13 a \left (a+b x^4\right )^{13/4}}+\frac {4 x}{39 a^2 \left (a+b x^4\right )^{9/4}}+\frac {32 x}{195 a^3 \left (a+b x^4\right )^{5/4}}+\frac {128 \int \frac {1}{\left (a+b x^4\right )^{5/4}} \, dx}{195 a^3}\\ &=\frac {x}{13 a \left (a+b x^4\right )^{13/4}}+\frac {4 x}{39 a^2 \left (a+b x^4\right )^{9/4}}+\frac {32 x}{195 a^3 \left (a+b x^4\right )^{5/4}}+\frac {128 x}{195 a^4 \sqrt [4]{a+b x^4}}\\ \end {align*}

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Mathematica [A]
time = 0.54, size = 51, normalized size = 0.66 \begin {gather*} \frac {195 a^3 x+468 a^2 b x^5+416 a b^2 x^9+128 b^3 x^{13}}{195 a^4 \left (a+b x^4\right )^{13/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^(-17/4),x]

[Out]

(195*a^3*x + 468*a^2*b*x^5 + 416*a*b^2*x^9 + 128*b^3*x^13)/(195*a^4*(a + b*x^4)^(13/4))

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Maple [A]
time = 0.16, size = 48, normalized size = 0.62

method	result	size
gosper	\(\frac {x \left (128 b^{3} x^{12}+416 a \,b^{2} x^{8}+468 a^{2} b \,x^{4}+195 a^{3}\right )}{195 \left (b \,x^{4}+a \right )^{\frac {13}{4}} a^{4}}\)	\(48\)
trager	\(\frac {x \left (128 b^{3} x^{12}+416 a \,b^{2} x^{8}+468 a^{2} b \,x^{4}+195 a^{3}\right )}{195 \left (b \,x^{4}+a \right )^{\frac {13}{4}} a^{4}}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^4+a)^(17/4),x,method=_RETURNVERBOSE)

[Out]

1/195*x*(128*b^3*x^12+416*a*b^2*x^8+468*a^2*b*x^4+195*a^3)/(b*x^4+a)^(13/4)/a^4

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Maxima [A]
time = 0.30, size = 67, normalized size = 0.87 \begin {gather*} -\frac {{\left (15 \, b^{3} - \frac {65 \, {\left (b x^{4} + a\right )} b^{2}}{x^{4}} + \frac {117 \, {\left (b x^{4} + a\right )}^{2} b}{x^{8}} - \frac {195 \, {\left (b x^{4} + a\right )}^{3}}{x^{12}}\right )} x^{13}}{195 \, {\left (b x^{4} + a\right )}^{\frac {13}{4}} a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a)^(17/4),x, algorithm="maxima")

[Out]

-1/195*(15*b^3 - 65*(b*x^4 + a)*b^2/x^4 + 117*(b*x^4 + a)^2*b/x^8 - 195*(b*x^4 + a)^3/x^12)*x^13/((b*x^4 + a)^
(13/4)*a^4)

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Fricas [A]
time = 0.40, size = 91, normalized size = 1.18 \begin {gather*} \frac {{\left (128 \, b^{3} x^{13} + 416 \, a b^{2} x^{9} + 468 \, a^{2} b x^{5} + 195 \, a^{3} x\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{195 \, {\left (a^{4} b^{4} x^{16} + 4 \, a^{5} b^{3} x^{12} + 6 \, a^{6} b^{2} x^{8} + 4 \, a^{7} b x^{4} + a^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a)^(17/4),x, algorithm="fricas")

[Out]

1/195*(128*b^3*x^13 + 416*a*b^2*x^9 + 468*a^2*b*x^5 + 195*a^3*x)*(b*x^4 + a)^(3/4)/(a^4*b^4*x^16 + 4*a^5*b^3*x
^12 + 6*a^6*b^2*x^8 + 4*a^7*b*x^4 + a^8)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1550 vs. \(2 (70) = 140\).
time = 2.80, size = 1550, normalized size = 20.13 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**4+a)**(17/4),x)

[Out]

585*a**14*x*gamma(1/4)/(256*a**(73/4)*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 1536*a**(69/4)*b*x**4*(1 + b*x**4/a)
**(1/4)*gamma(17/4) + 3840*a**(65/4)*b**2*x**8*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 5120*a**(61/4)*b**3*x**12*(
1 + b*x**4/a)**(1/4)*gamma(17/4) + 3840*a**(57/4)*b**4*x**16*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 1536*a**(53/4
)*b**5*x**20*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 256*a**(49/4)*b**6*x**24*(1 + b*x**4/a)**(1/4)*gamma(17/4)) +
3159*a**13*b*x**5*gamma(1/4)/(256*a**(73/4)*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 1536*a**(69/4)*b*x**4*(1 + b*
x**4/a)**(1/4)*gamma(17/4) + 3840*a**(65/4)*b**2*x**8*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 5120*a**(61/4)*b**3*
x**12*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 3840*a**(57/4)*b**4*x**16*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 1536*a
**(53/4)*b**5*x**20*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 256*a**(49/4)*b**6*x**24*(1 + b*x**4/a)**(1/4)*gamma(1
7/4)) + 7215*a**12*b**2*x**9*gamma(1/4)/(256*a**(73/4)*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 1536*a**(69/4)*b*x*
*4*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 3840*a**(65/4)*b**2*x**8*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 5120*a**(6
1/4)*b**3*x**12*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 3840*a**(57/4)*b**4*x**16*(1 + b*x**4/a)**(1/4)*gamma(17/4
) + 1536*a**(53/4)*b**5*x**20*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 256*a**(49/4)*b**6*x**24*(1 + b*x**4/a)**(1/
4)*gamma(17/4)) + 8925*a**11*b**3*x**13*gamma(1/4)/(256*a**(73/4)*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 1536*a**
(69/4)*b*x**4*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 3840*a**(65/4)*b**2*x**8*(1 + b*x**4/a)**(1/4)*gamma(17/4) +
5120*a**(61/4)*b**3*x**12*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 3840*a**(57/4)*b**4*x**16*(1 + b*x**4/a)**(1/4)
*gamma(17/4) + 1536*a**(53/4)*b**5*x**20*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 256*a**(49/4)*b**6*x**24*(1 + b*x
**4/a)**(1/4)*gamma(17/4)) + 6300*a**10*b**4*x**17*gamma(1/4)/(256*a**(73/4)*(1 + b*x**4/a)**(1/4)*gamma(17/4)
+ 1536*a**(69/4)*b*x**4*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 3840*a**(65/4)*b**2*x**8*(1 + b*x**4/a)**(1/4)*ga
mma(17/4) + 5120*a**(61/4)*b**3*x**12*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 3840*a**(57/4)*b**4*x**16*(1 + b*x**
4/a)**(1/4)*gamma(17/4) + 1536*a**(53/4)*b**5*x**20*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 256*a**(49/4)*b**6*x**
24*(1 + b*x**4/a)**(1/4)*gamma(17/4)) + 2400*a**9*b**5*x**21*gamma(1/4)/(256*a**(73/4)*(1 + b*x**4/a)**(1/4)*g
amma(17/4) + 1536*a**(69/4)*b*x**4*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 3840*a**(65/4)*b**2*x**8*(1 + b*x**4/a)
**(1/4)*gamma(17/4) + 5120*a**(61/4)*b**3*x**12*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 3840*a**(57/4)*b**4*x**16*
(1 + b*x**4/a)**(1/4)*gamma(17/4) + 1536*a**(53/4)*b**5*x**20*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 256*a**(49/4
)*b**6*x**24*(1 + b*x**4/a)**(1/4)*gamma(17/4)) + 384*a**8*b**6*x**25*gamma(1/4)/(256*a**(73/4)*(1 + b*x**4/a)
**(1/4)*gamma(17/4) + 1536*a**(69/4)*b*x**4*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 3840*a**(65/4)*b**2*x**8*(1 +
b*x**4/a)**(1/4)*gamma(17/4) + 5120*a**(61/4)*b**3*x**12*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 3840*a**(57/4)*b*
*4*x**16*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 1536*a**(53/4)*b**5*x**20*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 256
*a**(49/4)*b**6*x**24*(1 + b*x**4/a)**(1/4)*gamma(17/4))

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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/(b*x^4+a)^(17/4),x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(-17/4), x)

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Mupad [B]
time = 1.08, size = 61, normalized size = 0.79 \begin {gather*} \frac {128\,x}{195\,a^4\,{\left (b\,x^4+a\right )}^{1/4}}+\frac {32\,x}{195\,a^3\,{\left (b\,x^4+a\right )}^{5/4}}+\frac {4\,x}{39\,a^2\,{\left (b\,x^4+a\right )}^{9/4}}+\frac {x}{13\,a\,{\left (b\,x^4+a\right )}^{13/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*x^4)^(17/4),x)

[Out]

(128*x)/(195*a^4*(a + b*x^4)^(1/4)) + (32*x)/(195*a^3*(a + b*x^4)^(5/4)) + (4*x)/(39*a^2*(a + b*x^4)^(9/4)) +
x/(13*a*(a + b*x^4)^(13/4))

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