3.12.73 \(\int \frac {1}{(a+b x^4)^{13/4}} \, dx\) [1173]

Optimal. Leaf size=58 \[ \frac {x}{9 a \left (a+b x^4\right )^{9/4}}+\frac {8 x}{45 a^2 \left (a+b x^4\right )^{5/4}}+\frac {32 x}{45 a^3 \sqrt [4]{a+b x^4}} \]

[Out]

1/9*x/a/(b*x^4+a)^(9/4)+8/45*x/a^2/(b*x^4+a)^(5/4)+32/45*x/a^3/(b*x^4+a)^(1/4)

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Rubi [A]
time = 0.01, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, 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 {32 x}{45 a^3 \sqrt [4]{a+b x^4}}+\frac {8 x}{45 a^2 \left (a+b x^4\right )^{5/4}}+\frac {x}{9 a \left (a+b x^4\right )^{9/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(-13/4),x]

[Out]

x/(9*a*(a + b*x^4)^(9/4)) + (8*x)/(45*a^2*(a + b*x^4)^(5/4)) + (32*x)/(45*a^3*(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 )^{13/4}} \, dx &=\frac {x}{9 a \left (a+b x^4\right )^{9/4}}+\frac {8 \int \frac {1}{\left (a+b x^4\right )^{9/4}} \, dx}{9 a}\\ &=\frac {x}{9 a \left (a+b x^4\right )^{9/4}}+\frac {8 x}{45 a^2 \left (a+b x^4\right )^{5/4}}+\frac {32 \int \frac {1}{\left (a+b x^4\right )^{5/4}} \, dx}{45 a^2}\\ &=\frac {x}{9 a \left (a+b x^4\right )^{9/4}}+\frac {8 x}{45 a^2 \left (a+b x^4\right )^{5/4}}+\frac {32 x}{45 a^3 \sqrt [4]{a+b x^4}}\\ \end {align*}

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Mathematica [A]
time = 0.34, size = 40, normalized size = 0.69 \begin {gather*} \frac {45 a^2 x+72 a b x^5+32 b^2 x^9}{45 a^3 \left (a+b x^4\right )^{9/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^(-13/4),x]

[Out]

(45*a^2*x + 72*a*b*x^5 + 32*b^2*x^9)/(45*a^3*(a + b*x^4)^(9/4))

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Maple [A]
time = 0.19, size = 37, normalized size = 0.64

method	result	size
gosper	\(\frac {x \left (32 b^{2} x^{8}+72 a b \,x^{4}+45 a^{2}\right )}{45 \left (b \,x^{4}+a \right )^{\frac {9}{4}} a^{3}}\)	\(37\)
trager	\(\frac {x \left (32 b^{2} x^{8}+72 a b \,x^{4}+45 a^{2}\right )}{45 \left (b \,x^{4}+a \right )^{\frac {9}{4}} a^{3}}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^4+a)^(13/4),x,method=_RETURNVERBOSE)

[Out]

1/45*x*(32*b^2*x^8+72*a*b*x^4+45*a^2)/(b*x^4+a)^(9/4)/a^3

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Maxima [A]
time = 0.30, size = 50, normalized size = 0.86 \begin {gather*} \frac {{\left (5 \, b^{2} - \frac {18 \, {\left (b x^{4} + a\right )} b}{x^{4}} + \frac {45 \, {\left (b x^{4} + a\right )}^{2}}{x^{8}}\right )} x^{9}}{45 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}} a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a)^(13/4),x, algorithm="maxima")

[Out]

1/45*(5*b^2 - 18*(b*x^4 + a)*b/x^4 + 45*(b*x^4 + a)^2/x^8)*x^9/((b*x^4 + a)^(9/4)*a^3)

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Fricas [A]
time = 0.40, size = 69, normalized size = 1.19 \begin {gather*} \frac {{\left (32 \, b^{2} x^{9} + 72 \, a b x^{5} + 45 \, a^{2} x\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{45 \, {\left (a^{3} b^{3} x^{12} + 3 \, a^{4} b^{2} x^{8} + 3 \, a^{5} b x^{4} + a^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a)^(13/4),x, algorithm="fricas")

[Out]

1/45*(32*b^2*x^9 + 72*a*b*x^5 + 45*a^2*x)*(b*x^4 + a)^(3/4)/(a^3*b^3*x^12 + 3*a^4*b^2*x^8 + 3*a^5*b*x^4 + a^6)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 515 vs. \(2 (51) = 102\).
time = 1.37, size = 515, normalized size = 8.88 \begin {gather*} \frac {45 a^{5} x \Gamma \left (\frac {1}{4}\right )}{64 a^{\frac {33}{4}} \sqrt [4]{1 + \frac {b x^{4}}{a}} \Gamma \left (\frac {13}{4}\right ) + 192 a^{\frac {29}{4}} b x^{4} \sqrt [4]{1 + \frac {b x^{4}}{a}} \Gamma \left (\frac {13}{4}\right ) + 192 a^{\frac {25}{4}} b^{2} x^{8} \sqrt [4]{1 + \frac {b x^{4}}{a}} \Gamma \left (\frac {13}{4}\right ) + 64 a^{\frac {21}{4}} b^{3} x^{12} \sqrt [4]{1 + \frac {b x^{4}}{a}} \Gamma \left (\frac {13}{4}\right )} + \frac {117 a^{4} b x^{5} \Gamma \left (\frac {1}{4}\right )}{64 a^{\frac {33}{4}} \sqrt [4]{1 + \frac {b x^{4}}{a}} \Gamma \left (\frac {13}{4}\right ) + 192 a^{\frac {29}{4}} b x^{4} \sqrt [4]{1 + \frac {b x^{4}}{a}} \Gamma \left (\frac {13}{4}\right ) + 192 a^{\frac {25}{4}} b^{2} x^{8} \sqrt [4]{1 + \frac {b x^{4}}{a}} \Gamma \left (\frac {13}{4}\right ) + 64 a^{\frac {21}{4}} b^{3} x^{12} \sqrt [4]{1 + \frac {b x^{4}}{a}} \Gamma \left (\frac {13}{4}\right )} + \frac {104 a^{3} b^{2} x^{9} \Gamma \left (\frac {1}{4}\right )}{64 a^{\frac {33}{4}} \sqrt [4]{1 + \frac {b x^{4}}{a}} \Gamma \left (\frac {13}{4}\right ) + 192 a^{\frac {29}{4}} b x^{4} \sqrt [4]{1 + \frac {b x^{4}}{a}} \Gamma \left (\frac {13}{4}\right ) + 192 a^{\frac {25}{4}} b^{2} x^{8} \sqrt [4]{1 + \frac {b x^{4}}{a}} \Gamma \left (\frac {13}{4}\right ) + 64 a^{\frac {21}{4}} b^{3} x^{12} \sqrt [4]{1 + \frac {b x^{4}}{a}} \Gamma \left (\frac {13}{4}\right )} + \frac {32 a^{2} b^{3} x^{13} \Gamma \left (\frac {1}{4}\right )}{64 a^{\frac {33}{4}} \sqrt [4]{1 + \frac {b x^{4}}{a}} \Gamma \left (\frac {13}{4}\right ) + 192 a^{\frac {29}{4}} b x^{4} \sqrt [4]{1 + \frac {b x^{4}}{a}} \Gamma \left (\frac {13}{4}\right ) + 192 a^{\frac {25}{4}} b^{2} x^{8} \sqrt [4]{1 + \frac {b x^{4}}{a}} \Gamma \left (\frac {13}{4}\right ) + 64 a^{\frac {21}{4}} b^{3} x^{12} \sqrt [4]{1 + \frac {b x^{4}}{a}} \Gamma \left (\frac {13}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**4+a)**(13/4),x)

[Out]

45*a**5*x*gamma(1/4)/(64*a**(33/4)*(1 + b*x**4/a)**(1/4)*gamma(13/4) + 192*a**(29/4)*b*x**4*(1 + b*x**4/a)**(1
/4)*gamma(13/4) + 192*a**(25/4)*b**2*x**8*(1 + b*x**4/a)**(1/4)*gamma(13/4) + 64*a**(21/4)*b**3*x**12*(1 + b*x
**4/a)**(1/4)*gamma(13/4)) + 117*a**4*b*x**5*gamma(1/4)/(64*a**(33/4)*(1 + b*x**4/a)**(1/4)*gamma(13/4) + 192*
a**(29/4)*b*x**4*(1 + b*x**4/a)**(1/4)*gamma(13/4) + 192*a**(25/4)*b**2*x**8*(1 + b*x**4/a)**(1/4)*gamma(13/4)
+ 64*a**(21/4)*b**3*x**12*(1 + b*x**4/a)**(1/4)*gamma(13/4)) + 104*a**3*b**2*x**9*gamma(1/4)/(64*a**(33/4)*(1
+ b*x**4/a)**(1/4)*gamma(13/4) + 192*a**(29/4)*b*x**4*(1 + b*x**4/a)**(1/4)*gamma(13/4) + 192*a**(25/4)*b**2*
x**8*(1 + b*x**4/a)**(1/4)*gamma(13/4) + 64*a**(21/4)*b**3*x**12*(1 + b*x**4/a)**(1/4)*gamma(13/4)) + 32*a**2*
b**3*x**13*gamma(1/4)/(64*a**(33/4)*(1 + b*x**4/a)**(1/4)*gamma(13/4) + 192*a**(29/4)*b*x**4*(1 + b*x**4/a)**(
1/4)*gamma(13/4) + 192*a**(25/4)*b**2*x**8*(1 + b*x**4/a)**(1/4)*gamma(13/4) + 64*a**(21/4)*b**3*x**12*(1 + b*
x**4/a)**(1/4)*gamma(13/4))

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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)^(13/4),x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(-13/4), x)

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Mupad [B]
time = 1.07, size = 44, normalized size = 0.76 \begin {gather*} \frac {32\,x\,{\left (b\,x^4+a\right )}^2+5\,a^2\,x+8\,a\,x\,\left (b\,x^4+a\right )}{45\,a^3\,{\left (b\,x^4+a\right )}^{9/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*x^4)^(13/4),x)

[Out]

(32*x*(a + b*x^4)^2 + 5*a^2*x + 8*a*x*(a + b*x^4))/(45*a^3*(a + b*x^4)^(9/4))

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