3.11.17 \(\int x^{19} (a+b x^4)^{3/4} \, dx\) [1017]

Optimal. Leaf size=101 \[ \frac {a^4 \left (a+b x^4\right )^{7/4}}{7 b^5}-\frac {4 a^3 \left (a+b x^4\right )^{11/4}}{11 b^5}+\frac {2 a^2 \left (a+b x^4\right )^{15/4}}{5 b^5}-\frac {4 a \left (a+b x^4\right )^{19/4}}{19 b^5}+\frac {\left (a+b x^4\right )^{23/4}}{23 b^5} \]

[Out]

1/7*a^4*(b*x^4+a)^(7/4)/b^5-4/11*a^3*(b*x^4+a)^(11/4)/b^5+2/5*a^2*(b*x^4+a)^(15/4)/b^5-4/19*a*(b*x^4+a)^(19/4)
/b^5+1/23*(b*x^4+a)^(23/4)/b^5

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Rubi [A]
time = 0.04, antiderivative size = 101, 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 = {272, 45} \begin {gather*} \frac {a^4 \left (a+b x^4\right )^{7/4}}{7 b^5}-\frac {4 a^3 \left (a+b x^4\right )^{11/4}}{11 b^5}+\frac {2 a^2 \left (a+b x^4\right )^{15/4}}{5 b^5}+\frac {\left (a+b x^4\right )^{23/4}}{23 b^5}-\frac {4 a \left (a+b x^4\right )^{19/4}}{19 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^19*(a + b*x^4)^(3/4),x]

[Out]

(a^4*(a + b*x^4)^(7/4))/(7*b^5) - (4*a^3*(a + b*x^4)^(11/4))/(11*b^5) + (2*a^2*(a + b*x^4)^(15/4))/(5*b^5) - (
4*a*(a + b*x^4)^(19/4))/(19*b^5) + (a + b*x^4)^(23/4)/(23*b^5)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

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

Rubi steps

\begin {align*} \int x^{19} \left (a+b x^4\right )^{3/4} \, dx &=\frac {1}{4} \text {Subst}\left (\int x^4 (a+b x)^{3/4} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (\frac {a^4 (a+b x)^{3/4}}{b^4}-\frac {4 a^3 (a+b x)^{7/4}}{b^4}+\frac {6 a^2 (a+b x)^{11/4}}{b^4}-\frac {4 a (a+b x)^{15/4}}{b^4}+\frac {(a+b x)^{19/4}}{b^4}\right ) \, dx,x,x^4\right )\\ &=\frac {a^4 \left (a+b x^4\right )^{7/4}}{7 b^5}-\frac {4 a^3 \left (a+b x^4\right )^{11/4}}{11 b^5}+\frac {2 a^2 \left (a+b x^4\right )^{15/4}}{5 b^5}-\frac {4 a \left (a+b x^4\right )^{19/4}}{19 b^5}+\frac {\left (a+b x^4\right )^{23/4}}{23 b^5}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 61, normalized size = 0.60 \begin {gather*} \frac {\left (a+b x^4\right )^{7/4} \left (2048 a^4-3584 a^3 b x^4+4928 a^2 b^2 x^8-6160 a b^3 x^{12}+7315 b^4 x^{16}\right )}{168245 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^19*(a + b*x^4)^(3/4),x]

[Out]

((a + b*x^4)^(7/4)*(2048*a^4 - 3584*a^3*b*x^4 + 4928*a^2*b^2*x^8 - 6160*a*b^3*x^12 + 7315*b^4*x^16))/(168245*b
^5)

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Maple [A]
time = 0.14, size = 58, normalized size = 0.57

method	result	size
gosper	\(\frac {\left (b \,x^{4}+a \right )^{\frac {7}{4}} \left (7315 x^{16} b^{4}-6160 a \,b^{3} x^{12}+4928 a^{2} b^{2} x^{8}-3584 a^{3} b \,x^{4}+2048 a^{4}\right )}{168245 b^{5}}\)	\(58\)
trager	\(\frac {\left (7315 b^{5} x^{20}+1155 a \,b^{4} x^{16}-1232 a^{2} b^{3} x^{12}+1344 a^{3} b^{2} x^{8}-1536 a^{4} b \,x^{4}+2048 a^{5}\right ) \left (b \,x^{4}+a \right )^{\frac {3}{4}}}{168245 b^{5}}\)	\(69\)
risch	\(\frac {\left (7315 b^{5} x^{20}+1155 a \,b^{4} x^{16}-1232 a^{2} b^{3} x^{12}+1344 a^{3} b^{2} x^{8}-1536 a^{4} b \,x^{4}+2048 a^{5}\right ) \left (b \,x^{4}+a \right )^{\frac {3}{4}}}{168245 b^{5}}\)	\(69\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^19*(b*x^4+a)^(3/4),x,method=_RETURNVERBOSE)

[Out]

1/168245*(b*x^4+a)^(7/4)*(7315*b^4*x^16-6160*a*b^3*x^12+4928*a^2*b^2*x^8-3584*a^3*b*x^4+2048*a^4)/b^5

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Maxima [A]
time = 0.29, size = 81, normalized size = 0.80 \begin {gather*} \frac {{\left (b x^{4} + a\right )}^{\frac {23}{4}}}{23 \, b^{5}} - \frac {4 \, {\left (b x^{4} + a\right )}^{\frac {19}{4}} a}{19 \, b^{5}} + \frac {2 \, {\left (b x^{4} + a\right )}^{\frac {15}{4}} a^{2}}{5 \, b^{5}} - \frac {4 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}} a^{3}}{11 \, b^{5}} + \frac {{\left (b x^{4} + a\right )}^{\frac {7}{4}} a^{4}}{7 \, b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^19*(b*x^4+a)^(3/4),x, algorithm="maxima")

[Out]

1/23*(b*x^4 + a)^(23/4)/b^5 - 4/19*(b*x^4 + a)^(19/4)*a/b^5 + 2/5*(b*x^4 + a)^(15/4)*a^2/b^5 - 4/11*(b*x^4 + a
)^(11/4)*a^3/b^5 + 1/7*(b*x^4 + a)^(7/4)*a^4/b^5

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Fricas [A]
time = 0.36, size = 68, normalized size = 0.67 \begin {gather*} \frac {{\left (7315 \, b^{5} x^{20} + 1155 \, a b^{4} x^{16} - 1232 \, a^{2} b^{3} x^{12} + 1344 \, a^{3} b^{2} x^{8} - 1536 \, a^{4} b x^{4} + 2048 \, a^{5}\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{168245 \, b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^19*(b*x^4+a)^(3/4),x, algorithm="fricas")

[Out]

1/168245*(7315*b^5*x^20 + 1155*a*b^4*x^16 - 1232*a^2*b^3*x^12 + 1344*a^3*b^2*x^8 - 1536*a^4*b*x^4 + 2048*a^5)*
(b*x^4 + a)^(3/4)/b^5

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Sympy [A]
time = 1.55, size = 136, normalized size = 1.35 \begin {gather*} \begin {cases} \frac {2048 a^{5} \left (a + b x^{4}\right )^{\frac {3}{4}}}{168245 b^{5}} - \frac {1536 a^{4} x^{4} \left (a + b x^{4}\right )^{\frac {3}{4}}}{168245 b^{4}} + \frac {192 a^{3} x^{8} \left (a + b x^{4}\right )^{\frac {3}{4}}}{24035 b^{3}} - \frac {16 a^{2} x^{12} \left (a + b x^{4}\right )^{\frac {3}{4}}}{2185 b^{2}} + \frac {3 a x^{16} \left (a + b x^{4}\right )^{\frac {3}{4}}}{437 b} + \frac {x^{20} \left (a + b x^{4}\right )^{\frac {3}{4}}}{23} & \text {for}\: b \neq 0 \\\frac {a^{\frac {3}{4}} x^{20}}{20} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**19*(b*x**4+a)**(3/4),x)

[Out]

Piecewise((2048*a**5*(a + b*x**4)**(3/4)/(168245*b**5) - 1536*a**4*x**4*(a + b*x**4)**(3/4)/(168245*b**4) + 19
2*a**3*x**8*(a + b*x**4)**(3/4)/(24035*b**3) - 16*a**2*x**12*(a + b*x**4)**(3/4)/(2185*b**2) + 3*a*x**16*(a +
b*x**4)**(3/4)/(437*b) + x**20*(a + b*x**4)**(3/4)/23, Ne(b, 0)), (a**(3/4)*x**20/20, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (81) = 162\).
time = 1.53, size = 163, normalized size = 1.61 \begin {gather*} \frac {\frac {23 \, {\left (1155 \, {\left (b x^{4} + a\right )}^{\frac {19}{4}} - 5852 \, {\left (b x^{4} + a\right )}^{\frac {15}{4}} a + 11970 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}} a^{2} - 12540 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} a^{3} + 7315 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} a^{4}\right )} a}{b^{4}} + \frac {5 \, {\left (4389 \, {\left (b x^{4} + a\right )}^{\frac {23}{4}} - 26565 \, {\left (b x^{4} + a\right )}^{\frac {19}{4}} a + 67298 \, {\left (b x^{4} + a\right )}^{\frac {15}{4}} a^{2} - 91770 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}} a^{3} + 72105 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} a^{4} - 33649 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} a^{5}\right )}}{b^{4}}}{504735 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^19*(b*x^4+a)^(3/4),x, algorithm="giac")

[Out]

1/504735*(23*(1155*(b*x^4 + a)^(19/4) - 5852*(b*x^4 + a)^(15/4)*a + 11970*(b*x^4 + a)^(11/4)*a^2 - 12540*(b*x^
4 + a)^(7/4)*a^3 + 7315*(b*x^4 + a)^(3/4)*a^4)*a/b^4 + 5*(4389*(b*x^4 + a)^(23/4) - 26565*(b*x^4 + a)^(19/4)*a
+ 67298*(b*x^4 + a)^(15/4)*a^2 - 91770*(b*x^4 + a)^(11/4)*a^3 + 72105*(b*x^4 + a)^(7/4)*a^4 - 33649*(b*x^4 +
a)^(3/4)*a^5)/b^4)/b

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Mupad [B]
time = 1.13, size = 66, normalized size = 0.65 \begin {gather*} {\left (b\,x^4+a\right )}^{3/4}\,\left (\frac {x^{20}}{23}+\frac {2048\,a^5}{168245\,b^5}+\frac {3\,a\,x^{16}}{437\,b}-\frac {1536\,a^4\,x^4}{168245\,b^4}+\frac {192\,a^3\,x^8}{24035\,b^3}-\frac {16\,a^2\,x^{12}}{2185\,b^2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^19*(a + b*x^4)^(3/4),x)

[Out]

(a + b*x^4)^(3/4)*(x^20/23 + (2048*a^5)/(168245*b^5) + (3*a*x^16)/(437*b) - (1536*a^4*x^4)/(168245*b^4) + (192
*a^3*x^8)/(24035*b^3) - (16*a^2*x^12)/(2185*b^2))

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