3.11.18 \(\int x^{15} (a+b x^4)^{3/4} \, dx\) [1018]

Optimal. Leaf size=80 \[ -\frac {a^3 \left (a+b x^4\right )^{7/4}}{7 b^4}+\frac {3 a^2 \left (a+b x^4\right )^{11/4}}{11 b^4}-\frac {a \left (a+b x^4\right )^{15/4}}{5 b^4}+\frac {\left (a+b x^4\right )^{19/4}}{19 b^4} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 80, 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^3 \left (a+b x^4\right )^{7/4}}{7 b^4}+\frac {3 a^2 \left (a+b x^4\right )^{11/4}}{11 b^4}+\frac {\left (a+b x^4\right )^{19/4}}{19 b^4}-\frac {a \left (a+b x^4\right )^{15/4}}{5 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^15*(a + b*x^4)^(3/4),x]

[Out]

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

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

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Mathematica [A]
time = 0.03, size = 50, normalized size = 0.62 \begin {gather*} \frac {\left (a+b x^4\right )^{7/4} \left (-128 a^3+224 a^2 b x^4-308 a b^2 x^8+385 b^3 x^{12}\right )}{7315 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^15*(a + b*x^4)^(3/4),x]

[Out]

((a + b*x^4)^(7/4)*(-128*a^3 + 224*a^2*b*x^4 - 308*a*b^2*x^8 + 385*b^3*x^12))/(7315*b^4)

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Maple [A]
time = 0.15, size = 47, normalized size = 0.59

method	result	size
gosper	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {7}{4}} \left (-385 b^{3} x^{12}+308 a \,b^{2} x^{8}-224 a^{2} b \,x^{4}+128 a^{3}\right )}{7315 b^{4}}\)	\(47\)
trager	\(-\frac {\left (-385 x^{16} b^{4}-77 a \,b^{3} x^{12}+84 a^{2} b^{2} x^{8}-96 a^{3} b \,x^{4}+128 a^{4}\right ) \left (b \,x^{4}+a \right )^{\frac {3}{4}}}{7315 b^{4}}\)	\(58\)
risch	\(-\frac {\left (-385 x^{16} b^{4}-77 a \,b^{3} x^{12}+84 a^{2} b^{2} x^{8}-96 a^{3} b \,x^{4}+128 a^{4}\right ) \left (b \,x^{4}+a \right )^{\frac {3}{4}}}{7315 b^{4}}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^15*(b*x^4+a)^(3/4),x,method=_RETURNVERBOSE)

[Out]

-1/7315*(b*x^4+a)^(7/4)*(-385*b^3*x^12+308*a*b^2*x^8-224*a^2*b*x^4+128*a^3)/b^4

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Maxima [A]
time = 0.29, size = 64, normalized size = 0.80 \begin {gather*} \frac {{\left (b x^{4} + a\right )}^{\frac {19}{4}}}{19 \, b^{4}} - \frac {{\left (b x^{4} + a\right )}^{\frac {15}{4}} a}{5 \, b^{4}} + \frac {3 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}} a^{2}}{11 \, b^{4}} - \frac {{\left (b x^{4} + a\right )}^{\frac {7}{4}} a^{3}}{7 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15*(b*x^4+a)^(3/4),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.35, size = 57, normalized size = 0.71 \begin {gather*} \frac {{\left (385 \, b^{4} x^{16} + 77 \, a b^{3} x^{12} - 84 \, a^{2} b^{2} x^{8} + 96 \, a^{3} b x^{4} - 128 \, a^{4}\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{7315 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15*(b*x^4+a)^(3/4),x, algorithm="fricas")

[Out]

1/7315*(385*b^4*x^16 + 77*a*b^3*x^12 - 84*a^2*b^2*x^8 + 96*a^3*b*x^4 - 128*a^4)*(b*x^4 + a)^(3/4)/b^4

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Sympy [A]
time = 1.06, size = 110, normalized size = 1.38 \begin {gather*} \begin {cases} - \frac {128 a^{4} \left (a + b x^{4}\right )^{\frac {3}{4}}}{7315 b^{4}} + \frac {96 a^{3} x^{4} \left (a + b x^{4}\right )^{\frac {3}{4}}}{7315 b^{3}} - \frac {12 a^{2} x^{8} \left (a + b x^{4}\right )^{\frac {3}{4}}}{1045 b^{2}} + \frac {a x^{12} \left (a + b x^{4}\right )^{\frac {3}{4}}}{95 b} + \frac {x^{16} \left (a + b x^{4}\right )^{\frac {3}{4}}}{19} & \text {for}\: b \neq 0 \\\frac {a^{\frac {3}{4}} x^{16}}{16} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**15*(b*x**4+a)**(3/4),x)

[Out]

Piecewise((-128*a**4*(a + b*x**4)**(3/4)/(7315*b**4) + 96*a**3*x**4*(a + b*x**4)**(3/4)/(7315*b**3) - 12*a**2*
x**8*(a + b*x**4)**(3/4)/(1045*b**2) + a*x**12*(a + b*x**4)**(3/4)/(95*b) + x**16*(a + b*x**4)**(3/4)/19, Ne(b
, 0)), (a**(3/4)*x**16/16, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (64) = 128\).
time = 1.15, size = 134, normalized size = 1.68 \begin {gather*} \frac {\frac {19 \, {\left (77 \, {\left (b x^{4} + a\right )}^{\frac {15}{4}} - 315 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}} a + 495 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} a^{2} - 385 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} a^{3}\right )} a}{b^{3}} + \frac {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}}{b^{3}}}{21945 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15*(b*x^4+a)^(3/4),x, algorithm="giac")

[Out]

1/21945*(19*(77*(b*x^4 + a)^(15/4) - 315*(b*x^4 + a)^(11/4)*a + 495*(b*x^4 + a)^(7/4)*a^2 - 385*(b*x^4 + a)^(3
/4)*a^3)*a/b^3 + (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)/b^3)/b

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Mupad [B]
time = 1.09, size = 55, normalized size = 0.69 \begin {gather*} {\left (b\,x^4+a\right )}^{3/4}\,\left (\frac {x^{16}}{19}-\frac {128\,a^4}{7315\,b^4}+\frac {a\,x^{12}}{95\,b}+\frac {96\,a^3\,x^4}{7315\,b^3}-\frac {12\,a^2\,x^8}{1045\,b^2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^15*(a + b*x^4)^(3/4),x)

[Out]

(a + b*x^4)^(3/4)*(x^16/19 - (128*a^4)/(7315*b^4) + (a*x^12)/(95*b) + (96*a^3*x^4)/(7315*b^3) - (12*a^2*x^8)/(
1045*b^2))

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