3.1047 \(\int x^{19} (a+b x^4)^{5/4} \, dx\)

Optimal. Leaf size=101 \[ \frac {a^4 \left (a+b x^4\right )^{9/4}}{9 b^5}-\frac {4 a^3 \left (a+b x^4\right )^{13/4}}{13 b^5}+\frac {6 a^2 \left (a+b x^4\right )^{17/4}}{17 b^5}+\frac {\left (a+b x^4\right )^{25/4}}{25 b^5}-\frac {4 a \left (a+b x^4\right )^{21/4}}{21 b^5} \]

[Out]

1/9*a^4*(b*x^4+a)^(9/4)/b^5-4/13*a^3*(b*x^4+a)^(13/4)/b^5+6/17*a^2*(b*x^4+a)^(17/4)/b^5-4/21*a*(b*x^4+a)^(21/4
)/b^5+1/25*(b*x^4+a)^(25/4)/b^5

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Rubi [A]  time = 0.05, 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 = {266, 43} \[ \frac {6 a^2 \left (a+b x^4\right )^{17/4}}{17 b^5}-\frac {4 a^3 \left (a+b x^4\right )^{13/4}}{13 b^5}+\frac {a^4 \left (a+b x^4\right )^{9/4}}{9 b^5}+\frac {\left (a+b x^4\right )^{25/4}}{25 b^5}-\frac {4 a \left (a+b x^4\right )^{21/4}}{21 b^5} \]

Antiderivative was successfully verified.

[In]

Int[x^19*(a + b*x^4)^(5/4),x]

[Out]

(a^4*(a + b*x^4)^(9/4))/(9*b^5) - (4*a^3*(a + b*x^4)^(13/4))/(13*b^5) + (6*a^2*(a + b*x^4)^(17/4))/(17*b^5) -
(4*a*(a + b*x^4)^(21/4))/(21*b^5) + (a + b*x^4)^(25/4)/(25*b^5)

Rule 43

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 266

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 )^{5/4} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int x^4 (a+b x)^{5/4} \, dx,x,x^4\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \left (\frac {a^4 (a+b x)^{5/4}}{b^4}-\frac {4 a^3 (a+b x)^{9/4}}{b^4}+\frac {6 a^2 (a+b x)^{13/4}}{b^4}-\frac {4 a (a+b x)^{17/4}}{b^4}+\frac {(a+b x)^{21/4}}{b^4}\right ) \, dx,x,x^4\right )\\ &=\frac {a^4 \left (a+b x^4\right )^{9/4}}{9 b^5}-\frac {4 a^3 \left (a+b x^4\right )^{13/4}}{13 b^5}+\frac {6 a^2 \left (a+b x^4\right )^{17/4}}{17 b^5}-\frac {4 a \left (a+b x^4\right )^{21/4}}{21 b^5}+\frac {\left (a+b x^4\right )^{25/4}}{25 b^5}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 61, normalized size = 0.60 \[ \frac {\left (a+b x^4\right )^{9/4} \left (2048 a^4-4608 a^3 b x^4+7488 a^2 b^2 x^8-10608 a b^3 x^{12}+13923 b^4 x^{16}\right )}{348075 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^19*(a + b*x^4)^(5/4),x]

[Out]

((a + b*x^4)^(9/4)*(2048*a^4 - 4608*a^3*b*x^4 + 7488*a^2*b^2*x^8 - 10608*a*b^3*x^12 + 13923*b^4*x^16))/(348075
*b^5)

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fricas [A]  time = 0.87, size = 79, normalized size = 0.78 \[ \frac {{\left (13923 \, b^{6} x^{24} + 17238 \, a b^{5} x^{20} + 195 \, a^{2} b^{4} x^{16} - 240 \, a^{3} b^{3} x^{12} + 320 \, a^{4} b^{2} x^{8} - 512 \, a^{5} b x^{4} + 2048 \, a^{6}\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{348075 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^19*(b*x^4+a)^(5/4),x, algorithm="fricas")

[Out]

1/348075*(13923*b^6*x^24 + 17238*a*b^5*x^20 + 195*a^2*b^4*x^16 - 240*a^3*b^3*x^12 + 320*a^4*b^2*x^8 - 512*a^5*
b*x^4 + 2048*a^6)*(b*x^4 + a)^(1/4)/b^5

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giac [A]  time = 0.20, size = 71, normalized size = 0.70 \[ \frac {13923 \, {\left (b x^{4} + a\right )}^{\frac {25}{4}} - 66300 \, {\left (b x^{4} + a\right )}^{\frac {21}{4}} a + 122850 \, {\left (b x^{4} + a\right )}^{\frac {17}{4}} a^{2} - 107100 \, {\left (b x^{4} + a\right )}^{\frac {13}{4}} a^{3} + 38675 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}} a^{4}}{348075 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^19*(b*x^4+a)^(5/4),x, algorithm="giac")

[Out]

1/348075*(13923*(b*x^4 + a)^(25/4) - 66300*(b*x^4 + a)^(21/4)*a + 122850*(b*x^4 + a)^(17/4)*a^2 - 107100*(b*x^
4 + a)^(13/4)*a^3 + 38675*(b*x^4 + a)^(9/4)*a^4)/b^5

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maple [A]  time = 0.01, size = 58, normalized size = 0.57 \[ \frac {\left (b \,x^{4}+a \right )^{\frac {9}{4}} \left (13923 x^{16} b^{4}-10608 a \,x^{12} b^{3}+7488 a^{2} x^{8} b^{2}-4608 a^{3} x^{4} b +2048 a^{4}\right )}{348075 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^19*(b*x^4+a)^(5/4),x)

[Out]

1/348075*(b*x^4+a)^(9/4)*(13923*b^4*x^16-10608*a*b^3*x^12+7488*a^2*b^2*x^8-4608*a^3*b*x^4+2048*a^4)/b^5

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maxima [A]  time = 1.35, size = 81, normalized size = 0.80 \[ \frac {{\left (b x^{4} + a\right )}^{\frac {25}{4}}}{25 \, b^{5}} - \frac {4 \, {\left (b x^{4} + a\right )}^{\frac {21}{4}} a}{21 \, b^{5}} + \frac {6 \, {\left (b x^{4} + a\right )}^{\frac {17}{4}} a^{2}}{17 \, b^{5}} - \frac {4 \, {\left (b x^{4} + a\right )}^{\frac {13}{4}} a^{3}}{13 \, b^{5}} + \frac {{\left (b x^{4} + a\right )}^{\frac {9}{4}} a^{4}}{9 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^19*(b*x^4+a)^(5/4),x, algorithm="maxima")

[Out]

1/25*(b*x^4 + a)^(25/4)/b^5 - 4/21*(b*x^4 + a)^(21/4)*a/b^5 + 6/17*(b*x^4 + a)^(17/4)*a^2/b^5 - 4/13*(b*x^4 +
a)^(13/4)*a^3/b^5 + 1/9*(b*x^4 + a)^(9/4)*a^4/b^5

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mupad [B]  time = 1.18, size = 75, normalized size = 0.74 \[ {\left (b\,x^4+a\right )}^{1/4}\,\left (\frac {26\,a\,x^{20}}{525}+\frac {b\,x^{24}}{25}+\frac {2048\,a^6}{348075\,b^5}-\frac {512\,a^5\,x^4}{348075\,b^4}+\frac {64\,a^4\,x^8}{69615\,b^3}-\frac {16\,a^3\,x^{12}}{23205\,b^2}+\frac {a^2\,x^{16}}{1785\,b}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^19*(a + b*x^4)^(5/4),x)

[Out]

(a + b*x^4)^(1/4)*((26*a*x^20)/525 + (b*x^24)/25 + (2048*a^6)/(348075*b^5) - (512*a^5*x^4)/(348075*b^4) + (64*
a^4*x^8)/(69615*b^3) - (16*a^3*x^12)/(23205*b^2) + (a^2*x^16)/(1785*b))

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sympy [A]  time = 87.31, size = 156, normalized size = 1.54 \[ \begin {cases} \frac {2048 a^{6} \sqrt [4]{a + b x^{4}}}{348075 b^{5}} - \frac {512 a^{5} x^{4} \sqrt [4]{a + b x^{4}}}{348075 b^{4}} + \frac {64 a^{4} x^{8} \sqrt [4]{a + b x^{4}}}{69615 b^{3}} - \frac {16 a^{3} x^{12} \sqrt [4]{a + b x^{4}}}{23205 b^{2}} + \frac {a^{2} x^{16} \sqrt [4]{a + b x^{4}}}{1785 b} + \frac {26 a x^{20} \sqrt [4]{a + b x^{4}}}{525} + \frac {b x^{24} \sqrt [4]{a + b x^{4}}}{25} & \text {for}\: b \neq 0 \\\frac {a^{\frac {5}{4}} x^{20}}{20} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**19*(b*x**4+a)**(5/4),x)

[Out]

Piecewise((2048*a**6*(a + b*x**4)**(1/4)/(348075*b**5) - 512*a**5*x**4*(a + b*x**4)**(1/4)/(348075*b**4) + 64*
a**4*x**8*(a + b*x**4)**(1/4)/(69615*b**3) - 16*a**3*x**12*(a + b*x**4)**(1/4)/(23205*b**2) + a**2*x**16*(a +
b*x**4)**(1/4)/(1785*b) + 26*a*x**20*(a + b*x**4)**(1/4)/525 + b*x**24*(a + b*x**4)**(1/4)/25, Ne(b, 0)), (a**
(5/4)*x**20/20, True))

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