3.11.50 \(\int x^7 (a+b x^4)^{5/4} \, dx\) [1050]

Optimal. Leaf size=38 \[ -\frac {a \left (a+b x^4\right )^{9/4}}{9 b^2}+\frac {\left (a+b x^4\right )^{13/4}}{13 b^2} \]

[Out]

-1/9*a*(b*x^4+a)^(9/4)/b^2+1/13*(b*x^4+a)^(13/4)/b^2

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Rubi [A]
time = 0.02, antiderivative size = 38, 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 {\left (a+b x^4\right )^{13/4}}{13 b^2}-\frac {a \left (a+b x^4\right )^{9/4}}{9 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^7*(a + b*x^4)^(5/4),x]

[Out]

-1/9*(a*(a + b*x^4)^(9/4))/b^2 + (a + b*x^4)^(13/4)/(13*b^2)

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^7 \left (a+b x^4\right )^{5/4} \, dx &=\frac {1}{4} \text {Subst}\left (\int x (a+b x)^{5/4} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (-\frac {a (a+b x)^{5/4}}{b}+\frac {(a+b x)^{9/4}}{b}\right ) \, dx,x,x^4\right )\\ &=-\frac {a \left (a+b x^4\right )^{9/4}}{9 b^2}+\frac {\left (a+b x^4\right )^{13/4}}{13 b^2}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 28, normalized size = 0.74 \begin {gather*} \frac {\left (a+b x^4\right )^{9/4} \left (-4 a+9 b x^4\right )}{117 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7*(a + b*x^4)^(5/4),x]

[Out]

((a + b*x^4)^(9/4)*(-4*a + 9*b*x^4))/(117*b^2)

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Maple [A]
time = 0.15, size = 25, normalized size = 0.66

method	result	size
gosper	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {9}{4}} \left (-9 b \,x^{4}+4 a \right )}{117 b^{2}}\)	\(25\)
trager	\(-\frac {\left (-9 b^{3} x^{12}-14 a \,b^{2} x^{8}-a^{2} b \,x^{4}+4 a^{3}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{117 b^{2}}\)	\(47\)
risch	\(-\frac {\left (-9 b^{3} x^{12}-14 a \,b^{2} x^{8}-a^{2} b \,x^{4}+4 a^{3}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{117 b^{2}}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(b*x^4+a)^(5/4),x,method=_RETURNVERBOSE)

[Out]

-1/117*(b*x^4+a)^(9/4)*(-9*b*x^4+4*a)/b^2

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Maxima [A]
time = 0.29, size = 30, normalized size = 0.79 \begin {gather*} \frac {{\left (b x^{4} + a\right )}^{\frac {13}{4}}}{13 \, b^{2}} - \frac {{\left (b x^{4} + a\right )}^{\frac {9}{4}} a}{9 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x^4+a)^(5/4),x, algorithm="maxima")

[Out]

1/13*(b*x^4 + a)^(13/4)/b^2 - 1/9*(b*x^4 + a)^(9/4)*a/b^2

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Fricas [A]
time = 0.38, size = 45, normalized size = 1.18 \begin {gather*} \frac {{\left (9 \, b^{3} x^{12} + 14 \, a b^{2} x^{8} + a^{2} b x^{4} - 4 \, a^{3}\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{117 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x^4+a)^(5/4),x, algorithm="fricas")

[Out]

1/117*(9*b^3*x^12 + 14*a*b^2*x^8 + a^2*b*x^4 - 4*a^3)*(b*x^4 + a)^(1/4)/b^2

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (31) = 62\).
time = 0.66, size = 85, normalized size = 2.24 \begin {gather*} \begin {cases} - \frac {4 a^{3} \sqrt [4]{a + b x^{4}}}{117 b^{2}} + \frac {a^{2} x^{4} \sqrt [4]{a + b x^{4}}}{117 b} + \frac {14 a x^{8} \sqrt [4]{a + b x^{4}}}{117} + \frac {b x^{12} \sqrt [4]{a + b x^{4}}}{13} & \text {for}\: b \neq 0 \\\frac {a^{\frac {5}{4}} x^{8}}{8} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(b*x**4+a)**(5/4),x)

[Out]

Piecewise((-4*a**3*(a + b*x**4)**(1/4)/(117*b**2) + a**2*x**4*(a + b*x**4)**(1/4)/(117*b) + 14*a*x**8*(a + b*x
**4)**(1/4)/117 + b*x**12*(a + b*x**4)**(1/4)/13, Ne(b, 0)), (a**(5/4)*x**8/8, True))

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Giac [A]
time = 1.35, size = 29, normalized size = 0.76 \begin {gather*} \frac {9 \, {\left (b x^{4} + a\right )}^{\frac {13}{4}} - 13 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}} a}{117 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x^4+a)^(5/4),x, algorithm="giac")

[Out]

1/117*(9*(b*x^4 + a)^(13/4) - 13*(b*x^4 + a)^(9/4)*a)/b^2

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Mupad [B]
time = 1.11, size = 42, normalized size = 1.11 \begin {gather*} {\left (b\,x^4+a\right )}^{1/4}\,\left (\frac {14\,a\,x^8}{117}+\frac {b\,x^{12}}{13}-\frac {4\,a^3}{117\,b^2}+\frac {a^2\,x^4}{117\,b}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(a + b*x^4)^(5/4),x)

[Out]

(a + b*x^4)^(1/4)*((14*a*x^8)/117 + (b*x^12)/13 - (4*a^3)/(117*b^2) + (a^2*x^4)/(117*b))

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