3.11.20 \(\int x^7 (a+b x^4)^{3/4} \, dx\) [1020]

Optimal. Leaf size=38 \[ -\frac {a \left (a+b x^4\right )^{7/4}}{7 b^2}+\frac {\left (a+b x^4\right )^{11/4}}{11 b^2} \]

[Out]

-1/7*a*(b*x^4+a)^(7/4)/b^2+1/11*(b*x^4+a)^(11/4)/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 )^{11/4}}{11 b^2}-\frac {a \left (a+b x^4\right )^{7/4}}{7 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^7*(a + b*x^4)^(3/4),x]

[Out]

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

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Mathematica [A]
time = 0.03, size = 39, normalized size = 1.03 \begin {gather*} \frac {\left (a+b x^4\right )^{3/4} \left (-4 a^2+3 a b x^4+7 b^2 x^8\right )}{77 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7*(a + b*x^4)^(3/4),x]

[Out]

((a + b*x^4)^(3/4)*(-4*a^2 + 3*a*b*x^4 + 7*b^2*x^8))/(77*b^2)

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

method	result	size
gosper	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {7}{4}} \left (-7 b \,x^{4}+4 a \right )}{77 b^{2}}\)	\(25\)
trager	\(-\frac {\left (-7 b^{2} x^{8}-3 a b \,x^{4}+4 a^{2}\right ) \left (b \,x^{4}+a \right )^{\frac {3}{4}}}{77 b^{2}}\)	\(36\)
risch	\(-\frac {\left (-7 b^{2} x^{8}-3 a b \,x^{4}+4 a^{2}\right ) \left (b \,x^{4}+a \right )^{\frac {3}{4}}}{77 b^{2}}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(b*x^4+a)^(3/4),x,method=_RETURNVERBOSE)

[Out]

-1/77*(b*x^4+a)^(7/4)*(-7*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 {11}{4}}}{11 \, b^{2}} - \frac {{\left (b x^{4} + a\right )}^{\frac {7}{4}} a}{7 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x^4+a)^(3/4),x, algorithm="maxima")

[Out]

1/11*(b*x^4 + a)^(11/4)/b^2 - 1/7*(b*x^4 + a)^(7/4)*a/b^2

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Fricas [A]
time = 0.35, size = 35, normalized size = 0.92 \begin {gather*} \frac {{\left (7 \, b^{2} x^{8} + 3 \, a b x^{4} - 4 \, a^{2}\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{77 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x^4+a)^(3/4),x, algorithm="fricas")

[Out]

1/77*(7*b^2*x^8 + 3*a*b*x^4 - 4*a^2)*(b*x^4 + a)^(3/4)/b^2

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (31) = 62\).
time = 0.41, size = 65, normalized size = 1.71 \begin {gather*} \begin {cases} - \frac {4 a^{2} \left (a + b x^{4}\right )^{\frac {3}{4}}}{77 b^{2}} + \frac {3 a x^{4} \left (a + b x^{4}\right )^{\frac {3}{4}}}{77 b} + \frac {x^{8} \left (a + b x^{4}\right )^{\frac {3}{4}}}{11} & \text {for}\: b \neq 0 \\\frac {a^{\frac {3}{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)**(3/4),x)

[Out]

Piecewise((-4*a**2*(a + b*x**4)**(3/4)/(77*b**2) + 3*a*x**4*(a + b*x**4)**(3/4)/(77*b) + x**8*(a + b*x**4)**(3
/4)/11, Ne(b, 0)), (a**(3/4)*x**8/8, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (30) = 60\).
time = 1.30, size = 78, normalized size = 2.05 \begin {gather*} \frac {\frac {11 \, {\left (3 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} - 7 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} a\right )} a}{b} + \frac {21 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}} - 66 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} a + 77 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} a^{2}}{b}}{231 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x^4+a)^(3/4),x, algorithm="giac")

[Out]

1/231*(11*(3*(b*x^4 + a)^(7/4) - 7*(b*x^4 + a)^(3/4)*a)*a/b + (21*(b*x^4 + a)^(11/4) - 66*(b*x^4 + a)^(7/4)*a
+ 77*(b*x^4 + a)^(3/4)*a^2)/b)/b

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Mupad [B]
time = 1.08, size = 33, normalized size = 0.87 \begin {gather*} {\left (b\,x^4+a\right )}^{3/4}\,\left (\frac {x^8}{11}-\frac {4\,a^2}{77\,b^2}+\frac {3\,a\,x^4}{77\,b}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(a + b*x^4)^(3/4),x)

[Out]

(a + b*x^4)^(3/4)*(x^8/11 - (4*a^2)/(77*b^2) + (3*a*x^4)/(77*b))

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