3.691 \(\int x^3 (a+b x^2)^{4/3} \, dx\)

Optimal. Leaf size=38 \[ \frac{3 \left (a+b x^2\right )^{10/3}}{20 b^2}-\frac{3 a \left (a+b x^2\right )^{7/3}}{14 b^2} \]

[Out]

(-3*a*(a + b*x^2)^(7/3))/(14*b^2) + (3*(a + b*x^2)^(10/3))/(20*b^2)

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Rubi [A]  time = 0.0242434, antiderivative size = 38, normalized size of antiderivative = 1., 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} \[ \frac{3 \left (a+b x^2\right )^{10/3}}{20 b^2}-\frac{3 a \left (a+b x^2\right )^{7/3}}{14 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x^2)^(4/3),x]

[Out]

(-3*a*(a + b*x^2)^(7/3))/(14*b^2) + (3*(a + b*x^2)^(10/3))/(20*b^2)

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]]

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])

Rubi steps

\begin{align*} \int x^3 \left (a+b x^2\right )^{4/3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x (a+b x)^{4/3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^{4/3}}{b}+\frac{(a+b x)^{7/3}}{b}\right ) \, dx,x,x^2\right )\\ &=-\frac{3 a \left (a+b x^2\right )^{7/3}}{14 b^2}+\frac{3 \left (a+b x^2\right )^{10/3}}{20 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0140855, size = 28, normalized size = 0.74 \[ \frac{3 \left (a+b x^2\right )^{7/3} \left (7 b x^2-3 a\right )}{140 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x^2)^(4/3),x]

[Out]

(3*(a + b*x^2)^(7/3)*(-3*a + 7*b*x^2))/(140*b^2)

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Maple [A]  time = 0.003, size = 25, normalized size = 0.7 \begin{align*} -{\frac{-21\,b{x}^{2}+9\,a}{140\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x^2+a)^(4/3),x)

[Out]

-3/140*(b*x^2+a)^(7/3)*(-7*b*x^2+3*a)/b^2

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Maxima [A]  time = 2.11117, size = 41, normalized size = 1.08 \begin{align*} \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{10}{3}}}{20 \, b^{2}} - \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} a}{14 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^2+a)^(4/3),x, algorithm="maxima")

[Out]

3/20*(b*x^2 + a)^(10/3)/b^2 - 3/14*(b*x^2 + a)^(7/3)*a/b^2

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Fricas [A]  time = 1.59479, size = 103, normalized size = 2.71 \begin{align*} \frac{3 \,{\left (7 \, b^{3} x^{6} + 11 \, a b^{2} x^{4} + a^{2} b x^{2} - 3 \, a^{3}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{140 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^2+a)^(4/3),x, algorithm="fricas")

[Out]

3/140*(7*b^3*x^6 + 11*a*b^2*x^4 + a^2*b*x^2 - 3*a^3)*(b*x^2 + a)^(1/3)/b^2

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Sympy [A]  time = 2.32836, size = 88, normalized size = 2.32 \begin{align*} \begin{cases} - \frac{9 a^{3} \sqrt [3]{a + b x^{2}}}{140 b^{2}} + \frac{3 a^{2} x^{2} \sqrt [3]{a + b x^{2}}}{140 b} + \frac{33 a x^{4} \sqrt [3]{a + b x^{2}}}{140} + \frac{3 b x^{6} \sqrt [3]{a + b x^{2}}}{20} & \text{for}\: b \neq 0 \\\frac{a^{\frac{4}{3}} x^{4}}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x**2+a)**(4/3),x)

[Out]

Piecewise((-9*a**3*(a + b*x**2)**(1/3)/(140*b**2) + 3*a**2*x**2*(a + b*x**2)**(1/3)/(140*b) + 33*a*x**4*(a + b
*x**2)**(1/3)/140 + 3*b*x**6*(a + b*x**2)**(1/3)/20, Ne(b, 0)), (a**(4/3)*x**4/4, True))

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Giac [B]  time = 1.78949, size = 105, normalized size = 2.76 \begin{align*} \frac{3 \,{\left (\frac{5 \,{\left (4 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} - 7 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} a\right )} a}{b} + \frac{14 \,{\left (b x^{2} + a\right )}^{\frac{10}{3}} - 40 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} a + 35 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} a^{2}}{b}\right )}}{280 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^2+a)^(4/3),x, algorithm="giac")

[Out]

3/280*(5*(4*(b*x^2 + a)^(7/3) - 7*(b*x^2 + a)^(4/3)*a)*a/b + (14*(b*x^2 + a)^(10/3) - 40*(b*x^2 + a)^(7/3)*a +
35*(b*x^2 + a)^(4/3)*a^2)/b)/b