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]

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

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.93, size = 45, normalized size = 1.18 \[ \frac {3 \, {\left (7 \, b^{3} x^{6} + 11 \, a b^{2} x^{4} + a^{2} b x^{2} - 3 \, a^{3}\right )} {\left (b x^{2} + a\right )}^{\frac {1}{3}}}{140 \, b^{2}} \]

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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giac [A]  time = 0.69, size = 29, normalized size = 0.76 \[ \frac {3 \, {\left (7 \, {\left (b x^{2} + a\right )}^{\frac {10}{3}} - 10 \, {\left (b x^{2} + a\right )}^{\frac {7}{3}} a\right )}}{140 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/140*(7*(b*x^2 + a)^(10/3) - 10*(b*x^2 + a)^(7/3)*a)/b^2

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maple [A]  time = 0.01, size = 25, normalized size = 0.66 \[ -\frac {3 \left (b \,x^{2}+a \right )^{\frac {7}{3}} \left (-7 b \,x^{2}+3 a \right )}{140 b^{2}} \]

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 = 1.32, size = 30, normalized size = 0.79 \[ \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {10}{3}}}{20 \, b^{2}} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{3}} a}{14 \, b^{2}} \]

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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mupad [B]  time = 5.05, size = 42, normalized size = 1.11 \[ {\left (b\,x^2+a\right )}^{1/3}\,\left (\frac {33\,a\,x^4}{140}+\frac {3\,b\,x^6}{20}-\frac {9\,a^3}{140\,b^2}+\frac {3\,a^2\,x^2}{140\,b}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*x^2)^(4/3),x)

[Out]

(a + b*x^2)^(1/3)*((33*a*x^4)/140 + (3*b*x^6)/20 - (9*a^3)/(140*b^2) + (3*a^2*x^2)/(140*b))

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sympy [A]  time = 2.54, size = 88, normalized size = 2.32 \[ \begin {cases} - \frac {9 a^{3} \sqrt [3]{a + b x^{2}}}{140 b^{2}} + \frac {3 a^{2} x^{2} \sqrt [3]{a + b x^{2}}}{140 b} + \frac {33 a x^{4} \sqrt [3]{a + b x^{2}}}{140} + \frac {3 b x^{6} \sqrt [3]{a + b x^{2}}}{20} & \text {for}\: b \neq 0 \\\frac {a^{\frac {4}{3}} x^{4}}{4} & \text {otherwise} \end {cases} \]

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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