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

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 = 0.0702891, 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 \[ \frac{3 \left (a+b x^2\right )^{10/3}}{20 b^2}-\frac{3 a \left (a+b x^2\right )^{7/3}}{14 b^2} \]

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)

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Rubi in Sympy [A]  time = 7.84642, size = 34, normalized size = 0.89 \[ - \frac{3 a \left (a + b x^{2}\right )^{\frac{7}{3}}}{14 b^{2}} + \frac{3 \left (a + b x^{2}\right )^{\frac{10}{3}}}{20 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(b*x**2+a)**(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)

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Mathematica [A]  time = 0.0350308, 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 = 0.008, size = 25, normalized size = 0.7 \[ -{\frac{-21\,b{x}^{2}+9\,a}{140\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{3}}}} \]

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 = 1.35679, size = 41, normalized size = 1.08 \[ \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{10}{3}}}{20 \, b^{2}} - \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} a}{14 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(4/3)*x^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 = 0.206714, size = 61, normalized size = 1.61 \[ \frac{3 \,{\left (7 \, b^{3} x^{6} + 11 \, a b^{2} x^{4} + a^{2} b x^{2} - 3 \, a^{3}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{140 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(4/3)*x^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 = 7.49349, 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)/2
0, Ne(b, 0)), (a**(4/3)*x**4/4, True))

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GIAC/XCAS [A]  time = 0.216017, size = 105, normalized size = 2.76 \[ \frac{3 \,{\left (\frac{5 \,{\left (4 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} - 7 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} a\right )} a}{b} + \frac{14 \,{\left (b x^{2} + a\right )}^{\frac{10}{3}} - 40 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} a + 35 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} a^{2}}{b}\right )}}{280 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(4/3)*x^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