3.373 \(\int x \sqrt [3]{a+b x} \, dx\)

Optimal. Leaf size=34 \[ \frac{3 (a+b x)^{7/3}}{7 b^2}-\frac{3 a (a+b x)^{4/3}}{4 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0252399, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{3 (a+b x)^{7/3}}{7 b^2}-\frac{3 a (a+b x)^{4/3}}{4 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x*(a + b*x)^(1/3),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.89396, size = 31, normalized size = 0.91 \[ - \frac{3 a \left (a + b x\right )^{\frac{4}{3}}}{4 b^{2}} + \frac{3 \left (a + b x\right )^{\frac{7}{3}}}{7 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(b*x+a)**(1/3),x)

[Out]

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

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Mathematica [A]  time = 0.0129513, size = 34, normalized size = 1. \[ \frac{3 \sqrt [3]{a+b x} \left (-3 a^2+a b x+4 b^2 x^2\right )}{28 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(a + b*x)^(1/3),x]

[Out]

(3*(a + b*x)^(1/3)*(-3*a^2 + a*b*x + 4*b^2*x^2))/(28*b^2)

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Maple [A]  time = 0.003, size = 21, normalized size = 0.6 \[ -{\frac{-12\,bx+9\,a}{28\,{b}^{2}} \left ( bx+a \right ) ^{{\frac{4}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x+a)^(1/3),x)

[Out]

-3/28*(b*x+a)^(4/3)*(-4*b*x+3*a)/b^2

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Maxima [A]  time = 1.35778, size = 35, normalized size = 1.03 \[ \frac{3 \,{\left (b x + a\right )}^{\frac{7}{3}}}{7 \, b^{2}} - \frac{3 \,{\left (b x + a\right )}^{\frac{4}{3}} a}{4 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/3)*x,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.204301, size = 41, normalized size = 1.21 \[ \frac{3 \,{\left (4 \, b^{2} x^{2} + a b x - 3 \, a^{2}\right )}{\left (b x + a\right )}^{\frac{1}{3}}}{28 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/3)*x,x, algorithm="fricas")

[Out]

3/28*(4*b^2*x^2 + a*b*x - 3*a^2)*(b*x + a)^(1/3)/b^2

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Sympy [A]  time = 3.70959, size = 202, normalized size = 5.94 \[ - \frac{9 a^{\frac{13}{3}} \sqrt [3]{1 + \frac{b x}{a}}}{28 a^{2} b^{2} + 28 a b^{3} x} + \frac{9 a^{\frac{13}{3}}}{28 a^{2} b^{2} + 28 a b^{3} x} - \frac{6 a^{\frac{10}{3}} b x \sqrt [3]{1 + \frac{b x}{a}}}{28 a^{2} b^{2} + 28 a b^{3} x} + \frac{9 a^{\frac{10}{3}} b x}{28 a^{2} b^{2} + 28 a b^{3} x} + \frac{15 a^{\frac{7}{3}} b^{2} x^{2} \sqrt [3]{1 + \frac{b x}{a}}}{28 a^{2} b^{2} + 28 a b^{3} x} + \frac{12 a^{\frac{4}{3}} b^{3} x^{3} \sqrt [3]{1 + \frac{b x}{a}}}{28 a^{2} b^{2} + 28 a b^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(b*x+a)**(1/3),x)

[Out]

-9*a**(13/3)*(1 + b*x/a)**(1/3)/(28*a**2*b**2 + 28*a*b**3*x) + 9*a**(13/3)/(28*a
**2*b**2 + 28*a*b**3*x) - 6*a**(10/3)*b*x*(1 + b*x/a)**(1/3)/(28*a**2*b**2 + 28*
a*b**3*x) + 9*a**(10/3)*b*x/(28*a**2*b**2 + 28*a*b**3*x) + 15*a**(7/3)*b**2*x**2
*(1 + b*x/a)**(1/3)/(28*a**2*b**2 + 28*a*b**3*x) + 12*a**(4/3)*b**3*x**3*(1 + b*
x/a)**(1/3)/(28*a**2*b**2 + 28*a*b**3*x)

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GIAC/XCAS [A]  time = 0.204856, size = 34, normalized size = 1. \[ \frac{3 \,{\left (4 \,{\left (b x + a\right )}^{\frac{7}{3}} - 7 \,{\left (b x + a\right )}^{\frac{4}{3}} a\right )}}{28 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/3)*x,x, algorithm="giac")

[Out]

3/28*(4*(b*x + a)^(7/3) - 7*(b*x + a)^(4/3)*a)/b^2