3.931 \(\int (\frac{x^4}{b \sqrt{x^3+3 \sin (a+b x)}}+\frac{x^2 \cos (a+b x)}{\sqrt{x^3+3 \sin (a+b x)}}+\frac{4 x \sqrt{x^3+3 \sin (a+b x)}}{3 b}) \, dx\)

Optimal. Leaf size=26 \[ \frac{2 x^2 \sqrt{3 \sin (a+b x)+x^3}}{3 b} \]

[Out]

(2*x^2*Sqrt[x^3 + 3*Sin[a + b*x]])/(3*b)

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Rubi [F]  time = 0.810739, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \left (\frac{x^4}{b \sqrt{x^3+3 \sin (a+b x)}}+\frac{x^2 \cos (a+b x)}{\sqrt{x^3+3 \sin (a+b x)}}+\frac{4 x \sqrt{x^3+3 \sin (a+b x)}}{3 b}\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^4/(b*Sqrt[x^3 + 3*Sin[a + b*x]]) + (x^2*Cos[a + b*x])/Sqrt[x^3 + 3*Sin[a + b*x]] + (4*x*Sqrt[x^3 + 3*Sin
[a + b*x]])/(3*b),x]

[Out]

Defer[Int][x^4/Sqrt[x^3 + 3*Sin[a + b*x]], x]/b + Defer[Int][(x^2*Cos[a + b*x])/Sqrt[x^3 + 3*Sin[a + b*x]], x]
+ (4*Defer[Int][x*Sqrt[x^3 + 3*Sin[a + b*x]], x])/(3*b)

Rubi steps

\begin{align*} \int \left (\frac{x^4}{b \sqrt{x^3+3 \sin (a+b x)}}+\frac{x^2 \cos (a+b x)}{\sqrt{x^3+3 \sin (a+b x)}}+\frac{4 x \sqrt{x^3+3 \sin (a+b x)}}{3 b}\right ) \, dx &=\frac{\int \frac{x^4}{\sqrt{x^3+3 \sin (a+b x)}} \, dx}{b}+\frac{4 \int x \sqrt{x^3+3 \sin (a+b x)} \, dx}{3 b}+\int \frac{x^2 \cos (a+b x)}{\sqrt{x^3+3 \sin (a+b x)}} \, dx\\ \end{align*}

Mathematica [A]  time = 0.430944, size = 26, normalized size = 1. \[ \frac{2 x^2 \sqrt{3 \sin (a+b x)+x^3}}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(b*Sqrt[x^3 + 3*Sin[a + b*x]]) + (x^2*Cos[a + b*x])/Sqrt[x^3 + 3*Sin[a + b*x]] + (4*x*Sqrt[x^3 +
3*Sin[a + b*x]])/(3*b),x]

[Out]

(2*x^2*Sqrt[x^3 + 3*Sin[a + b*x]])/(3*b)

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Maple [A]  time = 0.324, size = 28, normalized size = 1.1 \begin{align*}{\frac{\sqrt{2}{x}^{2}}{3\,b}\sqrt{2\,{x}^{3}+6\,\sin \left ( bx+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/b/(x^3+3*sin(b*x+a))^(1/2)+x^2*cos(b*x+a)/(x^3+3*sin(b*x+a))^(1/2)+4/3*x*(x^3+3*sin(b*x+a))^(1/2)/b,x)

[Out]

1/3*(2*x^3+6*sin(b*x+a))^(1/2)/b*2^(1/2)*x^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\sqrt{x^{3} + 3 \, \sin \left (b x + a\right )} b} + \frac{x^{2} \cos \left (b x + a\right )}{\sqrt{x^{3} + 3 \, \sin \left (b x + a\right )}} + \frac{4 \, \sqrt{x^{3} + 3 \, \sin \left (b x + a\right )} x}{3 \, b}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/b/(x^3+3*sin(b*x+a))^(1/2)+x^2*cos(b*x+a)/(x^3+3*sin(b*x+a))^(1/2)+4/3*x*(x^3+3*sin(b*x+a))^(1/2
)/b,x, algorithm="maxima")

[Out]

integrate(x^4/(sqrt(x^3 + 3*sin(b*x + a))*b) + x^2*cos(b*x + a)/sqrt(x^3 + 3*sin(b*x + a)) + 4/3*sqrt(x^3 + 3*
sin(b*x + a))*x/b, x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/b/(x^3+3*sin(b*x+a))^(1/2)+x^2*cos(b*x+a)/(x^3+3*sin(b*x+a))^(1/2)+4/3*x*(x^3+3*sin(b*x+a))^(1/2
)/b,x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{7 x^{4}}{\sqrt{x^{3} + 3 \sin{\left (a + b x \right )}}}\, dx + \int \frac{12 x \sin{\left (a + b x \right )}}{\sqrt{x^{3} + 3 \sin{\left (a + b x \right )}}}\, dx + \int \frac{3 b x^{2} \cos{\left (a + b x \right )}}{\sqrt{x^{3} + 3 \sin{\left (a + b x \right )}}}\, dx}{3 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/b/(x**3+3*sin(b*x+a))**(1/2)+x**2*cos(b*x+a)/(x**3+3*sin(b*x+a))**(1/2)+4/3*x*(x**3+3*sin(b*x+a
))**(1/2)/b,x)

[Out]

(Integral(7*x**4/sqrt(x**3 + 3*sin(a + b*x)), x) + Integral(12*x*sin(a + b*x)/sqrt(x**3 + 3*sin(a + b*x)), x)
+ Integral(3*b*x**2*cos(a + b*x)/sqrt(x**3 + 3*sin(a + b*x)), x))/(3*b)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\sqrt{x^{3} + 3 \, \sin \left (b x + a\right )} b} + \frac{x^{2} \cos \left (b x + a\right )}{\sqrt{x^{3} + 3 \, \sin \left (b x + a\right )}} + \frac{4 \, \sqrt{x^{3} + 3 \, \sin \left (b x + a\right )} x}{3 \, b}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/b/(x^3+3*sin(b*x+a))^(1/2)+x^2*cos(b*x+a)/(x^3+3*sin(b*x+a))^(1/2)+4/3*x*(x^3+3*sin(b*x+a))^(1/2
)/b,x, algorithm="giac")

[Out]

integrate(x^4/(sqrt(x^3 + 3*sin(b*x + a))*b) + x^2*cos(b*x + a)/sqrt(x^3 + 3*sin(b*x + a)) + 4/3*sqrt(x^3 + 3*
sin(b*x + a))*x/b, x)