3.46 \(\int (2+3 x) \sin (5 x) \, dx\)

Optimal. Leaf size=22 \[ \frac{3}{25} \sin (5 x)-\frac{1}{5} (3 x+2) \cos (5 x) \]

[Out]

-((2 + 3*x)*Cos[5*x])/5 + (3*Sin[5*x])/25

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Rubi [A]  time = 0.0213374, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{3}{25} \sin (5 x)-\frac{1}{5} (3 x+2) \cos (5 x) \]

Antiderivative was successfully verified.

[In]  Int[(2 + 3*x)*Sin[5*x],x]

[Out]

-((2 + 3*x)*Cos[5*x])/5 + (3*Sin[5*x])/25

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Rubi in Sympy [A]  time = 1.15993, size = 20, normalized size = 0.91 \[ - \left (\frac{3 x}{5} + \frac{2}{5}\right ) \cos{\left (5 x \right )} + \frac{3 \sin{\left (5 x \right )}}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)*sin(5*x),x)

[Out]

-(3*x/5 + 2/5)*cos(5*x) + 3*sin(5*x)/25

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Mathematica [A]  time = 0.00817653, size = 26, normalized size = 1.18 \[ \frac{3}{25} \sin (5 x)-\frac{3}{5} x \cos (5 x)-\frac{2}{5} \cos (5 x) \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + 3*x)*Sin[5*x],x]

[Out]

(-2*Cos[5*x])/5 - (3*x*Cos[5*x])/5 + (3*Sin[5*x])/25

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Maple [A]  time = 0.019, size = 21, normalized size = 1. \[ -{\frac{2\,\cos \left ( 5\,x \right ) }{5}}+{\frac{3\,\sin \left ( 5\,x \right ) }{25}}-{\frac{3\,\cos \left ( 5\,x \right ) x}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)*sin(5*x),x)

[Out]

-2/5*cos(5*x)+3/25*sin(5*x)-3/5*cos(5*x)*x

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Maxima [A]  time = 1.3251, size = 27, normalized size = 1.23 \[ -\frac{3}{5} \, x \cos \left (5 \, x\right ) - \frac{2}{5} \, \cos \left (5 \, x\right ) + \frac{3}{25} \, \sin \left (5 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)*sin(5*x),x, algorithm="maxima")

[Out]

-3/5*x*cos(5*x) - 2/5*cos(5*x) + 3/25*sin(5*x)

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Fricas [A]  time = 0.215428, size = 24, normalized size = 1.09 \[ -\frac{1}{5} \,{\left (3 \, x + 2\right )} \cos \left (5 \, x\right ) + \frac{3}{25} \, \sin \left (5 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)*sin(5*x),x, algorithm="fricas")

[Out]

-1/5*(3*x + 2)*cos(5*x) + 3/25*sin(5*x)

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Sympy [A]  time = 0.212422, size = 26, normalized size = 1.18 \[ - \frac{3 x \cos{\left (5 x \right )}}{5} + \frac{3 \sin{\left (5 x \right )}}{25} - \frac{2 \cos{\left (5 x \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+3*x)*sin(5*x),x)

[Out]

-3*x*cos(5*x)/5 + 3*sin(5*x)/25 - 2*cos(5*x)/5

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GIAC/XCAS [A]  time = 0.217808, size = 24, normalized size = 1.09 \[ -\frac{1}{5} \,{\left (3 \, x + 2\right )} \cos \left (5 \, x\right ) + \frac{3}{25} \, \sin \left (5 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)*sin(5*x),x, algorithm="giac")

[Out]

-1/5*(3*x + 2)*cos(5*x) + 3/25*sin(5*x)