3.188 \(\int (e+f x) \sin (a+b \sqrt{c+d x}) \, dx\)

Optimal. Leaf size=185 \[ \frac{2 (d e-c f) \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d^2}+\frac{6 f (c+d x) \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d^2}-\frac{12 f \sin \left (a+b \sqrt{c+d x}\right )}{b^4 d^2}+\frac{12 f \sqrt{c+d x} \cos \left (a+b \sqrt{c+d x}\right )}{b^3 d^2}-\frac{2 \sqrt{c+d x} (d e-c f) \cos \left (a+b \sqrt{c+d x}\right )}{b d^2}-\frac{2 f (c+d x)^{3/2} \cos \left (a+b \sqrt{c+d x}\right )}{b d^2} \]

[Out]

(12*f*Sqrt[c + d*x]*Cos[a + b*Sqrt[c + d*x]])/(b^3*d^2) - (2*(d*e - c*f)*Sqrt[c + d*x]*Cos[a + b*Sqrt[c + d*x]
])/(b*d^2) - (2*f*(c + d*x)^(3/2)*Cos[a + b*Sqrt[c + d*x]])/(b*d^2) - (12*f*Sin[a + b*Sqrt[c + d*x]])/(b^4*d^2
) + (2*(d*e - c*f)*Sin[a + b*Sqrt[c + d*x]])/(b^2*d^2) + (6*f*(c + d*x)*Sin[a + b*Sqrt[c + d*x]])/(b^2*d^2)

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Rubi [A]  time = 0.159242, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {3431, 3296, 2637} \[ \frac{2 (d e-c f) \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d^2}+\frac{6 f (c+d x) \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d^2}-\frac{12 f \sin \left (a+b \sqrt{c+d x}\right )}{b^4 d^2}+\frac{12 f \sqrt{c+d x} \cos \left (a+b \sqrt{c+d x}\right )}{b^3 d^2}-\frac{2 \sqrt{c+d x} (d e-c f) \cos \left (a+b \sqrt{c+d x}\right )}{b d^2}-\frac{2 f (c+d x)^{3/2} \cos \left (a+b \sqrt{c+d x}\right )}{b d^2} \]

Antiderivative was successfully verified.

[In]

Int[(e + f*x)*Sin[a + b*Sqrt[c + d*x]],x]

[Out]

(12*f*Sqrt[c + d*x]*Cos[a + b*Sqrt[c + d*x]])/(b^3*d^2) - (2*(d*e - c*f)*Sqrt[c + d*x]*Cos[a + b*Sqrt[c + d*x]
])/(b*d^2) - (2*f*(c + d*x)^(3/2)*Cos[a + b*Sqrt[c + d*x]])/(b*d^2) - (12*f*Sin[a + b*Sqrt[c + d*x]])/(b^4*d^2
) + (2*(d*e - c*f)*Sin[a + b*Sqrt[c + d*x]])/(b^2*d^2) + (6*f*(c + d*x)*Sin[a + b*Sqrt[c + d*x]])/(b^2*d^2)

Rule 3431

Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_.), x_Symbol] :
> Dist[1/(n*f), Subst[Int[ExpandIntegrand[(a + b*Sin[c + d*x])^p, x^(1/n - 1)*(g - (e*h)/f + (h*x^(1/n))/f)^m,
x], x], x, (e + f*x)^n], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IGtQ[p, 0] && IntegerQ[1/n]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int (e+f x) \sin \left (a+b \sqrt{c+d x}\right ) \, dx &=\frac{2 \operatorname{Subst}\left (\int \left (\frac{(d e-c f) x \sin (a+b x)}{d}+\frac{f x^3 \sin (a+b x)}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{d}\\ &=\frac{(2 f) \operatorname{Subst}\left (\int x^3 \sin (a+b x) \, dx,x,\sqrt{c+d x}\right )}{d^2}+\frac{(2 (d e-c f)) \operatorname{Subst}\left (\int x \sin (a+b x) \, dx,x,\sqrt{c+d x}\right )}{d^2}\\ &=-\frac{2 (d e-c f) \sqrt{c+d x} \cos \left (a+b \sqrt{c+d x}\right )}{b d^2}-\frac{2 f (c+d x)^{3/2} \cos \left (a+b \sqrt{c+d x}\right )}{b d^2}+\frac{(6 f) \operatorname{Subst}\left (\int x^2 \cos (a+b x) \, dx,x,\sqrt{c+d x}\right )}{b d^2}+\frac{(2 (d e-c f)) \operatorname{Subst}\left (\int \cos (a+b x) \, dx,x,\sqrt{c+d x}\right )}{b d^2}\\ &=-\frac{2 (d e-c f) \sqrt{c+d x} \cos \left (a+b \sqrt{c+d x}\right )}{b d^2}-\frac{2 f (c+d x)^{3/2} \cos \left (a+b \sqrt{c+d x}\right )}{b d^2}+\frac{2 (d e-c f) \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d^2}+\frac{6 f (c+d x) \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d^2}-\frac{(12 f) \operatorname{Subst}\left (\int x \sin (a+b x) \, dx,x,\sqrt{c+d x}\right )}{b^2 d^2}\\ &=\frac{12 f \sqrt{c+d x} \cos \left (a+b \sqrt{c+d x}\right )}{b^3 d^2}-\frac{2 (d e-c f) \sqrt{c+d x} \cos \left (a+b \sqrt{c+d x}\right )}{b d^2}-\frac{2 f (c+d x)^{3/2} \cos \left (a+b \sqrt{c+d x}\right )}{b d^2}+\frac{2 (d e-c f) \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d^2}+\frac{6 f (c+d x) \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d^2}-\frac{(12 f) \operatorname{Subst}\left (\int \cos (a+b x) \, dx,x,\sqrt{c+d x}\right )}{b^3 d^2}\\ &=\frac{12 f \sqrt{c+d x} \cos \left (a+b \sqrt{c+d x}\right )}{b^3 d^2}-\frac{2 (d e-c f) \sqrt{c+d x} \cos \left (a+b \sqrt{c+d x}\right )}{b d^2}-\frac{2 f (c+d x)^{3/2} \cos \left (a+b \sqrt{c+d x}\right )}{b d^2}-\frac{12 f \sin \left (a+b \sqrt{c+d x}\right )}{b^4 d^2}+\frac{2 (d e-c f) \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d^2}+\frac{6 f (c+d x) \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d^2}\\ \end{align*}

Mathematica [A]  time = 0.415749, size = 85, normalized size = 0.46 \[ \frac{2 \sin \left (a+b \sqrt{c+d x}\right ) \left (b^2 (2 c f+d (e+3 f x))-6 f\right )-2 b \sqrt{c+d x} \left (b^2 d (e+f x)-6 f\right ) \cos \left (a+b \sqrt{c+d x}\right )}{b^4 d^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(e + f*x)*Sin[a + b*Sqrt[c + d*x]],x]

[Out]

(-2*b*Sqrt[c + d*x]*(-6*f + b^2*d*(e + f*x))*Cos[a + b*Sqrt[c + d*x]] + 2*(-6*f + b^2*(2*c*f + d*(e + 3*f*x)))
*Sin[a + b*Sqrt[c + d*x]])/(b^4*d^2)

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Maple [B]  time = 0.008, size = 366, normalized size = 2. \begin{align*} 2\,{\frac{1}{{d}^{2}{b}^{2}} \left ( -cf \left ( \sin \left ( a+b\sqrt{dx+c} \right ) - \left ( a+b\sqrt{dx+c} \right ) \cos \left ( a+b\sqrt{dx+c} \right ) \right ) +de \left ( \sin \left ( a+b\sqrt{dx+c} \right ) - \left ( a+b\sqrt{dx+c} \right ) \cos \left ( a+b\sqrt{dx+c} \right ) \right ) -acf\cos \left ( a+b\sqrt{dx+c} \right ) +ade\cos \left ( a+b\sqrt{dx+c} \right ) +{\frac{f \left ( - \left ( a+b\sqrt{dx+c} \right ) ^{3}\cos \left ( a+b\sqrt{dx+c} \right ) +3\, \left ( a+b\sqrt{dx+c} \right ) ^{2}\sin \left ( a+b\sqrt{dx+c} \right ) -6\,\sin \left ( a+b\sqrt{dx+c} \right ) +6\, \left ( a+b\sqrt{dx+c} \right ) \cos \left ( a+b\sqrt{dx+c} \right ) \right ) }{{b}^{2}}}-3\,{\frac{af \left ( - \left ( a+b\sqrt{dx+c} \right ) ^{2}\cos \left ( a+b\sqrt{dx+c} \right ) +2\,\cos \left ( a+b\sqrt{dx+c} \right ) +2\, \left ( a+b\sqrt{dx+c} \right ) \sin \left ( a+b\sqrt{dx+c} \right ) \right ) }{{b}^{2}}}+3\,{\frac{{a}^{2}f \left ( \sin \left ( a+b\sqrt{dx+c} \right ) - \left ( a+b\sqrt{dx+c} \right ) \cos \left ( a+b\sqrt{dx+c} \right ) \right ) }{{b}^{2}}}+{\frac{{a}^{3}f\cos \left ( a+b\sqrt{dx+c} \right ) }{{b}^{2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)*sin(a+b*(d*x+c)^(1/2)),x)

[Out]

2/d^2/b^2*(-c*f*(sin(a+b*(d*x+c)^(1/2))-(a+b*(d*x+c)^(1/2))*cos(a+b*(d*x+c)^(1/2)))+d*e*(sin(a+b*(d*x+c)^(1/2)
)-(a+b*(d*x+c)^(1/2))*cos(a+b*(d*x+c)^(1/2)))-a*c*f*cos(a+b*(d*x+c)^(1/2))+a*d*e*cos(a+b*(d*x+c)^(1/2))+1/b^2*
f*(-(a+b*(d*x+c)^(1/2))^3*cos(a+b*(d*x+c)^(1/2))+3*(a+b*(d*x+c)^(1/2))^2*sin(a+b*(d*x+c)^(1/2))-6*sin(a+b*(d*x
+c)^(1/2))+6*(a+b*(d*x+c)^(1/2))*cos(a+b*(d*x+c)^(1/2)))-3/b^2*a*f*(-(a+b*(d*x+c)^(1/2))^2*cos(a+b*(d*x+c)^(1/
2))+2*cos(a+b*(d*x+c)^(1/2))+2*(a+b*(d*x+c)^(1/2))*sin(a+b*(d*x+c)^(1/2)))+3/b^2*a^2*f*(sin(a+b*(d*x+c)^(1/2))
-(a+b*(d*x+c)^(1/2))*cos(a+b*(d*x+c)^(1/2)))+1/b^2*a^3*f*cos(a+b*(d*x+c)^(1/2)))

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Maxima [B]  time = 1.02153, size = 470, normalized size = 2.54 \begin{align*} \frac{2 \,{\left (a e \cos \left (\sqrt{d x + c} b + a\right ) - \frac{a c f \cos \left (\sqrt{d x + c} b + a\right )}{d} -{\left ({\left (\sqrt{d x + c} b + a\right )} \cos \left (\sqrt{d x + c} b + a\right ) - \sin \left (\sqrt{d x + c} b + a\right )\right )} e + \frac{{\left ({\left (\sqrt{d x + c} b + a\right )} \cos \left (\sqrt{d x + c} b + a\right ) - \sin \left (\sqrt{d x + c} b + a\right )\right )} c f}{d} + \frac{a^{3} f \cos \left (\sqrt{d x + c} b + a\right )}{b^{2} d} - \frac{3 \,{\left ({\left (\sqrt{d x + c} b + a\right )} \cos \left (\sqrt{d x + c} b + a\right ) - \sin \left (\sqrt{d x + c} b + a\right )\right )} a^{2} f}{b^{2} d} + \frac{3 \,{\left ({\left ({\left (\sqrt{d x + c} b + a\right )}^{2} - 2\right )} \cos \left (\sqrt{d x + c} b + a\right ) - 2 \,{\left (\sqrt{d x + c} b + a\right )} \sin \left (\sqrt{d x + c} b + a\right )\right )} a f}{b^{2} d} - \frac{{\left ({\left ({\left (\sqrt{d x + c} b + a\right )}^{3} - 6 \, \sqrt{d x + c} b - 6 \, a\right )} \cos \left (\sqrt{d x + c} b + a\right ) - 3 \,{\left ({\left (\sqrt{d x + c} b + a\right )}^{2} - 2\right )} \sin \left (\sqrt{d x + c} b + a\right )\right )} f}{b^{2} d}\right )}}{b^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*sin(a+b*(d*x+c)^(1/2)),x, algorithm="maxima")

[Out]

2*(a*e*cos(sqrt(d*x + c)*b + a) - a*c*f*cos(sqrt(d*x + c)*b + a)/d - ((sqrt(d*x + c)*b + a)*cos(sqrt(d*x + c)*
b + a) - sin(sqrt(d*x + c)*b + a))*e + ((sqrt(d*x + c)*b + a)*cos(sqrt(d*x + c)*b + a) - sin(sqrt(d*x + c)*b +
a))*c*f/d + a^3*f*cos(sqrt(d*x + c)*b + a)/(b^2*d) - 3*((sqrt(d*x + c)*b + a)*cos(sqrt(d*x + c)*b + a) - sin(
sqrt(d*x + c)*b + a))*a^2*f/(b^2*d) + 3*(((sqrt(d*x + c)*b + a)^2 - 2)*cos(sqrt(d*x + c)*b + a) - 2*(sqrt(d*x
+ c)*b + a)*sin(sqrt(d*x + c)*b + a))*a*f/(b^2*d) - (((sqrt(d*x + c)*b + a)^3 - 6*sqrt(d*x + c)*b - 6*a)*cos(s
qrt(d*x + c)*b + a) - 3*((sqrt(d*x + c)*b + a)^2 - 2)*sin(sqrt(d*x + c)*b + a))*f/(b^2*d))/(b^2*d)

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Fricas [A]  time = 1.68919, size = 208, normalized size = 1.12 \begin{align*} -\frac{2 \,{\left ({\left (b^{3} d f x + b^{3} d e - 6 \, b f\right )} \sqrt{d x + c} \cos \left (\sqrt{d x + c} b + a\right ) -{\left (3 \, b^{2} d f x + b^{2} d e + 2 \,{\left (b^{2} c - 3\right )} f\right )} \sin \left (\sqrt{d x + c} b + a\right )\right )}}{b^{4} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*sin(a+b*(d*x+c)^(1/2)),x, algorithm="fricas")

[Out]

-2*((b^3*d*f*x + b^3*d*e - 6*b*f)*sqrt(d*x + c)*cos(sqrt(d*x + c)*b + a) - (3*b^2*d*f*x + b^2*d*e + 2*(b^2*c -
3)*f)*sin(sqrt(d*x + c)*b + a))/(b^4*d^2)

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Sympy [A]  time = 0.741943, size = 231, normalized size = 1.25 \begin{align*} \begin{cases} \left (e x + \frac{f x^{2}}{2}\right ) \sin{\left (a \right )} & \text{for}\: b = 0 \wedge d = 0 \\\left (e x + \frac{f x^{2}}{2}\right ) \sin{\left (a + b \sqrt{c} \right )} & \text{for}\: d = 0 \\\left (e x + \frac{f x^{2}}{2}\right ) \sin{\left (a \right )} & \text{for}\: b = 0 \\- \frac{2 e \sqrt{c + d x} \cos{\left (a + b \sqrt{c + d x} \right )}}{b d} - \frac{2 f x \sqrt{c + d x} \cos{\left (a + b \sqrt{c + d x} \right )}}{b d} + \frac{4 c f \sin{\left (a + b \sqrt{c + d x} \right )}}{b^{2} d^{2}} + \frac{2 e \sin{\left (a + b \sqrt{c + d x} \right )}}{b^{2} d} + \frac{6 f x \sin{\left (a + b \sqrt{c + d x} \right )}}{b^{2} d} + \frac{12 f \sqrt{c + d x} \cos{\left (a + b \sqrt{c + d x} \right )}}{b^{3} d^{2}} - \frac{12 f \sin{\left (a + b \sqrt{c + d x} \right )}}{b^{4} d^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*sin(a+b*(d*x+c)**(1/2)),x)

[Out]

Piecewise(((e*x + f*x**2/2)*sin(a), Eq(b, 0) & Eq(d, 0)), ((e*x + f*x**2/2)*sin(a + b*sqrt(c)), Eq(d, 0)), ((e
*x + f*x**2/2)*sin(a), Eq(b, 0)), (-2*e*sqrt(c + d*x)*cos(a + b*sqrt(c + d*x))/(b*d) - 2*f*x*sqrt(c + d*x)*cos
(a + b*sqrt(c + d*x))/(b*d) + 4*c*f*sin(a + b*sqrt(c + d*x))/(b**2*d**2) + 2*e*sin(a + b*sqrt(c + d*x))/(b**2*
d) + 6*f*x*sin(a + b*sqrt(c + d*x))/(b**2*d) + 12*f*sqrt(c + d*x)*cos(a + b*sqrt(c + d*x))/(b**3*d**2) - 12*f*
sin(a + b*sqrt(c + d*x))/(b**4*d**2), True))

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Giac [B]  time = 1.299, size = 714, normalized size = 3.86 \begin{align*} -\frac{2 \,{\left (\frac{{\left ({\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right )} \cos \left (-{\left (\sqrt{d x + c} b + a\right )} \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) + a \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) - a\right ) + \frac{b \sin \left (-{\left (\sqrt{d x + c} b + a\right )} \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) + a \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) - a\right )}{\mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right )}\right )} e}{b^{2}} - \frac{f{\left (\frac{{\left ({\left (\sqrt{d x + c} b + a\right )} b^{2} c - a b^{2} c -{\left (\sqrt{d x + c} b + a\right )}^{3} + 3 \,{\left (\sqrt{d x + c} b + a\right )}^{2} a - 3 \,{\left (\sqrt{d x + c} b + a\right )} a^{2} + a^{3} + 6 \, \sqrt{d x + c} b\right )} \cos \left (-{\left (\sqrt{d x + c} b + a\right )} \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) + a \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) - a\right )}{b} + \frac{{\left (b^{2} c \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) - 3 \,{\left (\sqrt{d x + c} b + a\right )}^{2} \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) + 6 \,{\left (\sqrt{d x + c} b + a\right )} a \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) - 3 \, a^{2} \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) + 6 \, \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right )\right )} \sin \left (-{\left (\sqrt{d x + c} b + a\right )} \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) + a \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) - a\right )}{b}\right )}}{b^{2} d}\right )}}{b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*sin(a+b*(d*x+c)^(1/2)),x, algorithm="giac")

[Out]

-2*((((sqrt(d*x + c)*b + a)*b - a*b)*cos(-(sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a*sgn((sq
rt(d*x + c)*b + a)*b - a*b) - a) + b*sin(-(sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a*sgn((sq
rt(d*x + c)*b + a)*b - a*b) - a)/sgn((sqrt(d*x + c)*b + a)*b - a*b))*e/b^2 - f*(((sqrt(d*x + c)*b + a)*b^2*c -
a*b^2*c - (sqrt(d*x + c)*b + a)^3 + 3*(sqrt(d*x + c)*b + a)^2*a - 3*(sqrt(d*x + c)*b + a)*a^2 + a^3 + 6*sqrt(
d*x + c)*b)*cos(-(sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a*sgn((sqrt(d*x + c)*b + a)*b - a*
b) - a)/b + (b^2*c*sgn((sqrt(d*x + c)*b + a)*b - a*b) - 3*(sqrt(d*x + c)*b + a)^2*sgn((sqrt(d*x + c)*b + a)*b
- a*b) + 6*(sqrt(d*x + c)*b + a)*a*sgn((sqrt(d*x + c)*b + a)*b - a*b) - 3*a^2*sgn((sqrt(d*x + c)*b + a)*b - a*
b) + 6*sgn((sqrt(d*x + c)*b + a)*b - a*b))*sin(-(sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a*s
gn((sqrt(d*x + c)*b + a)*b - a*b) - a)/b)/(b^2*d))/(b*d)