3.203 \(\int (e+f x) \sin (a+\frac{b}{(c+d x)^{3/2}}) \, dx\)

Optimal. Leaf size=251 \[ -\frac{i e^{i a} (c+d x) \left (-\frac{i b}{(c+d x)^{3/2}}\right )^{2/3} (d e-c f) \text{Gamma}\left (-\frac{2}{3},-\frac{i b}{(c+d x)^{3/2}}\right )}{3 d^2}+\frac{i e^{-i a} (c+d x) \left (\frac{i b}{(c+d x)^{3/2}}\right )^{2/3} (d e-c f) \text{Gamma}\left (-\frac{2}{3},\frac{i b}{(c+d x)^{3/2}}\right )}{3 d^2}-\frac{i e^{i a} f (c+d x)^2 \left (-\frac{i b}{(c+d x)^{3/2}}\right )^{4/3} \text{Gamma}\left (-\frac{4}{3},-\frac{i b}{(c+d x)^{3/2}}\right )}{3 d^2}+\frac{i e^{-i a} f (c+d x)^2 \left (\frac{i b}{(c+d x)^{3/2}}\right )^{4/3} \text{Gamma}\left (-\frac{4}{3},\frac{i b}{(c+d x)^{3/2}}\right )}{3 d^2} \]

[Out]

((-I/3)*E^(I*a)*f*(((-I)*b)/(c + d*x)^(3/2))^(4/3)*(c + d*x)^2*Gamma[-4/3, ((-I)*b)/(c + d*x)^(3/2)])/d^2 + ((
I/3)*f*((I*b)/(c + d*x)^(3/2))^(4/3)*(c + d*x)^2*Gamma[-4/3, (I*b)/(c + d*x)^(3/2)])/(d^2*E^(I*a)) - ((I/3)*E^
(I*a)*(d*e - c*f)*(((-I)*b)/(c + d*x)^(3/2))^(2/3)*(c + d*x)*Gamma[-2/3, ((-I)*b)/(c + d*x)^(3/2)])/d^2 + ((I/
3)*(d*e - c*f)*((I*b)/(c + d*x)^(3/2))^(2/3)*(c + d*x)*Gamma[-2/3, (I*b)/(c + d*x)^(3/2)])/(d^2*E^(I*a))

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Rubi [A]  time = 0.225731, antiderivative size = 251, 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 = {3433, 3423, 2218} \[ -\frac{i e^{i a} (c+d x) \left (-\frac{i b}{(c+d x)^{3/2}}\right )^{2/3} (d e-c f) \text{Gamma}\left (-\frac{2}{3},-\frac{i b}{(c+d x)^{3/2}}\right )}{3 d^2}+\frac{i e^{-i a} (c+d x) \left (\frac{i b}{(c+d x)^{3/2}}\right )^{2/3} (d e-c f) \text{Gamma}\left (-\frac{2}{3},\frac{i b}{(c+d x)^{3/2}}\right )}{3 d^2}-\frac{i e^{i a} f (c+d x)^2 \left (-\frac{i b}{(c+d x)^{3/2}}\right )^{4/3} \text{Gamma}\left (-\frac{4}{3},-\frac{i b}{(c+d x)^{3/2}}\right )}{3 d^2}+\frac{i e^{-i a} f (c+d x)^2 \left (\frac{i b}{(c+d x)^{3/2}}\right )^{4/3} \text{Gamma}\left (-\frac{4}{3},\frac{i b}{(c+d x)^{3/2}}\right )}{3 d^2} \]

Antiderivative was successfully verified.

[In]

Int[(e + f*x)*Sin[a + b/(c + d*x)^(3/2)],x]

[Out]

((-I/3)*E^(I*a)*f*(((-I)*b)/(c + d*x)^(3/2))^(4/3)*(c + d*x)^2*Gamma[-4/3, ((-I)*b)/(c + d*x)^(3/2)])/d^2 + ((
I/3)*f*((I*b)/(c + d*x)^(3/2))^(4/3)*(c + d*x)^2*Gamma[-4/3, (I*b)/(c + d*x)^(3/2)])/(d^2*E^(I*a)) - ((I/3)*E^
(I*a)*(d*e - c*f)*(((-I)*b)/(c + d*x)^(3/2))^(2/3)*(c + d*x)*Gamma[-2/3, ((-I)*b)/(c + d*x)^(3/2)])/d^2 + ((I/
3)*(d*e - c*f)*((I*b)/(c + d*x)^(3/2))^(2/3)*(c + d*x)*Gamma[-2/3, (I*b)/(c + d*x)^(3/2)])/(d^2*E^(I*a))

Rule 3433

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

Rule 3423

Int[((e_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Dist[I/2, Int[(e*x)^m*E^(-(c*I) - d*I*x^n),
x], x] - Dist[I/2, Int[(e*x)^m*E^(c*I + d*I*x^n), x], x] /; FreeQ[{c, d, e, m, n}, x]

Rule 2218

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Simp[(F^a*(e + f*
x)^(m + 1)*Gamma[(m + 1)/n, -(b*(c + d*x)^n*Log[F])])/(f*n*(-(b*(c + d*x)^n*Log[F]))^((m + 1)/n)), x] /; FreeQ
[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

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

Mathematica [B]  time = 2.65297, size = 835, normalized size = 3.33 \[ \frac{9 i f \cos (a) \left (\frac{2 \text{Gamma}\left (\frac{2}{3},-\frac{i b}{(c+d x)^{3/2}}\right )}{3 \left (-\frac{i b}{(c+d x)^{3/2}}\right )^{2/3} (c+d x)}-\frac{2 \text{Gamma}\left (\frac{2}{3},\frac{i b}{(c+d x)^{3/2}}\right )}{3 \left (\frac{i b}{(c+d x)^{3/2}}\right )^{2/3} (c+d x)}\right ) b^2}{8 d^2}-\frac{9 f \left (\frac{2 \text{Gamma}\left (\frac{2}{3},-\frac{i b}{(c+d x)^{3/2}}\right )}{3 \left (-\frac{i b}{(c+d x)^{3/2}}\right )^{2/3} (c+d x)}+\frac{2 \text{Gamma}\left (\frac{2}{3},\frac{i b}{(c+d x)^{3/2}}\right )}{3 \left (\frac{i b}{(c+d x)^{3/2}}\right )^{2/3} (c+d x)}\right ) \sin (a) b^2}{8 d^2}+\frac{3 e \cos (a) \left (\frac{2 \text{Gamma}\left (\frac{1}{3},-\frac{i b}{(c+d x)^{3/2}}\right )}{3 \sqrt [3]{-\frac{i b}{(c+d x)^{3/2}}} \sqrt{c+d x}}+\frac{2 \text{Gamma}\left (\frac{1}{3},\frac{i b}{(c+d x)^{3/2}}\right )}{3 \sqrt [3]{\frac{i b}{(c+d x)^{3/2}}} \sqrt{c+d x}}\right ) b}{4 d}-\frac{3 c f \cos (a) \left (\frac{2 \text{Gamma}\left (\frac{1}{3},-\frac{i b}{(c+d x)^{3/2}}\right )}{3 \sqrt [3]{-\frac{i b}{(c+d x)^{3/2}}} \sqrt{c+d x}}+\frac{2 \text{Gamma}\left (\frac{1}{3},\frac{i b}{(c+d x)^{3/2}}\right )}{3 \sqrt [3]{\frac{i b}{(c+d x)^{3/2}}} \sqrt{c+d x}}\right ) b}{4 d^2}+\frac{3 i e \left (\frac{2 \text{Gamma}\left (\frac{1}{3},-\frac{i b}{(c+d x)^{3/2}}\right )}{3 \sqrt [3]{-\frac{i b}{(c+d x)^{3/2}}} \sqrt{c+d x}}-\frac{2 \text{Gamma}\left (\frac{1}{3},\frac{i b}{(c+d x)^{3/2}}\right )}{3 \sqrt [3]{\frac{i b}{(c+d x)^{3/2}}} \sqrt{c+d x}}\right ) \sin (a) b}{4 d}-\frac{3 i c f \left (\frac{2 \text{Gamma}\left (\frac{1}{3},-\frac{i b}{(c+d x)^{3/2}}\right )}{3 \sqrt [3]{-\frac{i b}{(c+d x)^{3/2}}} \sqrt{c+d x}}-\frac{2 \text{Gamma}\left (\frac{1}{3},\frac{i b}{(c+d x)^{3/2}}\right )}{3 \sqrt [3]{\frac{i b}{(c+d x)^{3/2}}} \sqrt{c+d x}}\right ) \sin (a) b}{4 d^2}+\frac{e (c+d x) \cos \left (\frac{b}{(c+d x)^{3/2}}\right ) \sin (a)}{d}+\frac{f \sqrt{c+d x} \cos \left (\frac{b}{(c+d x)^{3/2}}\right ) \left (\sin (a) (c+d x)^{3/2}-2 c \sin (a) \sqrt{c+d x}+3 b \cos (a)\right )}{2 d^2}+\frac{e (c+d x) \cos (a) \sin \left (\frac{b}{(c+d x)^{3/2}}\right )}{d}+\frac{f \sqrt{c+d x} \left (\cos (a) (c+d x)^{3/2}-2 c \cos (a) \sqrt{c+d x}-3 b \sin (a)\right ) \sin \left (\frac{b}{(c+d x)^{3/2}}\right )}{2 d^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(e + f*x)*Sin[a + b/(c + d*x)^(3/2)],x]

[Out]

(3*b*e*Cos[a]*((2*Gamma[1/3, ((-I)*b)/(c + d*x)^(3/2)])/(3*(((-I)*b)/(c + d*x)^(3/2))^(1/3)*Sqrt[c + d*x]) + (
2*Gamma[1/3, (I*b)/(c + d*x)^(3/2)])/(3*((I*b)/(c + d*x)^(3/2))^(1/3)*Sqrt[c + d*x])))/(4*d) - (3*b*c*f*Cos[a]
*((2*Gamma[1/3, ((-I)*b)/(c + d*x)^(3/2)])/(3*(((-I)*b)/(c + d*x)^(3/2))^(1/3)*Sqrt[c + d*x]) + (2*Gamma[1/3,
(I*b)/(c + d*x)^(3/2)])/(3*((I*b)/(c + d*x)^(3/2))^(1/3)*Sqrt[c + d*x])))/(4*d^2) + (((9*I)/8)*b^2*f*Cos[a]*((
2*Gamma[2/3, ((-I)*b)/(c + d*x)^(3/2)])/(3*(((-I)*b)/(c + d*x)^(3/2))^(2/3)*(c + d*x)) - (2*Gamma[2/3, (I*b)/(
c + d*x)^(3/2)])/(3*((I*b)/(c + d*x)^(3/2))^(2/3)*(c + d*x))))/d^2 + (e*(c + d*x)*Cos[b/(c + d*x)^(3/2)]*Sin[a
])/d + (((3*I)/4)*b*e*((2*Gamma[1/3, ((-I)*b)/(c + d*x)^(3/2)])/(3*(((-I)*b)/(c + d*x)^(3/2))^(1/3)*Sqrt[c + d
*x]) - (2*Gamma[1/3, (I*b)/(c + d*x)^(3/2)])/(3*((I*b)/(c + d*x)^(3/2))^(1/3)*Sqrt[c + d*x]))*Sin[a])/d - (((3
*I)/4)*b*c*f*((2*Gamma[1/3, ((-I)*b)/(c + d*x)^(3/2)])/(3*(((-I)*b)/(c + d*x)^(3/2))^(1/3)*Sqrt[c + d*x]) - (2
*Gamma[1/3, (I*b)/(c + d*x)^(3/2)])/(3*((I*b)/(c + d*x)^(3/2))^(1/3)*Sqrt[c + d*x]))*Sin[a])/d^2 - (9*b^2*f*((
2*Gamma[2/3, ((-I)*b)/(c + d*x)^(3/2)])/(3*(((-I)*b)/(c + d*x)^(3/2))^(2/3)*(c + d*x)) + (2*Gamma[2/3, (I*b)/(
c + d*x)^(3/2)])/(3*((I*b)/(c + d*x)^(3/2))^(2/3)*(c + d*x)))*Sin[a])/(8*d^2) + (f*Sqrt[c + d*x]*Cos[b/(c + d*
x)^(3/2)]*(3*b*Cos[a] - 2*c*Sqrt[c + d*x]*Sin[a] + (c + d*x)^(3/2)*Sin[a]))/(2*d^2) + (e*(c + d*x)*Cos[a]*Sin[
b/(c + d*x)^(3/2)])/d + (f*Sqrt[c + d*x]*(-2*c*Sqrt[c + d*x]*Cos[a] + (c + d*x)^(3/2)*Cos[a] - 3*b*Sin[a])*Sin
[b/(c + d*x)^(3/2)])/(2*d^2)

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \begin{align*} \int \left ( fx+e \right ) \sin \left ( a+{b \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)*sin(a+b/(d*x+c)^(3/2)),x)

[Out]

int((f*x+e)*sin(a+b/(d*x+c)^(3/2)),x)

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Maxima [B]  time = 2.07836, size = 1623, normalized size = 6.47 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*sin(a+b/(d*x+c)^(3/2)),x, algorithm="maxima")

[Out]

1/8*(2*(4*(d*x + c)^(3/2)*(abs(b)/(d*x + c)^(3/2))^(1/3)*sin(((d*x + c)^(3/2)*a + b)/(d*x + c)^(3/2)) + (((gam
ma(1/3, I*b/(d*x + c)^(3/2)) + gamma(1/3, -I*b/(d*x + c)^(3/2)))*cos(1/6*pi + 1/3*arctan2(0, b)) + (gamma(1/3,
I*b/(d*x + c)^(3/2)) + gamma(1/3, -I*b/(d*x + c)^(3/2)))*cos(-1/6*pi + 1/3*arctan2(0, b)) + (-I*gamma(1/3, I*
b/(d*x + c)^(3/2)) + I*gamma(1/3, -I*b/(d*x + c)^(3/2)))*sin(1/6*pi + 1/3*arctan2(0, b)) + (I*gamma(1/3, I*b/(
d*x + c)^(3/2)) - I*gamma(1/3, -I*b/(d*x + c)^(3/2)))*sin(-1/6*pi + 1/3*arctan2(0, b)))*cos(a) + ((-I*gamma(1/
3, I*b/(d*x + c)^(3/2)) + I*gamma(1/3, -I*b/(d*x + c)^(3/2)))*cos(1/6*pi + 1/3*arctan2(0, b)) + (-I*gamma(1/3,
I*b/(d*x + c)^(3/2)) + I*gamma(1/3, -I*b/(d*x + c)^(3/2)))*cos(-1/6*pi + 1/3*arctan2(0, b)) - (gamma(1/3, I*b
/(d*x + c)^(3/2)) + gamma(1/3, -I*b/(d*x + c)^(3/2)))*sin(1/6*pi + 1/3*arctan2(0, b)) + (gamma(1/3, I*b/(d*x +
c)^(3/2)) + gamma(1/3, -I*b/(d*x + c)^(3/2)))*sin(-1/6*pi + 1/3*arctan2(0, b)))*sin(a))*b)*e/(sqrt(d*x + c)*(
abs(b)/(d*x + c)^(3/2))^(1/3)) - 2*(4*(d*x + c)^(3/2)*(abs(b)/(d*x + c)^(3/2))^(1/3)*sin(((d*x + c)^(3/2)*a +
b)/(d*x + c)^(3/2)) + (((gamma(1/3, I*b/(d*x + c)^(3/2)) + gamma(1/3, -I*b/(d*x + c)^(3/2)))*cos(1/6*pi + 1/3*
arctan2(0, b)) + (gamma(1/3, I*b/(d*x + c)^(3/2)) + gamma(1/3, -I*b/(d*x + c)^(3/2)))*cos(-1/6*pi + 1/3*arctan
2(0, b)) + (-I*gamma(1/3, I*b/(d*x + c)^(3/2)) + I*gamma(1/3, -I*b/(d*x + c)^(3/2)))*sin(1/6*pi + 1/3*arctan2(
0, b)) + (I*gamma(1/3, I*b/(d*x + c)^(3/2)) - I*gamma(1/3, -I*b/(d*x + c)^(3/2)))*sin(-1/6*pi + 1/3*arctan2(0,
b)))*cos(a) + ((-I*gamma(1/3, I*b/(d*x + c)^(3/2)) + I*gamma(1/3, -I*b/(d*x + c)^(3/2)))*cos(1/6*pi + 1/3*arc
tan2(0, b)) + (-I*gamma(1/3, I*b/(d*x + c)^(3/2)) + I*gamma(1/3, -I*b/(d*x + c)^(3/2)))*cos(-1/6*pi + 1/3*arct
an2(0, b)) - (gamma(1/3, I*b/(d*x + c)^(3/2)) + gamma(1/3, -I*b/(d*x + c)^(3/2)))*sin(1/6*pi + 1/3*arctan2(0,
b)) + (gamma(1/3, I*b/(d*x + c)^(3/2)) + gamma(1/3, -I*b/(d*x + c)^(3/2)))*sin(-1/6*pi + 1/3*arctan2(0, b)))*s
in(a))*b)*c*f/(sqrt(d*x + c)*d*(abs(b)/(d*x + c)^(3/2))^(1/3)) + (4*(d*x + c)^3*(abs(b)/(d*x + c)^(3/2))^(2/3)
*sin(((d*x + c)^(3/2)*a + b)/(d*x + c)^(3/2)) + 12*(d*x + c)^(3/2)*b*(abs(b)/(d*x + c)^(3/2))^(2/3)*cos(((d*x
+ c)^(3/2)*a + b)/(d*x + c)^(3/2)) + (((-3*I*gamma(2/3, I*b/(d*x + c)^(3/2)) + 3*I*gamma(2/3, -I*b/(d*x + c)^(
3/2)))*cos(1/3*pi + 2/3*arctan2(0, b)) + (-3*I*gamma(2/3, I*b/(d*x + c)^(3/2)) + 3*I*gamma(2/3, -I*b/(d*x + c)
^(3/2)))*cos(-1/3*pi + 2/3*arctan2(0, b)) - 3*(gamma(2/3, I*b/(d*x + c)^(3/2)) + gamma(2/3, -I*b/(d*x + c)^(3/
2)))*sin(1/3*pi + 2/3*arctan2(0, b)) + 3*(gamma(2/3, I*b/(d*x + c)^(3/2)) + gamma(2/3, -I*b/(d*x + c)^(3/2)))*
sin(-1/3*pi + 2/3*arctan2(0, b)))*cos(a) - (3*(gamma(2/3, I*b/(d*x + c)^(3/2)) + gamma(2/3, -I*b/(d*x + c)^(3/
2)))*cos(1/3*pi + 2/3*arctan2(0, b)) + 3*(gamma(2/3, I*b/(d*x + c)^(3/2)) + gamma(2/3, -I*b/(d*x + c)^(3/2)))*
cos(-1/3*pi + 2/3*arctan2(0, b)) - (3*I*gamma(2/3, I*b/(d*x + c)^(3/2)) - 3*I*gamma(2/3, -I*b/(d*x + c)^(3/2))
)*sin(1/3*pi + 2/3*arctan2(0, b)) - (-3*I*gamma(2/3, I*b/(d*x + c)^(3/2)) + 3*I*gamma(2/3, -I*b/(d*x + c)^(3/2
)))*sin(-1/3*pi + 2/3*arctan2(0, b)))*sin(a))*b^2)*f/((d*x + c)*d*(abs(b)/(d*x + c)^(3/2))^(2/3)))/d

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Fricas [A]  time = 2.04125, size = 825, normalized size = 3.29 \begin{align*} -\frac{3 \, \left (i \, b\right )^{\frac{1}{3}} b f e^{\left (-i \, a\right )} \Gamma \left (\frac{2}{3}, \frac{i \, \sqrt{d x + c} b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + 3 \, \left (-i \, b\right )^{\frac{1}{3}} b f e^{\left (i \, a\right )} \Gamma \left (\frac{2}{3}, -\frac{i \, \sqrt{d x + c} b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) -{\left (-2 i \, d e + 2 i \, c f\right )} \left (i \, b\right )^{\frac{2}{3}} e^{\left (-i \, a\right )} \Gamma \left (\frac{1}{3}, \frac{i \, \sqrt{d x + c} b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) -{\left (2 i \, d e - 2 i \, c f\right )} \left (-i \, b\right )^{\frac{2}{3}} e^{\left (i \, a\right )} \Gamma \left (\frac{1}{3}, -\frac{i \, \sqrt{d x + c} b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - 6 \, \sqrt{d x + c} b f \cos \left (\frac{a d^{2} x^{2} + 2 \, a c d x + a c^{2} + \sqrt{d x + c} b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - 2 \,{\left (d^{2} f x^{2} + 2 \, d^{2} e x + 2 \, c d e - c^{2} f\right )} \sin \left (\frac{a d^{2} x^{2} + 2 \, a c d x + a c^{2} + \sqrt{d x + c} b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{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)^(3/2)),x, algorithm="fricas")

[Out]

-1/4*(3*(I*b)^(1/3)*b*f*e^(-I*a)*gamma(2/3, I*sqrt(d*x + c)*b/(d^2*x^2 + 2*c*d*x + c^2)) + 3*(-I*b)^(1/3)*b*f*
e^(I*a)*gamma(2/3, -I*sqrt(d*x + c)*b/(d^2*x^2 + 2*c*d*x + c^2)) - (-2*I*d*e + 2*I*c*f)*(I*b)^(2/3)*e^(-I*a)*g
amma(1/3, I*sqrt(d*x + c)*b/(d^2*x^2 + 2*c*d*x + c^2)) - (2*I*d*e - 2*I*c*f)*(-I*b)^(2/3)*e^(I*a)*gamma(1/3, -
I*sqrt(d*x + c)*b/(d^2*x^2 + 2*c*d*x + c^2)) - 6*sqrt(d*x + c)*b*f*cos((a*d^2*x^2 + 2*a*c*d*x + a*c^2 + sqrt(d
*x + c)*b)/(d^2*x^2 + 2*c*d*x + c^2)) - 2*(d^2*f*x^2 + 2*d^2*e*x + 2*c*d*e - c^2*f)*sin((a*d^2*x^2 + 2*a*c*d*x
+ a*c^2 + sqrt(d*x + c)*b)/(d^2*x^2 + 2*c*d*x + c^2)))/d^2

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*sin(a+b/(d*x+c)**(3/2)),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*sin(a+b/(d*x+c)^(3/2)),x, algorithm="giac")

[Out]

integrate((f*x + e)*sin(a + b/(d*x + c)^(3/2)), x)