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

Optimal. Leaf size=390 \[ -\frac{2 i e^{i a} f (c+d x)^2 \left (-\frac{i b}{(c+d x)^{3/2}}\right )^{4/3} (d e-c f) \text{Gamma}\left (-\frac{4}{3},-\frac{i b}{(c+d x)^{3/2}}\right )}{3 d^3}+\frac{2 i e^{-i a} f (c+d x)^2 \left (\frac{i b}{(c+d x)^{3/2}}\right )^{4/3} (d e-c f) \text{Gamma}\left (-\frac{4}{3},\frac{i b}{(c+d x)^{3/2}}\right )}{3 d^3}-\frac{i e^{i a} (c+d x) \left (-\frac{i b}{(c+d x)^{3/2}}\right )^{2/3} (d e-c f)^2 \text{Gamma}\left (-\frac{2}{3},-\frac{i b}{(c+d x)^{3/2}}\right )}{3 d^3}+\frac{i e^{-i a} (c+d x) \left (\frac{i b}{(c+d x)^{3/2}}\right )^{2/3} (d e-c f)^2 \text{Gamma}\left (-\frac{2}{3},\frac{i b}{(c+d x)^{3/2}}\right )}{3 d^3}+\frac{b^2 f^2 \sin (a) \text{CosIntegral}\left (\frac{b}{(c+d x)^{3/2}}\right )}{3 d^3}+\frac{b^2 f^2 \cos (a) \text{Si}\left (\frac{b}{(c+d x)^{3/2}}\right )}{3 d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{(c+d x)^{3/2}}\right )}{3 d^3}+\frac{b f^2 (c+d x)^{3/2} \cos \left (a+\frac{b}{(c+d x)^{3/2}}\right )}{3 d^3} \]

[Out]

(b*f^2*(c + d*x)^(3/2)*Cos[a + b/(c + d*x)^(3/2)])/(3*d^3) - (((2*I)/3)*E^(I*a)*f*(d*e - c*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^3 + (((2*I)/3)*f*(d*e - c*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^3*E^(I*a)) - ((I/3)*E^(I*a)*(d*e - c*f)^
2*(((-I)*b)/(c + d*x)^(3/2))^(2/3)*(c + d*x)*Gamma[-2/3, ((-I)*b)/(c + d*x)^(3/2)])/d^3 + ((I/3)*(d*e - c*f)^2
*((I*b)/(c + d*x)^(3/2))^(2/3)*(c + d*x)*Gamma[-2/3, (I*b)/(c + d*x)^(3/2)])/(d^3*E^(I*a)) + (b^2*f^2*CosInteg
ral[b/(c + d*x)^(3/2)]*Sin[a])/(3*d^3) + (f^2*(c + d*x)^3*Sin[a + b/(c + d*x)^(3/2)])/(3*d^3) + (b^2*f^2*Cos[a
]*SinIntegral[b/(c + d*x)^(3/2)])/(3*d^3)

________________________________________________________________________________________

Rubi [A]  time = 0.428222, antiderivative size = 390, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {3433, 3423, 2218, 3379, 3297, 3303, 3299, 3302} \[ -\frac{2 i e^{i a} f (c+d x)^2 \left (-\frac{i b}{(c+d x)^{3/2}}\right )^{4/3} (d e-c f) \text{Gamma}\left (-\frac{4}{3},-\frac{i b}{(c+d x)^{3/2}}\right )}{3 d^3}+\frac{2 i e^{-i a} f (c+d x)^2 \left (\frac{i b}{(c+d x)^{3/2}}\right )^{4/3} (d e-c f) \text{Gamma}\left (-\frac{4}{3},\frac{i b}{(c+d x)^{3/2}}\right )}{3 d^3}-\frac{i e^{i a} (c+d x) \left (-\frac{i b}{(c+d x)^{3/2}}\right )^{2/3} (d e-c f)^2 \text{Gamma}\left (-\frac{2}{3},-\frac{i b}{(c+d x)^{3/2}}\right )}{3 d^3}+\frac{i e^{-i a} (c+d x) \left (\frac{i b}{(c+d x)^{3/2}}\right )^{2/3} (d e-c f)^2 \text{Gamma}\left (-\frac{2}{3},\frac{i b}{(c+d x)^{3/2}}\right )}{3 d^3}+\frac{b^2 f^2 \sin (a) \text{CosIntegral}\left (\frac{b}{(c+d x)^{3/2}}\right )}{3 d^3}+\frac{b^2 f^2 \cos (a) \text{Si}\left (\frac{b}{(c+d x)^{3/2}}\right )}{3 d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{(c+d x)^{3/2}}\right )}{3 d^3}+\frac{b f^2 (c+d x)^{3/2} \cos \left (a+\frac{b}{(c+d x)^{3/2}}\right )}{3 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*f^2*(c + d*x)^(3/2)*Cos[a + b/(c + d*x)^(3/2)])/(3*d^3) - (((2*I)/3)*E^(I*a)*f*(d*e - c*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^3 + (((2*I)/3)*f*(d*e - c*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^3*E^(I*a)) - ((I/3)*E^(I*a)*(d*e - c*f)^
2*(((-I)*b)/(c + d*x)^(3/2))^(2/3)*(c + d*x)*Gamma[-2/3, ((-I)*b)/(c + d*x)^(3/2)])/d^3 + ((I/3)*(d*e - c*f)^2
*((I*b)/(c + d*x)^(3/2))^(2/3)*(c + d*x)*Gamma[-2/3, (I*b)/(c + d*x)^(3/2)])/(d^3*E^(I*a)) + (b^2*f^2*CosInteg
ral[b/(c + d*x)^(3/2)]*Sin[a])/(3*d^3) + (f^2*(c + d*x)^3*Sin[a + b/(c + d*x)^(3/2)])/(3*d^3) + (b^2*f^2*Cos[a
]*SinIntegral[b/(c + d*x)^(3/2)])/(3*d^3)

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]

Rule 3379

Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Sin[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simpl
ify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rule 3297

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

Rule 3303

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

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rubi steps

\begin{align*} \int (e+f x)^2 \sin \left (a+\frac{b}{(c+d x)^{3/2}}\right ) \, dx &=\frac{2 \operatorname{Subst}\left (\int \left ((d e-c f)^2 x \sin \left (a+\frac{b}{x^3}\right )-2 f (-d e+c f) x^3 \sin \left (a+\frac{b}{x^3}\right )+f^2 x^5 \sin \left (a+\frac{b}{x^3}\right )\right ) \, dx,x,\sqrt{c+d x}\right )}{d^3}\\ &=\frac{\left (2 f^2\right ) \operatorname{Subst}\left (\int x^5 \sin \left (a+\frac{b}{x^3}\right ) \, dx,x,\sqrt{c+d x}\right )}{d^3}+\frac{(4 f (d e-c f)) \operatorname{Subst}\left (\int x^3 \sin \left (a+\frac{b}{x^3}\right ) \, dx,x,\sqrt{c+d x}\right )}{d^3}+\frac{\left (2 (d e-c f)^2\right ) \operatorname{Subst}\left (\int x \sin \left (a+\frac{b}{x^3}\right ) \, dx,x,\sqrt{c+d x}\right )}{d^3}\\ &=-\frac{\left (2 f^2\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^3} \, dx,x,\frac{1}{(c+d x)^{3/2}}\right )}{3 d^3}+\frac{(2 i f (d e-c f)) \operatorname{Subst}\left (\int e^{-i a-\frac{i b}{x^3}} x^3 \, dx,x,\sqrt{c+d x}\right )}{d^3}-\frac{(2 i f (d e-c f)) \operatorname{Subst}\left (\int e^{i a+\frac{i b}{x^3}} x^3 \, dx,x,\sqrt{c+d x}\right )}{d^3}+\frac{\left (i (d e-c f)^2\right ) \operatorname{Subst}\left (\int e^{-i a-\frac{i b}{x^3}} x \, dx,x,\sqrt{c+d x}\right )}{d^3}-\frac{\left (i (d e-c f)^2\right ) \operatorname{Subst}\left (\int e^{i a+\frac{i b}{x^3}} x \, dx,x,\sqrt{c+d x}\right )}{d^3}\\ &=-\frac{2 i e^{i a} f (d e-c 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^3}+\frac{2 i e^{-i a} f (d e-c 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^3}-\frac{i e^{i a} (d e-c f)^2 \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^3}+\frac{i e^{-i a} (d e-c f)^2 \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^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{(c+d x)^{3/2}}\right )}{3 d^3}-\frac{\left (b f^2\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^2} \, dx,x,\frac{1}{(c+d x)^{3/2}}\right )}{3 d^3}\\ &=\frac{b f^2 (c+d x)^{3/2} \cos \left (a+\frac{b}{(c+d x)^{3/2}}\right )}{3 d^3}-\frac{2 i e^{i a} f (d e-c 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^3}+\frac{2 i e^{-i a} f (d e-c 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^3}-\frac{i e^{i a} (d e-c f)^2 \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^3}+\frac{i e^{-i a} (d e-c f)^2 \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^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{(c+d x)^{3/2}}\right )}{3 d^3}+\frac{\left (b^2 f^2\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x} \, dx,x,\frac{1}{(c+d x)^{3/2}}\right )}{3 d^3}\\ &=\frac{b f^2 (c+d x)^{3/2} \cos \left (a+\frac{b}{(c+d x)^{3/2}}\right )}{3 d^3}-\frac{2 i e^{i a} f (d e-c 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^3}+\frac{2 i e^{-i a} f (d e-c 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^3}-\frac{i e^{i a} (d e-c f)^2 \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^3}+\frac{i e^{-i a} (d e-c f)^2 \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^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{(c+d x)^{3/2}}\right )}{3 d^3}+\frac{\left (b^2 f^2 \cos (a)\right ) \operatorname{Subst}\left (\int \frac{\sin (b x)}{x} \, dx,x,\frac{1}{(c+d x)^{3/2}}\right )}{3 d^3}+\frac{\left (b^2 f^2 \sin (a)\right ) \operatorname{Subst}\left (\int \frac{\cos (b x)}{x} \, dx,x,\frac{1}{(c+d x)^{3/2}}\right )}{3 d^3}\\ &=\frac{b f^2 (c+d x)^{3/2} \cos \left (a+\frac{b}{(c+d x)^{3/2}}\right )}{3 d^3}-\frac{2 i e^{i a} f (d e-c 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^3}+\frac{2 i e^{-i a} f (d e-c 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^3}-\frac{i e^{i a} (d e-c f)^2 \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^3}+\frac{i e^{-i a} (d e-c f)^2 \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^3}+\frac{b^2 f^2 \text{Ci}\left (\frac{b}{(c+d x)^{3/2}}\right ) \sin (a)}{3 d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{(c+d x)^{3/2}}\right )}{3 d^3}+\frac{b^2 f^2 \cos (a) \text{Si}\left (\frac{b}{(c+d x)^{3/2}}\right )}{3 d^3}\\ \end{align*}

Mathematica [A]  time = 2.25228, size = 463, normalized size = 1.19 \[ \frac{i \left ((\cos (a)-i \sin (a)) \left (4 f (c+d x)^2 \left (\frac{i b}{(c+d x)^{3/2}}\right )^{4/3} (d e-c f) \text{Gamma}\left (-\frac{4}{3},\frac{i b}{(c+d x)^{3/2}}\right )+2 (c+d x) \left (\frac{i b}{(c+d x)^{3/2}}\right )^{2/3} (d e-c f)^2 \text{Gamma}\left (-\frac{2}{3},\frac{i b}{(c+d x)^{3/2}}\right )-i b f^2 \left (i b \text{Ei}\left (-\frac{i b}{(c+d x)^{3/2}}\right )+(c+d x)^{3/2} \left (\cos \left (\frac{b}{(c+d x)^{3/2}}\right )-i \sin \left (\frac{b}{(c+d x)^{3/2}}\right )\right )\right )+f^2 (c+d x)^3 \left (\cos \left (\frac{b}{(c+d x)^{3/2}}\right )-i \sin \left (\frac{b}{(c+d x)^{3/2}}\right )\right )\right )-(\cos (a)+i \sin (a)) \left (4 f (c+d x)^2 \left (-\frac{i b}{(c+d x)^{3/2}}\right )^{4/3} (d e-c f) \text{Gamma}\left (-\frac{4}{3},-\frac{i b}{(c+d x)^{3/2}}\right )+2 (c+d x) \left (-\frac{i b}{(c+d x)^{3/2}}\right )^{2/3} (d e-c f)^2 \text{Gamma}\left (-\frac{2}{3},-\frac{i b}{(c+d x)^{3/2}}\right )+b^2 f^2 \text{Ei}\left (\frac{i b}{(c+d x)^{3/2}}\right )+f^2 (c+d x)^3 \left (\cos \left (\frac{b}{(c+d x)^{3/2}}\right )+i \sin \left (\frac{b}{(c+d x)^{3/2}}\right )\right )+i b f^2 (c+d x)^{3/2} \left (\cos \left (\frac{b}{(c+d x)^{3/2}}\right )+i \sin \left (\frac{b}{(c+d x)^{3/2}}\right )\right )\right )\right )}{6 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \begin{align*} \int \left ( fx+e \right ) ^{2}\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)^2*sin(a+b/(d*x+c)^(3/2)),x)

[Out]

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

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Maxima [B]  time = 2.92936, size = 2916, normalized size = 7.48 \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)^2*sin(a+b/(d*x+c)^(3/2)),x, algorithm="maxima")

[Out]

1/12*(3*(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)) + (((ga
mma(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^2/(sqrt(d*x + c
)*(abs(b)/(d*x + c)^(3/2))^(1/3)) - 6*(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*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)))*sin(1/6*pi + 1/3*arcta
n2(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*a
rctan2(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)*c*e*f/(sqrt(d*x + c)*d*(abs(b)/(d*x + c)^(3/2))^(1/3)) + 3*(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*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)*c^2*f^2/(sqrt(d*x + c)*d^2*(abs(b)/(d*x + c)^(3/2))^(1/3)) + 2*
(2*(d*x + c)^3*sin(((d*x + c)^(3/2)*a + b)/(d*x + c)^(3/2)) + 2*(d*x + c)^(3/2)*b*cos(((d*x + c)^(3/2)*a + b)/
(d*x + c)^(3/2)) + ((-I*Ei(I*b/(d*x + c)^(3/2)) + I*Ei(-I*b/(d*x + c)^(3/2)))*cos(a) + (Ei(I*b/(d*x + c)^(3/2)
) + Ei(-I*b/(d*x + c)^(3/2)))*sin(a))*b^2)*f^2/d^2 + 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)))*c
os(-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*p
i + 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*p
i + 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)*e*f/((d*x + c)*d*(abs(b)/(d*x + c)^(3/2))^(2/3)) - 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*gam
ma(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*g
amma(2/3, -I*b/(d*x + c)^(3/2)))*sin(-1/3*pi + 2/3*arctan2(0, b)))*sin(a))*b^2)*c*f^2/((d*x + c)*d^2*(abs(b)/(
d*x + c)^(3/2))^(2/3)))/d

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Fricas [A]  time = 2.79872, size = 1211, normalized size = 3.11 \begin{align*} \frac{-i \, b^{2} f^{2}{\rm Ei}\left (\frac{i \, \sqrt{d x + c} b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) e^{\left (i \, a\right )} + i \, b^{2} f^{2}{\rm Ei}\left (-\frac{i \, \sqrt{d x + c} b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) e^{\left (-i \, a\right )} +{\left (-3 i \, d^{2} e^{2} + 6 i \, c d e f - 3 i \, c^{2} f^{2}\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 (3 i \, d^{2} e^{2} - 6 i \, c d e f + 3 i \, c^{2} f^{2}\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 ) - 9 \,{\left (b d e f - b c f^{2}\right )} \left (i \, b\right )^{\frac{1}{3}} 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 ) - 9 \,{\left (b d e f - b c f^{2}\right )} \left (-i \, b\right )^{\frac{1}{3}} 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 ) + 2 \,{\left (b d f^{2} x + 9 \, b d e f - 8 \, b c f^{2}\right )} \sqrt{d x + c} \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^{3} f^{2} x^{3} + 3 \, d^{3} e f x^{2} + 3 \, d^{3} e^{2} x + 3 \, c d^{2} e^{2} - 3 \, c^{2} d e f + c^{3} f^{2}\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 )}{6 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(-I*b^2*f^2*Ei(I*sqrt(d*x + c)*b/(d^2*x^2 + 2*c*d*x + c^2))*e^(I*a) + I*b^2*f^2*Ei(-I*sqrt(d*x + c)*b/(d^2
*x^2 + 2*c*d*x + c^2))*e^(-I*a) + (-3*I*d^2*e^2 + 6*I*c*d*e*f - 3*I*c^2*f^2)*(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)) + (3*I*d^2*e^2 - 6*I*c*d*e*f + 3*I*c^2*f^2)*(-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)) - 9*(b*d*e*f - b*c*f^2)*(I*b)^(1/3)*e^(-I*a)*gamma(2/3
, I*sqrt(d*x + c)*b/(d^2*x^2 + 2*c*d*x + c^2)) - 9*(b*d*e*f - b*c*f^2)*(-I*b)^(1/3)*e^(I*a)*gamma(2/3, -I*sqrt
(d*x + c)*b/(d^2*x^2 + 2*c*d*x + c^2)) + 2*(b*d*f^2*x + 9*b*d*e*f - 8*b*c*f^2)*sqrt(d*x + c)*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^3*f^2*x^3 + 3*d^3*e*f*x^2 + 3*d^3*e^2*x
+ 3*c*d^2*e^2 - 3*c^2*d*e*f + c^3*f^2)*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^3

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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)**2*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 )}^{2} \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)^2*sin(a+b/(d*x+c)^(3/2)),x, algorithm="giac")

[Out]

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