3.3.2 \(\int (e+f x)^2 \sin (a+\frac {b}{(c+d x)^{3/2}}) \, dx\) [202]

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

[Out]

1/3*b*f^2*(d*x+c)^(3/2)*cos(a+b/(d*x+c)^(3/2))/d^3-2/3*I*exp(I*a)*f*(-c*f+d*e)*(-I*b/(d*x+c)^(3/2))^(4/3)*(d*x
+c)^2*GAMMA(-4/3,-I*b/(d*x+c)^(3/2))/d^3+2/3*I*f*(-c*f+d*e)*(I*b/(d*x+c)^(3/2))^(4/3)*(d*x+c)^2*GAMMA(-4/3,I*b
/(d*x+c)^(3/2))/d^3/exp(I*a)-1/3*I*exp(I*a)*(-c*f+d*e)^2*(-I*b/(d*x+c)^(3/2))^(2/3)*(d*x+c)*GAMMA(-2/3,-I*b/(d
*x+c)^(3/2))/d^3+1/3*I*(-c*f+d*e)^2*(I*b/(d*x+c)^(3/2))^(2/3)*(d*x+c)*GAMMA(-2/3,I*b/(d*x+c)^(3/2))/d^3/exp(I*
a)+1/3*b^2*f^2*cos(a)*Si(b/(d*x+c)^(3/2))/d^3+1/3*b^2*f^2*Ci(b/(d*x+c)^(3/2))*sin(a)/d^3+1/3*f^2*(d*x+c)^3*sin
(a+b/(d*x+c)^(3/2))/d^3

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Rubi [A]
time = 0.29, antiderivative size = 390, normalized size of antiderivative = 1.00, 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 = {3514, 3504, 2250, 3460, 3378, 3384, 3380, 3383} \begin {gather*} -\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} \end {gather*}

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 2250

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

Rule 3378

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 3380

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 3383

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]

Rule 3384

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 3460

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 3504

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 3514

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]

Rubi steps

\begin {align*} \int (e+f x)^2 \sin \left (a+\frac {b}{(c+d x)^{3/2}}\right ) \, dx &=\frac {2 \text {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 ) \text {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)) \text {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 ) \text {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 ) \text {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)) \text {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)) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 3.10, size = 518, normalized size = 1.33 \begin {gather*} \frac {e^{-i a} \left (b e^{-\frac {i b}{(c+d x)^{3/2}}} f \sqrt {c+d x} (9 d e-8 c f+d f x)+i e^{-\frac {i b}{(c+d x)^{3/2}}} (c+d x) \left (c^2 f^2-c d f (3 e+f x)+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right )+i b^2 f^2 \text {Ei}\left (-\frac {i b}{(c+d x)^{3/2}}\right )-3 i (d e-c f)^2 \left (\frac {i b}{(c+d x)^{3/2}}\right )^{2/3} (c+d x) \Gamma \left (\frac {1}{3},\frac {i b}{(c+d x)^{3/2}}\right )+9 b f (-d e+c f) \sqrt [3]{\frac {i b}{(c+d x)^{3/2}}} \sqrt {c+d x} \Gamma \left (\frac {2}{3},\frac {i b}{(c+d x)^{3/2}}\right )\right )-i (\cos (a)+i \sin (a)) \left (b^2 f^2 \text {Ei}\left (\frac {i b}{(c+d x)^{3/2}}\right )+\sqrt {c+d x} \left (-3 (d e-c f)^2 \left (-\frac {i b}{(c+d x)^{3/2}}\right )^{2/3} \sqrt {c+d x} \Gamma \left (\frac {1}{3},-\frac {i b}{(c+d x)^{3/2}}\right )+9 i b f (-d e+c f) \sqrt [3]{-\frac {i b}{(c+d x)^{3/2}}} \Gamma \left (\frac {2}{3},-\frac {i b}{(c+d x)^{3/2}}\right )+\left (i b f (9 d e-8 c f+d f x)+\sqrt {c+d x} \left (c^2 f^2-c d f (3 e+f x)+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right )\right ) \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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(((b*f*Sqrt[c + d*x]*(9*d*e - 8*c*f + d*f*x))/E^((I*b)/(c + d*x)^(3/2)) + (I*(c + d*x)*(c^2*f^2 - c*d*f*(3*e +
f*x) + d^2*(3*e^2 + 3*e*f*x + f^2*x^2)))/E^((I*b)/(c + d*x)^(3/2)) + I*b^2*f^2*ExpIntegralEi[((-I)*b)/(c + d*
x)^(3/2)] - (3*I)*(d*e - c*f)^2*((I*b)/(c + d*x)^(3/2))^(2/3)*(c + d*x)*Gamma[1/3, (I*b)/(c + d*x)^(3/2)] + 9*
b*f*(-(d*e) + c*f)*((I*b)/(c + d*x)^(3/2))^(1/3)*Sqrt[c + d*x]*Gamma[2/3, (I*b)/(c + d*x)^(3/2)])/E^(I*a) - I*
(Cos[a] + I*Sin[a])*(b^2*f^2*ExpIntegralEi[(I*b)/(c + d*x)^(3/2)] + Sqrt[c + d*x]*(-3*(d*e - c*f)^2*(((-I)*b)/
(c + d*x)^(3/2))^(2/3)*Sqrt[c + d*x]*Gamma[1/3, ((-I)*b)/(c + d*x)^(3/2)] + (9*I)*b*f*(-(d*e) + c*f)*(((-I)*b)
/(c + d*x)^(3/2))^(1/3)*Gamma[2/3, ((-I)*b)/(c + d*x)^(3/2)] + (I*b*f*(9*d*e - 8*c*f + d*f*x) + Sqrt[c + d*x]*
(c^2*f^2 - c*d*f*(3*e + f*x) + d^2*(3*e^2 + 3*e*f*x + f^2*x^2)))*(Cos[b/(c + d*x)^(3/2)] + I*Sin[b/(c + d*x)^(
3/2)]))))/(6*d^3)

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \left (f x +e \right )^{2} \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {3}{2}}}\right )\, dx\]

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] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 994 vs. \(2 (300) = 600\).
time = 0.78, size = 994, normalized size = 2.55 \begin {gather*} \text {Too large to display} \end {gather*}

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)*(b/(d*x + c)^(3/2))^(1/3)*sin(((d*x + c)^(3/2)*a + b)/(d*x + c)^(3/2)) + (((sqrt(3)
- I)*gamma(1/3, I*b/(d*x + c)^(3/2)) + (sqrt(3) + I)*gamma(1/3, -I*b/(d*x + c)^(3/2)))*cos(a) + ((-I*sqrt(3)
- 1)*gamma(1/3, I*b/(d*x + c)^(3/2)) + (I*sqrt(3) - 1)*gamma(1/3, -I*b/(d*x + c)^(3/2)))*sin(a))*b)*c^2*f^2/(s
qrt(d*x + c)*d^2*(b/(d*x + c)^(3/2))^(1/3)) - 6*(4*(d*x + c)^(3/2)*(b/(d*x + c)^(3/2))^(1/3)*sin(((d*x + c)^(3
/2)*a + b)/(d*x + c)^(3/2)) + (((sqrt(3) - I)*gamma(1/3, I*b/(d*x + c)^(3/2)) + (sqrt(3) + I)*gamma(1/3, -I*b/
(d*x + c)^(3/2)))*cos(a) + ((-I*sqrt(3) - 1)*gamma(1/3, I*b/(d*x + c)^(3/2)) + (I*sqrt(3) - 1)*gamma(1/3, -I*b
/(d*x + c)^(3/2)))*sin(a))*b)*c*f*e/(sqrt(d*x + c)*d*(b/(d*x + c)^(3/2))^(1/3)) + 3*(4*(d*x + c)^(3/2)*(b/(d*x
+ c)^(3/2))^(1/3)*sin(((d*x + c)^(3/2)*a + b)/(d*x + c)^(3/2)) + (((sqrt(3) - I)*gamma(1/3, I*b/(d*x + c)^(3/
2)) + (sqrt(3) + I)*gamma(1/3, -I*b/(d*x + c)^(3/2)))*cos(a) + ((-I*sqrt(3) - 1)*gamma(1/3, I*b/(d*x + c)^(3/2
)) + (I*sqrt(3) - 1)*gamma(1/3, -I*b/(d*x + c)^(3/2)))*sin(a))*b)*e^2/(sqrt(d*x + c)*(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*(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*(b/(d*x + c)^(3/2))^(2/3)*cos(((d*x + c)^(3/2)*a +
b)/(d*x + c)^(3/2)) - 3*(((sqrt(3) + I)*gamma(2/3, I*b/(d*x + c)^(3/2)) + (sqrt(3) - I)*gamma(2/3, -I*b/(d*x
+ c)^(3/2)))*cos(a) + ((-I*sqrt(3) + 1)*gamma(2/3, I*b/(d*x + c)^(3/2)) + (I*sqrt(3) + 1)*gamma(2/3, -I*b/(d*x
+ c)^(3/2)))*sin(a))*b^2)*c*f^2/((d*x + c)*d^2*(b/(d*x + c)^(3/2))^(2/3)) + 3*(4*(d*x + c)^3*(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*(b/(d*x + c)^(3/2))^(2/3)*cos(((
d*x + c)^(3/2)*a + b)/(d*x + c)^(3/2)) - 3*(((sqrt(3) + I)*gamma(2/3, I*b/(d*x + c)^(3/2)) + (sqrt(3) - I)*gam
ma(2/3, -I*b/(d*x + c)^(3/2)))*cos(a) + ((-I*sqrt(3) + 1)*gamma(2/3, I*b/(d*x + c)^(3/2)) + (I*sqrt(3) + 1)*ga
mma(2/3, -I*b/(d*x + c)^(3/2)))*sin(a))*b^2)*f*e/((d*x + c)*d*(b/(d*x + c)^(3/2))^(2/3)))/d

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Fricas [A]
time = 0.17, size = 510, normalized size = 1.31 \begin {gather*} \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 )} - 3 \, {\left (i \, c^{2} f^{2} - 2 i \, c d f e + i \, d^{2} e^{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 ) - 3 \, {\left (-i \, c^{2} f^{2} + 2 i \, c d f e - i \, d^{2} e^{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 c f^{2} - b d f e\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 c f^{2} - b d f e\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 - 8 \, b c f^{2} + 9 \, b d f e\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} + c^{3} f^{2} + 3 \, {\left (d^{3} x + c d^{2}\right )} e^{2} + 3 \, {\left (d^{3} f x^{2} - c^{2} d f\right )} e\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 {gather*}

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*c^2*f^2 - 2*I*c*d*f*e + I*d^2*e^2)*(I*b)^(2/3)*e^(-I*a)*gamma(1/3, I*sq
rt(d*x + c)*b/(d^2*x^2 + 2*c*d*x + c^2)) - 3*(-I*c^2*f^2 + 2*I*c*d*f*e - I*d^2*e^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)) + 9*(b*c*f^2 - b*d*f*e)*(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*c*f^2 - b*d*f*e)*(-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 - 8*b*c*f^2 + 9*b*d*f*e)*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 + c^3*f^2 + 3*(d^3*x + c*d^2)*e^2
+ 3*(d^3*f*x^2 - c^2*d*f)*e)*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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (e + f x\right )^{2} \sin {\left (a + \frac {b}{c \sqrt {c + d x} + d x \sqrt {c + d x}} \right )}\, dx \end {gather*}

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]

Integral((e + f*x)**2*sin(a + b/(c*sqrt(c + d*x) + d*x*sqrt(c + d*x))), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{3/2}}\right )\,{\left (e+f\,x\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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