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

Optimal result

Integrand size = 22, antiderivative size = 390 \[ \int (e+f x)^2 \sin \left (a+\frac {b}{(c+d x)^{3/2}}\right ) \, dx=\frac {b f^2 (c+d x)^{3/2} \cos \left (a+\frac {b}{(c+d x)^{3/2}}\right )}{3 d^3}-\frac {2 i e^{i a} f (d e-c f) \left (-\frac {i b}{(c+d x)^{3/2}}\right )^{4/3} (c+d x)^2 \Gamma \left (-\frac {4}{3},-\frac {i b}{(c+d x)^{3/2}}\right )}{3 d^3}+\frac {2 i e^{-i a} f (d e-c f) \left (\frac {i b}{(c+d x)^{3/2}}\right )^{4/3} (c+d x)^2 \Gamma \left (-\frac {4}{3},\frac {i b}{(c+d x)^{3/2}}\right )}{3 d^3}-\frac {i e^{i a} (d e-c f)^2 \left (-\frac {i b}{(c+d x)^{3/2}}\right )^{2/3} (c+d x) \Gamma \left (-\frac {2}{3},-\frac {i b}{(c+d x)^{3/2}}\right )}{3 d^3}+\frac {i e^{-i a} (d e-c f)^2 \left (\frac {i b}{(c+d x)^{3/2}}\right )^{2/3} (c+d x) \Gamma \left (-\frac {2}{3},\frac {i b}{(c+d x)^{3/2}}\right )}{3 d^3}+\frac {b^2 f^2 \operatorname {CosIntegral}\left (\frac {b}{(c+d x)^{3/2}}\right ) \sin (a)}{3 d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^{3/2}}\right )}{3 d^3}+\frac {b^2 f^2 \cos (a) \text {Si}\left (\frac {b}{(c+d x)^{3/2}}\right )}{3 d^3} \] Output:

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*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+1/3*b^2*f^2*cos(a)*Si(b/(d*x+c)^(3/2))/d^3
 

Mathematica [A] (verified)

Time = 2.07 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.19 \[ \int (e+f x)^2 \sin \left (a+\frac {b}{(c+d x)^{3/2}}\right ) \, dx=\frac {i \left ((\cos (a)-i \sin (a)) \left (b^2 f^2 \operatorname {ExpIntegralEi}\left (-\frac {i b}{(c+d x)^{3/2}}\right )+4 f (d e-c f) \left (\frac {i b}{(c+d x)^{3/2}}\right )^{4/3} (c+d x)^2 \Gamma \left (-\frac {4}{3},\frac {i b}{(c+d x)^{3/2}}\right )+2 (d e-c f)^2 \left (\frac {i b}{(c+d x)^{3/2}}\right )^{2/3} (c+d x) \Gamma \left (-\frac {2}{3},\frac {i b}{(c+d x)^{3/2}}\right )-i b f^2 (c+d x)^{3/2} \left (\cos \left (\frac {b}{(c+d x)^{3/2}}\right )-i \sin \left (\frac {b}{(c+d x)^{3/2}}\right )\right )+f^2 (c+d x)^3 \left (\cos \left (\frac {b}{(c+d x)^{3/2}}\right )-i \sin \left (\frac {b}{(c+d x)^{3/2}}\right )\right )\right )-(\cos (a)+i \sin (a)) \left (b^2 f^2 \operatorname {ExpIntegralEi}\left (\frac {i b}{(c+d x)^{3/2}}\right )+4 f (d e-c f) \left (-\frac {i b}{(c+d x)^{3/2}}\right )^{4/3} (c+d x)^2 \Gamma \left (-\frac {4}{3},-\frac {i b}{(c+d x)^{3/2}}\right )+2 (d e-c f)^2 \left (-\frac {i b}{(c+d x)^{3/2}}\right )^{2/3} (c+d x) \Gamma \left (-\frac {2}{3},-\frac {i b}{(c+d x)^{3/2}}\right )+i b f^2 (c+d x)^{3/2} \left (\cos \left (\frac {b}{(c+d x)^{3/2}}\right )+i \sin \left (\frac {b}{(c+d x)^{3/2}}\right )\right )+f^2 (c+d x)^3 \left (\cos \left (\frac {b}{(c+d x)^{3/2}}\right )+i \sin \left (\frac {b}{(c+d x)^{3/2}}\right )\right )\right )\right )}{6 d^3} \] Input:

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

Output:

((I/6)*((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)])) - (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
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 371, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3914, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3914

\(\displaystyle \frac {2 \int \left (f^2 \sin \left (a+\frac {b}{(c+d x)^{3/2}}\right ) (c+d x)^{5/2}+2 f (d e-c f) \sin \left (a+\frac {b}{(c+d x)^{3/2}}\right ) (c+d x)^{3/2}+(d e-c f)^2 \sin \left (a+\frac {b}{(c+d x)^{3/2}}\right ) \sqrt {c+d x}\right )d\sqrt {c+d x}}{d^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \left (\frac {1}{6} b^2 f^2 \sin (a) \operatorname {CosIntegral}\left (\frac {b}{(c+d x)^{3/2}}\right )+\frac {1}{6} b^2 f^2 \cos (a) \text {Si}\left (\frac {b}{(c+d x)^{3/2}}\right )-\frac {1}{3} i e^{i a} f (c+d x)^2 \left (-\frac {i b}{(c+d x)^{3/2}}\right )^{4/3} (d e-c f) \Gamma \left (-\frac {4}{3},-\frac {i b}{(c+d x)^{3/2}}\right )+\frac {1}{3} i e^{-i a} f (c+d x)^2 \left (\frac {i b}{(c+d x)^{3/2}}\right )^{4/3} (d e-c f) \Gamma \left (-\frac {4}{3},\frac {i b}{(c+d x)^{3/2}}\right )-\frac {1}{6} i e^{i a} (c+d x) \left (-\frac {i b}{(c+d x)^{3/2}}\right )^{2/3} (d e-c f)^2 \Gamma \left (-\frac {2}{3},-\frac {i b}{(c+d x)^{3/2}}\right )+\frac {1}{6} i e^{-i a} (c+d x) \left (\frac {i b}{(c+d x)^{3/2}}\right )^{2/3} (d e-c f)^2 \Gamma \left (-\frac {2}{3},\frac {i b}{(c+d x)^{3/2}}\right )+\frac {1}{6} f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^{3/2}}\right )+\frac {1}{6} b f^2 (c+d x)^{3/2} \cos \left (a+\frac {b}{(c+d x)^{3/2}}\right )\right )}{d^3}\)

Input:

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

Output:

(2*((b*f^2*(c + d*x)^(3/2)*Cos[a + b/(c + d*x)^(3/2)])/6 - (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)] + ((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)])/E^(I*a) - (I/6)*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)] + ((I/6)*(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)])/E^(I*a) + (b^2*f^2*CosIntegra 
l[b/(c + d*x)^(3/2)]*Sin[a])/6 + (f^2*(c + d*x)^3*Sin[a + b/(c + d*x)^(3/2 
)])/6 + (b^2*f^2*Cos[a]*SinIntegral[b/(c + d*x)^(3/2)])/6))/d^3
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3914

Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f 
_.)*(x_))^(n_)])^(p_.), x_Symbol] :> Module[{k = If[FractionQ[n], Denominat 
or[n], 1]}, Simp[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]
 

Maple [F]

\[\int \left (f x +e \right )^{2} \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {3}{2}}}\right )d x\]

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 584, normalized size of antiderivative = 1.50 \[ \int (e+f x)^2 \sin \left (a+\frac {b}{(c+d x)^{3/2}}\right ) \, dx=\frac {2 \, b^{2} f^{2} \operatorname {Ci}\left (\frac {\sqrt {d x + c} b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) \sin \left (a\right ) + 2 \, b^{2} f^{2} \cos \left (a\right ) \operatorname {Si}\left (\frac {\sqrt {d x + c} b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - 3 \, {\left ({\left (i \, d^{2} e^{2} - 2 i \, c d e f + i \, c^{2} f^{2}\right )} \cos \left (a\right ) + {\left (d^{2} e^{2} - 2 \, c d e f + c^{2} f^{2}\right )} \sin \left (a\right )\right )} \left (i \, b\right )^{\frac {2}{3}} \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 ({\left (-i \, d^{2} e^{2} + 2 i \, c d e f - i \, c^{2} f^{2}\right )} \cos \left (a\right ) + {\left (d^{2} e^{2} - 2 \, c d e f + c^{2} f^{2}\right )} \sin \left (a\right )\right )} \left (-i \, b\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{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 ) - 9 \, {\left ({\left (b d e f - b c f^{2}\right )} \cos \left (a\right ) + {\left (-i \, b d e f + i \, b c f^{2}\right )} \sin \left (a\right )\right )} \left (i \, b\right )^{\frac {1}{3}} \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 ({\left (b d e f - b c f^{2}\right )} \cos \left (a\right ) + {\left (i \, b d e f - i \, b c f^{2}\right )} \sin \left (a\right )\right )} \left (-i \, b\right )^{\frac {1}{3}} \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 (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}} \] Input:

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

Output:

1/6*(2*b^2*f^2*cos_integral(sqrt(d*x + c)*b/(d^2*x^2 + 2*c*d*x + c^2))*sin 
(a) + 2*b^2*f^2*cos(a)*sin_integral(sqrt(d*x + c)*b/(d^2*x^2 + 2*c*d*x + c 
^2)) - 3*((I*d^2*e^2 - 2*I*c*d*e*f + I*c^2*f^2)*cos(a) + (d^2*e^2 - 2*c*d* 
e*f + c^2*f^2)*sin(a))*(I*b)^(2/3)*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 + 2*I*c*d*e*f - I*c^2*f^2)*cos(a) + (d^2 
*e^2 - 2*c*d*e*f + c^2*f^2)*sin(a))*(-I*b)^(2/3)*gamma(1/3, -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)*sq 
rt(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)) - 9*((b*d*e*f - b*c*f^2)*cos(a) + (-I*b*d*e*f + I*b*c*f 
^2)*sin(a))*(I*b)^(1/3)*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)*cos(a) + (I*b*d*e*f - I*b*c*f^2)*sin(a))*(- 
I*b)^(1/3)*gamma(2/3, -I*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
 

Sympy [F]

\[ \int (e+f x)^2 \sin \left (a+\frac {b}{(c+d x)^{3/2}}\right ) \, dx=\int \left (e + f x\right )^{2} \sin {\left (a + \frac {b}{c \sqrt {c + d x} + d x \sqrt {c + d x}} \right )}\, dx \] Input:

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

Output:

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

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 993 vs. \(2 (296) = 592\).

Time = 0.58 (sec) , antiderivative size = 993, normalized size of antiderivative = 2.55 \[ \int (e+f x)^2 \sin \left (a+\frac {b}{(c+d x)^{3/2}}\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

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)*e^2/(sqrt(d*x + c)*(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*e*f/(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)*c^2*f^2/(sqrt(d*x + c)*d^2*(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 ...
 

Giac [F]

\[ \int (e+f x)^2 \sin \left (a+\frac {b}{(c+d x)^{3/2}}\right ) \, dx=\int { {\left (f x + e\right )}^{2} \sin \left (a + \frac {b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right ) \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int (e+f x)^2 \sin \left (a+\frac {b}{(c+d x)^{3/2}}\right ) \, dx=\int \sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{3/2}}\right )\,{\left (e+f\,x\right )}^2 \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int (e+f x)^2 \sin \left (a+\frac {b}{(c+d x)^{3/2}}\right ) \, dx=\left (\int \sin \left (\frac {\sqrt {d x +c}\, a c +\sqrt {d x +c}\, a d x +b}{\sqrt {d x +c}\, c +\sqrt {d x +c}\, d x}\right )d x \right ) e^{2}+\left (\int \sin \left (\frac {\sqrt {d x +c}\, a c +\sqrt {d x +c}\, a d x +b}{\sqrt {d x +c}\, c +\sqrt {d x +c}\, d x}\right ) x^{2}d x \right ) f^{2}+2 \left (\int \sin \left (\frac {\sqrt {d x +c}\, a c +\sqrt {d x +c}\, a d x +b}{\sqrt {d x +c}\, c +\sqrt {d x +c}\, d x}\right ) x d x \right ) e f \] Input:

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

Output:

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