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

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

[Out]

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

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Rubi [A]  time = 0.23, antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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 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 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 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]

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*}

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Mathematica [B]  time = 2.77, size = 835, normalized size = 3.33 \[ \frac {9 i f \cos (a) \left (\frac {2 \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 \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 \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 \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 \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 \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 \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 \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 \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 \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 \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 \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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fricas [A]  time = 0.72, size = 330, normalized size = 1.31 \[ -\frac {3 \, \left (i \, b\right )^{\frac {1}{3}} b f e^{\left (-i \, a\right )} \Gamma \left (\frac {2}{3}, \frac {i \, \sqrt {d x + c} b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + 3 \, \left (-i \, b\right )^{\frac {1}{3}} b f e^{\left (i \, a\right )} \Gamma \left (\frac {2}{3}, -\frac {i \, \sqrt {d x + c} b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - {\left (-2 i \, d e + 2 i \, c f\right )} \left (i \, b\right )^{\frac {2}{3}} e^{\left (-i \, a\right )} \Gamma \left (\frac {1}{3}, \frac {i \, \sqrt {d x + c} b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - {\left (2 i \, d e - 2 i \, c f\right )} \left (-i \, b\right )^{\frac {2}{3}} e^{\left (i \, a\right )} \Gamma \left (\frac {1}{3}, -\frac {i \, \sqrt {d x + c} b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - 6 \, \sqrt {d x + c} b f \cos \left (\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + \sqrt {d x + c} b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - 2 \, {\left (d^{2} f x^{2} + 2 \, d^{2} e x + 2 \, c d e - c^{2} f\right )} \sin \left (\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + \sqrt {d x + c} b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{4 \, d^{2}} \]

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

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)

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[ \int \left (f x +e \right ) \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {3}{2}}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)*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 = 2.00, size = 508, normalized size = 2.02 \[ \frac {\frac {2 \, {\left (4 \, {\left (d x + c\right )}^{\frac {3}{2}} \left (\frac {b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )^{\frac {1}{3}} \sin \left (\frac {{\left (d x + c\right )}^{\frac {3}{2}} a + b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right ) + {\left ({\left ({\left (\sqrt {3} - i\right )} \Gamma \left (\frac {1}{3}, \frac {i \, b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right ) + {\left (\sqrt {3} + i\right )} \Gamma \left (\frac {1}{3}, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )\right )} \cos \relax (a) + {\left ({\left (-i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {1}{3}, \frac {i \, b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right ) + {\left (i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {1}{3}, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )\right )} \sin \relax (a)\right )} b\right )} e}{\sqrt {d x + c} \left (\frac {b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )^{\frac {1}{3}}} - \frac {2 \, {\left (4 \, {\left (d x + c\right )}^{\frac {3}{2}} \left (\frac {b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )^{\frac {1}{3}} \sin \left (\frac {{\left (d x + c\right )}^{\frac {3}{2}} a + b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right ) + {\left ({\left ({\left (\sqrt {3} - i\right )} \Gamma \left (\frac {1}{3}, \frac {i \, b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right ) + {\left (\sqrt {3} + i\right )} \Gamma \left (\frac {1}{3}, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )\right )} \cos \relax (a) + {\left ({\left (-i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {1}{3}, \frac {i \, b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right ) + {\left (i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {1}{3}, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )\right )} \sin \relax (a)\right )} b\right )} c f}{\sqrt {d x + c} d \left (\frac {b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )^{\frac {1}{3}}} + \frac {{\left (4 \, {\left (d x + c\right )}^{3} \left (\frac {b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )^{\frac {2}{3}} \sin \left (\frac {{\left (d x + c\right )}^{\frac {3}{2}} a + b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right ) + 12 \, {\left (d x + c\right )}^{\frac {3}{2}} b \left (\frac {b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )^{\frac {2}{3}} \cos \left (\frac {{\left (d x + c\right )}^{\frac {3}{2}} a + b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right ) - {\left ({\left ({\left (3 \, \sqrt {3} + 3 i\right )} \Gamma \left (\frac {2}{3}, \frac {i \, b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right ) + {\left (3 \, \sqrt {3} - 3 i\right )} \Gamma \left (\frac {2}{3}, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )\right )} \cos \relax (a) + 3 \, {\left ({\left (-i \, \sqrt {3} + 1\right )} \Gamma \left (\frac {2}{3}, \frac {i \, b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right ) + {\left (i \, \sqrt {3} + 1\right )} \Gamma \left (\frac {2}{3}, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )\right )} \sin \relax (a)\right )} b^{2}\right )} f}{{\left (d x + c\right )} d \left (\frac {b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )^{\frac {2}{3}}}}{8 \, d} \]

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)*(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/(sqrt(d*x
+ c)*(b/(d*x + c)^(3/2))^(1/3)) - 2*(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/(sqrt(d*x + c)*d*(b/(d*x + c)^(3/2))^(1/3)) + (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) + 3*I)*gamma(2/3, I*b/(d*x + c)^(3/2)) + (3*sqrt(3) - 3*I)*gamma(2
/3, -I*b/(d*x + c)^(3/2)))*cos(a) + 3*((-I*sqrt(3) + 1)*gamma(2/3, I*b/(d*x + c)^(3/2)) + (I*sqrt(3) + 1)*gamm
a(2/3, -I*b/(d*x + c)^(3/2)))*sin(a))*b^2)*f/((d*x + c)*d*(b/(d*x + c)^(3/2))^(2/3)))/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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]

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

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