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

Optimal. Leaf size=291 \[ \frac {i e^{i a} (c+d x) (d e-c f) \Gamma \left (\frac {2}{3},-i b (c+d x)^{3/2}\right )}{3 d^2 \left (-i b (c+d x)^{3/2}\right )^{2/3}}-\frac {i e^{-i a} (c+d x) (d e-c f) \Gamma \left (\frac {2}{3},i b (c+d x)^{3/2}\right )}{3 d^2 \left (i b (c+d x)^{3/2}\right )^{2/3}}-\frac {2 f \sqrt {c+d x} \cos \left (a+b (c+d x)^{3/2}\right )}{3 b d^2}-\frac {e^{i a} f \sqrt {c+d x} \Gamma \left (\frac {1}{3},-i b (c+d x)^{3/2}\right )}{9 b d^2 \sqrt [3]{-i b (c+d x)^{3/2}}}-\frac {e^{-i a} f \sqrt {c+d x} \Gamma \left (\frac {1}{3},i b (c+d x)^{3/2}\right )}{9 b d^2 \sqrt [3]{i b (c+d x)^{3/2}}} \]

[Out]

1/3*I*exp(I*a)*(-c*f+d*e)*(d*x+c)*GAMMA(2/3,-I*b*(d*x+c)^(3/2))/d^2/(-I*b*(d*x+c)^(3/2))^(2/3)-1/3*I*(-c*f+d*e
)*(d*x+c)*GAMMA(2/3,I*b*(d*x+c)^(3/2))/d^2/exp(I*a)/(I*b*(d*x+c)^(3/2))^(2/3)-2/3*f*cos(a+b*(d*x+c)^(3/2))*(d*
x+c)^(1/2)/b/d^2-1/9*exp(I*a)*f*GAMMA(1/3,-I*b*(d*x+c)^(3/2))*(d*x+c)^(1/2)/b/d^2/(-I*b*(d*x+c)^(3/2))^(1/3)-1
/9*f*GAMMA(1/3,I*b*(d*x+c)^(3/2))*(d*x+c)^(1/2)/b/d^2/exp(I*a)/(I*b*(d*x+c)^(3/2))^(1/3)

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Rubi [A]  time = 0.20, antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3433, 3389, 2218, 3385, 3356, 2208} \[ \frac {i e^{i a} (c+d x) (d e-c f) \text {Gamma}\left (\frac {2}{3},-i b (c+d x)^{3/2}\right )}{3 d^2 \left (-i b (c+d x)^{3/2}\right )^{2/3}}-\frac {i e^{-i a} (c+d x) (d e-c f) \text {Gamma}\left (\frac {2}{3},i b (c+d x)^{3/2}\right )}{3 d^2 \left (i b (c+d x)^{3/2}\right )^{2/3}}-\frac {e^{i a} f \sqrt {c+d x} \text {Gamma}\left (\frac {1}{3},-i b (c+d x)^{3/2}\right )}{9 b d^2 \sqrt [3]{-i b (c+d x)^{3/2}}}-\frac {e^{-i a} f \sqrt {c+d x} \text {Gamma}\left (\frac {1}{3},i b (c+d x)^{3/2}\right )}{9 b d^2 \sqrt [3]{i b (c+d x)^{3/2}}}-\frac {2 f \sqrt {c+d x} \cos \left (a+b (c+d x)^{3/2}\right )}{3 b d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2208

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> -Simp[(F^a*(c + d*x)*Gamma[1/n, -(b*(c + d*x)
^n*Log[F])])/(d*n*(-(b*(c + d*x)^n*Log[F]))^(1/n)), x] /; FreeQ[{F, a, b, c, d, n}, x] &&  !IntegerQ[2/n]

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 3356

Int[Cos[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)], x_Symbol] :> Dist[1/2, Int[E^(-(c*I) - d*I*(e + f*x)^n), x],
x] + Dist[1/2, Int[E^(c*I + d*I*(e + f*x)^n), x], x] /; FreeQ[{c, d, e, f}, x] && IGtQ[n, 2]

Rule 3385

Int[((e_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> -Simp[(e^(n - 1)*(e*x)^(m - n + 1)*Cos[c + d
*x^n])/(d*n), x] + Dist[(e^n*(m - n + 1))/(d*n), Int[(e*x)^(m - n)*Cos[c + d*x^n], x], x] /; FreeQ[{c, d, e},
x] && IGtQ[n, 0] && LtQ[n, m + 1]

Rule 3389

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}, x] && IGtQ[n, 0]

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+b (c+d x)^{3/2}\right ) \, dx &=\frac {2 \operatorname {Subst}\left (\int \left ((d e-c f) x \sin \left (a+b x^3\right )+f x^3 \sin \left (a+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+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+b x^3\right ) \, dx,x,\sqrt {c+d x}\right )}{d^2}\\ &=-\frac {2 f \sqrt {c+d x} \cos \left (a+b (c+d x)^{3/2}\right )}{3 b d^2}+\frac {(2 f) \operatorname {Subst}\left (\int \cos \left (a+b x^3\right ) \, dx,x,\sqrt {c+d x}\right )}{3 b d^2}+\frac {(i (d e-c f)) \operatorname {Subst}\left (\int e^{-i a-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+i b x^3} x \, dx,x,\sqrt {c+d x}\right )}{d^2}\\ &=-\frac {2 f \sqrt {c+d x} \cos \left (a+b (c+d x)^{3/2}\right )}{3 b d^2}+\frac {i e^{i a} (d e-c f) (c+d x) \Gamma \left (\frac {2}{3},-i b (c+d x)^{3/2}\right )}{3 d^2 \left (-i b (c+d x)^{3/2}\right )^{2/3}}-\frac {i e^{-i a} (d e-c f) (c+d x) \Gamma \left (\frac {2}{3},i b (c+d x)^{3/2}\right )}{3 d^2 \left (i b (c+d x)^{3/2}\right )^{2/3}}+\frac {f \operatorname {Subst}\left (\int e^{-i a-i b x^3} \, dx,x,\sqrt {c+d x}\right )}{3 b d^2}+\frac {f \operatorname {Subst}\left (\int e^{i a+i b x^3} \, dx,x,\sqrt {c+d x}\right )}{3 b d^2}\\ &=-\frac {2 f \sqrt {c+d x} \cos \left (a+b (c+d x)^{3/2}\right )}{3 b d^2}-\frac {e^{i a} f \sqrt {c+d x} \Gamma \left (\frac {1}{3},-i b (c+d x)^{3/2}\right )}{9 b d^2 \sqrt [3]{-i b (c+d x)^{3/2}}}-\frac {e^{-i a} f \sqrt {c+d x} \Gamma \left (\frac {1}{3},i b (c+d x)^{3/2}\right )}{9 b d^2 \sqrt [3]{i b (c+d x)^{3/2}}}+\frac {i e^{i a} (d e-c f) (c+d x) \Gamma \left (\frac {2}{3},-i b (c+d x)^{3/2}\right )}{3 d^2 \left (-i b (c+d x)^{3/2}\right )^{2/3}}-\frac {i e^{-i a} (d e-c f) (c+d x) \Gamma \left (\frac {2}{3},i b (c+d x)^{3/2}\right )}{3 d^2 \left (i b (c+d x)^{3/2}\right )^{2/3}}\\ \end {align*}

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Mathematica [B]  time = 2.63, size = 705, normalized size = 2.42 \[ \frac {2 f \sin (a) \sqrt {c+d x} \sin \left (b (c+d x)^{3/2}\right )}{3 b d^2}-\frac {2 f \cos (a) \sqrt {c+d x} \cos \left (b (c+d x)^{3/2}\right )}{3 b d^2}+\frac {f \cos (a) \left (-\frac {2 \sqrt {c+d x} \Gamma \left (\frac {1}{3},-i b (c+d x)^{3/2}\right )}{3 \sqrt [3]{-i b (c+d x)^{3/2}}}-\frac {2 \sqrt {c+d x} \Gamma \left (\frac {1}{3},i b (c+d x)^{3/2}\right )}{3 \sqrt [3]{i b (c+d x)^{3/2}}}\right )}{6 b d^2}+\frac {i c f \cos (a) \left (\frac {2 (c+d x) \Gamma \left (\frac {2}{3},i b (c+d x)^{3/2}\right )}{3 \left (i b (c+d x)^{3/2}\right )^{2/3}}-\frac {2 (c+d x) \Gamma \left (\frac {2}{3},-i b (c+d x)^{3/2}\right )}{3 \left (-i b (c+d x)^{3/2}\right )^{2/3}}\right )}{2 d^2}+\frac {i f \sin (a) \left (\frac {2 \sqrt {c+d x} \Gamma \left (\frac {1}{3},i b (c+d x)^{3/2}\right )}{3 \sqrt [3]{i b (c+d x)^{3/2}}}-\frac {2 \sqrt {c+d x} \Gamma \left (\frac {1}{3},-i b (c+d x)^{3/2}\right )}{3 \sqrt [3]{-i b (c+d x)^{3/2}}}\right )}{6 b d^2}-\frac {c f \sin (a) \left (-\frac {2 (c+d x) \Gamma \left (\frac {2}{3},-i b (c+d x)^{3/2}\right )}{3 \left (-i b (c+d x)^{3/2}\right )^{2/3}}-\frac {2 (c+d x) \Gamma \left (\frac {2}{3},i b (c+d x)^{3/2}\right )}{3 \left (i b (c+d x)^{3/2}\right )^{2/3}}\right )}{2 d^2}-\frac {i e \cos (a) \left (\frac {2 (c+d x) \Gamma \left (\frac {2}{3},i b (c+d x)^{3/2}\right )}{3 \left (i b (c+d x)^{3/2}\right )^{2/3}}-\frac {2 (c+d x) \Gamma \left (\frac {2}{3},-i b (c+d x)^{3/2}\right )}{3 \left (-i b (c+d x)^{3/2}\right )^{2/3}}\right )}{2 d}+\frac {e \sin (a) \left (-\frac {2 (c+d x) \Gamma \left (\frac {2}{3},-i b (c+d x)^{3/2}\right )}{3 \left (-i b (c+d x)^{3/2}\right )^{2/3}}-\frac {2 (c+d x) \Gamma \left (\frac {2}{3},i b (c+d x)^{3/2}\right )}{3 \left (i b (c+d x)^{3/2}\right )^{2/3}}\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.90, size = 185, normalized size = 0.64 \[ \frac {i \, \left (i \, b\right )^{\frac {2}{3}} f e^{\left (-i \, a\right )} \Gamma \left (\frac {1}{3}, {\left (i \, b d x + i \, b c\right )} \sqrt {d x + c}\right ) - i \, \left (-i \, b\right )^{\frac {2}{3}} f e^{\left (i \, a\right )} \Gamma \left (\frac {1}{3}, {\left (-i \, b d x - i \, b c\right )} \sqrt {d x + c}\right ) - 6 \, \sqrt {d x + c} b f \cos \left ({\left (b d x + b c\right )} \sqrt {d x + c} + a\right ) - 3 \, {\left (b d e - b c f\right )} \left (i \, b\right )^{\frac {1}{3}} e^{\left (-i \, a\right )} \Gamma \left (\frac {2}{3}, {\left (i \, b d x + i \, b c\right )} \sqrt {d x + c}\right ) - 3 \, {\left (b d e - b c f\right )} \left (-i \, b\right )^{\frac {1}{3}} e^{\left (i \, a\right )} \Gamma \left (\frac {2}{3}, {\left (-i \, b d x - i \, b c\right )} \sqrt {d x + c}\right )}{9 \, b^{2} d^{2}} \]

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]

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

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

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((d*x + c)^(3/2)*b + a), x)

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

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 [A]  time = 0.98, size = 375, normalized size = 1.29 \[ -\frac {\frac {3 \, \left ({\left (d x + c\right )}^{\frac {3}{2}} b\right )^{\frac {1}{3}} {\left ({\left ({\left (\sqrt {3} + i\right )} \Gamma \left (\frac {2}{3}, i \, {\left (d x + c\right )}^{\frac {3}{2}} b\right ) + {\left (\sqrt {3} - i\right )} \Gamma \left (\frac {2}{3}, -i \, {\left (d x + c\right )}^{\frac {3}{2}} b\right )\right )} \cos \relax (a) - {\left ({\left (i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {2}{3}, i \, {\left (d x + c\right )}^{\frac {3}{2}} b\right ) + {\left (-i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {2}{3}, -i \, {\left (d x + c\right )}^{\frac {3}{2}} b\right )\right )} \sin \relax (a)\right )} e}{\sqrt {d x + c} b} - \frac {3 \, \left ({\left (d x + c\right )}^{\frac {3}{2}} b\right )^{\frac {1}{3}} {\left ({\left ({\left (\sqrt {3} + i\right )} \Gamma \left (\frac {2}{3}, i \, {\left (d x + c\right )}^{\frac {3}{2}} b\right ) + {\left (\sqrt {3} - i\right )} \Gamma \left (\frac {2}{3}, -i \, {\left (d x + c\right )}^{\frac {3}{2}} b\right )\right )} \cos \relax (a) - {\left ({\left (i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {2}{3}, i \, {\left (d x + c\right )}^{\frac {3}{2}} b\right ) + {\left (-i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {2}{3}, -i \, {\left (d x + c\right )}^{\frac {3}{2}} b\right )\right )} \sin \relax (a)\right )} c f}{\sqrt {d x + c} b d} + \frac {{\left (12 \, \left ({\left (d x + c\right )}^{\frac {3}{2}} b\right )^{\frac {1}{3}} \sqrt {d x + c} \cos \left ({\left (d x + c\right )}^{\frac {3}{2}} b + a\right ) + \sqrt {d x + c} {\left ({\left ({\left (\sqrt {3} - i\right )} \Gamma \left (\frac {1}{3}, i \, {\left (d x + c\right )}^{\frac {3}{2}} b\right ) + {\left (\sqrt {3} + i\right )} \Gamma \left (\frac {1}{3}, -i \, {\left (d x + c\right )}^{\frac {3}{2}} b\right )\right )} \cos \relax (a) + {\left ({\left (-i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {1}{3}, i \, {\left (d x + c\right )}^{\frac {3}{2}} b\right ) + {\left (i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {1}{3}, -i \, {\left (d x + c\right )}^{\frac {3}{2}} b\right )\right )} \sin \relax (a)\right )}\right )} f}{\left ({\left (d x + c\right )}^{\frac {3}{2}} b\right )^{\frac {1}{3}} b d}}{18 \, 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]

-1/18*(3*((d*x + c)^(3/2)*b)^(1/3)*(((sqrt(3) + I)*gamma(2/3, I*(d*x + c)^(3/2)*b) + (sqrt(3) - I)*gamma(2/3,
-I*(d*x + c)^(3/2)*b))*cos(a) - ((I*sqrt(3) - 1)*gamma(2/3, I*(d*x + c)^(3/2)*b) + (-I*sqrt(3) - 1)*gamma(2/3,
-I*(d*x + c)^(3/2)*b))*sin(a))*e/(sqrt(d*x + c)*b) - 3*((d*x + c)^(3/2)*b)^(1/3)*(((sqrt(3) + I)*gamma(2/3, I
*(d*x + c)^(3/2)*b) + (sqrt(3) - I)*gamma(2/3, -I*(d*x + c)^(3/2)*b))*cos(a) - ((I*sqrt(3) - 1)*gamma(2/3, I*(
d*x + c)^(3/2)*b) + (-I*sqrt(3) - 1)*gamma(2/3, -I*(d*x + c)^(3/2)*b))*sin(a))*c*f/(sqrt(d*x + c)*b*d) + (12*(
(d*x + c)^(3/2)*b)^(1/3)*sqrt(d*x + c)*cos((d*x + c)^(3/2)*b + a) + sqrt(d*x + c)*(((sqrt(3) - I)*gamma(1/3, I
*(d*x + c)^(3/2)*b) + (sqrt(3) + I)*gamma(1/3, -I*(d*x + c)^(3/2)*b))*cos(a) + ((-I*sqrt(3) - 1)*gamma(1/3, I*
(d*x + c)^(3/2)*b) + (I*sqrt(3) - 1)*gamma(1/3, -I*(d*x + c)^(3/2)*b))*sin(a)))*f/(((d*x + c)^(3/2)*b)^(1/3)*b
*d))/d

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

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 \[ \int \left (e + f x\right ) \sin {\left (a + b c \sqrt {c + d x} + b d x \sqrt {c + d x} \right )}\, dx \]

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) + b*d*x*sqrt(c + d*x)), x)

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