Integrand size = 16, antiderivative size = 193 \[ \int \frac {\sin (a+b x)}{(c+d x)^{7/2}} \, dx=-\frac {4 b \cos (a+b x)}{15 d^2 (c+d x)^{3/2}}-\frac {8 b^{5/2} \sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{15 d^{7/2}}+\frac {8 b^{5/2} \sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{15 d^{7/2}}-\frac {2 \sin (a+b x)}{5 d (c+d x)^{5/2}}+\frac {8 b^2 \sin (a+b x)}{15 d^3 \sqrt {c+d x}} \] Output:
-4/15*b*cos(b*x+a)/d^2/(d*x+c)^(3/2)-8/15*b^(5/2)*2^(1/2)*Pi^(1/2)*cos(a-b *c/d)*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))/d^(7/2)+8/1 5*b^(5/2)*2^(1/2)*Pi^(1/2)*FresnelS(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c)^(1/2) /d^(1/2))*sin(a-b*c/d)/d^(7/2)-2/5*sin(b*x+a)/d/(d*x+c)^(5/2)+8/15*b^2*sin (b*x+a)/d^3/(d*x+c)^(1/2)
Result contains complex when optimal does not.
Time = 0.57 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.08 \[ \int \frac {\sin (a+b x)}{(c+d x)^{7/2}} \, dx=-\frac {i \left (b (c+d x) \left (2 e^{i \left (a-\frac {b c}{d}\right )} \left (e^{\frac {i b (c+d x)}{d}} (-i d+2 b (c+d x))-2 i d \left (-\frac {i b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {i b (c+d x)}{d}\right )\right )-i e^{-i (a+b x)} \left (2 d-4 i b (c+d x)+4 d e^{\frac {i b (c+d x)}{d}} \left (\frac {i b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},\frac {i b (c+d x)}{d}\right )\right )\right )-6 i d^2 \sin (a+b x)\right )}{15 d^3 (c+d x)^{5/2}} \] Input:
Integrate[Sin[a + b*x]/(c + d*x)^(7/2),x]
Output:
((-1/15*I)*(b*(c + d*x)*(2*E^(I*(a - (b*c)/d))*(E^((I*b*(c + d*x))/d)*((-I )*d + 2*b*(c + d*x)) - (2*I)*d*(((-I)*b*(c + d*x))/d)^(3/2)*Gamma[1/2, ((- I)*b*(c + d*x))/d]) - (I*(2*d - (4*I)*b*(c + d*x) + 4*d*E^((I*b*(c + d*x)) /d)*((I*b*(c + d*x))/d)^(3/2)*Gamma[1/2, (I*b*(c + d*x))/d]))/E^(I*(a + b* x))) - (6*I)*d^2*Sin[a + b*x]))/(d^3*(c + d*x)^(5/2))
Time = 0.96 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.07, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {3042, 3778, 3042, 3778, 25, 3042, 3778, 3042, 3787, 3042, 3785, 3786, 3832, 3833}
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.
| \(\displaystyle \int \frac {\sin (a+b x)}{(c+d x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (a+b x)}{(c+d x)^{7/2}}dx\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle \frac {2 b \int \frac {\cos (a+b x)}{(c+d x)^{5/2}}dx}{5 d}-\frac {2 \sin (a+b x)}{5 d (c+d x)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b \int \frac {\sin \left (a+b x+\frac {\pi }{2}\right )}{(c+d x)^{5/2}}dx}{5 d}-\frac {2 \sin (a+b x)}{5 d (c+d x)^{5/2}}\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle \frac {2 b \left (\frac {2 b \int -\frac {\sin (a+b x)}{(c+d x)^{3/2}}dx}{3 d}-\frac {2 \cos (a+b x)}{3 d (c+d x)^{3/2}}\right )}{5 d}-\frac {2 \sin (a+b x)}{5 d (c+d x)^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 b \left (-\frac {2 b \int \frac {\sin (a+b x)}{(c+d x)^{3/2}}dx}{3 d}-\frac {2 \cos (a+b x)}{3 d (c+d x)^{3/2}}\right )}{5 d}-\frac {2 \sin (a+b x)}{5 d (c+d x)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b \left (-\frac {2 b \int \frac {\sin (a+b x)}{(c+d x)^{3/2}}dx}{3 d}-\frac {2 \cos (a+b x)}{3 d (c+d x)^{3/2}}\right )}{5 d}-\frac {2 \sin (a+b x)}{5 d (c+d x)^{5/2}}\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle \frac {2 b \left (-\frac {2 b \left (\frac {2 b \int \frac {\cos (a+b x)}{\sqrt {c+d x}}dx}{d}-\frac {2 \sin (a+b x)}{d \sqrt {c+d x}}\right )}{3 d}-\frac {2 \cos (a+b x)}{3 d (c+d x)^{3/2}}\right )}{5 d}-\frac {2 \sin (a+b x)}{5 d (c+d x)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b \left (-\frac {2 b \left (\frac {2 b \int \frac {\sin \left (a+b x+\frac {\pi }{2}\right )}{\sqrt {c+d x}}dx}{d}-\frac {2 \sin (a+b x)}{d \sqrt {c+d x}}\right )}{3 d}-\frac {2 \cos (a+b x)}{3 d (c+d x)^{3/2}}\right )}{5 d}-\frac {2 \sin (a+b x)}{5 d (c+d x)^{5/2}}\) |
| \(\Big \downarrow \) 3787 |
| \(\displaystyle \frac {2 b \left (-\frac {2 b \left (\frac {2 b \left (\cos \left (a-\frac {b c}{d}\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx-\sin \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx\right )}{d}-\frac {2 \sin (a+b x)}{d \sqrt {c+d x}}\right )}{3 d}-\frac {2 \cos (a+b x)}{3 d (c+d x)^{3/2}}\right )}{5 d}-\frac {2 \sin (a+b x)}{5 d (c+d x)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b \left (-\frac {2 b \left (\frac {2 b \left (\cos \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x+\frac {\pi }{2}\right )}{\sqrt {c+d x}}dx-\sin \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx\right )}{d}-\frac {2 \sin (a+b x)}{d \sqrt {c+d x}}\right )}{3 d}-\frac {2 \cos (a+b x)}{3 d (c+d x)^{3/2}}\right )}{5 d}-\frac {2 \sin (a+b x)}{5 d (c+d x)^{5/2}}\) |
| \(\Big \downarrow \) 3785 |
| \(\displaystyle \frac {2 b \left (-\frac {2 b \left (\frac {2 b \left (\frac {2 \cos \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}-\sin \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx\right )}{d}-\frac {2 \sin (a+b x)}{d \sqrt {c+d x}}\right )}{3 d}-\frac {2 \cos (a+b x)}{3 d (c+d x)^{3/2}}\right )}{5 d}-\frac {2 \sin (a+b x)}{5 d (c+d x)^{5/2}}\) |
| \(\Big \downarrow \) 3786 |
| \(\displaystyle \frac {2 b \left (-\frac {2 b \left (\frac {2 b \left (\frac {2 \cos \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}-\frac {2 \sin \left (a-\frac {b c}{d}\right ) \int \sin \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}\right )}{d}-\frac {2 \sin (a+b x)}{d \sqrt {c+d x}}\right )}{3 d}-\frac {2 \cos (a+b x)}{3 d (c+d x)^{3/2}}\right )}{5 d}-\frac {2 \sin (a+b x)}{5 d (c+d x)^{5/2}}\) |
| \(\Big \downarrow \) 3832 |
| \(\displaystyle \frac {2 b \left (-\frac {2 b \left (\frac {2 b \left (\frac {2 \cos \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}-\frac {\sqrt {2 \pi } \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{\sqrt {b} \sqrt {d}}\right )}{d}-\frac {2 \sin (a+b x)}{d \sqrt {c+d x}}\right )}{3 d}-\frac {2 \cos (a+b x)}{3 d (c+d x)^{3/2}}\right )}{5 d}-\frac {2 \sin (a+b x)}{5 d (c+d x)^{5/2}}\) |
| \(\Big \downarrow \) 3833 |
| \(\displaystyle \frac {2 b \left (-\frac {2 b \left (\frac {2 b \left (\frac {\sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{\sqrt {b} \sqrt {d}}-\frac {\sqrt {2 \pi } \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{\sqrt {b} \sqrt {d}}\right )}{d}-\frac {2 \sin (a+b x)}{d \sqrt {c+d x}}\right )}{3 d}-\frac {2 \cos (a+b x)}{3 d (c+d x)^{3/2}}\right )}{5 d}-\frac {2 \sin (a+b x)}{5 d (c+d x)^{5/2}}\) |
Input:
Int[Sin[a + b*x]/(c + d*x)^(7/2),x]
Output:
(-2*Sin[a + b*x])/(5*d*(c + d*x)^(5/2)) + (2*b*((-2*Cos[a + b*x])/(3*d*(c + d*x)^(3/2)) - (2*b*((2*b*((Sqrt[2*Pi]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b] *Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(Sqrt[b]*Sqrt[d]) - (Sqrt[2*Pi]*Fresn elS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(Sqrt[b] *Sqrt[d])))/d - (2*Sin[a + b*x])/(d*Sqrt[c + d*x])))/(3*d)))/(5*d)
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] - Simp[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]
Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> S imp[2/d Subst[Int[Cos[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[2/d Subst[Int[Sin[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f }, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Cos [(d*e - c*f)/d] Int[Sin[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] + Simp[Sin[( d*e - c*f)/d] Int[Cos[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c, d , e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]
Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Time = 0.76 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.14
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{5 \left (d x +c \right )^{\frac {5}{2}}}+\frac {4 b \left (-\frac {\cos \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 b \left (-\frac {\sin \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{\sqrt {d x +c}}+\frac {b \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {a d -b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {a d -b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{3 d}\right )}{5 d}}{d}\) | \(220\) |
| default | \(\frac {-\frac {2 \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{5 \left (d x +c \right )^{\frac {5}{2}}}+\frac {4 b \left (-\frac {\cos \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 b \left (-\frac {\sin \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{\sqrt {d x +c}}+\frac {b \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {a d -b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {a d -b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{3 d}\right )}{5 d}}{d}\) | \(220\) |
Input:
int(sin(b*x+a)/(d*x+c)^(7/2),x,method=_RETURNVERBOSE)
Output:
2/d*(-1/5/(d*x+c)^(5/2)*sin(b*(d*x+c)/d+(a*d-b*c)/d)+2/5*b/d*(-1/3/(d*x+c) ^(3/2)*cos(b*(d*x+c)/d+(a*d-b*c)/d)-2/3*b/d*(-1/(d*x+c)^(1/2)*sin(b*(d*x+c )/d+(a*d-b*c)/d)+b/d*2^(1/2)*Pi^(1/2)/(b/d)^(1/2)*(cos((a*d-b*c)/d)*Fresne lC(2^(1/2)/Pi^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/d)-sin((a*d-b*c)/d)*Fresne lS(2^(1/2)/Pi^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/d)))))
Time = 0.10 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.54 \[ \int \frac {\sin (a+b x)}{(c+d x)^{7/2}} \, dx=-\frac {2 \, {\left (4 \, \sqrt {2} {\left (\pi b^{2} d^{3} x^{3} + 3 \, \pi b^{2} c d^{2} x^{2} + 3 \, \pi b^{2} c^{2} d x + \pi b^{2} c^{3}\right )} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 4 \, \sqrt {2} {\left (\pi b^{2} d^{3} x^{3} + 3 \, \pi b^{2} c d^{2} x^{2} + 3 \, \pi b^{2} c^{2} d x + \pi b^{2} c^{3}\right )} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + \sqrt {d x + c} {\left (2 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right ) - {\left (4 \, b^{2} d^{2} x^{2} + 8 \, b^{2} c d x + 4 \, b^{2} c^{2} - 3 \, d^{2}\right )} \sin \left (b x + a\right )\right )}\right )}}{15 \, {\left (d^{6} x^{3} + 3 \, c d^{5} x^{2} + 3 \, c^{2} d^{4} x + c^{3} d^{3}\right )}} \] Input:
integrate(sin(b*x+a)/(d*x+c)^(7/2),x, algorithm="fricas")
Output:
-2/15*(4*sqrt(2)*(pi*b^2*d^3*x^3 + 3*pi*b^2*c*d^2*x^2 + 3*pi*b^2*c^2*d*x + pi*b^2*c^3)*sqrt(b/(pi*d))*cos(-(b*c - a*d)/d)*fresnel_cos(sqrt(2)*sqrt(d *x + c)*sqrt(b/(pi*d))) - 4*sqrt(2)*(pi*b^2*d^3*x^3 + 3*pi*b^2*c*d^2*x^2 + 3*pi*b^2*c^2*d*x + pi*b^2*c^3)*sqrt(b/(pi*d))*fresnel_sin(sqrt(2)*sqrt(d* x + c)*sqrt(b/(pi*d)))*sin(-(b*c - a*d)/d) + sqrt(d*x + c)*(2*(b*d^2*x + b *c*d)*cos(b*x + a) - (4*b^2*d^2*x^2 + 8*b^2*c*d*x + 4*b^2*c^2 - 3*d^2)*sin (b*x + a)))/(d^6*x^3 + 3*c*d^5*x^2 + 3*c^2*d^4*x + c^3*d^3)
\[ \int \frac {\sin (a+b x)}{(c+d x)^{7/2}} \, dx=\int \frac {\sin {\left (a + b x \right )}}{\left (c + d x\right )^{\frac {7}{2}}}\, dx \] Input:
integrate(sin(b*x+a)/(d*x+c)**(7/2),x)
Output:
Integral(sin(a + b*x)/(c + d*x)**(7/2), x)
Result contains complex when optimal does not.
Time = 0.22 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.67 \[ \int \frac {\sin (a+b x)}{(c+d x)^{7/2}} \, dx=\frac {{\left ({\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, -\frac {i \, {\left (d x + c\right )} b}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) + {\left (\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) - \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, -\frac {i \, {\left (d x + c\right )} b}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right )\right )} \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {5}{2}}}{4 \, {\left (d x + c\right )}^{\frac {5}{2}} d} \] Input:
integrate(sin(b*x+a)/(d*x+c)^(7/2),x, algorithm="maxima")
Output:
1/4*(((I - 1)*sqrt(2)*gamma(-5/2, I*(d*x + c)*b/d) - (I + 1)*sqrt(2)*gamma (-5/2, -I*(d*x + c)*b/d))*cos(-(b*c - a*d)/d) + ((I + 1)*sqrt(2)*gamma(-5/ 2, I*(d*x + c)*b/d) - (I - 1)*sqrt(2)*gamma(-5/2, -I*(d*x + c)*b/d))*sin(- (b*c - a*d)/d))*((d*x + c)*b/d)^(5/2)/((d*x + c)^(5/2)*d)
\[ \int \frac {\sin (a+b x)}{(c+d x)^{7/2}} \, dx=\int { \frac {\sin \left (b x + a\right )}{{\left (d x + c\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(sin(b*x+a)/(d*x+c)^(7/2),x, algorithm="giac")
Output:
integrate(sin(b*x + a)/(d*x + c)^(7/2), x)
Timed out. \[ \int \frac {\sin (a+b x)}{(c+d x)^{7/2}} \, dx=\int \frac {\sin \left (a+b\,x\right )}{{\left (c+d\,x\right )}^{7/2}} \,d x \] Input:
int(sin(a + b*x)/(c + d*x)^(7/2),x)
Output:
int(sin(a + b*x)/(c + d*x)^(7/2), x)
\[ \int \frac {\sin (a+b x)}{(c+d x)^{7/2}} \, dx=\int \frac {\sin \left (b x +a \right )}{\sqrt {d x +c}\, c^{3}+3 \sqrt {d x +c}\, c^{2} d x +3 \sqrt {d x +c}\, c \,d^{2} x^{2}+\sqrt {d x +c}\, d^{3} x^{3}}d x \] Input:
int(sin(b*x+a)/(d*x+c)^(7/2),x)
Output:
int(sin(a + b*x)/(sqrt(c + d*x)*c**3 + 3*sqrt(c + d*x)*c**2*d*x + 3*sqrt(c + d*x)*c*d**2*x**2 + sqrt(c + d*x)*d**3*x**3),x)