Integrand size = 24, antiderivative size = 163 \[ \int \frac {x \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=-\frac {\sqrt {\arctan (a x)}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{4 a^2 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )}{12 a^2 c^2 \sqrt {c+a^2 c x^2}} \] Output:
-1/3*arctan(a*x)^(1/2)/a^2/c/(a^2*c*x^2+c)^(3/2)+1/8*2^(1/2)*Pi^(1/2)*(a^2 *x^2+1)^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))/a^2/c^2/(a^2*c* x^2+c)^(1/2)+1/72*6^(1/2)*Pi^(1/2)*(a^2*x^2+1)^(1/2)*FresnelC(6^(1/2)/Pi^( 1/2)*arctan(a*x)^(1/2))/a^2/c^2/(a^2*c*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.02 \[ \int \frac {x \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {-48 \arctan (a x)-i \left (1+a^2 x^2\right )^{3/2} \left (9 \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-i \arctan (a x)\right )-9 \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},i \arctan (a x)\right )+\sqrt {3} \left (\sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-3 i \arctan (a x)\right )-\sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},3 i \arctan (a x)\right )\right )\right )}{144 a^2 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \] Input:
Integrate[(x*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(5/2),x]
Output:
(-48*ArcTan[a*x] - I*(1 + a^2*x^2)^(3/2)*(9*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1 /2, (-I)*ArcTan[a*x]] - 9*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, I*ArcTan[a*x]] + Sqrt[3]*(Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-3*I)*ArcTan[a*x]] - Sqrt[I*Ar cTan[a*x]]*Gamma[1/2, (3*I)*ArcTan[a*x]])))/(144*a^2*c*(c + a^2*c*x^2)^(3/ 2)*Sqrt[ArcTan[a*x]])
Time = 0.65 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.83, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5465, 5440, 5439, 3042, 3793, 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.
\(\displaystyle \int \frac {x \sqrt {\arctan (a x)}}{\left (a^2 c x^2+c\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 5465 |
\(\displaystyle \frac {\int \frac {1}{\left (a^2 c x^2+c\right )^{5/2} \sqrt {\arctan (a x)}}dx}{6 a}-\frac {\sqrt {\arctan (a x)}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\) |
\(\Big \downarrow \) 5440 |
\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \frac {1}{\left (a^2 x^2+1\right )^{5/2} \sqrt {\arctan (a x)}}dx}{6 a c^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {\arctan (a x)}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\) |
\(\Big \downarrow \) 5439 |
\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \frac {1}{\left (a^2 x^2+1\right )^{3/2} \sqrt {\arctan (a x)}}d\arctan (a x)}{6 a^2 c^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {\arctan (a x)}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \frac {\sin \left (\arctan (a x)+\frac {\pi }{2}\right )^3}{\sqrt {\arctan (a x)}}d\arctan (a x)}{6 a^2 c^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {\arctan (a x)}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \left (\frac {\cos (3 \arctan (a x))}{4 \sqrt {\arctan (a x)}}+\frac {3}{4 \sqrt {a^2 x^2+1} \sqrt {\arctan (a x)}}\right )d\arctan (a x)}{6 a^2 c^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {\arctan (a x)}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {a^2 x^2+1} \left (\frac {3}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )\right )}{6 a^2 c^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {\arctan (a x)}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\) |
Input:
Int[(x*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(5/2),x]
Output:
-1/3*Sqrt[ArcTan[a*x]]/(a^2*c*(c + a^2*c*x^2)^(3/2)) + (Sqrt[1 + a^2*x^2]* ((3*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/2 + (Sqrt[Pi/6]*Fre snelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/2))/(6*a^2*c^2*Sqrt[c + a^2*c*x^2])
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_), x_ Symbol] :> Simp[d^q/c Subst[Int[(a + b*x)^p/Cos[x]^(2*(q + 1)), x], x, Ar cTan[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && ILtQ[2*( q + 1), 0] && (IntegerQ[q] || GtQ[d, 0])
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_), x_ Symbol] :> Simp[d^(q + 1/2)*(Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]) Int[(1 + c^2*x^2)^q*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && ILtQ[2*(q + 1), 0] && !(IntegerQ[q] || GtQ[d, 0])
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_ .), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*e*(q + 1))), x] - Simp[b*(p/(2*c*(q + 1))) Int[(d + e*x^2)^q*(a + b*ArcTan[c*x]) ^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]
\[\int \frac {x \sqrt {\arctan \left (a x \right )}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}d x\]
Input:
int(x*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^(5/2),x)
Output:
int(x*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^(5/2),x)
Exception generated. \[ \int \frac {x \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^(5/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x \sqrt {\operatorname {atan}{\left (a x \right )}}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(x*atan(a*x)**(1/2)/(a**2*c*x**2+c)**(5/2),x)
Output:
Integral(x*sqrt(atan(a*x))/(c*(a**2*x**2 + 1))**(5/2), x)
Exception generated. \[ \int \frac {x \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^(5/2),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
\[ \int \frac {x \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x \sqrt {\arctan \left (a x\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(x*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^(5/2),x, algorithm="giac")
Output:
integrate(x*sqrt(arctan(a*x))/(a^2*c*x^2 + c)^(5/2), x)
Timed out. \[ \int \frac {x \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x\,\sqrt {\mathrm {atan}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \] Input:
int((x*atan(a*x)^(1/2))/(c + a^2*c*x^2)^(5/2),x)
Output:
int((x*atan(a*x)^(1/2))/(c + a^2*c*x^2)^(5/2), x)
\[ \int \frac {x \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {\sqrt {c}\, \left (-2 \sqrt {a^{2} x^{2}+1}\, \sqrt {\mathit {atan} \left (a x \right )}+\left (\int \frac {\sqrt {a^{2} x^{2}+1}\, \sqrt {\mathit {atan} \left (a x \right )}}{\mathit {atan} \left (a x \right ) a^{6} x^{6}+3 \mathit {atan} \left (a x \right ) a^{4} x^{4}+3 \mathit {atan} \left (a x \right ) a^{2} x^{2}+\mathit {atan} \left (a x \right )}d x \right ) a^{5} x^{4}+2 \left (\int \frac {\sqrt {a^{2} x^{2}+1}\, \sqrt {\mathit {atan} \left (a x \right )}}{\mathit {atan} \left (a x \right ) a^{6} x^{6}+3 \mathit {atan} \left (a x \right ) a^{4} x^{4}+3 \mathit {atan} \left (a x \right ) a^{2} x^{2}+\mathit {atan} \left (a x \right )}d x \right ) a^{3} x^{2}+\left (\int \frac {\sqrt {a^{2} x^{2}+1}\, \sqrt {\mathit {atan} \left (a x \right )}}{\mathit {atan} \left (a x \right ) a^{6} x^{6}+3 \mathit {atan} \left (a x \right ) a^{4} x^{4}+3 \mathit {atan} \left (a x \right ) a^{2} x^{2}+\mathit {atan} \left (a x \right )}d x \right ) a \right )}{6 a^{2} c^{3} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )} \] Input:
int(x*atan(a*x)^(1/2)/(a^2*c*x^2+c)^(5/2),x)
Output:
(sqrt(c)*( - 2*sqrt(a**2*x**2 + 1)*sqrt(atan(a*x)) + int((sqrt(a**2*x**2 + 1)*sqrt(atan(a*x)))/(atan(a*x)*a**6*x**6 + 3*atan(a*x)*a**4*x**4 + 3*atan (a*x)*a**2*x**2 + atan(a*x)),x)*a**5*x**4 + 2*int((sqrt(a**2*x**2 + 1)*sqr t(atan(a*x)))/(atan(a*x)*a**6*x**6 + 3*atan(a*x)*a**4*x**4 + 3*atan(a*x)*a **2*x**2 + atan(a*x)),x)*a**3*x**2 + int((sqrt(a**2*x**2 + 1)*sqrt(atan(a* x)))/(atan(a*x)*a**6*x**6 + 3*atan(a*x)*a**4*x**4 + 3*atan(a*x)*a**2*x**2 + atan(a*x)),x)*a))/(6*a**2*c**3*(a**4*x**4 + 2*a**2*x**2 + 1))