Integrand size = 21, antiderivative size = 98 \[ \int e^{\arctan (a x)} \left (c+a^2 c x^2\right )^{3/2} \, dx=\frac {\left (\frac {1}{13}+\frac {5 i}{13}\right ) 2^{\frac {3}{2}-\frac {i}{2}} c (1-i a x)^{\frac {5}{2}+\frac {i}{2}} \sqrt {c+a^2 c x^2} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2}+\frac {i}{2},\frac {5}{2}+\frac {i}{2},\frac {7}{2}+\frac {i}{2},\frac {1}{2} (1-i a x)\right )}{a \sqrt {1+a^2 x^2}} \] Output:
(1/13+5/13*I)*2^(3/2-1/2*I)*c*(1-I*a*x)^(5/2+1/2*I)*(a^2*c*x^2+c)^(1/2)*hy pergeom([5/2+1/2*I, -3/2+1/2*I],[7/2+1/2*I],1/2-1/2*I*a*x)/a/(a^2*x^2+1)^( 1/2)
Time = 0.03 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00 \[ \int e^{\arctan (a x)} \left (c+a^2 c x^2\right )^{3/2} \, dx=\frac {\left (\frac {1}{13}+\frac {5 i}{13}\right ) 2^{\frac {3}{2}-\frac {i}{2}} c (1-i a x)^{\frac {5}{2}+\frac {i}{2}} \sqrt {c+a^2 c x^2} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2}+\frac {i}{2},\frac {5}{2}+\frac {i}{2},\frac {7}{2}+\frac {i}{2},\frac {1}{2} (1-i a x)\right )}{a \sqrt {1+a^2 x^2}} \] Input:
Integrate[E^ArcTan[a*x]*(c + a^2*c*x^2)^(3/2),x]
Output:
((1/13 + (5*I)/13)*2^(3/2 - I/2)*c*(1 - I*a*x)^(5/2 + I/2)*Sqrt[c + a^2*c* x^2]*Hypergeometric2F1[-3/2 + I/2, 5/2 + I/2, 7/2 + I/2, (1 - I*a*x)/2])/( a*Sqrt[1 + a^2*x^2])
Time = 0.53 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5599, 5596, 79}
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.
\(\displaystyle \int e^{\arctan (a x)} \left (a^2 c x^2+c\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 5599 |
\(\displaystyle \frac {c \sqrt {a^2 c x^2+c} \int e^{\arctan (a x)} \left (a^2 x^2+1\right )^{3/2}dx}{\sqrt {a^2 x^2+1}}\) |
\(\Big \downarrow \) 5596 |
\(\displaystyle \frac {c \sqrt {a^2 c x^2+c} \int (1-i a x)^{\frac {3}{2}+\frac {i}{2}} (i a x+1)^{\frac {3}{2}-\frac {i}{2}}dx}{\sqrt {a^2 x^2+1}}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {\left (\frac {1}{13}+\frac {5 i}{13}\right ) 2^{\frac {3}{2}-\frac {i}{2}} c (1-i a x)^{\frac {5}{2}+\frac {i}{2}} \sqrt {a^2 c x^2+c} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2}+\frac {i}{2},\frac {5}{2}+\frac {i}{2},\frac {7}{2}+\frac {i}{2},\frac {1}{2} (1-i a x)\right )}{a \sqrt {a^2 x^2+1}}\) |
Input:
Int[E^ArcTan[a*x]*(c + a^2*c*x^2)^(3/2),x]
Output:
((1/13 + (5*I)/13)*2^(3/2 - I/2)*c*(1 - I*a*x)^(5/2 + I/2)*Sqrt[c + a^2*c* x^2]*Hypergeometric2F1[-3/2 + I/2, 5/2 + I/2, 7/2 + I/2, (1 - I*a*x)/2])/( a*Sqrt[1 + a^2*x^2])
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 , m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[c^p Int[(1 - I*a*x)^(p + I*(n/2))*(1 + I*a*x)^(p - I*(n/2)), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[d, a^2*c] && (IntegerQ[p] || GtQ[c, 0])
Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> S imp[c^IntPart[p]*((c + d*x^2)^FracPart[p]/(1 + a^2*x^2)^FracPart[p]) Int[ (1 + a^2*x^2)^p*E^(n*ArcTan[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && E qQ[d, a^2*c] && !(IntegerQ[p] || GtQ[c, 0])
\[\int {\mathrm e}^{\arctan \left (a x \right )} \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}d x\]
Input:
int(exp(arctan(a*x))*(a^2*c*x^2+c)^(3/2),x)
Output:
int(exp(arctan(a*x))*(a^2*c*x^2+c)^(3/2),x)
\[ \int e^{\arctan (a x)} \left (c+a^2 c x^2\right )^{3/2} \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} e^{\left (\arctan \left (a x\right )\right )} \,d x } \] Input:
integrate(exp(arctan(a*x))*(a^2*c*x^2+c)^(3/2),x, algorithm="fricas")
Output:
integral((a^2*c*x^2 + c)^(3/2)*e^(arctan(a*x)), x)
\[ \int e^{\arctan (a x)} \left (c+a^2 c x^2\right )^{3/2} \, dx=\int \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} e^{\operatorname {atan}{\left (a x \right )}}\, dx \] Input:
integrate(exp(atan(a*x))*(a**2*c*x**2+c)**(3/2),x)
Output:
Integral((c*(a**2*x**2 + 1))**(3/2)*exp(atan(a*x)), x)
\[ \int e^{\arctan (a x)} \left (c+a^2 c x^2\right )^{3/2} \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} e^{\left (\arctan \left (a x\right )\right )} \,d x } \] Input:
integrate(exp(arctan(a*x))*(a^2*c*x^2+c)^(3/2),x, algorithm="maxima")
Output:
integrate((a^2*c*x^2 + c)^(3/2)*e^(arctan(a*x)), x)
Exception generated. \[ \int e^{\arctan (a x)} \left (c+a^2 c x^2\right )^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(exp(arctan(a*x))*(a^2*c*x^2+c)^(3/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int e^{\arctan (a x)} \left (c+a^2 c x^2\right )^{3/2} \, dx=\int {\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \] Input:
int(exp(atan(a*x))*(c + a^2*c*x^2)^(3/2),x)
Output:
int(exp(atan(a*x))*(c + a^2*c*x^2)^(3/2), x)
\[ \int e^{\arctan (a x)} \left (c+a^2 c x^2\right )^{3/2} \, dx=\sqrt {c}\, c \left (\left (\int e^{\mathit {atan} \left (a x \right )} \sqrt {a^{2} x^{2}+1}\, x^{2}d x \right ) a^{2}+\int e^{\mathit {atan} \left (a x \right )} \sqrt {a^{2} x^{2}+1}d x \right ) \] Input:
int(exp(atan(a*x))*(a^2*c*x^2+c)^(3/2),x)
Output:
sqrt(c)*c*(int(e**atan(a*x)*sqrt(a**2*x**2 + 1)*x**2,x)*a**2 + int(e**atan (a*x)*sqrt(a**2*x**2 + 1),x))