Integrand size = 23, antiderivative size = 176 \[ \int \frac {x^3 (a+b \arctan (c x))}{\sqrt {d+e x^2}} \, dx=-\frac {b x \sqrt {d+e x^2}}{6 c e}-\frac {d \sqrt {d+e x^2} (a+b \arctan (c x))}{e^2}+\frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{3 e^2}+\frac {b \sqrt {c^2 d-e} \left (2 c^2 d+e\right ) \arctan \left (\frac {\sqrt {c^2 d-e} x}{\sqrt {d+e x^2}}\right )}{3 c^3 e^2}+\frac {b \left (3 c^2 d+2 e\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{6 c^3 e^{3/2}} \]
1/3*(e*x^2+d)^(3/2)*(a+b*arctan(c*x))/e^2+1/6*b*(3*c^2*d+2*e)*arctanh(x*e^ (1/2)/(e*x^2+d)^(1/2))/c^3/e^(3/2)+1/3*b*(2*c^2*d+e)*arctan(x*(c^2*d-e)^(1 /2)/(e*x^2+d)^(1/2))*(c^2*d-e)^(1/2)/c^3/e^2-1/6*b*x*(e*x^2+d)^(1/2)/c/e-d *(a+b*arctan(c*x))*(e*x^2+d)^(1/2)/e^2
Result contains complex when optimal does not.
Time = 0.43 (sec) , antiderivative size = 377, normalized size of antiderivative = 2.14 \[ \int \frac {x^3 (a+b \arctan (c x))}{\sqrt {d+e x^2}} \, dx=\frac {-\frac {\sqrt {d+e x^2} \left (b e x+a c \left (4 d-2 e x^2\right )\right )}{c}+2 b \left (-2 d+e x^2\right ) \sqrt {d+e x^2} \arctan (c x)-\frac {i b \left (2 c^4 d^2-c^2 d e-e^2\right ) \log \left (\frac {12 i c^4 e^2 \left (c d-i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \sqrt {c^2 d-e} \left (-2 c^4 d^2+c^2 d e+e^2\right ) (i+c x)}\right )}{c^3 \sqrt {c^2 d-e}}+\frac {i b \left (2 c^4 d^2-c^2 d e-e^2\right ) \log \left (-\frac {12 i c^4 e^2 \left (c d+i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \sqrt {c^2 d-e} \left (-2 c^4 d^2+c^2 d e+e^2\right ) (-i+c x)}\right )}{c^3 \sqrt {c^2 d-e}}+\frac {b \sqrt {e} \left (3 c^2 d+2 e\right ) \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{c^3}}{6 e^2} \]
(-((Sqrt[d + e*x^2]*(b*e*x + a*c*(4*d - 2*e*x^2)))/c) + 2*b*(-2*d + e*x^2) *Sqrt[d + e*x^2]*ArcTan[c*x] - (I*b*(2*c^4*d^2 - c^2*d*e - e^2)*Log[((12*I )*c^4*e^2*(c*d - I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*Sqrt[c^2*d - e]*(-2*c^4*d^2 + c^2*d*e + e^2)*(I + c*x))])/(c^3*Sqrt[c^2*d - e]) + (I*b *(2*c^4*d^2 - c^2*d*e - e^2)*Log[((-12*I)*c^4*e^2*(c*d + I*e*x + Sqrt[c^2* d - e]*Sqrt[d + e*x^2]))/(b*Sqrt[c^2*d - e]*(-2*c^4*d^2 + c^2*d*e + e^2)*( -I + c*x))])/(c^3*Sqrt[c^2*d - e]) + (b*Sqrt[e]*(3*c^2*d + 2*e)*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/c^3)/(6*e^2)
Time = 0.43 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.03, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {5511, 27, 403, 398, 224, 219, 291, 216}
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.
| \(\displaystyle \int \frac {x^3 (a+b \arctan (c x))}{\sqrt {d+e x^2}} \, dx\) |
| \(\Big \downarrow \) 5511 |
| \(\displaystyle -b c \int -\frac {\left (2 d-e x^2\right ) \sqrt {e x^2+d}}{3 e^2 \left (c^2 x^2+1\right )}dx+\frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{3 e^2}-\frac {d \sqrt {d+e x^2} (a+b \arctan (c x))}{e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c \int \frac {\left (2 d-e x^2\right ) \sqrt {e x^2+d}}{c^2 x^2+1}dx}{3 e^2}+\frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{3 e^2}-\frac {d \sqrt {d+e x^2} (a+b \arctan (c x))}{e^2}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {b c \left (\frac {\int \frac {e \left (3 d c^2+2 e\right ) x^2+d \left (4 d c^2+e\right )}{\left (c^2 x^2+1\right ) \sqrt {e x^2+d}}dx}{2 c^2}-\frac {e x \sqrt {d+e x^2}}{2 c^2}\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{3 e^2}-\frac {d \sqrt {d+e x^2} (a+b \arctan (c x))}{e^2}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {b c \left (\frac {\frac {e \left (3 c^2 d+2 e\right ) \int \frac {1}{\sqrt {e x^2+d}}dx}{c^2}+\frac {2 \left (c^2 d-e\right ) \left (2 c^2 d+e\right ) \int \frac {1}{\left (c^2 x^2+1\right ) \sqrt {e x^2+d}}dx}{c^2}}{2 c^2}-\frac {e x \sqrt {d+e x^2}}{2 c^2}\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{3 e^2}-\frac {d \sqrt {d+e x^2} (a+b \arctan (c x))}{e^2}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b c \left (\frac {\frac {2 \left (c^2 d-e\right ) \left (2 c^2 d+e\right ) \int \frac {1}{\left (c^2 x^2+1\right ) \sqrt {e x^2+d}}dx}{c^2}+\frac {e \left (3 c^2 d+2 e\right ) \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{c^2}}{2 c^2}-\frac {e x \sqrt {d+e x^2}}{2 c^2}\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{3 e^2}-\frac {d \sqrt {d+e x^2} (a+b \arctan (c x))}{e^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b c \left (\frac {\frac {2 \left (c^2 d-e\right ) \left (2 c^2 d+e\right ) \int \frac {1}{\left (c^2 x^2+1\right ) \sqrt {e x^2+d}}dx}{c^2}+\frac {\sqrt {e} \left (3 c^2 d+2 e\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c^2}}{2 c^2}-\frac {e x \sqrt {d+e x^2}}{2 c^2}\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{3 e^2}-\frac {d \sqrt {d+e x^2} (a+b \arctan (c x))}{e^2}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b c \left (\frac {\frac {2 \left (c^2 d-e\right ) \left (2 c^2 d+e\right ) \int \frac {1}{1-\frac {\left (e-c^2 d\right ) x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{c^2}+\frac {\sqrt {e} \left (3 c^2 d+2 e\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c^2}}{2 c^2}-\frac {e x \sqrt {d+e x^2}}{2 c^2}\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{3 e^2}-\frac {d \sqrt {d+e x^2} (a+b \arctan (c x))}{e^2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{3 e^2}-\frac {d \sqrt {d+e x^2} (a+b \arctan (c x))}{e^2}+\frac {b c \left (\frac {\frac {2 \sqrt {c^2 d-e} \left (2 c^2 d+e\right ) \arctan \left (\frac {x \sqrt {c^2 d-e}}{\sqrt {d+e x^2}}\right )}{c^2}+\frac {\sqrt {e} \left (3 c^2 d+2 e\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c^2}}{2 c^2}-\frac {e x \sqrt {d+e x^2}}{2 c^2}\right )}{3 e^2}\) |
-((d*Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/e^2) + ((d + e*x^2)^(3/2)*(a + b *ArcTan[c*x]))/(3*e^2) + (b*c*(-1/2*(e*x*Sqrt[d + e*x^2])/c^2 + ((2*Sqrt[c ^2*d - e]*(2*c^2*d + e)*ArcTan[(Sqrt[c^2*d - e]*x)/Sqrt[d + e*x^2]])/c^2 + (Sqrt[e]*(3*c^2*d + 2*e)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/c^2)/(2*c^ 2)))/(3*e^2)
3.13.1.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) , x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ b Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} , x]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x _)^2)^(q_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^q, x]}, Sim p[(a + b*ArcTan[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/(1 + c^2 *x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && ((IGtQ[q, 0] && !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*q + 3, 0])) || (IGtQ[(m + 1)/2, 0] && !(ILtQ[q, 0] && GtQ[m + 2*q + 3, 0])) || (ILtQ[(m + 2*q + 1)/2, 0] && !ILt Q[(m - 1)/2, 0]))
\[\int \frac {x^{3} \left (a +b \arctan \left (c x \right )\right )}{\sqrt {e \,x^{2}+d}}d x\]
Time = 0.63 (sec) , antiderivative size = 882, normalized size of antiderivative = 5.01 \[ \int \frac {x^3 (a+b \arctan (c x))}{\sqrt {d+e x^2}} \, dx=\left [\frac {{\left (3 \, b c^{2} d + 2 \, b e\right )} \sqrt {e} \log \left (-2 \, e x^{2} - 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) + {\left (2 \, b c^{2} d + b e\right )} \sqrt {-c^{2} d + e} \log \left (\frac {{\left (c^{4} d^{2} - 8 \, c^{2} d e + 8 \, e^{2}\right )} x^{4} - 2 \, {\left (3 \, c^{2} d^{2} - 4 \, d e\right )} x^{2} + 4 \, {\left ({\left (c^{2} d - 2 \, e\right )} x^{3} - d x\right )} \sqrt {-c^{2} d + e} \sqrt {e x^{2} + d} + d^{2}}{c^{4} x^{4} + 2 \, c^{2} x^{2} + 1}\right ) + 2 \, {\left (2 \, a c^{3} e x^{2} - 4 \, a c^{3} d - b c^{2} e x + 2 \, {\left (b c^{3} e x^{2} - 2 \, b c^{3} d\right )} \arctan \left (c x\right )\right )} \sqrt {e x^{2} + d}}{12 \, c^{3} e^{2}}, \frac {2 \, {\left (2 \, b c^{2} d + b e\right )} \sqrt {c^{2} d - e} \arctan \left (\frac {\sqrt {c^{2} d - e} {\left ({\left (c^{2} d - 2 \, e\right )} x^{2} - d\right )} \sqrt {e x^{2} + d}}{2 \, {\left ({\left (c^{2} d e - e^{2}\right )} x^{3} + {\left (c^{2} d^{2} - d e\right )} x\right )}}\right ) + {\left (3 \, b c^{2} d + 2 \, b e\right )} \sqrt {e} \log \left (-2 \, e x^{2} - 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) + 2 \, {\left (2 \, a c^{3} e x^{2} - 4 \, a c^{3} d - b c^{2} e x + 2 \, {\left (b c^{3} e x^{2} - 2 \, b c^{3} d\right )} \arctan \left (c x\right )\right )} \sqrt {e x^{2} + d}}{12 \, c^{3} e^{2}}, -\frac {2 \, {\left (3 \, b c^{2} d + 2 \, b e\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - {\left (2 \, b c^{2} d + b e\right )} \sqrt {-c^{2} d + e} \log \left (\frac {{\left (c^{4} d^{2} - 8 \, c^{2} d e + 8 \, e^{2}\right )} x^{4} - 2 \, {\left (3 \, c^{2} d^{2} - 4 \, d e\right )} x^{2} + 4 \, {\left ({\left (c^{2} d - 2 \, e\right )} x^{3} - d x\right )} \sqrt {-c^{2} d + e} \sqrt {e x^{2} + d} + d^{2}}{c^{4} x^{4} + 2 \, c^{2} x^{2} + 1}\right ) - 2 \, {\left (2 \, a c^{3} e x^{2} - 4 \, a c^{3} d - b c^{2} e x + 2 \, {\left (b c^{3} e x^{2} - 2 \, b c^{3} d\right )} \arctan \left (c x\right )\right )} \sqrt {e x^{2} + d}}{12 \, c^{3} e^{2}}, \frac {{\left (2 \, b c^{2} d + b e\right )} \sqrt {c^{2} d - e} \arctan \left (\frac {\sqrt {c^{2} d - e} {\left ({\left (c^{2} d - 2 \, e\right )} x^{2} - d\right )} \sqrt {e x^{2} + d}}{2 \, {\left ({\left (c^{2} d e - e^{2}\right )} x^{3} + {\left (c^{2} d^{2} - d e\right )} x\right )}}\right ) - {\left (3 \, b c^{2} d + 2 \, b e\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) + {\left (2 \, a c^{3} e x^{2} - 4 \, a c^{3} d - b c^{2} e x + 2 \, {\left (b c^{3} e x^{2} - 2 \, b c^{3} d\right )} \arctan \left (c x\right )\right )} \sqrt {e x^{2} + d}}{6 \, c^{3} e^{2}}\right ] \]
[1/12*((3*b*c^2*d + 2*b*e)*sqrt(e)*log(-2*e*x^2 - 2*sqrt(e*x^2 + d)*sqrt(e )*x - d) + (2*b*c^2*d + b*e)*sqrt(-c^2*d + e)*log(((c^4*d^2 - 8*c^2*d*e + 8*e^2)*x^4 - 2*(3*c^2*d^2 - 4*d*e)*x^2 + 4*((c^2*d - 2*e)*x^3 - d*x)*sqrt( -c^2*d + e)*sqrt(e*x^2 + d) + d^2)/(c^4*x^4 + 2*c^2*x^2 + 1)) + 2*(2*a*c^3 *e*x^2 - 4*a*c^3*d - b*c^2*e*x + 2*(b*c^3*e*x^2 - 2*b*c^3*d)*arctan(c*x))* sqrt(e*x^2 + d))/(c^3*e^2), 1/12*(2*(2*b*c^2*d + b*e)*sqrt(c^2*d - e)*arct an(1/2*sqrt(c^2*d - e)*((c^2*d - 2*e)*x^2 - d)*sqrt(e*x^2 + d)/((c^2*d*e - e^2)*x^3 + (c^2*d^2 - d*e)*x)) + (3*b*c^2*d + 2*b*e)*sqrt(e)*log(-2*e*x^2 - 2*sqrt(e*x^2 + d)*sqrt(e)*x - d) + 2*(2*a*c^3*e*x^2 - 4*a*c^3*d - b*c^2 *e*x + 2*(b*c^3*e*x^2 - 2*b*c^3*d)*arctan(c*x))*sqrt(e*x^2 + d))/(c^3*e^2) , -1/12*(2*(3*b*c^2*d + 2*b*e)*sqrt(-e)*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) - (2*b*c^2*d + b*e)*sqrt(-c^2*d + e)*log(((c^4*d^2 - 8*c^2*d*e + 8*e^2)*x ^4 - 2*(3*c^2*d^2 - 4*d*e)*x^2 + 4*((c^2*d - 2*e)*x^3 - d*x)*sqrt(-c^2*d + e)*sqrt(e*x^2 + d) + d^2)/(c^4*x^4 + 2*c^2*x^2 + 1)) - 2*(2*a*c^3*e*x^2 - 4*a*c^3*d - b*c^2*e*x + 2*(b*c^3*e*x^2 - 2*b*c^3*d)*arctan(c*x))*sqrt(e*x ^2 + d))/(c^3*e^2), 1/6*((2*b*c^2*d + b*e)*sqrt(c^2*d - e)*arctan(1/2*sqrt (c^2*d - e)*((c^2*d - 2*e)*x^2 - d)*sqrt(e*x^2 + d)/((c^2*d*e - e^2)*x^3 + (c^2*d^2 - d*e)*x)) - (3*b*c^2*d + 2*b*e)*sqrt(-e)*arctan(sqrt(-e)*x/sqrt (e*x^2 + d)) + (2*a*c^3*e*x^2 - 4*a*c^3*d - b*c^2*e*x + 2*(b*c^3*e*x^2 - 2 *b*c^3*d)*arctan(c*x))*sqrt(e*x^2 + d))/(c^3*e^2)]
\[ \int \frac {x^3 (a+b \arctan (c x))}{\sqrt {d+e x^2}} \, dx=\int \frac {x^{3} \left (a + b \operatorname {atan}{\left (c x \right )}\right )}{\sqrt {d + e x^{2}}}\, dx \]
Exception generated. \[ \int \frac {x^3 (a+b \arctan (c x))}{\sqrt {d+e x^2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {x^3 (a+b \arctan (c x))}{\sqrt {d+e x^2}} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{3}}{\sqrt {e x^{2} + d}} \,d x } \]
Timed out. \[ \int \frac {x^3 (a+b \arctan (c x))}{\sqrt {d+e x^2}} \, dx=\int \frac {x^3\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{\sqrt {e\,x^2+d}} \,d x \]