Integrand size = 21, antiderivative size = 233 \[ \int x \left (d+e x^2\right )^{5/2} (a+b \arctan (c x)) \, dx=-\frac {b \left (19 c^4 d^2-22 c^2 d e+8 e^2\right ) x \sqrt {d+e x^2}}{112 c^5}-\frac {b \left (11 c^2 d-6 e\right ) x \left (d+e x^2\right )^{3/2}}{168 c^3}-\frac {b x \left (d+e x^2\right )^{5/2}}{42 c}+\frac {\left (d+e x^2\right )^{7/2} (a+b \arctan (c x))}{7 e}-\frac {b \left (c^2 d-e\right )^{7/2} \arctan \left (\frac {\sqrt {c^2 d-e} x}{\sqrt {d+e x^2}}\right )}{7 c^7 e}-\frac {b \left (35 c^6 d^3-70 c^4 d^2 e+56 c^2 d e^2-16 e^3\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{112 c^7 \sqrt {e}} \]
-1/168*b*(11*c^2*d-6*e)*x*(e*x^2+d)^(3/2)/c^3-1/42*b*x*(e*x^2+d)^(5/2)/c+1 /7*(e*x^2+d)^(7/2)*(a+b*arctan(c*x))/e-1/7*b*(c^2*d-e)^(7/2)*arctan(x*(c^2 *d-e)^(1/2)/(e*x^2+d)^(1/2))/c^7/e-1/112*b*(35*c^6*d^3-70*c^4*d^2*e+56*c^2 *d*e^2-16*e^3)*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/c^7/e^(1/2)-1/112*b*(19* c^4*d^2-22*c^2*d*e+8*e^2)*x*(e*x^2+d)^(1/2)/c^5
Result contains complex when optimal does not.
Time = 0.43 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.52 \[ \int x \left (d+e x^2\right )^{5/2} (a+b \arctan (c x)) \, dx=\frac {c^2 \sqrt {d+e x^2} \left (48 a c^5 \left (d+e x^2\right )^3-b e x \left (24 e^2-6 c^2 e \left (13 d+2 e x^2\right )+c^4 \left (87 d^2+38 d e x^2+8 e^2 x^4\right )\right )\right )+48 b c^7 \left (d+e x^2\right )^{7/2} \arctan (c x)-24 i b \left (c^2 d-e\right )^{7/2} \log \left (\frac {28 c^8 e \left (-i c d+e x-i \sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \left (c^2 d-e\right )^{9/2} (-i+c x)}\right )+24 i b \left (c^2 d-e\right )^{7/2} \log \left (\frac {28 c^8 e \left (i c d+e x+i \sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \left (c^2 d-e\right )^{9/2} (i+c x)}\right )+3 b \sqrt {e} \left (-35 c^6 d^3+70 c^4 d^2 e-56 c^2 d e^2+16 e^3\right ) \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{336 c^7 e} \]
(c^2*Sqrt[d + e*x^2]*(48*a*c^5*(d + e*x^2)^3 - b*e*x*(24*e^2 - 6*c^2*e*(13 *d + 2*e*x^2) + c^4*(87*d^2 + 38*d*e*x^2 + 8*e^2*x^4))) + 48*b*c^7*(d + e* x^2)^(7/2)*ArcTan[c*x] - (24*I)*b*(c^2*d - e)^(7/2)*Log[(28*c^8*e*((-I)*c* d + e*x - I*Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - e)^(9/2)*(-I + c *x))] + (24*I)*b*(c^2*d - e)^(7/2)*Log[(28*c^8*e*(I*c*d + e*x + I*Sqrt[c^2 *d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - e)^(9/2)*(I + c*x))] + 3*b*Sqrt[e]*( -35*c^6*d^3 + 70*c^4*d^2*e - 56*c^2*d*e^2 + 16*e^3)*Log[e*x + Sqrt[e]*Sqrt [d + e*x^2]])/(336*c^7*e)
Time = 0.52 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {5509, 318, 403, 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.
| \(\displaystyle \int x \left (d+e x^2\right )^{5/2} (a+b \arctan (c x)) \, dx\) |
| \(\Big \downarrow \) 5509 |
| \(\displaystyle \frac {\left (d+e x^2\right )^{7/2} (a+b \arctan (c x))}{7 e}-\frac {b c \int \frac {\left (e x^2+d\right )^{7/2}}{c^2 x^2+1}dx}{7 e}\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle \frac {\left (d+e x^2\right )^{7/2} (a+b \arctan (c x))}{7 e}-\frac {b c \left (\frac {\int \frac {\left (e x^2+d\right )^{3/2} \left (\left (11 c^2 d-6 e\right ) e x^2+d \left (6 c^2 d-e\right )\right )}{c^2 x^2+1}dx}{6 c^2}+\frac {e x \left (d+e x^2\right )^{5/2}}{6 c^2}\right )}{7 e}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\left (d+e x^2\right )^{7/2} (a+b \arctan (c x))}{7 e}-\frac {b c \left (\frac {\frac {\int \frac {3 \sqrt {e x^2+d} \left (e \left (19 d^2 c^4-22 d e c^2+8 e^2\right ) x^2+d \left (8 d^2 c^4-5 d e c^2+2 e^2\right )\right )}{c^2 x^2+1}dx}{4 c^2}+\frac {e x \left (11 c^2 d-6 e\right ) \left (d+e x^2\right )^{3/2}}{4 c^2}}{6 c^2}+\frac {e x \left (d+e x^2\right )^{5/2}}{6 c^2}\right )}{7 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (d+e x^2\right )^{7/2} (a+b \arctan (c x))}{7 e}-\frac {b c \left (\frac {\frac {3 \int \frac {\sqrt {e x^2+d} \left (e \left (19 d^2 c^4-22 d e c^2+8 e^2\right ) x^2+d \left (8 d^2 c^4-5 d e c^2+2 e^2\right )\right )}{c^2 x^2+1}dx}{4 c^2}+\frac {e x \left (11 c^2 d-6 e\right ) \left (d+e x^2\right )^{3/2}}{4 c^2}}{6 c^2}+\frac {e x \left (d+e x^2\right )^{5/2}}{6 c^2}\right )}{7 e}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\left (d+e x^2\right )^{7/2} (a+b \arctan (c x))}{7 e}-\frac {b c \left (\frac {\frac {3 \left (\frac {\int \frac {e \left (35 d^3 c^6-70 d^2 e c^4+56 d e^2 c^2-16 e^3\right ) x^2+d \left (16 d^3 c^6-29 d^2 e c^4+26 d e^2 c^2-8 e^3\right )}{\left (c^2 x^2+1\right ) \sqrt {e x^2+d}}dx}{2 c^2}+\frac {e x \left (19 c^4 d^2-22 c^2 d e+8 e^2\right ) \sqrt {d+e x^2}}{2 c^2}\right )}{4 c^2}+\frac {e x \left (11 c^2 d-6 e\right ) \left (d+e x^2\right )^{3/2}}{4 c^2}}{6 c^2}+\frac {e x \left (d+e x^2\right )^{5/2}}{6 c^2}\right )}{7 e}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {\left (d+e x^2\right )^{7/2} (a+b \arctan (c x))}{7 e}-\frac {b c \left (\frac {\frac {3 \left (\frac {\frac {16 \left (c^2 d-e\right )^4 \int \frac {1}{\left (c^2 x^2+1\right ) \sqrt {e x^2+d}}dx}{c^2}+\frac {e \left (35 c^6 d^3-70 c^4 d^2 e+56 c^2 d e^2-16 e^3\right ) \int \frac {1}{\sqrt {e x^2+d}}dx}{c^2}}{2 c^2}+\frac {e x \left (19 c^4 d^2-22 c^2 d e+8 e^2\right ) \sqrt {d+e x^2}}{2 c^2}\right )}{4 c^2}+\frac {e x \left (11 c^2 d-6 e\right ) \left (d+e x^2\right )^{3/2}}{4 c^2}}{6 c^2}+\frac {e x \left (d+e x^2\right )^{5/2}}{6 c^2}\right )}{7 e}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\left (d+e x^2\right )^{7/2} (a+b \arctan (c x))}{7 e}-\frac {b c \left (\frac {\frac {3 \left (\frac {\frac {16 \left (c^2 d-e\right )^4 \int \frac {1}{\left (c^2 x^2+1\right ) \sqrt {e x^2+d}}dx}{c^2}+\frac {e \left (35 c^6 d^3-70 c^4 d^2 e+56 c^2 d e^2-16 e^3\right ) \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{c^2}}{2 c^2}+\frac {e x \left (19 c^4 d^2-22 c^2 d e+8 e^2\right ) \sqrt {d+e x^2}}{2 c^2}\right )}{4 c^2}+\frac {e x \left (11 c^2 d-6 e\right ) \left (d+e x^2\right )^{3/2}}{4 c^2}}{6 c^2}+\frac {e x \left (d+e x^2\right )^{5/2}}{6 c^2}\right )}{7 e}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (d+e x^2\right )^{7/2} (a+b \arctan (c x))}{7 e}-\frac {b c \left (\frac {\frac {3 \left (\frac {\frac {16 \left (c^2 d-e\right )^4 \int \frac {1}{\left (c^2 x^2+1\right ) \sqrt {e x^2+d}}dx}{c^2}+\frac {\sqrt {e} \left (35 c^6 d^3-70 c^4 d^2 e+56 c^2 d e^2-16 e^3\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c^2}}{2 c^2}+\frac {e x \left (19 c^4 d^2-22 c^2 d e+8 e^2\right ) \sqrt {d+e x^2}}{2 c^2}\right )}{4 c^2}+\frac {e x \left (11 c^2 d-6 e\right ) \left (d+e x^2\right )^{3/2}}{4 c^2}}{6 c^2}+\frac {e x \left (d+e x^2\right )^{5/2}}{6 c^2}\right )}{7 e}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {\left (d+e x^2\right )^{7/2} (a+b \arctan (c x))}{7 e}-\frac {b c \left (\frac {\frac {3 \left (\frac {\frac {16 \left (c^2 d-e\right )^4 \int \frac {1}{1-\frac {\left (e-c^2 d\right ) x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{c^2}+\frac {\sqrt {e} \left (35 c^6 d^3-70 c^4 d^2 e+56 c^2 d e^2-16 e^3\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c^2}}{2 c^2}+\frac {e x \left (19 c^4 d^2-22 c^2 d e+8 e^2\right ) \sqrt {d+e x^2}}{2 c^2}\right )}{4 c^2}+\frac {e x \left (11 c^2 d-6 e\right ) \left (d+e x^2\right )^{3/2}}{4 c^2}}{6 c^2}+\frac {e x \left (d+e x^2\right )^{5/2}}{6 c^2}\right )}{7 e}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\left (d+e x^2\right )^{7/2} (a+b \arctan (c x))}{7 e}-\frac {b c \left (\frac {\frac {3 \left (\frac {\frac {16 \left (c^2 d-e\right )^{7/2} \arctan \left (\frac {x \sqrt {c^2 d-e}}{\sqrt {d+e x^2}}\right )}{c^2}+\frac {\sqrt {e} \left (35 c^6 d^3-70 c^4 d^2 e+56 c^2 d e^2-16 e^3\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c^2}}{2 c^2}+\frac {e x \left (19 c^4 d^2-22 c^2 d e+8 e^2\right ) \sqrt {d+e x^2}}{2 c^2}\right )}{4 c^2}+\frac {e x \left (11 c^2 d-6 e\right ) \left (d+e x^2\right )^{3/2}}{4 c^2}}{6 c^2}+\frac {e x \left (d+e x^2\right )^{5/2}}{6 c^2}\right )}{7 e}\) |
((d + e*x^2)^(7/2)*(a + b*ArcTan[c*x]))/(7*e) - (b*c*((e*x*(d + e*x^2)^(5/ 2))/(6*c^2) + (((11*c^2*d - 6*e)*e*x*(d + e*x^2)^(3/2))/(4*c^2) + (3*((e*( 19*c^4*d^2 - 22*c^2*d*e + 8*e^2)*x*Sqrt[d + e*x^2])/(2*c^2) + ((16*(c^2*d - e)^(7/2)*ArcTan[(Sqrt[c^2*d - e]*x)/Sqrt[d + e*x^2]])/c^2 + (Sqrt[e]*(35 *c^6*d^3 - 70*c^4*d^2*e + 56*c^2*d*e^2 - 16*e^3)*ArcTanh[(Sqrt[e]*x)/Sqrt[ d + e*x^2]])/c^2)/(2*c^2)))/(4*c^2))/(6*c^2)))/(7*e)
3.12.95.3.1 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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S imp[1/(b*(2*(p + q) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b *c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G tQ[q, 1] && NeQ[2*(p + q) + 1, 0] && !IGtQ[p, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
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_.))*(x_)*((d_.) + (e_.)*(x_)^2)^(q_.), x _Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])/(2*e*(q + 1))), x ] - Simp[b*(c/(2*e*(q + 1))) Int[(d + e*x^2)^(q + 1)/(1 + c^2*x^2), x], x ] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]
\[\int x \left (e \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \arctan \left (c x \right )\right )d x\]
Time = 8.40 (sec) , antiderivative size = 1562, normalized size of antiderivative = 6.70 \[ \int x \left (d+e x^2\right )^{5/2} (a+b \arctan (c x)) \, dx=\text {Too large to display} \]
[-1/672*(3*(35*b*c^6*d^3 - 70*b*c^4*d^2*e + 56*b*c^2*d*e^2 - 16*b*e^3)*sqr t(e)*log(-2*e*x^2 - 2*sqrt(e*x^2 + d)*sqrt(e)*x - d) + 24*(b*c^6*d^3 - 3*b *c^4*d^2*e + 3*b*c^2*d*e^2 - b*e^3)*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*( 48*a*c^7*e^3*x^6 + 144*a*c^7*d*e^2*x^4 - 8*b*c^6*e^3*x^5 + 144*a*c^7*d^2*e *x^2 + 48*a*c^7*d^3 - 2*(19*b*c^6*d*e^2 - 6*b*c^4*e^3)*x^3 - 3*(29*b*c^6*d ^2*e - 26*b*c^4*d*e^2 + 8*b*c^2*e^3)*x + 48*(b*c^7*e^3*x^6 + 3*b*c^7*d*e^2 *x^4 + 3*b*c^7*d^2*e*x^2 + b*c^7*d^3)*arctan(c*x))*sqrt(e*x^2 + d))/(c^7*e ), -1/672*(48*(b*c^6*d^3 - 3*b*c^4*d^2*e + 3*b*c^2*d*e^2 - b*e^3)*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*(35*b*c^6*d^3 - 70*b*c^4*d ^2*e + 56*b*c^2*d*e^2 - 16*b*e^3)*sqrt(e)*log(-2*e*x^2 - 2*sqrt(e*x^2 + d) *sqrt(e)*x - d) - 2*(48*a*c^7*e^3*x^6 + 144*a*c^7*d*e^2*x^4 - 8*b*c^6*e^3* x^5 + 144*a*c^7*d^2*e*x^2 + 48*a*c^7*d^3 - 2*(19*b*c^6*d*e^2 - 6*b*c^4*e^3 )*x^3 - 3*(29*b*c^6*d^2*e - 26*b*c^4*d*e^2 + 8*b*c^2*e^3)*x + 48*(b*c^7*e^ 3*x^6 + 3*b*c^7*d*e^2*x^4 + 3*b*c^7*d^2*e*x^2 + b*c^7*d^3)*arctan(c*x))*sq rt(e*x^2 + d))/(c^7*e), 1/336*(3*(35*b*c^6*d^3 - 70*b*c^4*d^2*e + 56*b*c^2 *d*e^2 - 16*b*e^3)*sqrt(-e)*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) - 12*(b*c^6 *d^3 - 3*b*c^4*d^2*e + 3*b*c^2*d*e^2 - b*e^3)*sqrt(-c^2*d + e)*log(((c^...
\[ \int x \left (d+e x^2\right )^{5/2} (a+b \arctan (c x)) \, dx=\int x \left (a + b \operatorname {atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac {5}{2}}\, dx \]
Exception generated. \[ \int x \left (d+e x^2\right )^{5/2} (a+b \arctan (c x)) \, dx=\text {Exception raised: ValueError} \]
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-c^2*d>0)', see `assume?` for m ore detail
\[ \int x \left (d+e x^2\right )^{5/2} (a+b \arctan (c x)) \, dx=\int { {\left (e x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arctan \left (c x\right ) + a\right )} x \,d x } \]
Timed out. \[ \int x \left (d+e x^2\right )^{5/2} (a+b \arctan (c x)) \, dx=\int x\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^{5/2} \,d x \]