3.1193 \(\int x^3 (d+e x^2)^{5/2} (a+b \tan ^{-1}(c x)) \, dx\)

Optimal. Leaf size=345 \[ \frac{\left (d+e x^2\right )^{9/2} \left (a+b \tan ^{-1}(c x)\right )}{9 e^2}-\frac{d \left (d+e x^2\right )^{7/2} \left (a+b \tan ^{-1}(c x)\right )}{7 e^2}-\frac{b x \left (69 c^4 d^2-520 c^2 d e+336 e^2\right ) \left (d+e x^2\right )^{3/2}}{12096 c^5 e}+\frac{b x \left (712 c^4 d^2 e+59 c^6 d^3-1104 c^2 d e^2+448 e^3\right ) \sqrt{d+e x^2}}{8064 c^7 e}+\frac{b \left (-3024 c^4 d^2 e^2+840 c^6 d^3 e+315 c^8 d^4+2880 c^2 d e^3-896 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{8064 c^9 e^{3/2}}+\frac{b \left (c^2 d-e\right )^{7/2} \left (2 c^2 d+7 e\right ) \tan ^{-1}\left (\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right )}{63 c^9 e^2}-\frac{b x \left (33 c^2 d-56 e\right ) \left (d+e x^2\right )^{5/2}}{3024 c^3 e}-\frac{b x \left (d+e x^2\right )^{7/2}}{72 c e} \]

[Out]

(b*(59*c^6*d^3 + 712*c^4*d^2*e - 1104*c^2*d*e^2 + 448*e^3)*x*Sqrt[d + e*x^2])/(8064*c^7*e) - (b*(69*c^4*d^2 -
520*c^2*d*e + 336*e^2)*x*(d + e*x^2)^(3/2))/(12096*c^5*e) - (b*(33*c^2*d - 56*e)*x*(d + e*x^2)^(5/2))/(3024*c^
3*e) - (b*x*(d + e*x^2)^(7/2))/(72*c*e) - (d*(d + e*x^2)^(7/2)*(a + b*ArcTan[c*x]))/(7*e^2) + ((d + e*x^2)^(9/
2)*(a + b*ArcTan[c*x]))/(9*e^2) + (b*(c^2*d - e)^(7/2)*(2*c^2*d + 7*e)*ArcTan[(Sqrt[c^2*d - e]*x)/Sqrt[d + e*x
^2]])/(63*c^9*e^2) + (b*(315*c^8*d^4 + 840*c^6*d^3*e - 3024*c^4*d^2*e^2 + 2880*c^2*d*e^3 - 896*e^4)*ArcTanh[(S
qrt[e]*x)/Sqrt[d + e*x^2]])/(8064*c^9*e^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.58331, antiderivative size = 345, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {266, 43, 4976, 12, 528, 523, 217, 206, 377, 203} \[ \frac{\left (d+e x^2\right )^{9/2} \left (a+b \tan ^{-1}(c x)\right )}{9 e^2}-\frac{d \left (d+e x^2\right )^{7/2} \left (a+b \tan ^{-1}(c x)\right )}{7 e^2}-\frac{b x \left (69 c^4 d^2-520 c^2 d e+336 e^2\right ) \left (d+e x^2\right )^{3/2}}{12096 c^5 e}+\frac{b x \left (712 c^4 d^2 e+59 c^6 d^3-1104 c^2 d e^2+448 e^3\right ) \sqrt{d+e x^2}}{8064 c^7 e}+\frac{b \left (-3024 c^4 d^2 e^2+840 c^6 d^3 e+315 c^8 d^4+2880 c^2 d e^3-896 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{8064 c^9 e^{3/2}}+\frac{b \left (c^2 d-e\right )^{7/2} \left (2 c^2 d+7 e\right ) \tan ^{-1}\left (\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right )}{63 c^9 e^2}-\frac{b x \left (33 c^2 d-56 e\right ) \left (d+e x^2\right )^{5/2}}{3024 c^3 e}-\frac{b x \left (d+e x^2\right )^{7/2}}{72 c e} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]),x]

[Out]

(b*(59*c^6*d^3 + 712*c^4*d^2*e - 1104*c^2*d*e^2 + 448*e^3)*x*Sqrt[d + e*x^2])/(8064*c^7*e) - (b*(69*c^4*d^2 -
520*c^2*d*e + 336*e^2)*x*(d + e*x^2)^(3/2))/(12096*c^5*e) - (b*(33*c^2*d - 56*e)*x*(d + e*x^2)^(5/2))/(3024*c^
3*e) - (b*x*(d + e*x^2)^(7/2))/(72*c*e) - (d*(d + e*x^2)^(7/2)*(a + b*ArcTan[c*x]))/(7*e^2) + ((d + e*x^2)^(9/
2)*(a + b*ArcTan[c*x]))/(9*e^2) + (b*(c^2*d - e)^(7/2)*(2*c^2*d + 7*e)*ArcTan[(Sqrt[c^2*d - e]*x)/Sqrt[d + e*x
^2]])/(63*c^9*e^2) + (b*(315*c^8*d^4 + 840*c^6*d^3*e - 3024*c^4*d^2*e^2 + 2880*c^2*d*e^3 - 896*e^4)*ArcTanh[(S
qrt[e]*x)/Sqrt[d + e*x^2]])/(8064*c^9*e^(3/2))

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 4976

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]}, Dist[a + b*ArcTan[c*x], u, x] - Dist[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] &&
!ILtQ[(m - 1)/2, 0]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 528

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[
(f*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(b*(n*(p + q + 1) + 1)), x] + Dist[1/(b*(n*(p + q + 1) + 1)), Int[(a +
b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e - a*f) + f*n*q*(b*c - a*d) + b*
d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1
, 0]

Rule 523

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

Rule 217

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]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int x^3 \left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right ) \, dx &=-\frac{d \left (d+e x^2\right )^{7/2} \left (a+b \tan ^{-1}(c x)\right )}{7 e^2}+\frac{\left (d+e x^2\right )^{9/2} \left (a+b \tan ^{-1}(c x)\right )}{9 e^2}-(b c) \int \frac{\left (d+e x^2\right )^{7/2} \left (-2 d+7 e x^2\right )}{63 e^2 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac{d \left (d+e x^2\right )^{7/2} \left (a+b \tan ^{-1}(c x)\right )}{7 e^2}+\frac{\left (d+e x^2\right )^{9/2} \left (a+b \tan ^{-1}(c x)\right )}{9 e^2}-\frac{(b c) \int \frac{\left (d+e x^2\right )^{7/2} \left (-2 d+7 e x^2\right )}{1+c^2 x^2} \, dx}{63 e^2}\\ &=-\frac{b x \left (d+e x^2\right )^{7/2}}{72 c e}-\frac{d \left (d+e x^2\right )^{7/2} \left (a+b \tan ^{-1}(c x)\right )}{7 e^2}+\frac{\left (d+e x^2\right )^{9/2} \left (a+b \tan ^{-1}(c x)\right )}{9 e^2}-\frac{b \int \frac{\left (d+e x^2\right )^{5/2} \left (-d \left (16 c^2 d+7 e\right )+\left (33 c^2 d-56 e\right ) e x^2\right )}{1+c^2 x^2} \, dx}{504 c e^2}\\ &=-\frac{b \left (33 c^2 d-56 e\right ) x \left (d+e x^2\right )^{5/2}}{3024 c^3 e}-\frac{b x \left (d+e x^2\right )^{7/2}}{72 c e}-\frac{d \left (d+e x^2\right )^{7/2} \left (a+b \tan ^{-1}(c x)\right )}{7 e^2}+\frac{\left (d+e x^2\right )^{9/2} \left (a+b \tan ^{-1}(c x)\right )}{9 e^2}-\frac{b \int \frac{\left (d+e x^2\right )^{3/2} \left (-d \left (96 c^4 d^2+75 c^2 d e-56 e^2\right )+e \left (69 c^4 d^2-520 c^2 d e+336 e^2\right ) x^2\right )}{1+c^2 x^2} \, dx}{3024 c^3 e^2}\\ &=-\frac{b \left (69 c^4 d^2-520 c^2 d e+336 e^2\right ) x \left (d+e x^2\right )^{3/2}}{12096 c^5 e}-\frac{b \left (33 c^2 d-56 e\right ) x \left (d+e x^2\right )^{5/2}}{3024 c^3 e}-\frac{b x \left (d+e x^2\right )^{7/2}}{72 c e}-\frac{d \left (d+e x^2\right )^{7/2} \left (a+b \tan ^{-1}(c x)\right )}{7 e^2}+\frac{\left (d+e x^2\right )^{9/2} \left (a+b \tan ^{-1}(c x)\right )}{9 e^2}-\frac{b \int \frac{\sqrt{d+e x^2} \left (-3 d \left (128 c^6 d^3+123 c^4 d^2 e-248 c^2 d e^2+112 e^3\right )-3 e \left (59 c^6 d^3+712 c^4 d^2 e-1104 c^2 d e^2+448 e^3\right ) x^2\right )}{1+c^2 x^2} \, dx}{12096 c^5 e^2}\\ &=\frac{b \left (59 c^6 d^3+712 c^4 d^2 e-1104 c^2 d e^2+448 e^3\right ) x \sqrt{d+e x^2}}{8064 c^7 e}-\frac{b \left (69 c^4 d^2-520 c^2 d e+336 e^2\right ) x \left (d+e x^2\right )^{3/2}}{12096 c^5 e}-\frac{b \left (33 c^2 d-56 e\right ) x \left (d+e x^2\right )^{5/2}}{3024 c^3 e}-\frac{b x \left (d+e x^2\right )^{7/2}}{72 c e}-\frac{d \left (d+e x^2\right )^{7/2} \left (a+b \tan ^{-1}(c x)\right )}{7 e^2}+\frac{\left (d+e x^2\right )^{9/2} \left (a+b \tan ^{-1}(c x)\right )}{9 e^2}-\frac{b \int \frac{-3 d \left (256 c^8 d^4+187 c^6 d^3 e-1208 c^4 d^2 e^2+1328 c^2 d e^3-448 e^4\right )-3 e \left (315 c^8 d^4+840 c^6 d^3 e-3024 c^4 d^2 e^2+2880 c^2 d e^3-896 e^4\right ) x^2}{\left (1+c^2 x^2\right ) \sqrt{d+e x^2}} \, dx}{24192 c^7 e^2}\\ &=\frac{b \left (59 c^6 d^3+712 c^4 d^2 e-1104 c^2 d e^2+448 e^3\right ) x \sqrt{d+e x^2}}{8064 c^7 e}-\frac{b \left (69 c^4 d^2-520 c^2 d e+336 e^2\right ) x \left (d+e x^2\right )^{3/2}}{12096 c^5 e}-\frac{b \left (33 c^2 d-56 e\right ) x \left (d+e x^2\right )^{5/2}}{3024 c^3 e}-\frac{b x \left (d+e x^2\right )^{7/2}}{72 c e}-\frac{d \left (d+e x^2\right )^{7/2} \left (a+b \tan ^{-1}(c x)\right )}{7 e^2}+\frac{\left (d+e x^2\right )^{9/2} \left (a+b \tan ^{-1}(c x)\right )}{9 e^2}+\frac{\left (b \left (c^2 d-e\right )^4 \left (2 c^2 d+7 e\right )\right ) \int \frac{1}{\left (1+c^2 x^2\right ) \sqrt{d+e x^2}} \, dx}{63 c^9 e^2}+\frac{\left (b \left (315 c^8 d^4+840 c^6 d^3 e-3024 c^4 d^2 e^2+2880 c^2 d e^3-896 e^4\right )\right ) \int \frac{1}{\sqrt{d+e x^2}} \, dx}{8064 c^9 e}\\ &=\frac{b \left (59 c^6 d^3+712 c^4 d^2 e-1104 c^2 d e^2+448 e^3\right ) x \sqrt{d+e x^2}}{8064 c^7 e}-\frac{b \left (69 c^4 d^2-520 c^2 d e+336 e^2\right ) x \left (d+e x^2\right )^{3/2}}{12096 c^5 e}-\frac{b \left (33 c^2 d-56 e\right ) x \left (d+e x^2\right )^{5/2}}{3024 c^3 e}-\frac{b x \left (d+e x^2\right )^{7/2}}{72 c e}-\frac{d \left (d+e x^2\right )^{7/2} \left (a+b \tan ^{-1}(c x)\right )}{7 e^2}+\frac{\left (d+e x^2\right )^{9/2} \left (a+b \tan ^{-1}(c x)\right )}{9 e^2}+\frac{\left (b \left (c^2 d-e\right )^4 \left (2 c^2 d+7 e\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-\left (-c^2 d+e\right ) x^2} \, dx,x,\frac{x}{\sqrt{d+e x^2}}\right )}{63 c^9 e^2}+\frac{\left (b \left (315 c^8 d^4+840 c^6 d^3 e-3024 c^4 d^2 e^2+2880 c^2 d e^3-896 e^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-e x^2} \, dx,x,\frac{x}{\sqrt{d+e x^2}}\right )}{8064 c^9 e}\\ &=\frac{b \left (59 c^6 d^3+712 c^4 d^2 e-1104 c^2 d e^2+448 e^3\right ) x \sqrt{d+e x^2}}{8064 c^7 e}-\frac{b \left (69 c^4 d^2-520 c^2 d e+336 e^2\right ) x \left (d+e x^2\right )^{3/2}}{12096 c^5 e}-\frac{b \left (33 c^2 d-56 e\right ) x \left (d+e x^2\right )^{5/2}}{3024 c^3 e}-\frac{b x \left (d+e x^2\right )^{7/2}}{72 c e}-\frac{d \left (d+e x^2\right )^{7/2} \left (a+b \tan ^{-1}(c x)\right )}{7 e^2}+\frac{\left (d+e x^2\right )^{9/2} \left (a+b \tan ^{-1}(c x)\right )}{9 e^2}+\frac{b \left (c^2 d-e\right )^{7/2} \left (2 c^2 d+7 e\right ) \tan ^{-1}\left (\frac{\sqrt{c^2 d-e} x}{\sqrt{d+e x^2}}\right )}{63 c^9 e^2}+\frac{b \left (315 c^8 d^4+840 c^6 d^3 e-3024 c^4 d^2 e^2+2880 c^2 d e^3-896 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{8064 c^9 e^{3/2}}\\ \end{align*}

Mathematica [C]  time = 0.869719, size = 470, normalized size = 1.36 \[ -\frac{c^2 \sqrt{d+e x^2} \left (384 a c^7 \left (2 d-7 e x^2\right ) \left (d+e x^2\right )^3+b e x \left (3 c^6 \left (558 d^2 e x^2+187 d^3+424 d e^2 x^4+112 e^3 x^6\right )-8 c^4 e \left (453 d^2+242 d e x^2+56 e^2 x^4\right )+48 c^2 e^2 \left (83 d+14 e x^2\right )-1344 e^3\right )\right )+3 b \sqrt{e} \left (3024 c^4 d^2 e^2-840 c^6 d^3 e-315 c^8 d^4-2880 c^2 d e^3+896 e^4\right ) \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right )+192 i b \left (c^2 d-e\right )^{7/2} \left (2 c^2 d+7 e\right ) \log \left (-\frac{252 i c^{10} e^2 \left (\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d-i e x\right )}{b (c x+i) \left (c^2 d-e\right )^{9/2} \left (2 c^2 d+7 e\right )}\right )-192 i b \left (c^2 d-e\right )^{7/2} \left (2 c^2 d+7 e\right ) \log \left (\frac{252 i c^{10} e^2 \left (\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d+i e x\right )}{b (c x-i) \left (c^2 d-e\right )^{9/2} \left (2 c^2 d+7 e\right )}\right )+384 b c^9 \tan ^{-1}(c x) \left (2 d-7 e x^2\right ) \left (d+e x^2\right )^{7/2}}{24192 c^9 e^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]),x]

[Out]

-(c^2*Sqrt[d + e*x^2]*(384*a*c^7*(2*d - 7*e*x^2)*(d + e*x^2)^3 + b*e*x*(-1344*e^3 + 48*c^2*e^2*(83*d + 14*e*x^
2) - 8*c^4*e*(453*d^2 + 242*d*e*x^2 + 56*e^2*x^4) + 3*c^6*(187*d^3 + 558*d^2*e*x^2 + 424*d*e^2*x^4 + 112*e^3*x
^6))) + 384*b*c^9*(2*d - 7*e*x^2)*(d + e*x^2)^(7/2)*ArcTan[c*x] + (192*I)*b*(c^2*d - e)^(7/2)*(2*c^2*d + 7*e)*
Log[((-252*I)*c^10*e^2*(c*d - I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - e)^(9/2)*(2*c^2*d + 7*e)*(
I + c*x))] - (192*I)*b*(c^2*d - e)^(7/2)*(2*c^2*d + 7*e)*Log[((252*I)*c^10*e^2*(c*d + I*e*x + Sqrt[c^2*d - e]*
Sqrt[d + e*x^2]))/(b*(c^2*d - e)^(9/2)*(2*c^2*d + 7*e)*(-I + c*x))] + 3*b*Sqrt[e]*(-315*c^8*d^4 - 840*c^6*d^3*
e + 3024*c^4*d^2*e^2 - 2880*c^2*d*e^3 + 896*e^4)*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/(24192*c^9*e^2)

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Maple [F]  time = 0.652, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( e{x}^{2}+d \right ) ^{{\frac{5}{2}}} \left ( a+b\arctan \left ( cx \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(e*x^2+d)^(5/2)*(a+b*arctan(c*x)),x)

[Out]

int(x^3*(e*x^2+d)^(5/2)*(a+b*arctan(c*x)),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x^2+d)^(5/2)*(a+b*arctan(c*x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x^2+d)^(5/2)*(a+b*arctan(c*x)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(e*x**2+d)**(5/2)*(a+b*atan(c*x)),x)

[Out]

Timed out

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Giac [B]  time = 1.93645, size = 1481, normalized size = 4.29 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x^2+d)^(5/2)*(a+b*arctan(c*x)),x, algorithm="giac")

[Out]

1/15*(3*(x^2*e + d)^(5/2) - 5*(x^2*e + d)^(3/2)*d)*a*d^2*e^(-2) + 1/240*(16*(3*(x^2*e + d)^(5/2) - 5*(x^2*e +
d)^(3/2)*d)*arctan(c*x)*e^(-2) - (2*sqrt(x^2*e + d)*x*(6*x^2/c^2 + (7*c^10*d*e^2 - 12*c^8*e^3)*e^(-3)/c^12) +
(15*c^4*d^2 + 20*c^2*d*e - 24*e^2)*e^(-3/2)*log((x*e^(1/2) - sqrt(x^2*e + d))^2)/c^6 + 16*(2*c^6*d^3*e^(1/2) -
c^4*d^2*e^(3/2) - 4*c^2*d*e^(5/2) + 3*e^(7/2))*arctan(1/2*((x*e^(1/2) - sqrt(x^2*e + d))^2*c^2 - c^2*d + 2*e)
*e^(-1/2)/sqrt(c^2*d - e))*e^(-5/2)/(sqrt(c^2*d - e)*c^6))*c)*b*d^2 + 1/241920*(768*(35*(x^2*e + d)^(9/2) - 13
5*(x^2*e + d)^(7/2)*d + 189*(x^2*e + d)^(5/2)*d^2 - 105*(x^2*e + d)^(3/2)*d^3)*a*e^(-4) + (768*(35*(x^2*e + d)
^(9/2) - 135*(x^2*e + d)^(7/2)*d + 189*(x^2*e + d)^(5/2)*d^2 - 105*(x^2*e + d)^(3/2)*d^3)*arctan(c*x)*e^(-4) -
(2*(2*(20*x^2*(42*x^2/c^2 + (15*c^28*d*e^11 - 56*c^26*e^12)*e^(-12)/c^30) - (423*c^28*d^2*e^10 + 520*c^26*d*e
^11 - 1680*c^24*e^12)*e^(-12)/c^30)*x^2 + 3*(551*c^28*d^3*e^9 + 584*c^26*d^2*e^10 + 880*c^24*d*e^11 - 2240*c^2
2*e^12)*e^(-12)/c^30)*sqrt(x^2*e + d)*x + 3*(1575*c^8*d^4 + 840*c^6*d^3*e + 1008*c^4*d^2*e^2 + 2880*c^2*d*e^3
- 4480*e^4)*e^(-7/2)*log((x*e^(1/2) - sqrt(x^2*e + d))^2)/c^10 + 768*(16*c^10*d^5*e^(1/2) - 8*c^8*d^4*e^(3/2)
- 2*c^6*d^3*e^(5/2) - c^4*d^2*e^(7/2) - 40*c^2*d*e^(9/2) + 35*e^(11/2))*arctan(1/2*((x*e^(1/2) - sqrt(x^2*e +
d))^2*c^2 - c^2*d + 2*e)*e^(-1/2)/sqrt(c^2*d - e))*e^(-9/2)/(sqrt(c^2*d - e)*c^10))*c)*b)*e^2 + 1/1680*(32*(15
*(x^2*e + d)^(7/2) - 42*(x^2*e + d)^(5/2)*d + 35*(x^2*e + d)^(3/2)*d^2)*a*d*e^(-3) + (32*(15*(x^2*e + d)^(7/2)
- 42*(x^2*e + d)^(5/2)*d + 35*(x^2*e + d)^(3/2)*d^2)*arctan(c*x)*e^(-3) - (2*(2*x^2*(20*x^2/c^2 + (11*c^18*d*
e^6 - 30*c^16*e^7)*e^(-7)/c^20) - (41*c^18*d^2*e^5 + 54*c^16*d*e^6 - 120*c^14*e^7)*e^(-7)/c^20)*sqrt(x^2*e + d
)*x - (105*c^6*d^3 + 70*c^4*d^2*e + 168*c^2*d*e^2 - 240*e^3)*e^(-5/2)*log((x*e^(1/2) - sqrt(x^2*e + d))^2)/c^8
- 32*(8*c^8*d^4*e^(1/2) - 4*c^6*d^3*e^(3/2) - c^4*d^2*e^(5/2) - 18*c^2*d*e^(7/2) + 15*e^(9/2))*arctan(1/2*((x
*e^(1/2) - sqrt(x^2*e + d))^2*c^2 - c^2*d + 2*e)*e^(-1/2)/sqrt(c^2*d - e))*e^(-7/2)/(sqrt(c^2*d - e)*c^8))*c)*
b*d)*e