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

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

[Out]

-1/12096*b*(69*c^4*d^2-520*c^2*d*e+336*e^2)*x*(e*x^2+d)^(3/2)/c^5/e-1/3024*b*(33*c^2*d-56*e)*x*(e*x^2+d)^(5/2)
/c^3/e-1/72*b*x*(e*x^2+d)^(7/2)/c/e-1/7*d*(e*x^2+d)^(7/2)*(a+b*arctan(c*x))/e^2+1/9*(e*x^2+d)^(9/2)*(a+b*arcta
n(c*x))/e^2+1/63*b*(c^2*d-e)^(7/2)*(2*c^2*d+7*e)*arctan(x*(c^2*d-e)^(1/2)/(e*x^2+d)^(1/2))/c^9/e^2+1/8064*b*(3
15*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(x*e^(1/2)/(e*x^2+d)^(1/2))/c^9/e^(3/
2)+1/8064*b*(59*c^6*d^3+712*c^4*d^2*e-1104*c^2*d*e^2+448*e^3)*x*(e*x^2+d)^(1/2)/c^7/e

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Rubi [A]  time = 0.58, antiderivative size = 345, normalized size of antiderivative = 1.00, 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 12

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

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 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])

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 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 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 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 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 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 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]))

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*}

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Mathematica [C]  time = 0.93, size = 470, normalized size = 1.36 \[ -\frac {c^2 \sqrt {d+e x^2} \left (384 a c^7 \left (2 d-7 e x^2\right ) \left (d+e x^2\right )^3+b e x \left (3 c^6 \left (187 d^3+558 d^2 e x^2+424 d e^2 x^4+112 e^3 x^6\right )-8 c^4 e \left (453 d^2+242 d e x^2+56 e^2 x^4\right )+48 c^2 e^2 \left (83 d+14 e x^2\right )-1344 e^3\right )\right )+384 b c^9 \tan ^{-1}(c x) \left (2 d-7 e x^2\right ) \left (d+e x^2\right )^{7/2}+192 i b \left (c^2 d-e\right )^{7/2} \left (2 c^2 d+7 e\right ) \log \left (-\frac {252 i c^{10} e^2 \left (\sqrt {c^2 d-e} \sqrt {d+e x^2}+c d-i e x\right )}{b (c x+i) \left (c^2 d-e\right )^{9/2} \left (2 c^2 d+7 e\right )}\right )-192 i b \left (c^2 d-e\right )^{7/2} \left (2 c^2 d+7 e\right ) \log \left (\frac {252 i c^{10} e^2 \left (\sqrt {c^2 d-e} \sqrt {d+e x^2}+c d+i e x\right )}{b (c x-i) \left (c^2 d-e\right )^{9/2} \left (2 c^2 d+7 e\right )}\right )+3 b \sqrt {e} \left (-315 c^8 d^4-840 c^6 d^3 e+3024 c^4 d^2 e^2-2880 c^2 d e^3+896 e^4\right ) \log \left (\sqrt {e} \sqrt {d+e x^2}+e x\right )}{24192 c^9 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/24192*(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 + 1
12*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]])/(c^9*e^2)

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fricas [A]  time = 49.89, size = 1978, normalized size = 5.73 \[ \text {result too large to display} \]

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]

[-1/48384*(3*(315*b*c^8*d^4 + 840*b*c^6*d^3*e - 3024*b*c^4*d^2*e^2 + 2880*b*c^2*d*e^3 - 896*b*e^4)*sqrt(e)*log
(-2*e*x^2 + 2*sqrt(e*x^2 + d)*sqrt(e)*x - d) + 192*(2*b*c^8*d^4 + b*c^6*d^3*e - 15*b*c^4*d^2*e^2 + 19*b*c^2*d*
e^3 - 7*b*e^4)*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*(2688*a*c^9*e^4*x^8
+ 7296*a*c^9*d*e^3*x^6 - 336*b*c^8*e^4*x^7 + 5760*a*c^9*d^2*e^2*x^4 + 384*a*c^9*d^3*e*x^2 - 768*a*c^9*d^4 - 8*
(159*b*c^8*d*e^3 - 56*b*c^6*e^4)*x^5 - 2*(837*b*c^8*d^2*e^2 - 968*b*c^6*d*e^3 + 336*b*c^4*e^4)*x^3 - 3*(187*b*
c^8*d^3*e - 1208*b*c^6*d^2*e^2 + 1328*b*c^4*d*e^3 - 448*b*c^2*e^4)*x + 384*(7*b*c^9*e^4*x^8 + 19*b*c^9*d*e^3*x
^6 + 15*b*c^9*d^2*e^2*x^4 + b*c^9*d^3*e*x^2 - 2*b*c^9*d^4)*arctan(c*x))*sqrt(e*x^2 + d))/(c^9*e^2), 1/48384*(3
84*(2*b*c^8*d^4 + b*c^6*d^3*e - 15*b*c^4*d^2*e^2 + 19*b*c^2*d*e^3 - 7*b*e^4)*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*(315*b*c^8*d^
4 + 840*b*c^6*d^3*e - 3024*b*c^4*d^2*e^2 + 2880*b*c^2*d*e^3 - 896*b*e^4)*sqrt(e)*log(-2*e*x^2 + 2*sqrt(e*x^2 +
d)*sqrt(e)*x - d) + 2*(2688*a*c^9*e^4*x^8 + 7296*a*c^9*d*e^3*x^6 - 336*b*c^8*e^4*x^7 + 5760*a*c^9*d^2*e^2*x^4
+ 384*a*c^9*d^3*e*x^2 - 768*a*c^9*d^4 - 8*(159*b*c^8*d*e^3 - 56*b*c^6*e^4)*x^5 - 2*(837*b*c^8*d^2*e^2 - 968*b
*c^6*d*e^3 + 336*b*c^4*e^4)*x^3 - 3*(187*b*c^8*d^3*e - 1208*b*c^6*d^2*e^2 + 1328*b*c^4*d*e^3 - 448*b*c^2*e^4)*
x + 384*(7*b*c^9*e^4*x^8 + 19*b*c^9*d*e^3*x^6 + 15*b*c^9*d^2*e^2*x^4 + b*c^9*d^3*e*x^2 - 2*b*c^9*d^4)*arctan(c
*x))*sqrt(e*x^2 + d))/(c^9*e^2), -1/24192*(3*(315*b*c^8*d^4 + 840*b*c^6*d^3*e - 3024*b*c^4*d^2*e^2 + 2880*b*c^
2*d*e^3 - 896*b*e^4)*sqrt(-e)*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) + 96*(2*b*c^8*d^4 + b*c^6*d^3*e - 15*b*c^4*d^
2*e^2 + 19*b*c^2*d*e^3 - 7*b*e^4)*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)) - (26
88*a*c^9*e^4*x^8 + 7296*a*c^9*d*e^3*x^6 - 336*b*c^8*e^4*x^7 + 5760*a*c^9*d^2*e^2*x^4 + 384*a*c^9*d^3*e*x^2 - 7
68*a*c^9*d^4 - 8*(159*b*c^8*d*e^3 - 56*b*c^6*e^4)*x^5 - 2*(837*b*c^8*d^2*e^2 - 968*b*c^6*d*e^3 + 336*b*c^4*e^4
)*x^3 - 3*(187*b*c^8*d^3*e - 1208*b*c^6*d^2*e^2 + 1328*b*c^4*d*e^3 - 448*b*c^2*e^4)*x + 384*(7*b*c^9*e^4*x^8 +
19*b*c^9*d*e^3*x^6 + 15*b*c^9*d^2*e^2*x^4 + b*c^9*d^3*e*x^2 - 2*b*c^9*d^4)*arctan(c*x))*sqrt(e*x^2 + d))/(c^9
*e^2), 1/24192*(192*(2*b*c^8*d^4 + b*c^6*d^3*e - 15*b*c^4*d^2*e^2 + 19*b*c^2*d*e^3 - 7*b*e^4)*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*(315*b*c^8*d^4 + 840*b*c^6*d^3*e - 3024*b*c^4*d^2*e^2 + 2880*b*c^2*d*e^3 - 896*b*e^4)*sqrt(-e)*arctan(sqrt
(-e)*x/sqrt(e*x^2 + d)) + (2688*a*c^9*e^4*x^8 + 7296*a*c^9*d*e^3*x^6 - 336*b*c^8*e^4*x^7 + 5760*a*c^9*d^2*e^2*
x^4 + 384*a*c^9*d^3*e*x^2 - 768*a*c^9*d^4 - 8*(159*b*c^8*d*e^3 - 56*b*c^6*e^4)*x^5 - 2*(837*b*c^8*d^2*e^2 - 96
8*b*c^6*d*e^3 + 336*b*c^4*e^4)*x^3 - 3*(187*b*c^8*d^3*e - 1208*b*c^6*d^2*e^2 + 1328*b*c^4*d*e^3 - 448*b*c^2*e^
4)*x + 384*(7*b*c^9*e^4*x^8 + 19*b*c^9*d*e^3*x^6 + 15*b*c^9*d^2*e^2*x^4 + b*c^9*d^3*e*x^2 - 2*b*c^9*d^4)*arcta
n(c*x))*sqrt(e*x^2 + d))/(c^9*e^2)]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

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]

sage0*x

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maple [F]  time = 1.15, size = 0, normalized size = 0.00 \[ \int x^{3} \left (e \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \arctan \left (c x \right )\right )\, dx \]

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]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{63} \, {\left (\frac {7 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{e} - \frac {2 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} d}{e^{2}}\right )} a + \frac {1}{2} \, b \int 2 \, {\left (e^{2} x^{7} + 2 \, d e x^{5} + d^{2} x^{3}\right )} \sqrt {e x^{2} + d} \arctan \left (c x\right )\,{d x} \]

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]

1/63*(7*(e*x^2 + d)^(7/2)*x^2/e - 2*(e*x^2 + d)^(7/2)*d/e^2)*a + 1/2*b*integrate(2*(e^2*x^7 + 2*d*e*x^5 + d^2*
x^3)*sqrt(e*x^2 + d)*arctan(c*x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \left (a + b \operatorname {atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac {5}{2}}\, dx \]

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]

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

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