3.12.83 \(\int x^3 (d+e x^2)^{3/2} (a+b \text {ArcTan}(c x)) \, dx\) [1183]

Optimal. Leaf size=279 \[ \frac {b \left (3 c^4 d^2+54 c^2 d e-40 e^2\right ) x \sqrt {d+e x^2}}{560 c^5 e}-\frac {b \left (13 c^2 d-30 e\right ) x \left (d+e x^2\right )^{3/2}}{840 c^3 e}-\frac {b x \left (d+e x^2\right )^{5/2}}{42 c e}-\frac {d \left (d+e x^2\right )^{5/2} (a+b \text {ArcTan}(c x))}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} (a+b \text {ArcTan}(c x))}{7 e^2}+\frac {b \left (c^2 d-e\right )^{5/2} \left (2 c^2 d+5 e\right ) \text {ArcTan}\left (\frac {\sqrt {c^2 d-e} x}{\sqrt {d+e x^2}}\right )}{35 c^7 e^2}+\frac {b \left (35 c^6 d^3+70 c^4 d^2 e-168 c^2 d e^2+80 e^3\right ) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{560 c^7 e^{3/2}} \]

[Out]

-1/840*b*(13*c^2*d-30*e)*x*(e*x^2+d)^(3/2)/c^3/e-1/42*b*x*(e*x^2+d)^(5/2)/c/e-1/5*d*(e*x^2+d)^(5/2)*(a+b*arcta
n(c*x))/e^2+1/7*(e*x^2+d)^(7/2)*(a+b*arctan(c*x))/e^2+1/35*b*(c^2*d-e)^(5/2)*(2*c^2*d+5*e)*arctan(x*(c^2*d-e)^
(1/2)/(e*x^2+d)^(1/2))/c^7/e^2+1/560*b*(35*c^6*d^3+70*c^4*d^2*e-168*c^2*d*e^2+80*e^3)*arctanh(x*e^(1/2)/(e*x^2
+d)^(1/2))/c^7/e^(3/2)+1/560*b*(3*c^4*d^2+54*c^2*d*e-40*e^2)*x*(e*x^2+d)^(1/2)/c^5/e

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Rubi [A]
time = 0.33, antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {272, 45, 5096, 12, 542, 537, 223, 212, 385, 209} \begin {gather*} \frac {\left (d+e x^2\right )^{7/2} (a+b \text {ArcTan}(c x))}{7 e^2}-\frac {d \left (d+e x^2\right )^{5/2} (a+b \text {ArcTan}(c x))}{5 e^2}+\frac {b \left (c^2 d-e\right )^{5/2} \left (2 c^2 d+5 e\right ) \text {ArcTan}\left (\frac {x \sqrt {c^2 d-e}}{\sqrt {d+e x^2}}\right )}{35 c^7 e^2}-\frac {b x \left (13 c^2 d-30 e\right ) \left (d+e x^2\right )^{3/2}}{840 c^3 e}+\frac {b x \left (3 c^4 d^2+54 c^2 d e-40 e^2\right ) \sqrt {d+e x^2}}{560 c^5 e}+\frac {b \left (35 c^6 d^3+70 c^4 d^2 e-168 c^2 d e^2+80 e^3\right ) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{560 c^7 e^{3/2}}-\frac {b x \left (d+e x^2\right )^{5/2}}{42 c e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(3*c^4*d^2 + 54*c^2*d*e - 40*e^2)*x*Sqrt[d + e*x^2])/(560*c^5*e) - (b*(13*c^2*d - 30*e)*x*(d + e*x^2)^(3/2)
)/(840*c^3*e) - (b*x*(d + e*x^2)^(5/2))/(42*c*e) - (d*(d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]))/(5*e^2) + ((d + e
*x^2)^(7/2)*(a + b*ArcTan[c*x]))/(7*e^2) + (b*(c^2*d - e)^(5/2)*(2*c^2*d + 5*e)*ArcTan[(Sqrt[c^2*d - e]*x)/Sqr
t[d + e*x^2]])/(35*c^7*e^2) + (b*(35*c^6*d^3 + 70*c^4*d^2*e - 168*c^2*d*e^2 + 80*e^3)*ArcTanh[(Sqrt[e]*x)/Sqrt
[d + e*x^2]])/(560*c^7*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 45

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 209

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

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

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 272

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 385

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 537

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 542

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 5096

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 )^{3/2} \left (a+b \tan ^{-1}(c x)\right ) \, dx &=-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \tan ^{-1}(c x)\right )}{7 e^2}-(b c) \int \frac {\left (d+e x^2\right )^{5/2} \left (-2 d+5 e x^2\right )}{35 e^2 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \tan ^{-1}(c x)\right )}{7 e^2}-\frac {(b c) \int \frac {\left (d+e x^2\right )^{5/2} \left (-2 d+5 e x^2\right )}{1+c^2 x^2} \, dx}{35 e^2}\\ &=-\frac {b x \left (d+e x^2\right )^{5/2}}{42 c e}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \tan ^{-1}(c x)\right )}{7 e^2}-\frac {b \int \frac {\left (d+e x^2\right )^{3/2} \left (-d \left (12 c^2 d+5 e\right )+\left (13 c^2 d-30 e\right ) e x^2\right )}{1+c^2 x^2} \, dx}{210 c e^2}\\ &=-\frac {b \left (13 c^2 d-30 e\right ) x \left (d+e x^2\right )^{3/2}}{840 c^3 e}-\frac {b x \left (d+e x^2\right )^{5/2}}{42 c e}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \tan ^{-1}(c x)\right )}{7 e^2}-\frac {b \int \frac {\sqrt {d+e x^2} \left (-3 d \left (16 c^4 d^2+11 c^2 d e-10 e^2\right )-3 e \left (3 c^4 d^2+54 c^2 d e-40 e^2\right ) x^2\right )}{1+c^2 x^2} \, dx}{840 c^3 e^2}\\ &=\frac {b \left (3 c^4 d^2+54 c^2 d e-40 e^2\right ) x \sqrt {d+e x^2}}{560 c^5 e}-\frac {b \left (13 c^2 d-30 e\right ) x \left (d+e x^2\right )^{3/2}}{840 c^3 e}-\frac {b x \left (d+e x^2\right )^{5/2}}{42 c e}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \tan ^{-1}(c x)\right )}{7 e^2}-\frac {b \int \frac {-3 d \left (32 c^6 d^3+19 c^4 d^2 e-74 c^2 d e^2+40 e^3\right )-3 e \left (35 c^6 d^3+70 c^4 d^2 e-168 c^2 d e^2+80 e^3\right ) x^2}{\left (1+c^2 x^2\right ) \sqrt {d+e x^2}} \, dx}{1680 c^5 e^2}\\ &=\frac {b \left (3 c^4 d^2+54 c^2 d e-40 e^2\right ) x \sqrt {d+e x^2}}{560 c^5 e}-\frac {b \left (13 c^2 d-30 e\right ) x \left (d+e x^2\right )^{3/2}}{840 c^3 e}-\frac {b x \left (d+e x^2\right )^{5/2}}{42 c e}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \tan ^{-1}(c x)\right )}{7 e^2}+\frac {\left (b \left (c^2 d-e\right )^3 \left (2 c^2 d+5 e\right )\right ) \int \frac {1}{\left (1+c^2 x^2\right ) \sqrt {d+e x^2}} \, dx}{35 c^7 e^2}+\frac {\left (b \left (35 c^6 d^3+70 c^4 d^2 e-168 c^2 d e^2+80 e^3\right )\right ) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{560 c^7 e}\\ &=\frac {b \left (3 c^4 d^2+54 c^2 d e-40 e^2\right ) x \sqrt {d+e x^2}}{560 c^5 e}-\frac {b \left (13 c^2 d-30 e\right ) x \left (d+e x^2\right )^{3/2}}{840 c^3 e}-\frac {b x \left (d+e x^2\right )^{5/2}}{42 c e}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \tan ^{-1}(c x)\right )}{7 e^2}+\frac {\left (b \left (c^2 d-e\right )^3 \left (2 c^2 d+5 e\right )\right ) \text {Subst}\left (\int \frac {1}{1-\left (-c^2 d+e\right ) x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{35 c^7 e^2}+\frac {\left (b \left (35 c^6 d^3+70 c^4 d^2 e-168 c^2 d e^2+80 e^3\right )\right ) \text {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{560 c^7 e}\\ &=\frac {b \left (3 c^4 d^2+54 c^2 d e-40 e^2\right ) x \sqrt {d+e x^2}}{560 c^5 e}-\frac {b \left (13 c^2 d-30 e\right ) x \left (d+e x^2\right )^{3/2}}{840 c^3 e}-\frac {b x \left (d+e x^2\right )^{5/2}}{42 c e}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}+\frac {\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 )^{5/2} \left (2 c^2 d+5 e\right ) \tan ^{-1}\left (\frac {\sqrt {c^2 d-e} x}{\sqrt {d+e x^2}}\right )}{35 c^7 e^2}+\frac {b \left (35 c^6 d^3+70 c^4 d^2 e-168 c^2 d e^2+80 e^3\right ) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{560 c^7 e^{3/2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.47, size = 418, normalized size = 1.50 \begin {gather*} -\frac {c^2 \sqrt {d+e x^2} \left (48 a c^5 \left (2 d-5 e x^2\right ) \left (d+e x^2\right )^2+b e x \left (120 e^2-6 c^2 e \left (37 d+10 e x^2\right )+c^4 \left (57 d^2+106 d e x^2+40 e^2 x^4\right )\right )\right )+48 b c^7 \left (2 d-5 e x^2\right ) \left (d+e x^2\right )^{5/2} \text {ArcTan}(c x)+24 i b \left (c^2 d-e\right )^{5/2} \left (2 c^2 d+5 e\right ) \log \left (-\frac {140 i c^8 e^2 \left (c d-i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \left (c^2 d-e\right )^{7/2} \left (2 c^2 d+5 e\right ) (i+c x)}\right )-24 i b \left (c^2 d-e\right )^{5/2} \left (2 c^2 d+5 e\right ) \log \left (\frac {140 i c^8 e^2 \left (c d+i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \left (c^2 d-e\right )^{7/2} \left (2 c^2 d+5 e\right ) (-i+c x)}\right )-3 b \sqrt {e} \left (35 c^6 d^3+70 c^4 d^2 e-168 c^2 d e^2+80 e^3\right ) \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{1680 c^7 e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/1680*(c^2*Sqrt[d + e*x^2]*(48*a*c^5*(2*d - 5*e*x^2)*(d + e*x^2)^2 + b*e*x*(120*e^2 - 6*c^2*e*(37*d + 10*e*x
^2) + c^4*(57*d^2 + 106*d*e*x^2 + 40*e^2*x^4))) + 48*b*c^7*(2*d - 5*e*x^2)*(d + e*x^2)^(5/2)*ArcTan[c*x] + (24
*I)*b*(c^2*d - e)^(5/2)*(2*c^2*d + 5*e)*Log[((-140*I)*c^8*e^2*(c*d - I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))
/(b*(c^2*d - e)^(7/2)*(2*c^2*d + 5*e)*(I + c*x))] - (24*I)*b*(c^2*d - e)^(5/2)*(2*c^2*d + 5*e)*Log[((140*I)*c^
8*e^2*(c*d + I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - e)^(7/2)*(2*c^2*d + 5*e)*(-I + c*x))] - 3*b
*Sqrt[e]*(35*c^6*d^3 + 70*c^4*d^2*e - 168*c^2*d*e^2 + 80*e^3)*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/(c^7*e^2)

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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int x^{3} \left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arctan \left (c x \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/35*(5*(x^2*e + d)^(5/2)*x^2*e^(-1) - 2*(x^2*e + d)^(5/2)*d*e^(-2))*a + 1/2*b*integrate(2*(x^5*e + d*x^3)*sqr
t(x^2*e + d)*arctan(c*x), x)

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Fricas [A]
time = 17.15, size = 783, normalized size = 2.81 \begin {gather*} \left [\frac {{\left (3 \, {\left (35 \, b c^{6} d^{3} + 70 \, b c^{4} d^{2} e - 168 \, b c^{2} d e^{2} + 80 \, b e^{3}\right )} e^{\frac {1}{2}} \log \left (-2 \, x^{2} e - 2 \, \sqrt {x^{2} e + d} x e^{\frac {1}{2}} - d\right ) + 24 \, {\left (2 \, b c^{6} d^{3} + b c^{4} d^{2} e - 8 \, b c^{2} d e^{2} + 5 \, b e^{3}\right )} \sqrt {-c^{2} d + e} \log \left (\frac {c^{4} d^{2} x^{4} - 6 \, c^{2} d^{2} x^{2} + 8 \, x^{4} e^{2} + 4 \, {\left (c^{2} d x^{3} - 2 \, x^{3} e - d x\right )} \sqrt {-c^{2} d + e} \sqrt {x^{2} e + d} + d^{2} - 8 \, {\left (c^{2} d x^{4} - d x^{2}\right )} e}{c^{4} x^{4} + 2 \, c^{2} x^{2} + 1}\right ) - 2 \, {\left (96 \, a c^{7} d^{3} - 48 \, {\left (5 \, b c^{7} x^{6} e^{3} + 8 \, b c^{7} d x^{4} e^{2} + b c^{7} d^{2} x^{2} e - 2 \, b c^{7} d^{3}\right )} \arctan \left (c x\right ) - 20 \, {\left (12 \, a c^{7} x^{6} - 2 \, b c^{6} x^{5} + 3 \, b c^{4} x^{3} - 6 \, b c^{2} x\right )} e^{3} - 2 \, {\left (192 \, a c^{7} d x^{4} - 53 \, b c^{6} d x^{3} + 111 \, b c^{4} d x\right )} e^{2} - 3 \, {\left (16 \, a c^{7} d^{2} x^{2} - 19 \, b c^{6} d^{2} x\right )} e\right )} \sqrt {x^{2} e + d}\right )} e^{\left (-2\right )}}{3360 \, c^{7}}, \frac {{\left (3 \, {\left (35 \, b c^{6} d^{3} + 70 \, b c^{4} d^{2} e - 168 \, b c^{2} d e^{2} + 80 \, b e^{3}\right )} e^{\frac {1}{2}} \log \left (-2 \, x^{2} e - 2 \, \sqrt {x^{2} e + d} x e^{\frac {1}{2}} - d\right ) + 48 \, {\left (2 \, b c^{6} d^{3} + b c^{4} d^{2} e - 8 \, b c^{2} d e^{2} + 5 \, b e^{3}\right )} \sqrt {c^{2} d - e} \arctan \left (\frac {{\left (c^{2} d x^{2} - 2 \, x^{2} e - d\right )} \sqrt {c^{2} d - e} \sqrt {x^{2} e + d}}{2 \, {\left (c^{2} d^{2} x - x^{3} e^{2} + {\left (c^{2} d x^{3} - d x\right )} e\right )}}\right ) - 2 \, {\left (96 \, a c^{7} d^{3} - 48 \, {\left (5 \, b c^{7} x^{6} e^{3} + 8 \, b c^{7} d x^{4} e^{2} + b c^{7} d^{2} x^{2} e - 2 \, b c^{7} d^{3}\right )} \arctan \left (c x\right ) - 20 \, {\left (12 \, a c^{7} x^{6} - 2 \, b c^{6} x^{5} + 3 \, b c^{4} x^{3} - 6 \, b c^{2} x\right )} e^{3} - 2 \, {\left (192 \, a c^{7} d x^{4} - 53 \, b c^{6} d x^{3} + 111 \, b c^{4} d x\right )} e^{2} - 3 \, {\left (16 \, a c^{7} d^{2} x^{2} - 19 \, b c^{6} d^{2} x\right )} e\right )} \sqrt {x^{2} e + d}\right )} e^{\left (-2\right )}}{3360 \, c^{7}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/3360*(3*(35*b*c^6*d^3 + 70*b*c^4*d^2*e - 168*b*c^2*d*e^2 + 80*b*e^3)*e^(1/2)*log(-2*x^2*e - 2*sqrt(x^2*e +
d)*x*e^(1/2) - d) + 24*(2*b*c^6*d^3 + b*c^4*d^2*e - 8*b*c^2*d*e^2 + 5*b*e^3)*sqrt(-c^2*d + e)*log((c^4*d^2*x^4
- 6*c^2*d^2*x^2 + 8*x^4*e^2 + 4*(c^2*d*x^3 - 2*x^3*e - d*x)*sqrt(-c^2*d + e)*sqrt(x^2*e + d) + d^2 - 8*(c^2*d
*x^4 - d*x^2)*e)/(c^4*x^4 + 2*c^2*x^2 + 1)) - 2*(96*a*c^7*d^3 - 48*(5*b*c^7*x^6*e^3 + 8*b*c^7*d*x^4*e^2 + b*c^
7*d^2*x^2*e - 2*b*c^7*d^3)*arctan(c*x) - 20*(12*a*c^7*x^6 - 2*b*c^6*x^5 + 3*b*c^4*x^3 - 6*b*c^2*x)*e^3 - 2*(19
2*a*c^7*d*x^4 - 53*b*c^6*d*x^3 + 111*b*c^4*d*x)*e^2 - 3*(16*a*c^7*d^2*x^2 - 19*b*c^6*d^2*x)*e)*sqrt(x^2*e + d)
)*e^(-2)/c^7, 1/3360*(3*(35*b*c^6*d^3 + 70*b*c^4*d^2*e - 168*b*c^2*d*e^2 + 80*b*e^3)*e^(1/2)*log(-2*x^2*e - 2*
sqrt(x^2*e + d)*x*e^(1/2) - d) + 48*(2*b*c^6*d^3 + b*c^4*d^2*e - 8*b*c^2*d*e^2 + 5*b*e^3)*sqrt(c^2*d - e)*arct
an(1/2*(c^2*d*x^2 - 2*x^2*e - d)*sqrt(c^2*d - e)*sqrt(x^2*e + d)/(c^2*d^2*x - x^3*e^2 + (c^2*d*x^3 - d*x)*e))
- 2*(96*a*c^7*d^3 - 48*(5*b*c^7*x^6*e^3 + 8*b*c^7*d*x^4*e^2 + b*c^7*d^2*x^2*e - 2*b*c^7*d^3)*arctan(c*x) - 20*
(12*a*c^7*x^6 - 2*b*c^6*x^5 + 3*b*c^4*x^3 - 6*b*c^2*x)*e^3 - 2*(192*a*c^7*d*x^4 - 53*b*c^6*d*x^3 + 111*b*c^4*d
*x)*e^2 - 3*(16*a*c^7*d^2*x^2 - 19*b*c^6*d^2*x)*e)*sqrt(x^2*e + d))*e^(-2)/c^7]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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