3.1173 \(\int x^3 \sqrt{d+e x^2} (a+b \tan ^{-1}(c x)) \, dx\)

Optimal. Leaf size=223 \[ \frac{\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}-\frac{d \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{3 e^2}+\frac{b \left (15 c^4 d^2+20 c^2 d e-24 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{120 c^5 e^{3/2}}+\frac{b \left (c^2 d-e\right )^{3/2} \left (2 c^2 d+3 e\right ) \tan ^{-1}\left (\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right )}{15 c^5 e^2}-\frac{b x \left (c^2 d-12 e\right ) \sqrt{d+e x^2}}{120 c^3 e}-\frac{b x \left (d+e x^2\right )^{3/2}}{20 c e} \]

[Out]

-(b*(c^2*d - 12*e)*x*Sqrt[d + e*x^2])/(120*c^3*e) - (b*x*(d + e*x^2)^(3/2))/(20*c*e) - (d*(d + e*x^2)^(3/2)*(a
+ b*ArcTan[c*x]))/(3*e^2) + ((d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]))/(5*e^2) + (b*(c^2*d - e)^(3/2)*(2*c^2*d +
3*e)*ArcTan[(Sqrt[c^2*d - e]*x)/Sqrt[d + e*x^2]])/(15*c^5*e^2) + (b*(15*c^4*d^2 + 20*c^2*d*e - 24*e^2)*ArcTan
h[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(120*c^5*e^(3/2))

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Rubi [A]  time = 0.367628, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}-\frac{d \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{3 e^2}+\frac{b \left (15 c^4 d^2+20 c^2 d e-24 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{120 c^5 e^{3/2}}+\frac{b \left (c^2 d-e\right )^{3/2} \left (2 c^2 d+3 e\right ) \tan ^{-1}\left (\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right )}{15 c^5 e^2}-\frac{b x \left (c^2 d-12 e\right ) \sqrt{d+e x^2}}{120 c^3 e}-\frac{b x \left (d+e x^2\right )^{3/2}}{20 c e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*(c^2*d - 12*e)*x*Sqrt[d + e*x^2])/(120*c^3*e) - (b*x*(d + e*x^2)^(3/2))/(20*c*e) - (d*(d + e*x^2)^(3/2)*(a
+ b*ArcTan[c*x]))/(3*e^2) + ((d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]))/(5*e^2) + (b*(c^2*d - e)^(3/2)*(2*c^2*d +
3*e)*ArcTan[(Sqrt[c^2*d - e]*x)/Sqrt[d + e*x^2]])/(15*c^5*e^2) + (b*(15*c^4*d^2 + 20*c^2*d*e - 24*e^2)*ArcTan
h[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(120*c^5*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 \sqrt{d+e x^2} \left (a+b \tan ^{-1}(c x)\right ) \, dx &=-\frac{d \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{3 e^2}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}-(b c) \int \frac{\left (d+e x^2\right )^{3/2} \left (-2 d+3 e x^2\right )}{15 e^2 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac{d \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{3 e^2}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}-\frac{(b c) \int \frac{\left (d+e x^2\right )^{3/2} \left (-2 d+3 e x^2\right )}{1+c^2 x^2} \, dx}{15 e^2}\\ &=-\frac{b x \left (d+e x^2\right )^{3/2}}{20 c e}-\frac{d \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{3 e^2}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}-\frac{b \int \frac{\sqrt{d+e x^2} \left (-d \left (8 c^2 d+3 e\right )+\left (c^2 d-12 e\right ) e x^2\right )}{1+c^2 x^2} \, dx}{60 c e^2}\\ &=-\frac{b \left (c^2 d-12 e\right ) x \sqrt{d+e x^2}}{120 c^3 e}-\frac{b x \left (d+e x^2\right )^{3/2}}{20 c e}-\frac{d \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{3 e^2}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}-\frac{b \int \frac{-d \left (16 c^4 d^2+7 c^2 d e-12 e^2\right )-e \left (15 c^4 d^2+20 c^2 d e-24 e^2\right ) x^2}{\left (1+c^2 x^2\right ) \sqrt{d+e x^2}} \, dx}{120 c^3 e^2}\\ &=-\frac{b \left (c^2 d-12 e\right ) x \sqrt{d+e x^2}}{120 c^3 e}-\frac{b x \left (d+e x^2\right )^{3/2}}{20 c e}-\frac{d \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{3 e^2}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}+\frac{\left (b \left (c^2 d-e\right )^2 \left (2 c^2 d+3 e\right )\right ) \int \frac{1}{\left (1+c^2 x^2\right ) \sqrt{d+e x^2}} \, dx}{15 c^5 e^2}+\frac{\left (b \left (15 c^4 d^2+20 c^2 d e-24 e^2\right )\right ) \int \frac{1}{\sqrt{d+e x^2}} \, dx}{120 c^5 e}\\ &=-\frac{b \left (c^2 d-12 e\right ) x \sqrt{d+e x^2}}{120 c^3 e}-\frac{b x \left (d+e x^2\right )^{3/2}}{20 c e}-\frac{d \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{3 e^2}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}+\frac{\left (b \left (c^2 d-e\right )^2 \left (2 c^2 d+3 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 )}{15 c^5 e^2}+\frac{\left (b \left (15 c^4 d^2+20 c^2 d e-24 e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-e x^2} \, dx,x,\frac{x}{\sqrt{d+e x^2}}\right )}{120 c^5 e}\\ &=-\frac{b \left (c^2 d-12 e\right ) x \sqrt{d+e x^2}}{120 c^3 e}-\frac{b x \left (d+e x^2\right )^{3/2}}{20 c e}-\frac{d \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{3 e^2}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}+\frac{b \left (c^2 d-e\right )^{3/2} \left (2 c^2 d+3 e\right ) \tan ^{-1}\left (\frac{\sqrt{c^2 d-e} x}{\sqrt{d+e x^2}}\right )}{15 c^5 e^2}+\frac{b \left (15 c^4 d^2+20 c^2 d e-24 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{120 c^5 e^{3/2}}\\ \end{align*}

Mathematica [C]  time = 0.491348, size = 391, normalized size = 1.75 \[ \frac{-c^2 \sqrt{d+e x^2} \left (8 a c^3 \left (2 d^2-d e x^2-3 e^2 x^4\right )+b e x \left (c^2 \left (7 d+6 e x^2\right )-12 e\right )\right )+b \sqrt{e} \left (15 c^4 d^2+20 c^2 d e-24 e^2\right ) \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right )-8 b c^5 \tan ^{-1}(c x) \sqrt{d+e x^2} \left (2 d^2-d e x^2-3 e^2 x^4\right )-4 i b \left (c^2 d-e\right )^{3/2} \left (2 c^2 d+3 e\right ) \log \left (-\frac{60 i c^6 e^2 \left (\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d-i e x\right )}{b (c x+i) \left (c^2 d-e\right )^{5/2} \left (2 c^2 d+3 e\right )}\right )+4 i b \left (c^2 d-e\right )^{3/2} \left (2 c^2 d+3 e\right ) \log \left (\frac{60 i c^6 e^2 \left (\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d+i e x\right )}{b (c x-i) \left (c^2 d-e\right )^{5/2} \left (2 c^2 d+3 e\right )}\right )}{120 c^5 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(c^2*Sqrt[d + e*x^2]*(8*a*c^3*(2*d^2 - d*e*x^2 - 3*e^2*x^4) + b*e*x*(-12*e + c^2*(7*d + 6*e*x^2)))) - 8*b*c^
5*Sqrt[d + e*x^2]*(2*d^2 - d*e*x^2 - 3*e^2*x^4)*ArcTan[c*x] - (4*I)*b*(c^2*d - e)^(3/2)*(2*c^2*d + 3*e)*Log[((
-60*I)*c^6*e^2*(c*d - I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - e)^(5/2)*(2*c^2*d + 3*e)*(I + c*x)
)] + (4*I)*b*(c^2*d - e)^(3/2)*(2*c^2*d + 3*e)*Log[((60*I)*c^6*e^2*(c*d + I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x
^2]))/(b*(c^2*d - e)^(5/2)*(2*c^2*d + 3*e)*(-I + c*x))] + b*Sqrt[e]*(15*c^4*d^2 + 20*c^2*d*e - 24*e^2)*Log[e*x
+ Sqrt[e]*Sqrt[d + e*x^2]])/(120*c^5*e^2)

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Maple [F]  time = 0.965, size = 0, normalized size = 0. \begin{align*} \int{x}^{3}\sqrt{e{x}^{2}+d} \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)^(1/2)*(a+b*arctan(c*x)),x)

[Out]

int(x^3*(e*x^2+d)^(1/2)*(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)^(1/2)*(a+b*arctan(c*x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 26.3063, size = 2657, normalized size = 11.91 \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)^(1/2)*(a+b*arctan(c*x)),x, algorithm="fricas")

[Out]

[-1/240*((15*b*c^4*d^2 + 20*b*c^2*d*e - 24*b*e^2)*sqrt(e)*log(-2*e*x^2 + 2*sqrt(e*x^2 + d)*sqrt(e)*x - d) + 4*
(2*b*c^4*d^2 + b*c^2*d*e - 3*b*e^2)*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
*(24*a*c^5*e^2*x^4 + 8*a*c^5*d*e*x^2 - 6*b*c^4*e^2*x^3 - 16*a*c^5*d^2 - (7*b*c^4*d*e - 12*b*c^2*e^2)*x + 8*(3*
b*c^5*e^2*x^4 + b*c^5*d*e*x^2 - 2*b*c^5*d^2)*arctan(c*x))*sqrt(e*x^2 + d))/(c^5*e^2), 1/240*(8*(2*b*c^4*d^2 +
b*c^2*d*e - 3*b*e^2)*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)) - (15*b*c^4*d^2 + 20*b*c^2*d*e - 24*b*e^2)*sqrt(e)*log(-2*e*x^2 + 2*sqrt(
e*x^2 + d)*sqrt(e)*x - d) + 2*(24*a*c^5*e^2*x^4 + 8*a*c^5*d*e*x^2 - 6*b*c^4*e^2*x^3 - 16*a*c^5*d^2 - (7*b*c^4*
d*e - 12*b*c^2*e^2)*x + 8*(3*b*c^5*e^2*x^4 + b*c^5*d*e*x^2 - 2*b*c^5*d^2)*arctan(c*x))*sqrt(e*x^2 + d))/(c^5*e
^2), -1/120*((15*b*c^4*d^2 + 20*b*c^2*d*e - 24*b*e^2)*sqrt(-e)*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) + 2*(2*b*c^4
*d^2 + b*c^2*d*e - 3*b*e^2)*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)) - (24*a*c^5
*e^2*x^4 + 8*a*c^5*d*e*x^2 - 6*b*c^4*e^2*x^3 - 16*a*c^5*d^2 - (7*b*c^4*d*e - 12*b*c^2*e^2)*x + 8*(3*b*c^5*e^2*
x^4 + b*c^5*d*e*x^2 - 2*b*c^5*d^2)*arctan(c*x))*sqrt(e*x^2 + d))/(c^5*e^2), 1/120*(4*(2*b*c^4*d^2 + b*c^2*d*e
- 3*b*e^2)*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)) - (15*b*c^4*d^2 + 20*b*c^2*d*e - 24*b*e^2)*sqrt(-e)*arctan(sqrt(-e)*x/sqrt(e*x^2 +
d)) + (24*a*c^5*e^2*x^4 + 8*a*c^5*d*e*x^2 - 6*b*c^4*e^2*x^3 - 16*a*c^5*d^2 - (7*b*c^4*d*e - 12*b*c^2*e^2)*x +
8*(3*b*c^5*e^2*x^4 + b*c^5*d*e*x^2 - 2*b*c^5*d^2)*arctan(c*x))*sqrt(e*x^2 + d))/(c^5*e^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(e*x**2+d)**(1/2)*(a+b*atan(c*x)),x)

[Out]

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

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Giac [A]  time = 1.32272, size = 362, normalized size = 1.62 \begin{align*} \frac{1}{15} \,{\left (3 \,{\left (x^{2} e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x^{2} e + d\right )}^{\frac{3}{2}} d\right )} a e^{\left (-2\right )} + \frac{1}{240} \,{\left (16 \,{\left (3 \,{\left (x^{2} e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x^{2} e + d\right )}^{\frac{3}{2}} d\right )} \arctan \left (c x\right ) e^{\left (-2\right )} -{\left (2 \, \sqrt{x^{2} e + d} x{\left (\frac{6 \, x^{2}}{c^{2}} + \frac{{\left (7 \, c^{10} d e^{2} - 12 \, c^{8} e^{3}\right )} e^{\left (-3\right )}}{c^{12}}\right )} + \frac{{\left (15 \, c^{4} d^{2} + 20 \, c^{2} d e - 24 \, e^{2}\right )} e^{\left (-\frac{3}{2}\right )} \log \left ({\left (x e^{\frac{1}{2}} - \sqrt{x^{2} e + d}\right )}^{2}\right )}{c^{6}} + \frac{16 \,{\left (2 \, c^{6} d^{3} e^{\frac{1}{2}} - c^{4} d^{2} e^{\frac{3}{2}} - 4 \, c^{2} d e^{\frac{5}{2}} + 3 \, e^{\frac{7}{2}}\right )} \arctan \left (\frac{{\left ({\left (x e^{\frac{1}{2}} - \sqrt{x^{2} e + d}\right )}^{2} c^{2} - c^{2} d + 2 \, e\right )} e^{\left (-\frac{1}{2}\right )}}{2 \, \sqrt{c^{2} d - e}}\right ) e^{\left (-\frac{5}{2}\right )}}{\sqrt{c^{2} d - e} c^{6}}\right )} c\right )} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x^2+d)^(1/2)*(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*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