3.343 \(\int x^4 \sqrt{-c+d x} \sqrt{c+d x} (a+b x^2) \, dx\)

Optimal. Leaf size=208 \[ \frac{x^3 (d x-c)^{3/2} (c+d x)^{3/2} \left (8 a d^2+5 b c^2\right )}{48 d^4}+\frac{c^4 x \sqrt{d x-c} \sqrt{c+d x} \left (8 a d^2+5 b c^2\right )}{128 d^6}+\frac{c^2 x (d x-c)^{3/2} (c+d x)^{3/2} \left (8 a d^2+5 b c^2\right )}{64 d^6}-\frac{c^6 \left (8 a d^2+5 b c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{64 d^7}+\frac{b x^5 (d x-c)^{3/2} (c+d x)^{3/2}}{8 d^2} \]

[Out]

(c^4*(5*b*c^2 + 8*a*d^2)*x*Sqrt[-c + d*x]*Sqrt[c + d*x])/(128*d^6) + (c^2*(5*b*c^2 + 8*a*d^2)*x*(-c + d*x)^(3/
2)*(c + d*x)^(3/2))/(64*d^6) + ((5*b*c^2 + 8*a*d^2)*x^3*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(48*d^4) + (b*x^5*(-
c + d*x)^(3/2)*(c + d*x)^(3/2))/(8*d^2) - (c^6*(5*b*c^2 + 8*a*d^2)*ArcTanh[Sqrt[-c + d*x]/Sqrt[c + d*x]])/(64*
d^7)

________________________________________________________________________________________

Rubi [A]  time = 0.148884, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.258, Rules used = {460, 100, 12, 90, 38, 63, 217, 206} \[ \frac{x^3 (d x-c)^{3/2} (c+d x)^{3/2} \left (8 a d^2+5 b c^2\right )}{48 d^4}+\frac{c^4 x \sqrt{d x-c} \sqrt{c+d x} \left (8 a d^2+5 b c^2\right )}{128 d^6}+\frac{c^2 x (d x-c)^{3/2} (c+d x)^{3/2} \left (8 a d^2+5 b c^2\right )}{64 d^6}-\frac{c^6 \left (8 a d^2+5 b c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{64 d^7}+\frac{b x^5 (d x-c)^{3/2} (c+d x)^{3/2}}{8 d^2} \]

Antiderivative was successfully verified.

[In]

Int[x^4*Sqrt[-c + d*x]*Sqrt[c + d*x]*(a + b*x^2),x]

[Out]

(c^4*(5*b*c^2 + 8*a*d^2)*x*Sqrt[-c + d*x]*Sqrt[c + d*x])/(128*d^6) + (c^2*(5*b*c^2 + 8*a*d^2)*x*(-c + d*x)^(3/
2)*(c + d*x)^(3/2))/(64*d^6) + ((5*b*c^2 + 8*a*d^2)*x^3*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(48*d^4) + (b*x^5*(-
c + d*x)^(3/2)*(c + d*x)^(3/2))/(8*d^2) - (c^6*(5*b*c^2 + 8*a*d^2)*ArcTanh[Sqrt[-c + d*x]/Sqrt[c + d*x]])/(64*
d^7)

Rule 460

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_.)*((c_) + (d_.)
*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1))/(b1*b2*e*
(m + n*(p + 1) + 1)), x] - Dist[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(b1*b2*(m + n*(p + 1) + 1)), I
nt[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] &&
EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 0]

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rule 12

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

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 38

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(x*(a + b*x)^m*(c + d*x)^m)/(2*m + 1)
, x] + Dist[(2*a*c*m)/(2*m + 1), Int[(a + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] &&
EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, 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])

Rubi steps

\begin{align*} \int x^4 \sqrt{-c+d x} \sqrt{c+d x} \left (a+b x^2\right ) \, dx &=\frac{b x^5 (-c+d x)^{3/2} (c+d x)^{3/2}}{8 d^2}-\frac{1}{8} \left (-8 a-\frac{5 b c^2}{d^2}\right ) \int x^4 \sqrt{-c+d x} \sqrt{c+d x} \, dx\\ &=\frac{\left (5 b c^2+8 a d^2\right ) x^3 (-c+d x)^{3/2} (c+d x)^{3/2}}{48 d^4}+\frac{b x^5 (-c+d x)^{3/2} (c+d x)^{3/2}}{8 d^2}+\frac{\left (5 b c^2+8 a d^2\right ) \int 3 c^2 x^2 \sqrt{-c+d x} \sqrt{c+d x} \, dx}{48 d^4}\\ &=\frac{\left (5 b c^2+8 a d^2\right ) x^3 (-c+d x)^{3/2} (c+d x)^{3/2}}{48 d^4}+\frac{b x^5 (-c+d x)^{3/2} (c+d x)^{3/2}}{8 d^2}+\frac{\left (c^2 \left (5 b c^2+8 a d^2\right )\right ) \int x^2 \sqrt{-c+d x} \sqrt{c+d x} \, dx}{16 d^4}\\ &=\frac{c^2 \left (5 b c^2+8 a d^2\right ) x (-c+d x)^{3/2} (c+d x)^{3/2}}{64 d^6}+\frac{\left (5 b c^2+8 a d^2\right ) x^3 (-c+d x)^{3/2} (c+d x)^{3/2}}{48 d^4}+\frac{b x^5 (-c+d x)^{3/2} (c+d x)^{3/2}}{8 d^2}+\frac{\left (c^2 \left (5 b c^2+8 a d^2\right )\right ) \int c^2 \sqrt{-c+d x} \sqrt{c+d x} \, dx}{64 d^6}\\ &=\frac{c^2 \left (5 b c^2+8 a d^2\right ) x (-c+d x)^{3/2} (c+d x)^{3/2}}{64 d^6}+\frac{\left (5 b c^2+8 a d^2\right ) x^3 (-c+d x)^{3/2} (c+d x)^{3/2}}{48 d^4}+\frac{b x^5 (-c+d x)^{3/2} (c+d x)^{3/2}}{8 d^2}+\frac{\left (c^4 \left (5 b c^2+8 a d^2\right )\right ) \int \sqrt{-c+d x} \sqrt{c+d x} \, dx}{64 d^6}\\ &=\frac{c^4 \left (5 b c^2+8 a d^2\right ) x \sqrt{-c+d x} \sqrt{c+d x}}{128 d^6}+\frac{c^2 \left (5 b c^2+8 a d^2\right ) x (-c+d x)^{3/2} (c+d x)^{3/2}}{64 d^6}+\frac{\left (5 b c^2+8 a d^2\right ) x^3 (-c+d x)^{3/2} (c+d x)^{3/2}}{48 d^4}+\frac{b x^5 (-c+d x)^{3/2} (c+d x)^{3/2}}{8 d^2}-\frac{\left (c^6 \left (5 b c^2+8 a d^2\right )\right ) \int \frac{1}{\sqrt{-c+d x} \sqrt{c+d x}} \, dx}{128 d^6}\\ &=\frac{c^4 \left (5 b c^2+8 a d^2\right ) x \sqrt{-c+d x} \sqrt{c+d x}}{128 d^6}+\frac{c^2 \left (5 b c^2+8 a d^2\right ) x (-c+d x)^{3/2} (c+d x)^{3/2}}{64 d^6}+\frac{\left (5 b c^2+8 a d^2\right ) x^3 (-c+d x)^{3/2} (c+d x)^{3/2}}{48 d^4}+\frac{b x^5 (-c+d x)^{3/2} (c+d x)^{3/2}}{8 d^2}-\frac{\left (c^6 \left (5 b c^2+8 a d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2 c+x^2}} \, dx,x,\sqrt{-c+d x}\right )}{64 d^7}\\ &=\frac{c^4 \left (5 b c^2+8 a d^2\right ) x \sqrt{-c+d x} \sqrt{c+d x}}{128 d^6}+\frac{c^2 \left (5 b c^2+8 a d^2\right ) x (-c+d x)^{3/2} (c+d x)^{3/2}}{64 d^6}+\frac{\left (5 b c^2+8 a d^2\right ) x^3 (-c+d x)^{3/2} (c+d x)^{3/2}}{48 d^4}+\frac{b x^5 (-c+d x)^{3/2} (c+d x)^{3/2}}{8 d^2}-\frac{\left (c^6 \left (5 b c^2+8 a d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt{-c+d x}}{\sqrt{c+d x}}\right )}{64 d^7}\\ &=\frac{c^4 \left (5 b c^2+8 a d^2\right ) x \sqrt{-c+d x} \sqrt{c+d x}}{128 d^6}+\frac{c^2 \left (5 b c^2+8 a d^2\right ) x (-c+d x)^{3/2} (c+d x)^{3/2}}{64 d^6}+\frac{\left (5 b c^2+8 a d^2\right ) x^3 (-c+d x)^{3/2} (c+d x)^{3/2}}{48 d^4}+\frac{b x^5 (-c+d x)^{3/2} (c+d x)^{3/2}}{8 d^2}-\frac{c^6 \left (5 b c^2+8 a d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{-c+d x}}{\sqrt{c+d x}}\right )}{64 d^7}\\ \end{align*}

Mathematica [A]  time = 0.214394, size = 161, normalized size = 0.77 \[ \frac{\sqrt{d x-c} \sqrt{c+d x} \left (3 \left (8 a c^5 d^2+5 b c^7\right ) \sin ^{-1}\left (\frac{d x}{c}\right )-d x \sqrt{1-\frac{d^2 x^2}{c^2}} \left (8 a d^2 \left (2 c^2 d^2 x^2+3 c^4-8 d^4 x^4\right )+b \left (10 c^4 d^2 x^2+8 c^2 d^4 x^4+15 c^6-48 d^6 x^6\right )\right )\right )}{384 d^7 \sqrt{1-\frac{d^2 x^2}{c^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*Sqrt[-c + d*x]*Sqrt[c + d*x]*(a + b*x^2),x]

[Out]

(Sqrt[-c + d*x]*Sqrt[c + d*x]*(-(d*x*Sqrt[1 - (d^2*x^2)/c^2]*(8*a*d^2*(3*c^4 + 2*c^2*d^2*x^2 - 8*d^4*x^4) + b*
(15*c^6 + 10*c^4*d^2*x^2 + 8*c^2*d^4*x^4 - 48*d^6*x^6))) + 3*(5*b*c^7 + 8*a*c^5*d^2)*ArcSin[(d*x)/c]))/(384*d^
7*Sqrt[1 - (d^2*x^2)/c^2])

________________________________________________________________________________________

Maple [C]  time = 0.022, size = 298, normalized size = 1.4 \begin{align*}{\frac{{\it csgn} \left ( d \right ) }{384\,{d}^{7}}\sqrt{dx-c}\sqrt{dx+c} \left ( 48\,{\it csgn} \left ( d \right ){x}^{7}b{d}^{7}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}+64\,{\it csgn} \left ( d \right ){x}^{5}a{d}^{7}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}-8\,{\it csgn} \left ( d \right ){x}^{5}b{c}^{2}{d}^{5}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}-16\,{\it csgn} \left ( d \right ){x}^{3}a{c}^{2}{d}^{5}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}-10\,{\it csgn} \left ( d \right ){x}^{3}b{c}^{4}{d}^{3}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}-24\,{\it csgn} \left ( d \right ){d}^{3}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}xa{c}^{4}-15\,{\it csgn} \left ( d \right ) d\sqrt{{d}^{2}{x}^{2}-{c}^{2}}xb{c}^{6}-24\,\ln \left ( \left ( \sqrt{{d}^{2}{x}^{2}-{c}^{2}}{\it csgn} \left ( d \right ) +dx \right ){\it csgn} \left ( d \right ) \right ) a{c}^{6}{d}^{2}-15\,\ln \left ( \left ( \sqrt{{d}^{2}{x}^{2}-{c}^{2}}{\it csgn} \left ( d \right ) +dx \right ){\it csgn} \left ( d \right ) \right ) b{c}^{8} \right ){\frac{1}{\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(b*x^2+a)*(d*x-c)^(1/2)*(d*x+c)^(1/2),x)

[Out]

1/384*(d*x-c)^(1/2)*(d*x+c)^(1/2)*(48*csgn(d)*x^7*b*d^7*(d^2*x^2-c^2)^(1/2)+64*csgn(d)*x^5*a*d^7*(d^2*x^2-c^2)
^(1/2)-8*csgn(d)*x^5*b*c^2*d^5*(d^2*x^2-c^2)^(1/2)-16*csgn(d)*x^3*a*c^2*d^5*(d^2*x^2-c^2)^(1/2)-10*csgn(d)*x^3
*b*c^4*d^3*(d^2*x^2-c^2)^(1/2)-24*csgn(d)*d^3*(d^2*x^2-c^2)^(1/2)*x*a*c^4-15*csgn(d)*d*(d^2*x^2-c^2)^(1/2)*x*b
*c^6-24*ln(((d^2*x^2-c^2)^(1/2)*csgn(d)+d*x)*csgn(d))*a*c^6*d^2-15*ln(((d^2*x^2-c^2)^(1/2)*csgn(d)+d*x)*csgn(d
))*b*c^8)*csgn(d)/(d^2*x^2-c^2)^(1/2)/d^7

________________________________________________________________________________________

Maxima [A]  time = 0.948575, size = 356, normalized size = 1.71 \begin{align*} \frac{{\left (d^{2} x^{2} - c^{2}\right )}^{\frac{3}{2}} b x^{5}}{8 \, d^{2}} + \frac{5 \,{\left (d^{2} x^{2} - c^{2}\right )}^{\frac{3}{2}} b c^{2} x^{3}}{48 \, d^{4}} + \frac{{\left (d^{2} x^{2} - c^{2}\right )}^{\frac{3}{2}} a x^{3}}{6 \, d^{2}} - \frac{5 \, b c^{8} \log \left (2 \, d^{2} x + 2 \, \sqrt{d^{2} x^{2} - c^{2}} \sqrt{d^{2}}\right )}{128 \, \sqrt{d^{2}} d^{6}} - \frac{a c^{6} \log \left (2 \, d^{2} x + 2 \, \sqrt{d^{2} x^{2} - c^{2}} \sqrt{d^{2}}\right )}{16 \, \sqrt{d^{2}} d^{4}} + \frac{5 \, \sqrt{d^{2} x^{2} - c^{2}} b c^{6} x}{128 \, d^{6}} + \frac{\sqrt{d^{2} x^{2} - c^{2}} a c^{4} x}{16 \, d^{4}} + \frac{5 \,{\left (d^{2} x^{2} - c^{2}\right )}^{\frac{3}{2}} b c^{4} x}{64 \, d^{6}} + \frac{{\left (d^{2} x^{2} - c^{2}\right )}^{\frac{3}{2}} a c^{2} x}{8 \, d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)*(d*x-c)^(1/2)*(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

1/8*(d^2*x^2 - c^2)^(3/2)*b*x^5/d^2 + 5/48*(d^2*x^2 - c^2)^(3/2)*b*c^2*x^3/d^4 + 1/6*(d^2*x^2 - c^2)^(3/2)*a*x
^3/d^2 - 5/128*b*c^8*log(2*d^2*x + 2*sqrt(d^2*x^2 - c^2)*sqrt(d^2))/(sqrt(d^2)*d^6) - 1/16*a*c^6*log(2*d^2*x +
2*sqrt(d^2*x^2 - c^2)*sqrt(d^2))/(sqrt(d^2)*d^4) + 5/128*sqrt(d^2*x^2 - c^2)*b*c^6*x/d^6 + 1/16*sqrt(d^2*x^2
- c^2)*a*c^4*x/d^4 + 5/64*(d^2*x^2 - c^2)^(3/2)*b*c^4*x/d^6 + 1/8*(d^2*x^2 - c^2)^(3/2)*a*c^2*x/d^4

________________________________________________________________________________________

Fricas [A]  time = 1.82932, size = 300, normalized size = 1.44 \begin{align*} \frac{{\left (48 \, b d^{7} x^{7} - 8 \,{\left (b c^{2} d^{5} - 8 \, a d^{7}\right )} x^{5} - 2 \,{\left (5 \, b c^{4} d^{3} + 8 \, a c^{2} d^{5}\right )} x^{3} - 3 \,{\left (5 \, b c^{6} d + 8 \, a c^{4} d^{3}\right )} x\right )} \sqrt{d x + c} \sqrt{d x - c} + 3 \,{\left (5 \, b c^{8} + 8 \, a c^{6} d^{2}\right )} \log \left (-d x + \sqrt{d x + c} \sqrt{d x - c}\right )}{384 \, d^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)*(d*x-c)^(1/2)*(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

1/384*((48*b*d^7*x^7 - 8*(b*c^2*d^5 - 8*a*d^7)*x^5 - 2*(5*b*c^4*d^3 + 8*a*c^2*d^5)*x^3 - 3*(5*b*c^6*d + 8*a*c^
4*d^3)*x)*sqrt(d*x + c)*sqrt(d*x - c) + 3*(5*b*c^8 + 8*a*c^6*d^2)*log(-d*x + sqrt(d*x + c)*sqrt(d*x - c)))/d^7

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \left (a + b x^{2}\right ) \sqrt{- c + d x} \sqrt{c + d x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(b*x**2+a)*(d*x-c)**(1/2)*(d*x+c)**(1/2),x)

[Out]

Integral(x**4*(a + b*x**2)*sqrt(-c + d*x)*sqrt(c + d*x), x)

________________________________________________________________________________________

Giac [A]  time = 1.35004, size = 394, normalized size = 1.89 \begin{align*} \frac{8 \,{\left (\frac{6 \, c^{6} \log \left ({\left | -\sqrt{d x + c} + \sqrt{d x - c} \right |}\right )}{d^{4}} +{\left ({\left (2 \,{\left ({\left (d x + c\right )}{\left (4 \,{\left (d x + c\right )}{\left (\frac{d x + c}{d^{4}} - \frac{5 \, c}{d^{4}}\right )} + \frac{39 \, c^{2}}{d^{4}}\right )} - \frac{37 \, c^{3}}{d^{4}}\right )}{\left (d x + c\right )} + \frac{31 \, c^{4}}{d^{4}}\right )}{\left (d x + c\right )} - \frac{3 \, c^{5}}{d^{4}}\right )} \sqrt{d x + c} \sqrt{d x - c}\right )} a +{\left (\frac{30 \, c^{8} \log \left ({\left | -\sqrt{d x + c} + \sqrt{d x - c} \right |}\right )}{d^{6}} +{\left ({\left (2 \,{\left ({\left (4 \,{\left ({\left (d x + c\right )}{\left (6 \,{\left (d x + c\right )}{\left (\frac{d x + c}{d^{6}} - \frac{7 \, c}{d^{6}}\right )} + \frac{125 \, c^{2}}{d^{6}}\right )} - \frac{205 \, c^{3}}{d^{6}}\right )}{\left (d x + c\right )} + \frac{795 \, c^{4}}{d^{6}}\right )}{\left (d x + c\right )} - \frac{449 \, c^{5}}{d^{6}}\right )}{\left (d x + c\right )} + \frac{251 \, c^{6}}{d^{6}}\right )}{\left (d x + c\right )} - \frac{15 \, c^{7}}{d^{6}}\right )} \sqrt{d x + c} \sqrt{d x - c}\right )} b}{384 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)*(d*x-c)^(1/2)*(d*x+c)^(1/2),x, algorithm="giac")

[Out]

1/384*(8*(6*c^6*log(abs(-sqrt(d*x + c) + sqrt(d*x - c)))/d^4 + ((2*((d*x + c)*(4*(d*x + c)*((d*x + c)/d^4 - 5*
c/d^4) + 39*c^2/d^4) - 37*c^3/d^4)*(d*x + c) + 31*c^4/d^4)*(d*x + c) - 3*c^5/d^4)*sqrt(d*x + c)*sqrt(d*x - c))
*a + (30*c^8*log(abs(-sqrt(d*x + c) + sqrt(d*x - c)))/d^6 + ((2*((4*((d*x + c)*(6*(d*x + c)*((d*x + c)/d^6 - 7
*c/d^6) + 125*c^2/d^6) - 205*c^3/d^6)*(d*x + c) + 795*c^4/d^6)*(d*x + c) - 449*c^5/d^6)*(d*x + c) + 251*c^6/d^
6)*(d*x + c) - 15*c^7/d^6)*sqrt(d*x + c)*sqrt(d*x - c))*b)/d