\(\int x^4 \sqrt {-c+d x} \sqrt {c+d x} (a+b x^2) \, dx\) [343]

Optimal result

Integrand size = 31, antiderivative size = 208 \[ \int x^4 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {c^4 \left (5 b c^2+8 a d^2\right ) x \sqrt {-c+d x} \sqrt {c+d x}}{128 d^6}+\frac {c^2 \left (5 b c^2+8 a d^2\right ) x (-c+d x)^{3/2} (c+d x)^{3/2}}{64 d^6}+\frac {\left (5 b c^2+8 a d^2\right ) x^3 (-c+d x)^{3/2} (c+d x)^{3/2}}{48 d^4}+\frac {b x^5 (-c+d x)^{3/2} (c+d x)^{3/2}}{8 d^2}-\frac {c^6 \left (5 b c^2+8 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right )}{64 d^7} \]

[Out]

1/64*c^2*(8*a*d^2+5*b*c^2)*x*(d*x-c)^(3/2)*(d*x+c)^(3/2)/d^6+1/48*(8*a*d^2+5*b*c^2)*x^3*(d*x-c)^(3/2)*(d*x+c)^
(3/2)/d^4+1/8*b*x^5*(d*x-c)^(3/2)*(d*x+c)^(3/2)/d^2-1/64*c^6*(8*a*d^2+5*b*c^2)*arctanh((d*x-c)^(1/2)/(d*x+c)^(
1/2))/d^7+1/128*c^4*(8*a*d^2+5*b*c^2)*x*(d*x-c)^(1/2)*(d*x+c)^(1/2)/d^6

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {471, 102, 12, 92, 38, 65, 223, 212} \[ \int x^4 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=-\frac {c^6 \left (8 a d^2+5 b c^2\right ) \text {arctanh}\left (\frac {\sqrt {d x-c}}{\sqrt {c+d x}}\right )}{64 d^7}+\frac {c^2 x (d x-c)^{3/2} (c+d x)^{3/2} \left (8 a d^2+5 b c^2\right )}{64 d^6}+\frac {x^3 (d x-c)^{3/2} (c+d x)^{3/2} \left (8 a d^2+5 b c^2\right )}{48 d^4}+\frac {c^4 x \sqrt {d x-c} \sqrt {c+d x} \left (8 a d^2+5 b c^2\right )}{128 d^6}+\frac {b x^5 (d x-c)^{3/2} (c+d x)^{3/2}}{8 d^2} \]

[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 12

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

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 65

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 92

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 102

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 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 471

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]

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \text {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] (verified)

Time = 0.46 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.68 \[ \int x^4 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {d x \sqrt {-c+d x} \sqrt {c+d x} \left (8 a d^2 \left (-3 c^4-2 c^2 d^2 x^2+8 d^4 x^4\right )-b \left (15 c^6+10 c^4 d^2 x^2+8 c^2 d^4 x^4-48 d^6 x^6\right )\right )-6 c^6 \left (5 b c^2+8 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right )}{384 d^7} \]

[In]

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

[Out]

(d*x*Sqrt[-c + d*x]*Sqrt[c + d*x]*(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)) - 6*c^6*(5*b*c^2 + 8*a*d^2)*ArcTanh[Sqrt[-c + d*x]/Sqrt[c + d*x]])/(384*d^7)

Maple [A] (verified)

Time = 4.19 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.88

method	result	size
risch	\(\frac {x \left (-48 b \,x^{6} d^{6}-64 a \,d^{6} x^{4}+8 b \,c^{2} d^{4} x^{4}+16 a \,c^{2} d^{4} x^{2}+10 b \,c^{4} d^{2} x^{2}+24 a \,c^{4} d^{2}+15 b \,c^{6}\right ) \left (-d x +c \right ) \sqrt {d x +c}}{384 d^{6} \sqrt {d x -c}}-\frac {c^{6} \left (8 a \,d^{2}+5 b \,c^{2}\right ) \ln \left (\frac {x \,d^{2}}{\sqrt {d^{2}}}+\sqrt {d^{2} x^{2}-c^{2}}\right ) \sqrt {\left (d x -c \right ) \left (d x +c \right )}}{128 d^{6} \sqrt {d^{2}}\, \sqrt {d x -c}\, \sqrt {d x +c}}\)	\(184\)
default	\(-\frac {\sqrt {d x -c}\, \sqrt {d x +c}\, \left (-48 \,\operatorname {csgn}\left (d \right ) b \,d^{7} x^{7} \sqrt {d^{2} x^{2}-c^{2}}-64 \,\operatorname {csgn}\left (d \right ) a \,d^{7} x^{5} \sqrt {d^{2} x^{2}-c^{2}}+8 \,\operatorname {csgn}\left (d \right ) b \,c^{2} d^{5} x^{5} \sqrt {d^{2} x^{2}-c^{2}}+16 \,\operatorname {csgn}\left (d \right ) a \,c^{2} d^{5} x^{3} \sqrt {d^{2} x^{2}-c^{2}}+10 \,\operatorname {csgn}\left (d \right ) b \,c^{4} d^{3} x^{3} \sqrt {d^{2} x^{2}-c^{2}}+24 \sqrt {d^{2} x^{2}-c^{2}}\, \operatorname {csgn}\left (d \right ) d^{3} a \,c^{4} x +15 \sqrt {d^{2} x^{2}-c^{2}}\, \operatorname {csgn}\left (d \right ) d b \,c^{6} x +24 \ln \left (\left (\sqrt {d^{2} x^{2}-c^{2}}\, \operatorname {csgn}\left (d \right )+d x \right ) \operatorname {csgn}\left (d \right )\right ) a \,c^{6} d^{2}+15 \ln \left (\left (\sqrt {d^{2} x^{2}-c^{2}}\, \operatorname {csgn}\left (d \right )+d x \right ) \operatorname {csgn}\left (d \right )\right ) b \,c^{8}\right ) \operatorname {csgn}\left (d \right )}{384 \sqrt {d^{2} x^{2}-c^{2}}\, d^{7}}\)	\(298\)

[In]

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

[Out]

1/384*x*(-48*b*d^6*x^6-64*a*d^6*x^4+8*b*c^2*d^4*x^4+16*a*c^2*d^4*x^2+10*b*c^4*d^2*x^2+24*a*c^4*d^2+15*b*c^6)*(
-d*x+c)*(d*x+c)^(1/2)/d^6/(d*x-c)^(1/2)-1/128*c^6*(8*a*d^2+5*b*c^2)/d^6*ln(x*d^2/(d^2)^(1/2)+(d^2*x^2-c^2)^(1/
2))/(d^2)^(1/2)*((d*x-c)*(d*x+c))^(1/2)/(d*x-c)^(1/2)/(d*x+c)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.66 \[ \int x^4 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {{\left (48 \, b d^{7} x^{7} - 8 \, {\left (b c^{2} d^{5} - 8 \, a d^{7}\right )} x^{5} - 2 \, {\left (5 \, b c^{4} d^{3} + 8 \, a c^{2} d^{5}\right )} x^{3} - 3 \, {\left (5 \, b c^{6} d + 8 \, a c^{4} d^{3}\right )} x\right )} \sqrt {d x + c} \sqrt {d x - c} + 3 \, {\left (5 \, b c^{8} + 8 \, a c^{6} d^{2}\right )} \log \left (-d x + \sqrt {d x + c} \sqrt {d x - c}\right )}{384 \, d^{7}} \]

[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]

\[ \int x^4 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\int x^{4} \left (a + b x^{2}\right ) \sqrt {- c + d x} \sqrt {c + d x}\, dx \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.18 \[ \int x^4 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {{\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} b x^{5}}{8 \, d^{2}} + \frac {5 \, {\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} b c^{2} x^{3}}{48 \, d^{4}} + \frac {{\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} a x^{3}}{6 \, d^{2}} - \frac {5 \, b c^{8} \log \left (2 \, d^{2} x + 2 \, \sqrt {d^{2} x^{2} - c^{2}} d\right )}{128 \, d^{7}} - \frac {a c^{6} \log \left (2 \, d^{2} x + 2 \, \sqrt {d^{2} x^{2} - c^{2}} d\right )}{16 \, d^{5}} + \frac {5 \, \sqrt {d^{2} x^{2} - c^{2}} b c^{6} x}{128 \, d^{6}} + \frac {\sqrt {d^{2} x^{2} - c^{2}} a c^{4} x}{16 \, d^{4}} + \frac {5 \, {\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} b c^{4} x}{64 \, d^{6}} + \frac {{\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} a c^{2} x}{8 \, d^{4}} \]

[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)*d)/d^7 - 1/16*a*c^6*log(2*d^2*x + 2*sqrt(d^2*x^2 - c^
2)*d)/d^5 + 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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 558 vs. \(2 (178) = 356\).

Time = 0.40 (sec) , antiderivative size = 558, normalized size of antiderivative = 2.68 \[ \int x^4 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {112 \, {\left ({\left ({\left (2 \, {\left (d x + c\right )} {\left (3 \, {\left (d x + c\right )} {\left (\frac {4 \, {\left (d x + c\right )}}{d^{4}} - \frac {21 \, c}{d^{4}}\right )} + \frac {133 \, c^{2}}{d^{4}}\right )} - \frac {295 \, c^{3}}{d^{4}}\right )} {\left (d x + c\right )} + \frac {195 \, c^{4}}{d^{4}}\right )} \sqrt {d x + c} \sqrt {d x - c} + \frac {90 \, c^{5} \log \left ({\left | -\sqrt {d x + c} + \sqrt {d x - c} \right |}\right )}{d^{4}}\right )} a c + 8 \, {\left ({\left ({\left (2 \, {\left ({\left (4 \, {\left (d x + c\right )} {\left (5 \, {\left (d x + c\right )} {\left (\frac {6 \, {\left (d x + c\right )}}{d^{6}} - \frac {43 \, c}{d^{6}}\right )} + \frac {661 \, c^{2}}{d^{6}}\right )} - \frac {4551 \, c^{3}}{d^{6}}\right )} {\left (d x + c\right )} + \frac {4781 \, c^{4}}{d^{6}}\right )} {\left (d x + c\right )} - \frac {6335 \, c^{5}}{d^{6}}\right )} {\left (d x + c\right )} + \frac {2835 \, c^{6}}{d^{6}}\right )} \sqrt {d x + c} \sqrt {d x - c} + \frac {1050 \, c^{7} \log \left ({\left | -\sqrt {d x + c} + \sqrt {d x - c} \right |}\right )}{d^{6}}\right )} b c + 56 \, {\left ({\left ({\left (2 \, {\left ({\left (d x + c\right )} {\left (4 \, {\left (d x + c\right )} {\left (\frac {5 \, {\left (d x + c\right )}}{d^{5}} - \frac {31 \, c}{d^{5}}\right )} + \frac {321 \, c^{2}}{d^{5}}\right )} - \frac {451 \, c^{3}}{d^{5}}\right )} {\left (d x + c\right )} + \frac {745 \, c^{4}}{d^{5}}\right )} {\left (d x + c\right )} - \frac {405 \, c^{5}}{d^{5}}\right )} \sqrt {d x + c} \sqrt {d x - c} - \frac {150 \, c^{6} \log \left ({\left | -\sqrt {d x + c} + \sqrt {d x - c} \right |}\right )}{d^{5}}\right )} a d + {\left ({\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (d x + c\right )} {\left (6 \, {\left (d x + c\right )} {\left (\frac {7 \, {\left (d x + c\right )}}{d^{7}} - \frac {57 \, c}{d^{7}}\right )} + \frac {1219 \, c^{2}}{d^{7}}\right )} - \frac {12463 \, c^{3}}{d^{7}}\right )} {\left (d x + c\right )} + \frac {64233 \, c^{4}}{d^{7}}\right )} {\left (d x + c\right )} - \frac {53963 \, c^{5}}{d^{7}}\right )} {\left (d x + c\right )} + \frac {59465 \, c^{6}}{d^{7}}\right )} {\left (d x + c\right )} - \frac {23205 \, c^{7}}{d^{7}}\right )} \sqrt {d x + c} \sqrt {d x - c} - \frac {7350 \, c^{8} \log \left ({\left | -\sqrt {d x + c} + \sqrt {d x - c} \right |}\right )}{d^{7}}\right )} b d}{13440 \, d} \]

[In]

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

[Out]

1/13440*(112*(((2*(d*x + c)*(3*(d*x + c)*(4*(d*x + c)/d^4 - 21*c/d^4) + 133*c^2/d^4) - 295*c^3/d^4)*(d*x + c)
+ 195*c^4/d^4)*sqrt(d*x + c)*sqrt(d*x - c) + 90*c^5*log(abs(-sqrt(d*x + c) + sqrt(d*x - c)))/d^4)*a*c + 8*(((2
*((4*(d*x + c)*(5*(d*x + c)*(6*(d*x + c)/d^6 - 43*c/d^6) + 661*c^2/d^6) - 4551*c^3/d^6)*(d*x + c) + 4781*c^4/d
^6)*(d*x + c) - 6335*c^5/d^6)*(d*x + c) + 2835*c^6/d^6)*sqrt(d*x + c)*sqrt(d*x - c) + 1050*c^7*log(abs(-sqrt(d
*x + c) + sqrt(d*x - c)))/d^6)*b*c + 56*(((2*((d*x + c)*(4*(d*x + c)*(5*(d*x + c)/d^5 - 31*c/d^5) + 321*c^2/d^
5) - 451*c^3/d^5)*(d*x + c) + 745*c^4/d^5)*(d*x + c) - 405*c^5/d^5)*sqrt(d*x + c)*sqrt(d*x - c) - 150*c^6*log(
abs(-sqrt(d*x + c) + sqrt(d*x - c)))/d^5)*a*d + (((2*((4*(5*(d*x + c)*(6*(d*x + c)*(7*(d*x + c)/d^7 - 57*c/d^7
) + 1219*c^2/d^7) - 12463*c^3/d^7)*(d*x + c) + 64233*c^4/d^7)*(d*x + c) - 53963*c^5/d^7)*(d*x + c) + 59465*c^6
/d^7)*(d*x + c) - 23205*c^7/d^7)*sqrt(d*x + c)*sqrt(d*x - c) - 7350*c^8*log(abs(-sqrt(d*x + c) + sqrt(d*x - c)
))/d^7)*b*d)/d

Mupad [B] (verification not implemented)

Time = 49.01 (sec) , antiderivative size = 2314, normalized size of antiderivative = 11.12 \[ \int x^4 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

((35*a*c^6*((c + d*x)^(1/2) - c^(1/2))^3)/(12*((-c)^(1/2) - (d*x - c)^(1/2))^3) - (a*c^6*((c + d*x)^(1/2) - c^
(1/2)))/(4*((-c)^(1/2) - (d*x - c)^(1/2))) + (757*a*c^6*((c + d*x)^(1/2) - c^(1/2))^5)/(4*((-c)^(1/2) - (d*x -
c)^(1/2))^5) + (7339*a*c^6*((c + d*x)^(1/2) - c^(1/2))^7)/(4*((-c)^(1/2) - (d*x - c)^(1/2))^7) + (41929*a*c^6
*((c + d*x)^(1/2) - c^(1/2))^9)/(6*((-c)^(1/2) - (d*x - c)^(1/2))^9) + (25661*a*c^6*((c + d*x)^(1/2) - c^(1/2)
)^11)/(2*((-c)^(1/2) - (d*x - c)^(1/2))^11) + (25661*a*c^6*((c + d*x)^(1/2) - c^(1/2))^13)/(2*((-c)^(1/2) - (d
*x - c)^(1/2))^13) + (41929*a*c^6*((c + d*x)^(1/2) - c^(1/2))^15)/(6*((-c)^(1/2) - (d*x - c)^(1/2))^15) + (733
9*a*c^6*((c + d*x)^(1/2) - c^(1/2))^17)/(4*((-c)^(1/2) - (d*x - c)^(1/2))^17) + (757*a*c^6*((c + d*x)^(1/2) -
c^(1/2))^19)/(4*((-c)^(1/2) - (d*x - c)^(1/2))^19) + (35*a*c^6*((c + d*x)^(1/2) - c^(1/2))^21)/(12*((-c)^(1/2)
- (d*x - c)^(1/2))^21) - (a*c^6*((c + d*x)^(1/2) - c^(1/2))^23)/(4*((-c)^(1/2) - (d*x - c)^(1/2))^23))/(d^5 -
(12*d^5*((c + d*x)^(1/2) - c^(1/2))^2)/((-c)^(1/2) - (d*x - c)^(1/2))^2 + (66*d^5*((c + d*x)^(1/2) - c^(1/2))
^4)/((-c)^(1/2) - (d*x - c)^(1/2))^4 - (220*d^5*((c + d*x)^(1/2) - c^(1/2))^6)/((-c)^(1/2) - (d*x - c)^(1/2))^
6 + (495*d^5*((c + d*x)^(1/2) - c^(1/2))^8)/((-c)^(1/2) - (d*x - c)^(1/2))^8 - (792*d^5*((c + d*x)^(1/2) - c^(
1/2))^10)/((-c)^(1/2) - (d*x - c)^(1/2))^10 + (924*d^5*((c + d*x)^(1/2) - c^(1/2))^12)/((-c)^(1/2) - (d*x - c)
^(1/2))^12 - (792*d^5*((c + d*x)^(1/2) - c^(1/2))^14)/((-c)^(1/2) - (d*x - c)^(1/2))^14 + (495*d^5*((c + d*x)^
(1/2) - c^(1/2))^16)/((-c)^(1/2) - (d*x - c)^(1/2))^16 - (220*d^5*((c + d*x)^(1/2) - c^(1/2))^18)/((-c)^(1/2)
- (d*x - c)^(1/2))^18 + (66*d^5*((c + d*x)^(1/2) - c^(1/2))^20)/((-c)^(1/2) - (d*x - c)^(1/2))^20 - (12*d^5*((
c + d*x)^(1/2) - c^(1/2))^22)/((-c)^(1/2) - (d*x - c)^(1/2))^22 + (d^5*((c + d*x)^(1/2) - c^(1/2))^24)/((-c)^(
1/2) - (d*x - c)^(1/2))^24) - ((5*b*c^8*((c + d*x)^(1/2) - c^(1/2)))/(32*((-c)^(1/2) - (d*x - c)^(1/2))) - (23
5*b*c^8*((c + d*x)^(1/2) - c^(1/2))^3)/(96*((-c)^(1/2) - (d*x - c)^(1/2))^3) + (1723*b*c^8*((c + d*x)^(1/2) -
c^(1/2))^5)/(96*((-c)^(1/2) - (d*x - c)^(1/2))^5) + (72283*b*c^8*((c + d*x)^(1/2) - c^(1/2))^7)/(32*((-c)^(1/2
) - (d*x - c)^(1/2))^7) + (848801*b*c^8*((c + d*x)^(1/2) - c^(1/2))^9)/(32*((-c)^(1/2) - (d*x - c)^(1/2))^9) +
(4181067*b*c^8*((c + d*x)^(1/2) - c^(1/2))^11)/(32*((-c)^(1/2) - (d*x - c)^(1/2))^11) + (10994181*b*c^8*((c +
d*x)^(1/2) - c^(1/2))^13)/(32*((-c)^(1/2) - (d*x - c)^(1/2))^13) + (17457599*b*c^8*((c + d*x)^(1/2) - c^(1/2)
)^15)/(32*((-c)^(1/2) - (d*x - c)^(1/2))^15) + (17457599*b*c^8*((c + d*x)^(1/2) - c^(1/2))^17)/(32*((-c)^(1/2)
- (d*x - c)^(1/2))^17) + (10994181*b*c^8*((c + d*x)^(1/2) - c^(1/2))^19)/(32*((-c)^(1/2) - (d*x - c)^(1/2))^1
9) + (4181067*b*c^8*((c + d*x)^(1/2) - c^(1/2))^21)/(32*((-c)^(1/2) - (d*x - c)^(1/2))^21) + (848801*b*c^8*((c
+ d*x)^(1/2) - c^(1/2))^23)/(32*((-c)^(1/2) - (d*x - c)^(1/2))^23) + (72283*b*c^8*((c + d*x)^(1/2) - c^(1/2))
^25)/(32*((-c)^(1/2) - (d*x - c)^(1/2))^25) + (1723*b*c^8*((c + d*x)^(1/2) - c^(1/2))^27)/(96*((-c)^(1/2) - (d
*x - c)^(1/2))^27) - (235*b*c^8*((c + d*x)^(1/2) - c^(1/2))^29)/(96*((-c)^(1/2) - (d*x - c)^(1/2))^29) + (5*b*
c^8*((c + d*x)^(1/2) - c^(1/2))^31)/(32*((-c)^(1/2) - (d*x - c)^(1/2))^31))/(d^7 - (16*d^7*((c + d*x)^(1/2) -
c^(1/2))^2)/((-c)^(1/2) - (d*x - c)^(1/2))^2 + (120*d^7*((c + d*x)^(1/2) - c^(1/2))^4)/((-c)^(1/2) - (d*x - c)
^(1/2))^4 - (560*d^7*((c + d*x)^(1/2) - c^(1/2))^6)/((-c)^(1/2) - (d*x - c)^(1/2))^6 + (1820*d^7*((c + d*x)^(1
/2) - c^(1/2))^8)/((-c)^(1/2) - (d*x - c)^(1/2))^8 - (4368*d^7*((c + d*x)^(1/2) - c^(1/2))^10)/((-c)^(1/2) - (
d*x - c)^(1/2))^10 + (8008*d^7*((c + d*x)^(1/2) - c^(1/2))^12)/((-c)^(1/2) - (d*x - c)^(1/2))^12 - (11440*d^7*
((c + d*x)^(1/2) - c^(1/2))^14)/((-c)^(1/2) - (d*x - c)^(1/2))^14 + (12870*d^7*((c + d*x)^(1/2) - c^(1/2))^16)
/((-c)^(1/2) - (d*x - c)^(1/2))^16 - (11440*d^7*((c + d*x)^(1/2) - c^(1/2))^18)/((-c)^(1/2) - (d*x - c)^(1/2))
^18 + (8008*d^7*((c + d*x)^(1/2) - c^(1/2))^20)/((-c)^(1/2) - (d*x - c)^(1/2))^20 - (4368*d^7*((c + d*x)^(1/2)
- c^(1/2))^22)/((-c)^(1/2) - (d*x - c)^(1/2))^22 + (1820*d^7*((c + d*x)^(1/2) - c^(1/2))^24)/((-c)^(1/2) - (d
*x - c)^(1/2))^24 - (560*d^7*((c + d*x)^(1/2) - c^(1/2))^26)/((-c)^(1/2) - (d*x - c)^(1/2))^26 + (120*d^7*((c
+ d*x)^(1/2) - c^(1/2))^28)/((-c)^(1/2) - (d*x - c)^(1/2))^28 - (16*d^7*((c + d*x)^(1/2) - c^(1/2))^30)/((-c)^
(1/2) - (d*x - c)^(1/2))^30 + (d^7*((c + d*x)^(1/2) - c^(1/2))^32)/((-c)^(1/2) - (d*x - c)^(1/2))^32) + (a*c^6
*atanh(((c + d*x)^(1/2) - c^(1/2))/((-c)^(1/2) - (d*x - c)^(1/2))))/(4*d^5) + (5*b*c^8*atanh(((c + d*x)^(1/2)
- c^(1/2))/((-c)^(1/2) - (d*x - c)^(1/2))))/(32*d^7)