Integrand size = 24, antiderivative size = 157 \[ \int \frac {x^4 \sqrt {c+d x^2}}{a+b x^2} \, dx=\frac {(b c-4 a d) x \sqrt {c+d x^2}}{8 b^2 d}+\frac {x^3 \sqrt {c+d x^2}}{4 b}+\frac {a^{3/2} \sqrt {b c-a d} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b^3}-\frac {\left (b^2 c^2+4 a b c d-8 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 b^3 d^{3/2}} \]
-1/8*(-8*a^2*d^2+4*a*b*c*d+b^2*c^2)*arctanh(x*d^(1/2)/(d*x^2+c)^(1/2))/b^3 /d^(3/2)+a^(3/2)*arctan(x*(-a*d+b*c)^(1/2)/a^(1/2)/(d*x^2+c)^(1/2))*(-a*d+ b*c)^(1/2)/b^3+1/8*(-4*a*d+b*c)*x*(d*x^2+c)^(1/2)/b^2/d+1/4*x^3*(d*x^2+c)^ (1/2)/b
Leaf count is larger than twice the leaf count of optimal. \(395\) vs. \(2(157)=314\).
Time = 1.91 (sec) , antiderivative size = 395, normalized size of antiderivative = 2.52 \[ \int \frac {x^4 \sqrt {c+d x^2}}{a+b x^2} \, dx=\frac {x \sqrt {c+d x^2} \left (b c-4 a d+2 b d x^2\right )}{8 b^2 d}+\frac {\sqrt {a} \sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} \left (-b c+a d-\sqrt {b} \sqrt {c} \sqrt {b c-a d}\right ) \arctan \left (\frac {\sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )}{b^3 d}+\frac {\sqrt {a} \left (-b c+a d+\sqrt {b} \sqrt {c} \sqrt {b c-a d}\right ) \sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )}{b^3 d}+\frac {\left (-b^2 c^2-4 a b c d+8 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{-\sqrt {c}+\sqrt {c+d x^2}}\right )}{4 b^3 d^{3/2}} \]
(x*Sqrt[c + d*x^2]*(b*c - 4*a*d + 2*b*d*x^2))/(8*b^2*d) + (Sqrt[a]*Sqrt[2* b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*(-(b*c) + a*d - Sqrt[b]*Sqr t[c]*Sqrt[b*c - a*d])*ArcTan[(Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b* c - a*d]]*x)/(Sqrt[a]*(-Sqrt[c] + Sqrt[c + d*x^2]))])/(b^3*d) + (Sqrt[a]*( -(b*c) + a*d + Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d])*Sqrt[2*b*c - a*d + 2*Sqrt[ b]*Sqrt[c]*Sqrt[b*c - a*d]]*ArcTan[(Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*S qrt[b*c - a*d]]*x)/(Sqrt[a]*(-Sqrt[c] + Sqrt[c + d*x^2]))])/(b^3*d) + ((-( b^2*c^2) - 4*a*b*c*d + 8*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/(-Sqrt[c] + Sqrt[c + d*x^2])])/(4*b^3*d^(3/2))
Time = 0.38 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {380, 444, 25, 398, 224, 219, 291, 218}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^4 \sqrt {c+d x^2}}{a+b x^2} \, dx\) |
| \(\Big \downarrow \) 380 |
| \(\displaystyle \frac {x^3 \sqrt {c+d x^2}}{4 b}-\frac {\int \frac {x^2 \left (3 a c-(b c-4 a d) x^2\right )}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{4 b}\) |
| \(\Big \downarrow \) 444 |
| \(\displaystyle \frac {x^3 \sqrt {c+d x^2}}{4 b}-\frac {-\frac {\int -\frac {\left (b^2 c^2+4 a b d c-8 a^2 d^2\right ) x^2+a c (b c-4 a d)}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{2 b d}-\frac {x \sqrt {c+d x^2} (b c-4 a d)}{2 b d}}{4 b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x^3 \sqrt {c+d x^2}}{4 b}-\frac {\frac {\int \frac {\left (b^2 c^2+4 a b d c-8 a^2 d^2\right ) x^2+a c (b c-4 a d)}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{2 b d}-\frac {x \sqrt {c+d x^2} (b c-4 a d)}{2 b d}}{4 b}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {x^3 \sqrt {c+d x^2}}{4 b}-\frac {\frac {\frac {\left (-8 a^2 d^2+4 a b c d+b^2 c^2\right ) \int \frac {1}{\sqrt {d x^2+c}}dx}{b}-\frac {8 a^2 d (b c-a d) \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{b}}{2 b d}-\frac {x \sqrt {c+d x^2} (b c-4 a d)}{2 b d}}{4 b}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {x^3 \sqrt {c+d x^2}}{4 b}-\frac {\frac {\frac {\left (-8 a^2 d^2+4 a b c d+b^2 c^2\right ) \int \frac {1}{1-\frac {d x^2}{d x^2+c}}d\frac {x}{\sqrt {d x^2+c}}}{b}-\frac {8 a^2 d (b c-a d) \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{b}}{2 b d}-\frac {x \sqrt {c+d x^2} (b c-4 a d)}{2 b d}}{4 b}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {x^3 \sqrt {c+d x^2}}{4 b}-\frac {\frac {\frac {\left (-8 a^2 d^2+4 a b c d+b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b \sqrt {d}}-\frac {8 a^2 d (b c-a d) \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{b}}{2 b d}-\frac {x \sqrt {c+d x^2} (b c-4 a d)}{2 b d}}{4 b}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {x^3 \sqrt {c+d x^2}}{4 b}-\frac {\frac {\frac {\left (-8 a^2 d^2+4 a b c d+b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b \sqrt {d}}-\frac {8 a^2 d (b c-a d) \int \frac {1}{a-\frac {(a d-b c) x^2}{d x^2+c}}d\frac {x}{\sqrt {d x^2+c}}}{b}}{2 b d}-\frac {x \sqrt {c+d x^2} (b c-4 a d)}{2 b d}}{4 b}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {x^3 \sqrt {c+d x^2}}{4 b}-\frac {\frac {\frac {\left (-8 a^2 d^2+4 a b c d+b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b \sqrt {d}}-\frac {8 a^{3/2} d \sqrt {b c-a d} \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b}}{2 b d}-\frac {x \sqrt {c+d x^2} (b c-4 a d)}{2 b d}}{4 b}\) |
(x^3*Sqrt[c + d*x^2])/(4*b) - (-1/2*((b*c - 4*a*d)*x*Sqrt[c + d*x^2])/(b*d ) + ((-8*a^(3/2)*d*Sqrt[b*c - a*d]*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqr t[c + d*x^2])])/b + ((b^2*c^2 + 4*a*b*c*d - 8*a^2*d^2)*ArcTanh[(Sqrt[d]*x) /Sqrt[c + d*x^2]])/(b*Sqrt[d]))/(2*b*d))/(4*b)
3.7.76.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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] && (Gt Q[a, 0] || LtQ[b, 0])
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]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b* (m + 2*(p + q) + 1))), x] - Simp[e^2/(b*(m + 2*(p + q) + 1)) Int[(e*x)^(m - 2)*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[a*c*(m - 1) + (a*d*(m - 1) - 2 *q*(b*c - a*d))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 0] && GtQ[m, 1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) , x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ b Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} , x]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q _.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[f*g*(g*x)^(m - 1)*(a + b*x^2)^ (p + 1)*((c + d*x^2)^(q + 1)/(b*d*(m + 2*(p + q + 1) + 1))), x] - Simp[g^2/ (b*d*(m + 2*(p + q + 1) + 1)) Int[(g*x)^(m - 2)*(a + b*x^2)^p*(c + d*x^2) ^q*Simp[a*f*c*(m - 1) + (a*f*d*(m + 2*q + 1) + b*(f*c*(m + 2*p + 1) - e*d*( m + 2*(p + q + 1) + 1)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && GtQ[m, 1]
Time = 3.02 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.87
| method | result | size |
| pseudoelliptic | \(-\frac {\frac {b \sqrt {d \,x^{2}+c}\, \left (-2 b d \,x^{2}+4 a d -b c \right ) x}{4 d}-\frac {\left (8 a^{2} d^{2}-4 a b c d -b^{2} c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right )}{4 d^{\frac {3}{2}}}+\frac {2 \left (a d -b c \right ) a^{2} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}}{2 b^{3}}\) | \(136\) |
| risch | \(-\frac {x \left (-2 b d \,x^{2}+4 a d -b c \right ) \sqrt {d \,x^{2}+c}}{8 d \,b^{2}}+\frac {\frac {\left (8 a^{2} d^{2}-4 a b c d -b^{2} c^{2}\right ) \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{b \sqrt {d}}+\frac {4 a^{2} d \left (a d -b c \right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}-\frac {4 a^{2} d \left (a d -b c \right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}}{8 d \,b^{2}}\) | \(425\) |
| default | \(\frac {\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4 d}-\frac {c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4 d}}{b}-\frac {a \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{b^{2}}+\frac {a^{2} \left (\sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}+\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {\frac {d \sqrt {-a b}}{b}+d \left (x -\frac {\sqrt {-a b}}{b}\right )}{\sqrt {d}}+\sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}\right )}{b}+\frac {\left (a d -b c \right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -b c}{b}}}\right )}{2 b^{2} \sqrt {-a b}}-\frac {a^{2} \left (\sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}-\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {-\frac {d \sqrt {-a b}}{b}+d \left (x +\frac {\sqrt {-a b}}{b}\right )}{\sqrt {d}}+\sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}\right )}{b}+\frac {\left (a d -b c \right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -b c}{b}}}\right )}{2 b^{2} \sqrt {-a b}}\) | \(767\) |
-1/2/b^3*(1/4*b*(d*x^2+c)^(1/2)*(-2*b*d*x^2+4*a*d-b*c)/d*x-1/4*(8*a^2*d^2- 4*a*b*c*d-b^2*c^2)/d^(3/2)*arctanh((d*x^2+c)^(1/2)/x/d^(1/2))+2*(a*d-b*c)* a^2/((a*d-b*c)*a)^(1/2)*arctanh((d*x^2+c)^(1/2)/x*a/((a*d-b*c)*a)^(1/2)))
Time = 0.37 (sec) , antiderivative size = 857, normalized size of antiderivative = 5.46 \[ \int \frac {x^4 \sqrt {c+d x^2}}{a+b x^2} \, dx=\left [\frac {4 \, \sqrt {-a b c + a^{2} d} a d^{2} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - {\left (b^{2} c^{2} + 4 \, a b c d - 8 \, a^{2} d^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (2 \, b^{2} d^{2} x^{3} + {\left (b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{16 \, b^{3} d^{2}}, \frac {2 \, \sqrt {-a b c + a^{2} d} a d^{2} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + {\left (b^{2} c^{2} + 4 \, a b c d - 8 \, a^{2} d^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (2 \, b^{2} d^{2} x^{3} + {\left (b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{8 \, b^{3} d^{2}}, \frac {8 \, \sqrt {a b c - a^{2} d} a d^{2} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) - {\left (b^{2} c^{2} + 4 \, a b c d - 8 \, a^{2} d^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (2 \, b^{2} d^{2} x^{3} + {\left (b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{16 \, b^{3} d^{2}}, \frac {4 \, \sqrt {a b c - a^{2} d} a d^{2} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) + {\left (b^{2} c^{2} + 4 \, a b c d - 8 \, a^{2} d^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (2 \, b^{2} d^{2} x^{3} + {\left (b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{8 \, b^{3} d^{2}}\right ] \]
[1/16*(4*sqrt(-a*b*c + a^2*d)*a*d^2*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2) *x^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2 + 4*((b*c - 2*a*d)*x^3 - a* c*x)*sqrt(-a*b*c + a^2*d)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - (b^2*c^2 + 4*a*b*c*d - 8*a^2*d^2)*sqrt(d)*log(-2*d*x^2 - 2*sqrt(d*x^2 + c) *sqrt(d)*x - c) + 2*(2*b^2*d^2*x^3 + (b^2*c*d - 4*a*b*d^2)*x)*sqrt(d*x^2 + c))/(b^3*d^2), 1/8*(2*sqrt(-a*b*c + a^2*d)*a*d^2*log(((b^2*c^2 - 8*a*b*c* d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2 + 4*((b*c - 2 *a*d)*x^3 - a*c*x)*sqrt(-a*b*c + a^2*d)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b* x^2 + a^2)) + (b^2*c^2 + 4*a*b*c*d - 8*a^2*d^2)*sqrt(-d)*arctan(sqrt(-d)*x /sqrt(d*x^2 + c)) + (2*b^2*d^2*x^3 + (b^2*c*d - 4*a*b*d^2)*x)*sqrt(d*x^2 + c))/(b^3*d^2), 1/16*(8*sqrt(a*b*c - a^2*d)*a*d^2*arctan(1/2*sqrt(a*b*c - a^2*d)*((b*c - 2*a*d)*x^2 - a*c)*sqrt(d*x^2 + c)/((a*b*c*d - a^2*d^2)*x^3 + (a*b*c^2 - a^2*c*d)*x)) - (b^2*c^2 + 4*a*b*c*d - 8*a^2*d^2)*sqrt(d)*log( -2*d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) + 2*(2*b^2*d^2*x^3 + (b^2*c*d - 4*a*b*d^2)*x)*sqrt(d*x^2 + c))/(b^3*d^2), 1/8*(4*sqrt(a*b*c - a^2*d)*a*d ^2*arctan(1/2*sqrt(a*b*c - a^2*d)*((b*c - 2*a*d)*x^2 - a*c)*sqrt(d*x^2 + c )/((a*b*c*d - a^2*d^2)*x^3 + (a*b*c^2 - a^2*c*d)*x)) + (b^2*c^2 + 4*a*b*c* d - 8*a^2*d^2)*sqrt(-d)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)) + (2*b^2*d^2*x^ 3 + (b^2*c*d - 4*a*b*d^2)*x)*sqrt(d*x^2 + c))/(b^3*d^2)]
\[ \int \frac {x^4 \sqrt {c+d x^2}}{a+b x^2} \, dx=\int \frac {x^{4} \sqrt {c + d x^{2}}}{a + b x^{2}}\, dx \]
\[ \int \frac {x^4 \sqrt {c+d x^2}}{a+b x^2} \, dx=\int { \frac {\sqrt {d x^{2} + c} x^{4}}{b x^{2} + a} \,d x } \]
Exception generated. \[ \int \frac {x^4 \sqrt {c+d x^2}}{a+b x^2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {x^4 \sqrt {c+d x^2}}{a+b x^2} \, dx=\int \frac {x^4\,\sqrt {d\,x^2+c}}{b\,x^2+a} \,d x \]