Integrand size = 24, antiderivative size = 93 \[ \int \frac {x^7 \sqrt {c+d x^4}}{a+b x^4} \, dx=-\frac {a \sqrt {c+d x^4}}{2 b^2}+\frac {\left (c+d x^4\right )^{3/2}}{6 b d}+\frac {a \sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{2 b^{5/2}} \] Output:
-1/2*a*(d*x^4+c)^(1/2)/b^2+1/6*(d*x^4+c)^(3/2)/b/d+1/2*a*(-a*d+b*c)^(1/2)* arctanh(b^(1/2)*(d*x^4+c)^(1/2)/(-a*d+b*c)^(1/2))/b^(5/2)
Time = 0.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.95 \[ \int \frac {x^7 \sqrt {c+d x^4}}{a+b x^4} \, dx=\frac {\sqrt {c+d x^4} \left (-3 a d+b \left (c+d x^4\right )\right )}{6 b^2 d}+\frac {a \sqrt {-b c+a d} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {-b c+a d}}\right )}{2 b^{5/2}} \] Input:
Integrate[(x^7*Sqrt[c + d*x^4])/(a + b*x^4),x]
Output:
(Sqrt[c + d*x^4]*(-3*a*d + b*(c + d*x^4)))/(6*b^2*d) + (a*Sqrt[-(b*c) + a* d]*ArcTan[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[-(b*c) + a*d]])/(2*b^(5/2))
Time = 0.39 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {948, 90, 60, 73, 221}
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.
| \(\displaystyle \int \frac {x^7 \sqrt {c+d x^4}}{a+b x^4} \, dx\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle \frac {1}{4} \int \frac {x^4 \sqrt {d x^4+c}}{b x^4+a}dx^4\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {1}{4} \left (\frac {2 \left (c+d x^4\right )^{3/2}}{3 b d}-\frac {a \int \frac {\sqrt {d x^4+c}}{b x^4+a}dx^4}{b}\right )\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{4} \left (\frac {2 \left (c+d x^4\right )^{3/2}}{3 b d}-\frac {a \left (\frac {(b c-a d) \int \frac {1}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^4}{b}+\frac {2 \sqrt {c+d x^4}}{b}\right )}{b}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{4} \left (\frac {2 \left (c+d x^4\right )^{3/2}}{3 b d}-\frac {a \left (\frac {2 (b c-a d) \int \frac {1}{\frac {b x^8}{d}+a-\frac {b c}{d}}d\sqrt {d x^4+c}}{b d}+\frac {2 \sqrt {c+d x^4}}{b}\right )}{b}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{4} \left (\frac {2 \left (c+d x^4\right )^{3/2}}{3 b d}-\frac {a \left (\frac {2 \sqrt {c+d x^4}}{b}-\frac {2 \sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{b^{3/2}}\right )}{b}\right )\) |
Input:
Int[(x^7*Sqrt[c + d*x^4])/(a + b*x^4),x]
Output:
((2*(c + d*x^4)^(3/2))/(3*b*d) - (a*((2*Sqrt[c + d*x^4])/b - (2*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[b*c - a*d]])/b^(3/2)))/b)/4
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.19 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.90
| method | result | size |
| pseudoelliptic | \(\frac {-\frac {\sqrt {d \,x^{4}+c}\, \left (-d b \,x^{4}+3 a d -c b \right )}{3}+\frac {a d \left (a d -c b \right ) \arctan \left (\frac {\sqrt {d \,x^{4}+c}\, b}{\sqrt {\left (a d -c b \right ) b}}\right )}{\sqrt {\left (a d -c b \right ) b}}}{2 b^{2} d}\) | \(84\) |
| risch | \(-\frac {\left (-d b \,x^{4}+3 a d -c b \right ) \sqrt {d \,x^{4}+c}}{6 d \,b^{2}}+\frac {a \left (a d -c b \right ) \left (-\frac {\ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 b \sqrt {-\frac {a d -c b}{b}}}-\frac {\ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 b \sqrt {-\frac {a d -c b}{b}}}\right )}{b^{2}}\) | \(363\) |
| elliptic | \(\frac {\left (d \,x^{4}+c \right )^{\frac {3}{2}}}{6 b d}-\frac {a \left (\sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}+\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {\frac {d \sqrt {-a b}}{b}+\left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{\sqrt {d}}+\sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}\right )}{b}+\frac {\left (a d -c b \right ) \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -c b}{b}}}\right )}{4 b^{2}}-\frac {a \left (\sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}-\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {-\frac {d \sqrt {-a b}}{b}+\left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{\sqrt {d}}+\sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}\right )}{b}+\frac {\left (a d -c b \right ) \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -c b}{b}}}\right )}{4 b^{2}}\) | \(702\) |
| default | \(\frac {\left (d \,x^{4}+c \right )^{\frac {3}{2}}}{6 b d}-\frac {a \left (\frac {\sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}+\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {\frac {d \sqrt {-a b}}{b}+\left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{\sqrt {d}}+\sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}\right )}{b}+\frac {\left (a d -c b \right ) \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -c b}{b}}}}{4 b}+\frac {\sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}-\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {-\frac {d \sqrt {-a b}}{b}+\left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{\sqrt {d}}+\sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}\right )}{b}+\frac {\left (a d -c b \right ) \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -c b}{b}}}}{4 b}\right )}{b}\) | \(707\) |
Input:
int(x^7*(d*x^4+c)^(1/2)/(b*x^4+a),x,method=_RETURNVERBOSE)
Output:
1/2/b^2*(-1/3*(d*x^4+c)^(1/2)*(-b*d*x^4+3*a*d-b*c)+a*d*(a*d-b*c)/((a*d-b*c )*b)^(1/2)*arctan((d*x^4+c)^(1/2)*b/((a*d-b*c)*b)^(1/2)))/d
Time = 0.09 (sec) , antiderivative size = 195, normalized size of antiderivative = 2.10 \[ \int \frac {x^7 \sqrt {c+d x^4}}{a+b x^4} \, dx=\left [\frac {3 \, a d \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x^{4} + 2 \, b c - a d + 2 \, \sqrt {d x^{4} + c} b \sqrt {\frac {b c - a d}{b}}}{b x^{4} + a}\right ) + 2 \, {\left (b d x^{4} + b c - 3 \, a d\right )} \sqrt {d x^{4} + c}}{12 \, b^{2} d}, \frac {3 \, a d \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x^{4} + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + {\left (b d x^{4} + b c - 3 \, a d\right )} \sqrt {d x^{4} + c}}{6 \, b^{2} d}\right ] \] Input:
integrate(x^7*(d*x^4+c)^(1/2)/(b*x^4+a),x, algorithm="fricas")
Output:
[1/12*(3*a*d*sqrt((b*c - a*d)/b)*log((b*d*x^4 + 2*b*c - a*d + 2*sqrt(d*x^4 + c)*b*sqrt((b*c - a*d)/b))/(b*x^4 + a)) + 2*(b*d*x^4 + b*c - 3*a*d)*sqrt (d*x^4 + c))/(b^2*d), 1/6*(3*a*d*sqrt(-(b*c - a*d)/b)*arctan(-sqrt(d*x^4 + c)*b*sqrt(-(b*c - a*d)/b)/(b*c - a*d)) + (b*d*x^4 + b*c - 3*a*d)*sqrt(d*x ^4 + c))/(b^2*d)]
Time = 8.18 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.31 \[ \int \frac {x^7 \sqrt {c+d x^4}}{a+b x^4} \, dx=\begin {cases} \frac {2 \left (- \frac {a d^{2} \sqrt {c + d x^{4}}}{4 b^{2}} + \frac {a d^{2} \left (a d - b c\right ) \operatorname {atan}{\left (\frac {\sqrt {c + d x^{4}}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{4 b^{3} \sqrt {\frac {a d - b c}{b}}} + \frac {d \left (c + d x^{4}\right )^{\frac {3}{2}}}{12 b}\right )}{d^{2}} & \text {for}\: d \neq 0 \\\sqrt {c} \left (- \frac {a \left (\begin {cases} \frac {x^{4}}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + b x^{4} \right )}}{b} & \text {otherwise} \end {cases}\right )}{4 b} + \frac {x^{4}}{4 b}\right ) & \text {otherwise} \end {cases} \] Input:
integrate(x**7*(d*x**4+c)**(1/2)/(b*x**4+a),x)
Output:
Piecewise((2*(-a*d**2*sqrt(c + d*x**4)/(4*b**2) + a*d**2*(a*d - b*c)*atan( sqrt(c + d*x**4)/sqrt((a*d - b*c)/b))/(4*b**3*sqrt((a*d - b*c)/b)) + d*(c + d*x**4)**(3/2)/(12*b))/d**2, Ne(d, 0)), (sqrt(c)*(-a*Piecewise((x**4/a, Eq(b, 0)), (log(a + b*x**4)/b, True))/(4*b) + x**4/(4*b)), True))
Exception generated. \[ \int \frac {x^7 \sqrt {c+d x^4}}{a+b x^4} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^7*(d*x^4+c)^(1/2)/(b*x^4+a),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Time = 0.13 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.03 \[ \int \frac {x^7 \sqrt {c+d x^4}}{a+b x^4} \, dx=-\frac {{\left (a b c - a^{2} d\right )} \arctan \left (\frac {\sqrt {d x^{4} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, \sqrt {-b^{2} c + a b d} b^{2}} + \frac {{\left (d x^{4} + c\right )}^{\frac {3}{2}} b^{2} d^{2} - 3 \, \sqrt {d x^{4} + c} a b d^{3}}{6 \, b^{3} d^{3}} \] Input:
integrate(x^7*(d*x^4+c)^(1/2)/(b*x^4+a),x, algorithm="giac")
Output:
-1/2*(a*b*c - a^2*d)*arctan(sqrt(d*x^4 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt( -b^2*c + a*b*d)*b^2) + 1/6*((d*x^4 + c)^(3/2)*b^2*d^2 - 3*sqrt(d*x^4 + c)* a*b*d^3)/(b^3*d^3)
Time = 3.25 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.94 \[ \int \frac {x^7 \sqrt {c+d x^4}}{a+b x^4} \, dx=\frac {{\left (d\,x^4+c\right )}^{3/2}}{6\,b\,d}-\frac {a\,\sqrt {d\,x^4+c}}{2\,b^2}+\frac {a\,\mathrm {atan}\left (\frac {a\,\sqrt {b}\,\sqrt {d\,x^4+c}\,\sqrt {a\,d-b\,c}}{a^2\,d-a\,b\,c}\right )\,\sqrt {a\,d-b\,c}}{2\,b^{5/2}} \] Input:
int((x^7*(c + d*x^4)^(1/2))/(a + b*x^4),x)
Output:
(c + d*x^4)^(3/2)/(6*b*d) - (a*(c + d*x^4)^(1/2))/(2*b^2) + (a*atan((a*b^( 1/2)*(c + d*x^4)^(1/2)*(a*d - b*c)^(1/2))/(a^2*d - a*b*c))*(a*d - b*c)^(1/ 2))/(2*b^(5/2))
Time = 0.25 (sec) , antiderivative size = 583, normalized size of antiderivative = 6.27 \[ \int \frac {x^7 \sqrt {c+d x^4}}{a+b x^4} \, dx=\frac {3 \sqrt {b}\, \sqrt {d \,x^{4}+c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d}\, \sqrt {b}\, \sqrt {d \,x^{4}+c}\, x^{2}+\sqrt {b}\, c +\sqrt {b}\, d \,x^{4}}{\sqrt {d \,x^{4}+c}\, \sqrt {a d -b c}+\sqrt {d}\, \sqrt {a d -b c}\, x^{2}}\right ) a c d +12 \sqrt {b}\, \sqrt {d \,x^{4}+c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d}\, \sqrt {b}\, \sqrt {d \,x^{4}+c}\, x^{2}+\sqrt {b}\, c +\sqrt {b}\, d \,x^{4}}{\sqrt {d \,x^{4}+c}\, \sqrt {a d -b c}+\sqrt {d}\, \sqrt {a d -b c}\, x^{2}}\right ) a \,d^{2} x^{4}+9 \sqrt {d}\, \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d}\, \sqrt {b}\, \sqrt {d \,x^{4}+c}\, x^{2}+\sqrt {b}\, c +\sqrt {b}\, d \,x^{4}}{\sqrt {d \,x^{4}+c}\, \sqrt {a d -b c}+\sqrt {d}\, \sqrt {a d -b c}\, x^{2}}\right ) a c d \,x^{2}+12 \sqrt {d}\, \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d}\, \sqrt {b}\, \sqrt {d \,x^{4}+c}\, x^{2}+\sqrt {b}\, c +\sqrt {b}\, d \,x^{4}}{\sqrt {d \,x^{4}+c}\, \sqrt {a d -b c}+\sqrt {d}\, \sqrt {a d -b c}\, x^{2}}\right ) a \,d^{2} x^{6}-9 \sqrt {d}\, \sqrt {d \,x^{4}+c}\, a b c d \,x^{2}-12 \sqrt {d}\, \sqrt {d \,x^{4}+c}\, a b \,d^{2} x^{6}+3 \sqrt {d}\, \sqrt {d \,x^{4}+c}\, b^{2} c^{2} x^{2}+7 \sqrt {d}\, \sqrt {d \,x^{4}+c}\, b^{2} c d \,x^{6}+4 \sqrt {d}\, \sqrt {d \,x^{4}+c}\, b^{2} d^{2} x^{10}-3 a b \,c^{2} d -15 a b c \,d^{2} x^{4}-12 a b \,d^{3} x^{8}+b^{2} c^{3}+6 b^{2} c^{2} d \,x^{4}+9 b^{2} c \,d^{2} x^{8}+4 b^{2} d^{3} x^{12}}{6 b^{3} d \left (\sqrt {d \,x^{4}+c}\, c +4 \sqrt {d \,x^{4}+c}\, d \,x^{4}+3 \sqrt {d}\, c \,x^{2}+4 \sqrt {d}\, d \,x^{6}\right )} \] Input:
int(x^7*(d*x^4+c)^(1/2)/(b*x^4+a),x)
Output:
(3*sqrt(b)*sqrt(c + d*x**4)*sqrt(a*d - b*c)*atan((sqrt(d)*sqrt(b)*sqrt(c + d*x**4)*x**2 + sqrt(b)*c + sqrt(b)*d*x**4)/(sqrt(c + d*x**4)*sqrt(a*d - b *c) + sqrt(d)*sqrt(a*d - b*c)*x**2))*a*c*d + 12*sqrt(b)*sqrt(c + d*x**4)*s qrt(a*d - b*c)*atan((sqrt(d)*sqrt(b)*sqrt(c + d*x**4)*x**2 + sqrt(b)*c + s qrt(b)*d*x**4)/(sqrt(c + d*x**4)*sqrt(a*d - b*c) + sqrt(d)*sqrt(a*d - b*c) *x**2))*a*d**2*x**4 + 9*sqrt(d)*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(d)*sqrt (b)*sqrt(c + d*x**4)*x**2 + sqrt(b)*c + sqrt(b)*d*x**4)/(sqrt(c + d*x**4)* sqrt(a*d - b*c) + sqrt(d)*sqrt(a*d - b*c)*x**2))*a*c*d*x**2 + 12*sqrt(d)*s qrt(b)*sqrt(a*d - b*c)*atan((sqrt(d)*sqrt(b)*sqrt(c + d*x**4)*x**2 + sqrt( b)*c + sqrt(b)*d*x**4)/(sqrt(c + d*x**4)*sqrt(a*d - b*c) + sqrt(d)*sqrt(a* d - b*c)*x**2))*a*d**2*x**6 - 9*sqrt(d)*sqrt(c + d*x**4)*a*b*c*d*x**2 - 12 *sqrt(d)*sqrt(c + d*x**4)*a*b*d**2*x**6 + 3*sqrt(d)*sqrt(c + d*x**4)*b**2* c**2*x**2 + 7*sqrt(d)*sqrt(c + d*x**4)*b**2*c*d*x**6 + 4*sqrt(d)*sqrt(c + d*x**4)*b**2*d**2*x**10 - 3*a*b*c**2*d - 15*a*b*c*d**2*x**4 - 12*a*b*d**3* x**8 + b**2*c**3 + 6*b**2*c**2*d*x**4 + 9*b**2*c*d**2*x**8 + 4*b**2*d**3*x **12)/(6*b**3*d*(sqrt(c + d*x**4)*c + 4*sqrt(c + d*x**4)*d*x**4 + 3*sqrt(d )*c*x**2 + 4*sqrt(d)*d*x**6))