\(\int \frac {x^7}{(a+b x^2) \sqrt {c+d x^2}} \, dx\) [950]

Optimal result

Integrand size = 24, antiderivative size = 140 \[ \int \frac {x^7}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\frac {\left (b^2 c^2+a b c d+a^2 d^2\right ) \sqrt {c+d x^2}}{b^3 d^3}-\frac {(2 b c+a d) \left (c+d x^2\right )^{3/2}}{3 b^2 d^3}+\frac {\left (c+d x^2\right )^{5/2}}{5 b d^3}+\frac {a^3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{b^{7/2} \sqrt {b c-a d}} \] Output:

(a^2*d^2+a*b*c*d+b^2*c^2)*(d*x^2+c)^(1/2)/b^3/d^3-1/3*(a*d+2*b*c)*(d*x^2+c 
)^(3/2)/b^2/d^3+1/5*(d*x^2+c)^(5/2)/b/d^3+a^3*arctanh(b^(1/2)*(d*x^2+c)^(1 
/2)/(-a*d+b*c)^(1/2))/b^(7/2)/(-a*d+b*c)^(1/2)
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.87 \[ \int \frac {x^7}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\frac {\sqrt {c+d x^2} \left (15 a^2 d^2-5 a b d \left (-2 c+d x^2\right )+b^2 \left (8 c^2-4 c d x^2+3 d^2 x^4\right )\right )}{15 b^3 d^3}-\frac {a^3 \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{b^{7/2} \sqrt {-b c+a d}} \] Input:

Integrate[x^7/((a + b*x^2)*Sqrt[c + d*x^2]),x]
 

Output:

(Sqrt[c + d*x^2]*(15*a^2*d^2 - 5*a*b*d*(-2*c + d*x^2) + b^2*(8*c^2 - 4*c*d 
*x^2 + 3*d^2*x^4)))/(15*b^3*d^3) - (a^3*ArcTan[(Sqrt[b]*Sqrt[c + d*x^2])/S 
qrt[-(b*c) + a*d]])/(b^(7/2)*Sqrt[-(b*c) + a*d])
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {354, 99, 2009}

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.

\(\displaystyle \int \frac {x^7}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx\)

\(\Big \downarrow \) 354

\(\displaystyle \frac {1}{2} \int \frac {x^6}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx^2\)

\(\Big \downarrow \) 99

\(\displaystyle \frac {1}{2} \int \left (-\frac {a^3}{b^3 \left (b x^2+a\right ) \sqrt {d x^2+c}}+\frac {\left (d x^2+c\right )^{3/2}}{b d^2}+\frac {(-2 b c-a d) \sqrt {d x^2+c}}{b^2 d^2}+\frac {b^2 c^2+a b d c+a^2 d^2}{b^3 d^2 \sqrt {d x^2+c}}\right )dx^2\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (\frac {2 a^3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{b^{7/2} \sqrt {b c-a d}}+\frac {2 \sqrt {c+d x^2} \left (a^2 d^2+a b c d+b^2 c^2\right )}{b^3 d^3}-\frac {2 \left (c+d x^2\right )^{3/2} (a d+2 b c)}{3 b^2 d^3}+\frac {2 \left (c+d x^2\right )^{5/2}}{5 b d^3}\right )\)

Input:

Int[x^7/((a + b*x^2)*Sqrt[c + d*x^2]),x]
 

Output:

((2*(b^2*c^2 + a*b*c*d + a^2*d^2)*Sqrt[c + d*x^2])/(b^3*d^3) - (2*(2*b*c + 
 a*d)*(c + d*x^2)^(3/2))/(3*b^2*d^3) + (2*(c + d*x^2)^(5/2))/(5*b*d^3) + ( 
2*a^3*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(b^(7/2)*Sqrt[b* 
c - a*d]))/2
 

Defintions of rubi rules used

rule 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | 
| (GtQ[m, 0] && GeQ[n, -1]))
 

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.77 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.86

method	result	size
pseudoelliptic	\(\frac {\left (\frac {\left (d^{2} x^{4}-\frac {4}{3} c d \,x^{2}+\frac {8}{3} c^{2}\right ) b^{2}}{5}+\frac {2 a \left (-\frac {x^{2} d}{2}+c \right ) d b}{3}+a^{2} d^{2}\right ) \sqrt {\left (a d -b c \right ) b}\, \sqrt {x^{2} d +c}-\arctan \left (\frac {\sqrt {x^{2} d +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right ) a^{3} d^{3}}{\sqrt {\left (a d -b c \right ) b}\, b^{3} d^{3}}\)	\(121\)
risch	\(\frac {\left (3 b^{2} d^{2} x^{4}-5 x^{2} a b \,d^{2}-4 x^{2} b^{2} c d +15 a^{2} d^{2}+10 a b c d +8 b^{2} c^{2}\right ) \sqrt {x^{2} d +c}}{15 d^{3} b^{3}}+\frac {a^{3} \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 b^{4} \sqrt {-\frac {a d -b c}{b}}}+\frac {a^{3} \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 b^{4} \sqrt {-\frac {a d -b c}{b}}}\)	\(377\)
default	\(\frac {\frac {x^{4} \sqrt {x^{2} d +c}}{5 d}-\frac {4 c \left (\frac {x^{2} \sqrt {x^{2} d +c}}{3 d}-\frac {2 c \sqrt {x^{2} d +c}}{3 d^{2}}\right )}{5 d}}{b}+\frac {a^{2} \sqrt {x^{2} d +c}}{b^{3} d}-\frac {a \left (\frac {x^{2} \sqrt {x^{2} d +c}}{3 d}-\frac {2 c \sqrt {x^{2} d +c}}{3 d^{2}}\right )}{b^{2}}+\frac {a^{3} \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 b^{4} \sqrt {-\frac {a d -b c}{b}}}+\frac {a^{3} \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 b^{4} \sqrt {-\frac {a d -b c}{b}}}\)	\(425\)

Input:

int(x^7/(b*x^2+a)/(d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

((1/5*(d^2*x^4-4/3*c*d*x^2+8/3*c^2)*b^2+2/3*a*(-1/2*x^2*d+c)*d*b+a^2*d^2)* 
((a*d-b*c)*b)^(1/2)*(d*x^2+c)^(1/2)-arctan((d*x^2+c)^(1/2)*b/((a*d-b*c)*b) 
^(1/2))*a^3*d^3)/((a*d-b*c)*b)^(1/2)/b^3/d^3
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 494, normalized size of antiderivative = 3.53 \[ \int \frac {x^7}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\left [\frac {15 \, \sqrt {b^{2} c - a b d} a^{3} d^{3} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} + 4 \, {\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {b^{2} c - a b d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, {\left (8 \, b^{4} c^{3} + 2 \, a b^{3} c^{2} d + 5 \, a^{2} b^{2} c d^{2} - 15 \, a^{3} b d^{3} + 3 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{4} - {\left (4 \, b^{4} c^{2} d + a b^{3} c d^{2} - 5 \, a^{2} b^{2} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{60 \, {\left (b^{5} c d^{3} - a b^{4} d^{4}\right )}}, \frac {15 \, \sqrt {-b^{2} c + a b d} a^{3} d^{3} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {-b^{2} c + a b d} \sqrt {d x^{2} + c}}{2 \, {\left (b^{2} c^{2} - a b c d + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )}}\right ) + 2 \, {\left (8 \, b^{4} c^{3} + 2 \, a b^{3} c^{2} d + 5 \, a^{2} b^{2} c d^{2} - 15 \, a^{3} b d^{3} + 3 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{4} - {\left (4 \, b^{4} c^{2} d + a b^{3} c d^{2} - 5 \, a^{2} b^{2} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{30 \, {\left (b^{5} c d^{3} - a b^{4} d^{4}\right )}}\right ] \] Input:

integrate(x^7/(b*x^2+a)/(d*x^2+c)^(1/2),x, algorithm="fricas")
 

Output:

[1/60*(15*sqrt(b^2*c - a*b*d)*a^3*d^3*log((b^2*d^2*x^4 + 8*b^2*c^2 - 8*a*b 
*c*d + a^2*d^2 + 2*(4*b^2*c*d - 3*a*b*d^2)*x^2 + 4*(b*d*x^2 + 2*b*c - a*d) 
*sqrt(b^2*c - a*b*d)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + 4*(8* 
b^4*c^3 + 2*a*b^3*c^2*d + 5*a^2*b^2*c*d^2 - 15*a^3*b*d^3 + 3*(b^4*c*d^2 - 
a*b^3*d^3)*x^4 - (4*b^4*c^2*d + a*b^3*c*d^2 - 5*a^2*b^2*d^3)*x^2)*sqrt(d*x 
^2 + c))/(b^5*c*d^3 - a*b^4*d^4), 1/30*(15*sqrt(-b^2*c + a*b*d)*a^3*d^3*ar 
ctan(-1/2*(b*d*x^2 + 2*b*c - a*d)*sqrt(-b^2*c + a*b*d)*sqrt(d*x^2 + c)/(b^ 
2*c^2 - a*b*c*d + (b^2*c*d - a*b*d^2)*x^2)) + 2*(8*b^4*c^3 + 2*a*b^3*c^2*d 
 + 5*a^2*b^2*c*d^2 - 15*a^3*b*d^3 + 3*(b^4*c*d^2 - a*b^3*d^3)*x^4 - (4*b^4 
*c^2*d + a*b^3*c*d^2 - 5*a^2*b^2*d^3)*x^2)*sqrt(d*x^2 + c))/(b^5*c*d^3 - a 
*b^4*d^4)]
 

Sympy [F]

\[ \int \frac {x^7}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\int \frac {x^{7}}{\left (a + b x^{2}\right ) \sqrt {c + d x^{2}}}\, dx \] Input:

integrate(x**7/(b*x**2+a)/(d*x**2+c)**(1/2),x)
 

Output:

Integral(x**7/((a + b*x**2)*sqrt(c + d*x**2)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^7}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^7/(b*x^2+a)/(d*x^2+c)^(1/2),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
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.20 \[ \int \frac {x^7}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=-\frac {a^{3} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b^{3}} + \frac {3 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{4} d^{12} - 10 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{4} c d^{12} + 15 \, \sqrt {d x^{2} + c} b^{4} c^{2} d^{12} - 5 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b^{3} d^{13} + 15 \, \sqrt {d x^{2} + c} a b^{3} c d^{13} + 15 \, \sqrt {d x^{2} + c} a^{2} b^{2} d^{14}}{15 \, b^{5} d^{15}} \] Input:

integrate(x^7/(b*x^2+a)/(d*x^2+c)^(1/2),x, algorithm="giac")
 

Output:

-a^3*arctan(sqrt(d*x^2 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)* 
b^3) + 1/15*(3*(d*x^2 + c)^(5/2)*b^4*d^12 - 10*(d*x^2 + c)^(3/2)*b^4*c*d^1 
2 + 15*sqrt(d*x^2 + c)*b^4*c^2*d^12 - 5*(d*x^2 + c)^(3/2)*a*b^3*d^13 + 15* 
sqrt(d*x^2 + c)*a*b^3*c*d^13 + 15*sqrt(d*x^2 + c)*a^2*b^2*d^14)/(b^5*d^15)
 

Mupad [B] (verification not implemented)

Time = 1.02 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.24 \[ \int \frac {x^7}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\left (\frac {3\,c^2}{b\,d^3}+\frac {\left (\frac {3\,c}{b\,d^3}+\frac {a\,d^4-b\,c\,d^3}{b^2\,d^6}\right )\,\left (a\,d^4-b\,c\,d^3\right )}{b\,d^3}\right )\,\sqrt {d\,x^2+c}-\left (\frac {c}{b\,d^3}+\frac {a\,d^4-b\,c\,d^3}{3\,b^2\,d^6}\right )\,{\left (d\,x^2+c\right )}^{3/2}+\frac {{\left (d\,x^2+c\right )}^{5/2}}{5\,b\,d^3}-\frac {a^3\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d\,x^2+c}}{\sqrt {a\,d-b\,c}}\right )}{b^{7/2}\,\sqrt {a\,d-b\,c}} \] Input:

int(x^7/((a + b*x^2)*(c + d*x^2)^(1/2)),x)
 

Output:

((3*c^2)/(b*d^3) + (((3*c)/(b*d^3) + (a*d^4 - b*c*d^3)/(b^2*d^6))*(a*d^4 - 
 b*c*d^3))/(b*d^3))*(c + d*x^2)^(1/2) - (c/(b*d^3) + (a*d^4 - b*c*d^3)/(3* 
b^2*d^6))*(c + d*x^2)^(3/2) + (c + d*x^2)^(5/2)/(5*b*d^3) - (a^3*atan((b^( 
1/2)*(c + d*x^2)^(1/2))/(a*d - b*c)^(1/2)))/(b^(7/2)*(a*d - b*c)^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.96 \[ \int \frac {x^7}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\frac {-15 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b x +b c +b d \,x^{2}}{\sqrt {b}\, \sqrt {d \,x^{2}+c}\, \sqrt {a d -b c}+\sqrt {d}\, \sqrt {b}\, \sqrt {a d -b c}\, x}\right ) a^{3} d^{3}+15 \sqrt {d \,x^{2}+c}\, a^{3} b \,d^{3}-5 \sqrt {d \,x^{2}+c}\, a^{2} b^{2} c \,d^{2}-5 \sqrt {d \,x^{2}+c}\, a^{2} b^{2} d^{3} x^{2}-2 \sqrt {d \,x^{2}+c}\, a \,b^{3} c^{2} d +\sqrt {d \,x^{2}+c}\, a \,b^{3} c \,d^{2} x^{2}+3 \sqrt {d \,x^{2}+c}\, a \,b^{3} d^{3} x^{4}-8 \sqrt {d \,x^{2}+c}\, b^{4} c^{3}+4 \sqrt {d \,x^{2}+c}\, b^{4} c^{2} d \,x^{2}-3 \sqrt {d \,x^{2}+c}\, b^{4} c \,d^{2} x^{4}}{15 b^{4} d^{3} \left (a d -b c \right )} \] Input:

int(x^7/(b*x^2+a)/(d*x^2+c)^(1/2),x)
 

Output:

( - 15*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(d)*sqrt(c + d*x**2)*b*x + b*c + 
b*d*x**2)/(sqrt(b)*sqrt(c + d*x**2)*sqrt(a*d - b*c) + sqrt(d)*sqrt(b)*sqrt 
(a*d - b*c)*x))*a**3*d**3 + 15*sqrt(c + d*x**2)*a**3*b*d**3 - 5*sqrt(c + d 
*x**2)*a**2*b**2*c*d**2 - 5*sqrt(c + d*x**2)*a**2*b**2*d**3*x**2 - 2*sqrt( 
c + d*x**2)*a*b**3*c**2*d + sqrt(c + d*x**2)*a*b**3*c*d**2*x**2 + 3*sqrt(c 
 + d*x**2)*a*b**3*d**3*x**4 - 8*sqrt(c + d*x**2)*b**4*c**3 + 4*sqrt(c + d* 
x**2)*b**4*c**2*d*x**2 - 3*sqrt(c + d*x**2)*b**4*c*d**2*x**4)/(15*b**4*d** 
3*(a*d - b*c))