\(\int \frac {x^7 (A+B x^2)}{\sqrt {b x^2+c x^4}} \, dx\) [195]

Optimal result

Integrand size = 26, antiderivative size = 176 \[ \int \frac {x^7 \left (A+B x^2\right )}{\sqrt {b x^2+c x^4}} \, dx=-\frac {5 b^2 (7 b B-8 A c) \sqrt {b x^2+c x^4}}{128 c^4}+\frac {5 b (7 b B-8 A c) x^2 \sqrt {b x^2+c x^4}}{192 c^3}-\frac {(7 b B-8 A c) x^4 \sqrt {b x^2+c x^4}}{48 c^2}+\frac {B x^6 \sqrt {b x^2+c x^4}}{8 c}+\frac {5 b^3 (7 b B-8 A c) \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{128 c^{9/2}} \] Output:

-5/128*b^2*(-8*A*c+7*B*b)*(c*x^4+b*x^2)^(1/2)/c^4+5/192*b*(-8*A*c+7*B*b)*x 
^2*(c*x^4+b*x^2)^(1/2)/c^3-1/48*(-8*A*c+7*B*b)*x^4*(c*x^4+b*x^2)^(1/2)/c^2 
+1/8*B*x^6*(c*x^4+b*x^2)^(1/2)/c+5/128*b^3*(-8*A*c+7*B*b)*arctanh(c^(1/2)* 
x^2/(c*x^4+b*x^2)^(1/2))/c^(9/2)
 

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.88 \[ \int \frac {x^7 \left (A+B x^2\right )}{\sqrt {b x^2+c x^4}} \, dx=\frac {x \left (\sqrt {c} \left (b+c x^2\right ) \left (-105 b^3 B x+16 c^3 x^5 \left (4 A+3 B x^2\right )-8 b c^2 x^3 \left (10 A+7 B x^2\right )+10 b^2 c x \left (12 A+7 B x^2\right )\right )+30 b^3 (7 b B-8 A c) \sqrt {b+c x^2} \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {b}+\sqrt {b+c x^2}}\right )\right )}{384 c^{9/2} \sqrt {x^2 \left (b+c x^2\right )}} \] Input:

Integrate[(x^7*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]
 

Output:

(x*(Sqrt[c]*(b + c*x^2)*(-105*b^3*B*x + 16*c^3*x^5*(4*A + 3*B*x^2) - 8*b*c 
^2*x^3*(10*A + 7*B*x^2) + 10*b^2*c*x*(12*A + 7*B*x^2)) + 30*b^3*(7*b*B - 8 
*A*c)*Sqrt[b + c*x^2]*ArcTanh[(Sqrt[c]*x)/(-Sqrt[b] + Sqrt[b + c*x^2])]))/ 
(384*c^(9/2)*Sqrt[x^2*(b + c*x^2)])
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1940, 1221, 1134, 1134, 1160, 1091, 219}

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.

\(\displaystyle \int \frac {x^7 \left (A+B x^2\right )}{\sqrt {b x^2+c x^4}} \, dx\)

\(\Big \downarrow \) 1940

\(\displaystyle \frac {1}{2} \int \frac {x^6 \left (B x^2+A\right )}{\sqrt {c x^4+b x^2}}dx^2\)

\(\Big \downarrow \) 1221

\(\displaystyle \frac {1}{2} \left (\frac {B x^6 \sqrt {b x^2+c x^4}}{4 c}-\frac {(7 b B-8 A c) \int \frac {x^6}{\sqrt {c x^4+b x^2}}dx^2}{8 c}\right )\)

\(\Big \downarrow \) 1134

\(\displaystyle \frac {1}{2} \left (\frac {B x^6 \sqrt {b x^2+c x^4}}{4 c}-\frac {(7 b B-8 A c) \left (\frac {x^4 \sqrt {b x^2+c x^4}}{3 c}-\frac {5 b \int \frac {x^4}{\sqrt {c x^4+b x^2}}dx^2}{6 c}\right )}{8 c}\right )\)

\(\Big \downarrow \) 1134

\(\displaystyle \frac {1}{2} \left (\frac {B x^6 \sqrt {b x^2+c x^4}}{4 c}-\frac {(7 b B-8 A c) \left (\frac {x^4 \sqrt {b x^2+c x^4}}{3 c}-\frac {5 b \left (\frac {x^2 \sqrt {b x^2+c x^4}}{2 c}-\frac {3 b \int \frac {x^2}{\sqrt {c x^4+b x^2}}dx^2}{4 c}\right )}{6 c}\right )}{8 c}\right )\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {1}{2} \left (\frac {B x^6 \sqrt {b x^2+c x^4}}{4 c}-\frac {(7 b B-8 A c) \left (\frac {x^4 \sqrt {b x^2+c x^4}}{3 c}-\frac {5 b \left (\frac {x^2 \sqrt {b x^2+c x^4}}{2 c}-\frac {3 b \left (\frac {\sqrt {b x^2+c x^4}}{c}-\frac {b \int \frac {1}{\sqrt {c x^4+b x^2}}dx^2}{2 c}\right )}{4 c}\right )}{6 c}\right )}{8 c}\right )\)

\(\Big \downarrow \) 1091

\(\displaystyle \frac {1}{2} \left (\frac {B x^6 \sqrt {b x^2+c x^4}}{4 c}-\frac {(7 b B-8 A c) \left (\frac {x^4 \sqrt {b x^2+c x^4}}{3 c}-\frac {5 b \left (\frac {x^2 \sqrt {b x^2+c x^4}}{2 c}-\frac {3 b \left (\frac {\sqrt {b x^2+c x^4}}{c}-\frac {b \int \frac {1}{1-c x^4}d\frac {x^2}{\sqrt {c x^4+b x^2}}}{c}\right )}{4 c}\right )}{6 c}\right )}{8 c}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{2} \left (\frac {B x^6 \sqrt {b x^2+c x^4}}{4 c}-\frac {(7 b B-8 A c) \left (\frac {x^4 \sqrt {b x^2+c x^4}}{3 c}-\frac {5 b \left (\frac {x^2 \sqrt {b x^2+c x^4}}{2 c}-\frac {3 b \left (\frac {\sqrt {b x^2+c x^4}}{c}-\frac {b \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{c^{3/2}}\right )}{4 c}\right )}{6 c}\right )}{8 c}\right )\)

Input:

Int[(x^7*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]
 

Output:

((B*x^6*Sqrt[b*x^2 + c*x^4])/(4*c) - ((7*b*B - 8*A*c)*((x^4*Sqrt[b*x^2 + c 
*x^4])/(3*c) - (5*b*((x^2*Sqrt[b*x^2 + c*x^4])/(2*c) - (3*b*(Sqrt[b*x^2 + 
c*x^4]/c - (b*ArcTanh[(Sqrt[c]*x^2)/Sqrt[b*x^2 + c*x^4]])/c^(3/2)))/(4*c)) 
)/(6*c)))/(8*c))/2
 

Defintions of rubi rules used

rule 219

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])
 

rule 1091

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[Int[1/(1 
 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{b, c}, x]
 

rule 1134

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

rule 1160

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

rule 1221

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1 
)/(c*(m + 2*p + 2))), x] + Simp[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c 
*f - b*g))/(c*e*(m + 2*p + 2))   Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x 
] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] 
 && NeQ[m + 2*p + 2, 0]
 

rule 1940

Int[(x_)^(m_.)*((b_.)*(x_)^(k_.) + (a_.)*(x_)^(j_))^(p_)*((c_) + (d_.)*(x_) 
^(n_))^(q_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1) 
*(a*x^Simplify[j/n] + b*x^Simplify[k/n])^p*(c + d*x)^q, x], x, x^n], x] /; 
FreeQ[{a, b, c, d, j, k, m, n, p, q}, x] &&  !IntegerQ[p] && NeQ[k, j] && I 
ntegerQ[Simplify[j/n]] && IntegerQ[Simplify[k/n]] && IntegerQ[Simplify[(m + 
 1)/n]] && NeQ[n^2, 1]
 

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.80

method	result	size
pseudoelliptic	\(\frac {\frac {5 \left (-\frac {1}{2} A \,b^{3} c +\frac {7}{16} B \,b^{4}\right ) \ln \left (\frac {2 c \,x^{2}+2 \sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \sqrt {c}+b}{\sqrt {c}}\right )}{16}+\frac {5 \left (b^{2} \left (\frac {7 B \,x^{2}}{12}+A \right ) c^{\frac {3}{2}}-\frac {2 \left (\frac {7 B \,x^{2}}{10}+A \right ) x^{2} b \,c^{\frac {5}{2}}}{3}+\frac {8 x^{4} \left (\frac {3 B \,x^{2}}{4}+A \right ) c^{\frac {7}{2}}}{15}-\frac {7 B \sqrt {c}\, b^{3}}{8}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{16}+\frac {5 \ln \left (2\right ) \left (A c -\frac {7 B b}{8}\right ) b^{3}}{32}}{c^{\frac {9}{2}}}\)	\(140\)
risch	\(\frac {x^{2} \left (48 B \,c^{3} x^{6}+64 A \,c^{3} x^{4}-56 B b \,c^{2} x^{4}-80 A b \,c^{2} x^{2}+70 x^{2} B \,b^{2} c +120 A \,b^{2} c -105 B \,b^{3}\right ) \left (c \,x^{2}+b \right )}{384 c^{4} \sqrt {x^{2} \left (c \,x^{2}+b \right )}}-\frac {5 b^{3} \left (8 A c -7 B b \right ) \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right ) x \sqrt {c \,x^{2}+b}}{128 c^{\frac {9}{2}} \sqrt {x^{2} \left (c \,x^{2}+b \right )}}\)	\(148\)
default	\(\frac {x \sqrt {c \,x^{2}+b}\, \left (48 B \,c^{\frac {9}{2}} \sqrt {c \,x^{2}+b}\, x^{7}+64 A \,c^{\frac {9}{2}} \sqrt {c \,x^{2}+b}\, x^{5}-56 B \,c^{\frac {7}{2}} \sqrt {c \,x^{2}+b}\, b \,x^{5}-80 A \,c^{\frac {7}{2}} \sqrt {c \,x^{2}+b}\, b \,x^{3}+70 B \,c^{\frac {5}{2}} \sqrt {c \,x^{2}+b}\, b^{2} x^{3}+120 A \,c^{\frac {5}{2}} \sqrt {c \,x^{2}+b}\, b^{2} x -105 B \,c^{\frac {3}{2}} \sqrt {c \,x^{2}+b}\, b^{3} x -120 A \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right ) b^{3} c^{2}+105 B \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right ) b^{4} c \right )}{384 \sqrt {c \,x^{4}+b \,x^{2}}\, c^{\frac {11}{2}}}\)	\(211\)

Input:

int(x^7*(B*x^2+A)/(c*x^4+b*x^2)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

5/16/c^(9/2)*((-1/2*A*b^3*c+7/16*B*b^4)*ln((2*c*x^2+2*(x^2*(c*x^2+b))^(1/2 
)*c^(1/2)+b)/c^(1/2))+(b^2*(7/12*B*x^2+A)*c^(3/2)-2/3*(7/10*B*x^2+A)*x^2*b 
*c^(5/2)+8/15*x^4*(3/4*B*x^2+A)*c^(7/2)-7/8*B*c^(1/2)*b^3)*(x^2*(c*x^2+b)) 
^(1/2)+1/2*ln(2)*(A*c-7/8*B*b)*b^3)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.56 \[ \int \frac {x^7 \left (A+B x^2\right )}{\sqrt {b x^2+c x^4}} \, dx=\left [-\frac {15 \, {\left (7 \, B b^{4} - 8 \, A b^{3} c\right )} \sqrt {c} \log \left (-2 \, c x^{2} - b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right ) - 2 \, {\left (48 \, B c^{4} x^{6} - 105 \, B b^{3} c + 120 \, A b^{2} c^{2} - 8 \, {\left (7 \, B b c^{3} - 8 \, A c^{4}\right )} x^{4} + 10 \, {\left (7 \, B b^{2} c^{2} - 8 \, A b c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{768 \, c^{5}}, -\frac {15 \, {\left (7 \, B b^{4} - 8 \, A b^{3} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-c}}{c x^{2} + b}\right ) - {\left (48 \, B c^{4} x^{6} - 105 \, B b^{3} c + 120 \, A b^{2} c^{2} - 8 \, {\left (7 \, B b c^{3} - 8 \, A c^{4}\right )} x^{4} + 10 \, {\left (7 \, B b^{2} c^{2} - 8 \, A b c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{384 \, c^{5}}\right ] \] Input:

integrate(x^7*(B*x^2+A)/(c*x^4+b*x^2)^(1/2),x, algorithm="fricas")
 

Output:

[-1/768*(15*(7*B*b^4 - 8*A*b^3*c)*sqrt(c)*log(-2*c*x^2 - b + 2*sqrt(c*x^4 
+ b*x^2)*sqrt(c)) - 2*(48*B*c^4*x^6 - 105*B*b^3*c + 120*A*b^2*c^2 - 8*(7*B 
*b*c^3 - 8*A*c^4)*x^4 + 10*(7*B*b^2*c^2 - 8*A*b*c^3)*x^2)*sqrt(c*x^4 + b*x 
^2))/c^5, -1/384*(15*(7*B*b^4 - 8*A*b^3*c)*sqrt(-c)*arctan(sqrt(c*x^4 + b* 
x^2)*sqrt(-c)/(c*x^2 + b)) - (48*B*c^4*x^6 - 105*B*b^3*c + 120*A*b^2*c^2 - 
 8*(7*B*b*c^3 - 8*A*c^4)*x^4 + 10*(7*B*b^2*c^2 - 8*A*b*c^3)*x^2)*sqrt(c*x^ 
4 + b*x^2))/c^5]
 

Sympy [A] (verification not implemented)

Time = 0.84 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.27 \[ \int \frac {x^7 \left (A+B x^2\right )}{\sqrt {b x^2+c x^4}} \, dx=\frac {\begin {cases} - \frac {5 b^{3} \left (A - \frac {7 B b}{8 c}\right ) \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {b x^{2} + c x^{4}} + 2 c x^{2} \right )}}{\sqrt {c}} & \text {for}\: \frac {b^{2}}{c} \neq 0 \\\frac {\left (\frac {b}{2 c} + x^{2}\right ) \log {\left (\frac {b}{2 c} + x^{2} \right )}}{\sqrt {c \left (\frac {b}{2 c} + x^{2}\right )^{2}}} & \text {otherwise} \end {cases}\right )}{16 c^{3}} + \sqrt {b x^{2} + c x^{4}} \left (\frac {B x^{6}}{4 c} + \frac {5 b^{2} \left (A - \frac {7 B b}{8 c}\right )}{8 c^{3}} - \frac {5 b x^{2} \left (A - \frac {7 B b}{8 c}\right )}{12 c^{2}} + \frac {x^{4} \left (A - \frac {7 B b}{8 c}\right )}{3 c}\right ) & \text {for}\: c \neq 0 \\\frac {\frac {2 A \left (b x^{2}\right )^{\frac {7}{2}}}{7 b^{3}} + \frac {2 B \left (b x^{2}\right )^{\frac {9}{2}}}{9 b^{4}}}{b} & \text {for}\: b \neq 0 \\\tilde {\infty } \left (\frac {A x^{8}}{4} + \frac {B x^{10}}{5}\right ) & \text {otherwise} \end {cases}}{2} \] Input:

integrate(x**7*(B*x**2+A)/(c*x**4+b*x**2)**(1/2),x)
 

Output:

Piecewise((-5*b**3*(A - 7*B*b/(8*c))*Piecewise((log(b + 2*sqrt(c)*sqrt(b*x 
**2 + c*x**4) + 2*c*x**2)/sqrt(c), Ne(b**2/c, 0)), ((b/(2*c) + x**2)*log(b 
/(2*c) + x**2)/sqrt(c*(b/(2*c) + x**2)**2), True))/(16*c**3) + sqrt(b*x**2 
 + c*x**4)*(B*x**6/(4*c) + 5*b**2*(A - 7*B*b/(8*c))/(8*c**3) - 5*b*x**2*(A 
 - 7*B*b/(8*c))/(12*c**2) + x**4*(A - 7*B*b/(8*c))/(3*c)), Ne(c, 0)), ((2* 
A*(b*x**2)**(7/2)/(7*b**3) + 2*B*(b*x**2)**(9/2)/(9*b**4))/b, Ne(b, 0)), ( 
zoo*(A*x**8/4 + B*x**10/5), True))/2
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.31 \[ \int \frac {x^7 \left (A+B x^2\right )}{\sqrt {b x^2+c x^4}} \, dx=\frac {1}{96} \, {\left (\frac {16 \, \sqrt {c x^{4} + b x^{2}} x^{4}}{c} - \frac {20 \, \sqrt {c x^{4} + b x^{2}} b x^{2}}{c^{2}} - \frac {15 \, b^{3} \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right )}{c^{\frac {7}{2}}} + \frac {30 \, \sqrt {c x^{4} + b x^{2}} b^{2}}{c^{3}}\right )} A + \frac {1}{768} \, {\left (\frac {96 \, \sqrt {c x^{4} + b x^{2}} x^{6}}{c} - \frac {112 \, \sqrt {c x^{4} + b x^{2}} b x^{4}}{c^{2}} + \frac {140 \, \sqrt {c x^{4} + b x^{2}} b^{2} x^{2}}{c^{3}} + \frac {105 \, b^{4} \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right )}{c^{\frac {9}{2}}} - \frac {210 \, \sqrt {c x^{4} + b x^{2}} b^{3}}{c^{4}}\right )} B \] Input:

integrate(x^7*(B*x^2+A)/(c*x^4+b*x^2)^(1/2),x, algorithm="maxima")
 

Output:

1/96*(16*sqrt(c*x^4 + b*x^2)*x^4/c - 20*sqrt(c*x^4 + b*x^2)*b*x^2/c^2 - 15 
*b^3*log(2*c*x^2 + b + 2*sqrt(c*x^4 + b*x^2)*sqrt(c))/c^(7/2) + 30*sqrt(c* 
x^4 + b*x^2)*b^2/c^3)*A + 1/768*(96*sqrt(c*x^4 + b*x^2)*x^6/c - 112*sqrt(c 
*x^4 + b*x^2)*b*x^4/c^2 + 140*sqrt(c*x^4 + b*x^2)*b^2*x^2/c^3 + 105*b^4*lo 
g(2*c*x^2 + b + 2*sqrt(c*x^4 + b*x^2)*sqrt(c))/c^(9/2) - 210*sqrt(c*x^4 + 
b*x^2)*b^3/c^4)*B
 

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.05 \[ \int \frac {x^7 \left (A+B x^2\right )}{\sqrt {b x^2+c x^4}} \, dx=\frac {1}{384} \, {\left (2 \, {\left (4 \, x^{2} {\left (\frac {6 \, B x^{2}}{c \mathrm {sgn}\left (x\right )} - \frac {7 \, B b c^{5} \mathrm {sgn}\left (x\right ) - 8 \, A c^{6} \mathrm {sgn}\left (x\right )}{c^{7}}\right )} + \frac {5 \, {\left (7 \, B b^{2} c^{4} \mathrm {sgn}\left (x\right ) - 8 \, A b c^{5} \mathrm {sgn}\left (x\right )\right )}}{c^{7}}\right )} x^{2} - \frac {15 \, {\left (7 \, B b^{3} c^{3} \mathrm {sgn}\left (x\right ) - 8 \, A b^{2} c^{4} \mathrm {sgn}\left (x\right )\right )}}{c^{7}}\right )} \sqrt {c x^{2} + b} x + \frac {5 \, {\left (7 \, B b^{4} \log \left ({\left | b \right |}\right ) - 8 \, A b^{3} c \log \left ({\left | b \right |}\right )\right )} \mathrm {sgn}\left (x\right )}{256 \, c^{\frac {9}{2}}} - \frac {5 \, {\left (7 \, B b^{4} - 8 \, A b^{3} c\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + b} \right |}\right )}{128 \, c^{\frac {9}{2}} \mathrm {sgn}\left (x\right )} \] Input:

integrate(x^7*(B*x^2+A)/(c*x^4+b*x^2)^(1/2),x, algorithm="giac")
 

Output:

1/384*(2*(4*x^2*(6*B*x^2/(c*sgn(x)) - (7*B*b*c^5*sgn(x) - 8*A*c^6*sgn(x))/ 
c^7) + 5*(7*B*b^2*c^4*sgn(x) - 8*A*b*c^5*sgn(x))/c^7)*x^2 - 15*(7*B*b^3*c^ 
3*sgn(x) - 8*A*b^2*c^4*sgn(x))/c^7)*sqrt(c*x^2 + b)*x + 5/256*(7*B*b^4*log 
(abs(b)) - 8*A*b^3*c*log(abs(b)))*sgn(x)/c^(9/2) - 5/128*(7*B*b^4 - 8*A*b^ 
3*c)*log(abs(-sqrt(c)*x + sqrt(c*x^2 + b)))/(c^(9/2)*sgn(x))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^7 \left (A+B x^2\right )}{\sqrt {b x^2+c x^4}} \, dx=\int \frac {x^7\,\left (B\,x^2+A\right )}{\sqrt {c\,x^4+b\,x^2}} \,d x \] Input:

int((x^7*(A + B*x^2))/(b*x^2 + c*x^4)^(1/2),x)
 

Output:

int((x^7*(A + B*x^2))/(b*x^2 + c*x^4)^(1/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.04 \[ \int \frac {x^7 \left (A+B x^2\right )}{\sqrt {b x^2+c x^4}} \, dx=\frac {120 \sqrt {c \,x^{2}+b}\, a \,b^{2} c^{2} x -80 \sqrt {c \,x^{2}+b}\, a b \,c^{3} x^{3}+64 \sqrt {c \,x^{2}+b}\, a \,c^{4} x^{5}-105 \sqrt {c \,x^{2}+b}\, b^{4} c x +70 \sqrt {c \,x^{2}+b}\, b^{3} c^{2} x^{3}-56 \sqrt {c \,x^{2}+b}\, b^{2} c^{3} x^{5}+48 \sqrt {c \,x^{2}+b}\, b \,c^{4} x^{7}-120 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+b}+\sqrt {c}\, x}{\sqrt {b}}\right ) a \,b^{3} c +105 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+b}+\sqrt {c}\, x}{\sqrt {b}}\right ) b^{5}}{384 c^{5}} \] Input:

int(x^7*(B*x^2+A)/(c*x^4+b*x^2)^(1/2),x)
 

Output:

(120*sqrt(b + c*x**2)*a*b**2*c**2*x - 80*sqrt(b + c*x**2)*a*b*c**3*x**3 + 
64*sqrt(b + c*x**2)*a*c**4*x**5 - 105*sqrt(b + c*x**2)*b**4*c*x + 70*sqrt( 
b + c*x**2)*b**3*c**2*x**3 - 56*sqrt(b + c*x**2)*b**2*c**3*x**5 + 48*sqrt( 
b + c*x**2)*b*c**4*x**7 - 120*sqrt(c)*log((sqrt(b + c*x**2) + sqrt(c)*x)/s 
qrt(b))*a*b**3*c + 105*sqrt(c)*log((sqrt(b + c*x**2) + sqrt(c)*x)/sqrt(b)) 
*b**5)/(384*c**5)