\(\int \frac {(a+b x^2)^2 (c+d x^2)^{3/2}}{x^8} \, dx\) [856]

Optimal result

Integrand size = 24, antiderivative size = 128 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^8} \, dx=-\frac {b^2 c \sqrt {c+d x^2}}{3 x^3}-\frac {4 b^2 d \sqrt {c+d x^2}}{3 x}-\frac {a^2 \left (c+d x^2\right )^{5/2}}{7 c x^7}-\frac {2 a (7 b c-a d) \left (c+d x^2\right )^{5/2}}{35 c^2 x^5}+b^2 d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right ) \] Output:

-1/3*b^2*c*(d*x^2+c)^(1/2)/x^3-4/3*b^2*d*(d*x^2+c)^(1/2)/x-1/7*a^2*(d*x^2+ 
c)^(5/2)/c/x^7-2/35*a*(-a*d+7*b*c)*(d*x^2+c)^(5/2)/c^2/x^5+b^2*d^(3/2)*arc 
tanh(d^(1/2)*x/(d*x^2+c)^(1/2))
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^8} \, dx=-\frac {\sqrt {c+d x^2} \left (42 a b c x^2 \left (c+d x^2\right )^2+3 a^2 \left (5 c-2 d x^2\right ) \left (c+d x^2\right )^2+35 b^2 c^2 x^4 \left (c+4 d x^2\right )\right )}{105 c^2 x^7}-b^2 d^{3/2} \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right ) \] Input:

Integrate[((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^8,x]
 

Output:

-1/105*(Sqrt[c + d*x^2]*(42*a*b*c*x^2*(c + d*x^2)^2 + 3*a^2*(5*c - 2*d*x^2 
)*(c + d*x^2)^2 + 35*b^2*c^2*x^4*(c + 4*d*x^2)))/(c^2*x^7) - b^2*d^(3/2)*L 
og[-(Sqrt[d]*x) + Sqrt[c + d*x^2]]
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {365, 358, 247, 247, 224, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^8} \, dx\)

\(\Big \downarrow \) 365

\(\displaystyle \frac {\int \frac {\left (7 b^2 c x^2+2 a (7 b c-a d)\right ) \left (d x^2+c\right )^{3/2}}{x^6}dx}{7 c}-\frac {a^2 \left (c+d x^2\right )^{5/2}}{7 c x^7}\)

\(\Big \downarrow \) 358

\(\displaystyle \frac {7 b^2 c \int \frac {\left (d x^2+c\right )^{3/2}}{x^4}dx-\frac {2 a \left (c+d x^2\right )^{5/2} (7 b c-a d)}{5 c x^5}}{7 c}-\frac {a^2 \left (c+d x^2\right )^{5/2}}{7 c x^7}\)

\(\Big \downarrow \) 247

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

\(\Big \downarrow \) 247

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

Input:

Int[((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^8,x]
 

Output:

-1/7*(a^2*(c + d*x^2)^(5/2))/(c*x^7) + ((-2*a*(7*b*c - a*d)*(c + d*x^2)^(5 
/2))/(5*c*x^5) + 7*b^2*c*(-1/3*(c + d*x^2)^(3/2)/x^3 + d*(-(Sqrt[c + d*x^2 
]/x) + Sqrt[d]*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])))/(7*c)
 

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 224

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 247

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1)))   Int[ 
(c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 
0] && LtQ[m, -1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, 
m, p, x]
 

rule 358

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x_ 
Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + S 
imp[d/e^2   Int[(e*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e 
, m, p}, x] && NeQ[b*c - a*d, 0] && EqQ[Simplify[m + 2*p + 3], 0] && NeQ[m, 
 -1]
 

rule 365

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

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.04

method	result	size
pseudoelliptic	\(\frac {7 b^{2} c^{2} d^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}}{x \sqrt {d}}\right ) x^{7}-\sqrt {x^{2} d +c}\, \left (\left (\frac {7}{3} b^{2} x^{4}+\frac {14}{5} a b \,x^{2}+a^{2}\right ) c^{3}+\frac {8 d \left (\frac {35}{6} b^{2} x^{4}+\frac {7}{2} a b \,x^{2}+a^{2}\right ) x^{2} c^{2}}{5}+\frac {a \,d^{2} x^{4} \left (14 b \,x^{2}+a \right ) c}{5}-\frac {2 a^{2} d^{3} x^{6}}{5}\right )}{7 c^{2} x^{7}}\)	\(133\)
risch	\(-\frac {\sqrt {x^{2} d +c}\, \left (-6 a^{2} d^{3} x^{6}+42 a b c \,d^{2} x^{6}+140 b^{2} c^{2} d \,x^{6}+3 a^{2} c \,d^{2} x^{4}+84 a b \,c^{2} d \,x^{4}+35 b^{2} c^{3} x^{4}+24 a^{2} c^{2} d \,x^{2}+42 a b \,c^{3} x^{2}+15 a^{2} c^{3}\right )}{105 x^{7} c^{2}}+b^{2} d^{\frac {3}{2}} \ln \left (\sqrt {d}\, x +\sqrt {x^{2} d +c}\right )\)	\(141\)
default	\(a^{2} \left (-\frac {\left (x^{2} d +c \right )^{\frac {5}{2}}}{7 c \,x^{7}}+\frac {2 d \left (x^{2} d +c \right )^{\frac {5}{2}}}{35 c^{2} x^{5}}\right )+b^{2} \left (-\frac {\left (x^{2} d +c \right )^{\frac {5}{2}}}{3 c \,x^{3}}+\frac {2 d \left (-\frac {\left (x^{2} d +c \right )^{\frac {5}{2}}}{c x}+\frac {4 d \left (\frac {x \left (x^{2} d +c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {x^{2} d +c}}{2}+\frac {c \ln \left (\sqrt {d}\, x +\sqrt {x^{2} d +c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{c}\right )}{3 c}\right )-\frac {2 a b \left (x^{2} d +c \right )^{\frac {5}{2}}}{5 c \,x^{5}}\)	\(164\)

Input:

int((b*x^2+a)^2*(d*x^2+c)^(3/2)/x^8,x,method=_RETURNVERBOSE)
 

Output:

1/7*(7*b^2*c^2*d^(3/2)*arctanh((d*x^2+c)^(1/2)/x/d^(1/2))*x^7-(d*x^2+c)^(1 
/2)*((7/3*b^2*x^4+14/5*a*b*x^2+a^2)*c^3+8/5*d*(35/6*b^2*x^4+7/2*a*b*x^2+a^ 
2)*x^2*c^2+1/5*a*d^2*x^4*(14*b*x^2+a)*c-2/5*a^2*d^3*x^6))/c^2/x^7
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.33 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^8} \, dx=\left [\frac {105 \, b^{2} c^{2} d^{\frac {3}{2}} x^{7} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) - 2 \, {\left (2 \, {\left (70 \, b^{2} c^{2} d + 21 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{6} + 15 \, a^{2} c^{3} + {\left (35 \, b^{2} c^{3} + 84 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} + 6 \, {\left (7 \, a b c^{3} + 4 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{210 \, c^{2} x^{7}}, -\frac {105 \, b^{2} c^{2} \sqrt {-d} d x^{7} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (2 \, {\left (70 \, b^{2} c^{2} d + 21 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{6} + 15 \, a^{2} c^{3} + {\left (35 \, b^{2} c^{3} + 84 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} + 6 \, {\left (7 \, a b c^{3} + 4 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{105 \, c^{2} x^{7}}\right ] \] Input:

integrate((b*x^2+a)^2*(d*x^2+c)^(3/2)/x^8,x, algorithm="fricas")
 

Output:

[1/210*(105*b^2*c^2*d^(3/2)*x^7*log(-2*d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x 
 - c) - 2*(2*(70*b^2*c^2*d + 21*a*b*c*d^2 - 3*a^2*d^3)*x^6 + 15*a^2*c^3 + 
(35*b^2*c^3 + 84*a*b*c^2*d + 3*a^2*c*d^2)*x^4 + 6*(7*a*b*c^3 + 4*a^2*c^2*d 
)*x^2)*sqrt(d*x^2 + c))/(c^2*x^7), -1/105*(105*b^2*c^2*sqrt(-d)*d*x^7*arct 
an(sqrt(-d)*x/sqrt(d*x^2 + c)) + (2*(70*b^2*c^2*d + 21*a*b*c*d^2 - 3*a^2*d 
^3)*x^6 + 15*a^2*c^3 + (35*b^2*c^3 + 84*a*b*c^2*d + 3*a^2*c*d^2)*x^4 + 6*( 
7*a*b*c^3 + 4*a^2*c^2*d)*x^2)*sqrt(d*x^2 + c))/(c^2*x^7)]
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 663 vs. \(2 (117) = 234\).

Time = 2.96 (sec) , antiderivative size = 663, normalized size of antiderivative = 5.18 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^8} \, dx=- \frac {15 a^{2} c^{6} d^{\frac {9}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{105 c^{5} d^{4} x^{6} + 210 c^{4} d^{5} x^{8} + 105 c^{3} d^{6} x^{10}} - \frac {33 a^{2} c^{5} d^{\frac {11}{2}} x^{2} \sqrt {\frac {c}{d x^{2}} + 1}}{105 c^{5} d^{4} x^{6} + 210 c^{4} d^{5} x^{8} + 105 c^{3} d^{6} x^{10}} - \frac {17 a^{2} c^{4} d^{\frac {13}{2}} x^{4} \sqrt {\frac {c}{d x^{2}} + 1}}{105 c^{5} d^{4} x^{6} + 210 c^{4} d^{5} x^{8} + 105 c^{3} d^{6} x^{10}} - \frac {3 a^{2} c^{3} d^{\frac {15}{2}} x^{6} \sqrt {\frac {c}{d x^{2}} + 1}}{105 c^{5} d^{4} x^{6} + 210 c^{4} d^{5} x^{8} + 105 c^{3} d^{6} x^{10}} - \frac {12 a^{2} c^{2} d^{\frac {17}{2}} x^{8} \sqrt {\frac {c}{d x^{2}} + 1}}{105 c^{5} d^{4} x^{6} + 210 c^{4} d^{5} x^{8} + 105 c^{3} d^{6} x^{10}} - \frac {8 a^{2} c d^{\frac {19}{2}} x^{10} \sqrt {\frac {c}{d x^{2}} + 1}}{105 c^{5} d^{4} x^{6} + 210 c^{4} d^{5} x^{8} + 105 c^{3} d^{6} x^{10}} - \frac {a^{2} d^{\frac {3}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{5 x^{4}} - \frac {a^{2} d^{\frac {5}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{15 c x^{2}} + \frac {2 a^{2} d^{\frac {7}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{15 c^{2}} - \frac {2 a b c \sqrt {d} \sqrt {\frac {c}{d x^{2}} + 1}}{5 x^{4}} - \frac {4 a b d^{\frac {3}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{5 x^{2}} - \frac {2 a b d^{\frac {5}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{5 c} - \frac {b^{2} \sqrt {c} d}{x \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {b^{2} c \sqrt {d} \sqrt {\frac {c}{d x^{2}} + 1}}{3 x^{2}} - \frac {b^{2} d^{\frac {3}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{3} + b^{2} d^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )} - \frac {b^{2} d^{2} x}{\sqrt {c} \sqrt {1 + \frac {d x^{2}}{c}}} \] Input:

integrate((b*x**2+a)**2*(d*x**2+c)**(3/2)/x**8,x)
 

Output:

-15*a**2*c**6*d**(9/2)*sqrt(c/(d*x**2) + 1)/(105*c**5*d**4*x**6 + 210*c**4 
*d**5*x**8 + 105*c**3*d**6*x**10) - 33*a**2*c**5*d**(11/2)*x**2*sqrt(c/(d* 
x**2) + 1)/(105*c**5*d**4*x**6 + 210*c**4*d**5*x**8 + 105*c**3*d**6*x**10) 
 - 17*a**2*c**4*d**(13/2)*x**4*sqrt(c/(d*x**2) + 1)/(105*c**5*d**4*x**6 + 
210*c**4*d**5*x**8 + 105*c**3*d**6*x**10) - 3*a**2*c**3*d**(15/2)*x**6*sqr 
t(c/(d*x**2) + 1)/(105*c**5*d**4*x**6 + 210*c**4*d**5*x**8 + 105*c**3*d**6 
*x**10) - 12*a**2*c**2*d**(17/2)*x**8*sqrt(c/(d*x**2) + 1)/(105*c**5*d**4* 
x**6 + 210*c**4*d**5*x**8 + 105*c**3*d**6*x**10) - 8*a**2*c*d**(19/2)*x**1 
0*sqrt(c/(d*x**2) + 1)/(105*c**5*d**4*x**6 + 210*c**4*d**5*x**8 + 105*c**3 
*d**6*x**10) - a**2*d**(3/2)*sqrt(c/(d*x**2) + 1)/(5*x**4) - a**2*d**(5/2) 
*sqrt(c/(d*x**2) + 1)/(15*c*x**2) + 2*a**2*d**(7/2)*sqrt(c/(d*x**2) + 1)/( 
15*c**2) - 2*a*b*c*sqrt(d)*sqrt(c/(d*x**2) + 1)/(5*x**4) - 4*a*b*d**(3/2)* 
sqrt(c/(d*x**2) + 1)/(5*x**2) - 2*a*b*d**(5/2)*sqrt(c/(d*x**2) + 1)/(5*c) 
- b**2*sqrt(c)*d/(x*sqrt(1 + d*x**2/c)) - b**2*c*sqrt(d)*sqrt(c/(d*x**2) + 
 1)/(3*x**2) - b**2*d**(3/2)*sqrt(c/(d*x**2) + 1)/3 + b**2*d**(3/2)*asinh( 
sqrt(d)*x/sqrt(c)) - b**2*d**2*x/(sqrt(c)*sqrt(1 + d*x**2/c))
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^8} \, dx=\frac {\sqrt {d x^{2} + c} b^{2} d^{2} x}{c} + b^{2} d^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right ) - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} d}{3 \, c x} - \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2}}{3 \, c x^{3}} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b}{5 \, c x^{5}} + \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} d}{35 \, c^{2} x^{5}} - \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2}}{7 \, c x^{7}} \] Input:

integrate((b*x^2+a)^2*(d*x^2+c)^(3/2)/x^8,x, algorithm="maxima")
 

Output:

sqrt(d*x^2 + c)*b^2*d^2*x/c + b^2*d^(3/2)*arcsinh(d*x/sqrt(c*d)) - 2/3*(d* 
x^2 + c)^(3/2)*b^2*d/(c*x) - 1/3*(d*x^2 + c)^(5/2)*b^2/(c*x^3) - 2/5*(d*x^ 
2 + c)^(5/2)*a*b/(c*x^5) + 2/35*(d*x^2 + c)^(5/2)*a^2*d/(c^2*x^5) - 1/7*(d 
*x^2 + c)^(5/2)*a^2/(c*x^7)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 581 vs. \(2 (106) = 212\).

Time = 0.15 (sec) , antiderivative size = 581, normalized size of antiderivative = 4.54 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^8} \, dx=-\frac {1}{2} \, b^{2} d^{\frac {3}{2}} \log \left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2}\right ) + \frac {4 \, {\left (105 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{12} b^{2} c d^{\frac {3}{2}} + 105 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{12} a b d^{\frac {5}{2}} - 525 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{10} b^{2} c^{2} d^{\frac {3}{2}} - 210 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{10} a b c d^{\frac {5}{2}} + 105 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{10} a^{2} d^{\frac {7}{2}} + 1120 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} b^{2} c^{3} d^{\frac {3}{2}} + 315 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a b c^{2} d^{\frac {5}{2}} + 105 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a^{2} c d^{\frac {7}{2}} - 1330 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} b^{2} c^{4} d^{\frac {3}{2}} - 420 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a b c^{3} d^{\frac {5}{2}} + 210 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a^{2} c^{2} d^{\frac {7}{2}} + 945 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b^{2} c^{5} d^{\frac {3}{2}} + 231 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a b c^{4} d^{\frac {5}{2}} + 42 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{2} c^{3} d^{\frac {7}{2}} - 385 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{2} c^{6} d^{\frac {3}{2}} - 42 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b c^{5} d^{\frac {5}{2}} + 21 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} c^{4} d^{\frac {7}{2}} + 70 \, b^{2} c^{7} d^{\frac {3}{2}} + 21 \, a b c^{6} d^{\frac {5}{2}} - 3 \, a^{2} c^{5} d^{\frac {7}{2}}\right )}}{105 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{7}} \] Input:

integrate((b*x^2+a)^2*(d*x^2+c)^(3/2)/x^8,x, algorithm="giac")
 

Output:

-1/2*b^2*d^(3/2)*log((sqrt(d)*x - sqrt(d*x^2 + c))^2) + 4/105*(105*(sqrt(d 
)*x - sqrt(d*x^2 + c))^12*b^2*c*d^(3/2) + 105*(sqrt(d)*x - sqrt(d*x^2 + c) 
)^12*a*b*d^(5/2) - 525*(sqrt(d)*x - sqrt(d*x^2 + c))^10*b^2*c^2*d^(3/2) - 
210*(sqrt(d)*x - sqrt(d*x^2 + c))^10*a*b*c*d^(5/2) + 105*(sqrt(d)*x - sqrt 
(d*x^2 + c))^10*a^2*d^(7/2) + 1120*(sqrt(d)*x - sqrt(d*x^2 + c))^8*b^2*c^3 
*d^(3/2) + 315*(sqrt(d)*x - sqrt(d*x^2 + c))^8*a*b*c^2*d^(5/2) + 105*(sqrt 
(d)*x - sqrt(d*x^2 + c))^8*a^2*c*d^(7/2) - 1330*(sqrt(d)*x - sqrt(d*x^2 + 
c))^6*b^2*c^4*d^(3/2) - 420*(sqrt(d)*x - sqrt(d*x^2 + c))^6*a*b*c^3*d^(5/2 
) + 210*(sqrt(d)*x - sqrt(d*x^2 + c))^6*a^2*c^2*d^(7/2) + 945*(sqrt(d)*x - 
 sqrt(d*x^2 + c))^4*b^2*c^5*d^(3/2) + 231*(sqrt(d)*x - sqrt(d*x^2 + c))^4* 
a*b*c^4*d^(5/2) + 42*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a^2*c^3*d^(7/2) - 385 
*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b^2*c^6*d^(3/2) - 42*(sqrt(d)*x - sqrt(d* 
x^2 + c))^2*a*b*c^5*d^(5/2) + 21*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a^2*c^4*d 
^(7/2) + 70*b^2*c^7*d^(3/2) + 21*a*b*c^6*d^(5/2) - 3*a^2*c^5*d^(7/2))/((sq 
rt(d)*x - sqrt(d*x^2 + c))^2 - c)^7
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^8} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2\,{\left (d\,x^2+c\right )}^{3/2}}{x^8} \,d x \] Input:

int(((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^8,x)
 

Output:

int(((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^8, x)
 

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.97 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^8} \, dx=\frac {-15 \sqrt {d \,x^{2}+c}\, a^{2} c^{3}-24 \sqrt {d \,x^{2}+c}\, a^{2} c^{2} d \,x^{2}-3 \sqrt {d \,x^{2}+c}\, a^{2} c \,d^{2} x^{4}+6 \sqrt {d \,x^{2}+c}\, a^{2} d^{3} x^{6}-42 \sqrt {d \,x^{2}+c}\, a b \,c^{3} x^{2}-84 \sqrt {d \,x^{2}+c}\, a b \,c^{2} d \,x^{4}-42 \sqrt {d \,x^{2}+c}\, a b c \,d^{2} x^{6}-35 \sqrt {d \,x^{2}+c}\, b^{2} c^{3} x^{4}-140 \sqrt {d \,x^{2}+c}\, b^{2} c^{2} d \,x^{6}+105 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d \,x^{2}+c}+\sqrt {d}\, x}{\sqrt {c}}\right ) b^{2} c^{2} d \,x^{7}-6 \sqrt {d}\, a^{2} d^{3} x^{7}-18 \sqrt {d}\, a b c \,d^{2} x^{7}+80 \sqrt {d}\, b^{2} c^{2} d \,x^{7}}{105 c^{2} x^{7}} \] Input:

int((b*x^2+a)^2*(d*x^2+c)^(3/2)/x^8,x)
 

Output:

( - 15*sqrt(c + d*x**2)*a**2*c**3 - 24*sqrt(c + d*x**2)*a**2*c**2*d*x**2 - 
 3*sqrt(c + d*x**2)*a**2*c*d**2*x**4 + 6*sqrt(c + d*x**2)*a**2*d**3*x**6 - 
 42*sqrt(c + d*x**2)*a*b*c**3*x**2 - 84*sqrt(c + d*x**2)*a*b*c**2*d*x**4 - 
 42*sqrt(c + d*x**2)*a*b*c*d**2*x**6 - 35*sqrt(c + d*x**2)*b**2*c**3*x**4 
- 140*sqrt(c + d*x**2)*b**2*c**2*d*x**6 + 105*sqrt(d)*log((sqrt(c + d*x**2 
) + sqrt(d)*x)/sqrt(c))*b**2*c**2*d*x**7 - 6*sqrt(d)*a**2*d**3*x**7 - 18*s 
qrt(d)*a*b*c*d**2*x**7 + 80*sqrt(d)*b**2*c**2*d*x**7)/(105*c**2*x**7)