\(\int \frac {(a+b x^2)^2}{x^7 (c+d x^2)^{3/2}} \, dx\) [659]

Optimal result

Integrand size = 24, antiderivative size = 190 \[ \int \frac {\left (a+b x^2\right )^2}{x^7 \left (c+d x^2\right )^{3/2}} \, dx=-\frac {d \left (24 b^2 c^2-5 a d (12 b c-7 a d)\right )}{16 c^4 \sqrt {c+d x^2}}-\frac {a^2}{6 c x^6 \sqrt {c+d x^2}}-\frac {a (12 b c-7 a d)}{24 c^2 x^4 \sqrt {c+d x^2}}-\frac {24 b^2 c^2-5 a d (12 b c-7 a d)}{48 c^3 x^2 \sqrt {c+d x^2}}+\frac {d \left (24 b^2 c^2-5 a d (12 b c-7 a d)\right ) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{16 c^{9/2}} \]

[Out]

1/16*d*(24*b^2*c^2-5*a*d*(-7*a*d+12*b*c))*arctanh((d*x^2+c)^(1/2)/c^(1/2))/c^(9/2)-1/16*d*(24*b^2*c^2-5*a*d*(-
7*a*d+12*b*c))/c^4/(d*x^2+c)^(1/2)-1/6*a^2/c/x^6/(d*x^2+c)^(1/2)-1/24*a*(-7*a*d+12*b*c)/c^2/x^4/(d*x^2+c)^(1/2
)+1/48*(-24*b^2*c^2+5*a*d*(-7*a*d+12*b*c))/c^3/x^2/(d*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {457, 91, 79, 44, 53, 65, 214} \[ \int \frac {\left (a+b x^2\right )^2}{x^7 \left (c+d x^2\right )^{3/2}} \, dx=-\frac {35 a^2 d^2-60 a b c d+24 b^2 c^2}{48 c^3 x^2 \sqrt {c+d x^2}}-\frac {a^2}{6 c x^6 \sqrt {c+d x^2}}+\frac {d \left (24 b^2 c^2-5 a d (12 b c-7 a d)\right ) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{16 c^{9/2}}-\frac {d \left (24 b^2 c^2-5 a d (12 b c-7 a d)\right )}{16 c^4 \sqrt {c+d x^2}}-\frac {a (12 b c-7 a d)}{24 c^2 x^4 \sqrt {c+d x^2}} \]

[In]

Int[(a + b*x^2)^2/(x^7*(c + d*x^2)^(3/2)),x]

[Out]

-1/16*(d*(24*b^2*c^2 - 5*a*d*(12*b*c - 7*a*d)))/(c^4*Sqrt[c + d*x^2]) - a^2/(6*c*x^6*Sqrt[c + d*x^2]) - (a*(12
*b*c - 7*a*d))/(24*c^2*x^4*Sqrt[c + d*x^2]) - (24*b^2*c^2 - 60*a*b*c*d + 35*a^2*d^2)/(48*c^3*x^2*Sqrt[c + d*x^
2]) + (d*(24*b^2*c^2 - 5*a*d*(12*b*c - 7*a*d))*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(16*c^(9/2))

Rule 44

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 53

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 91

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c - a*d
)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d*e - c*f)*(n + 1))), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 214

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]

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[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]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^2}{x^4 (c+d x)^{3/2}} \, dx,x,x^2\right ) \\ & = -\frac {a^2}{6 c x^6 \sqrt {c+d x^2}}+\frac {\text {Subst}\left (\int \frac {\frac {1}{2} a (12 b c-7 a d)+3 b^2 c x}{x^3 (c+d x)^{3/2}} \, dx,x,x^2\right )}{6 c} \\ & = -\frac {a^2}{6 c x^6 \sqrt {c+d x^2}}-\frac {a (12 b c-7 a d)}{24 c^2 x^4 \sqrt {c+d x^2}}+\frac {1}{48} \left (24 b^2-\frac {5 a d (12 b c-7 a d)}{c^2}\right ) \text {Subst}\left (\int \frac {1}{x^2 (c+d x)^{3/2}} \, dx,x,x^2\right ) \\ & = -\frac {a^2}{6 c x^6 \sqrt {c+d x^2}}-\frac {a (12 b c-7 a d)}{24 c^2 x^4 \sqrt {c+d x^2}}-\frac {24 b^2 c^2-60 a b c d+35 a^2 d^2}{48 c^3 x^2 \sqrt {c+d x^2}}+\frac {\left (d \left (-24 b^2+\frac {5 a d (12 b c-7 a d)}{c^2}\right )\right ) \text {Subst}\left (\int \frac {1}{x (c+d x)^{3/2}} \, dx,x,x^2\right )}{32 c} \\ & = -\frac {d \left (24 b^2-\frac {5 a d (12 b c-7 a d)}{c^2}\right )}{16 c^2 \sqrt {c+d x^2}}-\frac {a^2}{6 c x^6 \sqrt {c+d x^2}}-\frac {a (12 b c-7 a d)}{24 c^2 x^4 \sqrt {c+d x^2}}-\frac {24 b^2 c^2-60 a b c d+35 a^2 d^2}{48 c^3 x^2 \sqrt {c+d x^2}}+\frac {\left (d \left (-24 b^2+\frac {5 a d (12 b c-7 a d)}{c^2}\right )\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right )}{32 c^2} \\ & = -\frac {d \left (24 b^2-\frac {5 a d (12 b c-7 a d)}{c^2}\right )}{16 c^2 \sqrt {c+d x^2}}-\frac {a^2}{6 c x^6 \sqrt {c+d x^2}}-\frac {a (12 b c-7 a d)}{24 c^2 x^4 \sqrt {c+d x^2}}-\frac {24 b^2 c^2-60 a b c d+35 a^2 d^2}{48 c^3 x^2 \sqrt {c+d x^2}}+\frac {\left (-24 b^2+\frac {5 a d (12 b c-7 a d)}{c^2}\right ) \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{16 c^2} \\ & = -\frac {d \left (24 b^2-\frac {5 a d (12 b c-7 a d)}{c^2}\right )}{16 c^2 \sqrt {c+d x^2}}-\frac {a^2}{6 c x^6 \sqrt {c+d x^2}}-\frac {a (12 b c-7 a d)}{24 c^2 x^4 \sqrt {c+d x^2}}-\frac {24 b^2 c^2-60 a b c d+35 a^2 d^2}{48 c^3 x^2 \sqrt {c+d x^2}}+\frac {d \left (24 b^2-\frac {5 a d (12 b c-7 a d)}{c^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{16 c^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+b x^2\right )^2}{x^7 \left (c+d x^2\right )^{3/2}} \, dx=-\frac {24 b^2 c^2 x^4 \left (c+3 d x^2\right )+12 a b c x^2 \left (2 c^2-5 c d x^2-15 d^2 x^4\right )+a^2 \left (8 c^3-14 c^2 d x^2+35 c d^2 x^4+105 d^3 x^6\right )}{48 c^4 x^6 \sqrt {c+d x^2}}+\frac {d \left (24 b^2 c^2-60 a b c d+35 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{16 c^{9/2}} \]

[In]

Integrate[(a + b*x^2)^2/(x^7*(c + d*x^2)^(3/2)),x]

[Out]

-1/48*(24*b^2*c^2*x^4*(c + 3*d*x^2) + 12*a*b*c*x^2*(2*c^2 - 5*c*d*x^2 - 15*d^2*x^4) + a^2*(8*c^3 - 14*c^2*d*x^
2 + 35*c*d^2*x^4 + 105*d^3*x^6))/(c^4*x^6*Sqrt[c + d*x^2]) + (d*(24*b^2*c^2 - 60*a*b*c*d + 35*a^2*d^2)*ArcTanh
[Sqrt[c + d*x^2]/Sqrt[c]])/(16*c^(9/2))

Maple [A] (verified)

Time = 2.99 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.82

method	result	size
pseudoelliptic	\(\frac {-\frac {35 x^{4} \left (-\frac {36 b \,x^{2}}{7}+a \right ) d^{2} a \,c^{\frac {3}{2}}}{48}+\frac {7 d \,x^{2} \left (-\frac {36}{7} b^{2} x^{4}+\frac {30}{7} a b \,x^{2}+a^{2}\right ) c^{\frac {5}{2}}}{24}+\frac {\left (-b^{2} x^{4}-a b \,x^{2}-\frac {1}{3} a^{2}\right ) c^{\frac {7}{2}}}{2}+\frac {35 x^{6} \left (-a^{2} d^{2} \sqrt {c}+\sqrt {d \,x^{2}+c}\, \left (a^{2} d^{2}-\frac {12}{7} a b c d +\frac {24}{35} b^{2} c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{\sqrt {c}}\right )\right ) d}{16}}{x^{6} c^{\frac {9}{2}} \sqrt {d \,x^{2}+c}}\)	\(156\)
risch	\(-\frac {\sqrt {d \,x^{2}+c}\, \left (57 a^{2} d^{2} x^{4}-84 x^{4} a b c d +24 b^{2} c^{2} x^{4}-22 a^{2} c d \,x^{2}+24 a b \,c^{2} x^{2}+8 a^{2} c^{2}\right )}{48 c^{4} x^{6}}-\frac {d \left (-\frac {19 a^{2} d^{2}-28 a b c d +8 b^{2} c^{2}}{\sqrt {d \,x^{2}+c}}+c \left (35 a^{2} d^{2}-60 a b c d +24 b^{2} c^{2}\right ) \left (\frac {1}{c \sqrt {d \,x^{2}+c}}-\frac {\ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{c^{\frac {3}{2}}}\right )\right )}{16 c^{4}}\)	\(187\)
default	\(a^{2} \left (-\frac {1}{6 c \,x^{6} \sqrt {d \,x^{2}+c}}-\frac {7 d \left (-\frac {1}{4 c \,x^{4} \sqrt {d \,x^{2}+c}}-\frac {5 d \left (-\frac {1}{2 c \,x^{2} \sqrt {d \,x^{2}+c}}-\frac {3 d \left (\frac {1}{c \sqrt {d \,x^{2}+c}}-\frac {\ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{c^{\frac {3}{2}}}\right )}{2 c}\right )}{4 c}\right )}{6 c}\right )+b^{2} \left (-\frac {1}{2 c \,x^{2} \sqrt {d \,x^{2}+c}}-\frac {3 d \left (\frac {1}{c \sqrt {d \,x^{2}+c}}-\frac {\ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{c^{\frac {3}{2}}}\right )}{2 c}\right )+2 a b \left (-\frac {1}{4 c \,x^{4} \sqrt {d \,x^{2}+c}}-\frac {5 d \left (-\frac {1}{2 c \,x^{2} \sqrt {d \,x^{2}+c}}-\frac {3 d \left (\frac {1}{c \sqrt {d \,x^{2}+c}}-\frac {\ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{c^{\frac {3}{2}}}\right )}{2 c}\right )}{4 c}\right )\)	\(284\)

[In]

int((b*x^2+a)^2/x^7/(d*x^2+c)^(3/2),x,method=_RETURNVERBOSE)

[Out]

35/16/(d*x^2+c)^(1/2)/c^(9/2)*(-1/3*x^4*(-36/7*b*x^2+a)*d^2*a*c^(3/2)+2/15*d*x^2*(-36/7*b^2*x^4+30/7*a*b*x^2+a
^2)*c^(5/2)+8/35*(-b^2*x^4-a*b*x^2-1/3*a^2)*c^(7/2)+x^6*(-a^2*d^2*c^(1/2)+(d*x^2+c)^(1/2)*(a^2*d^2-12/7*a*b*c*
d+24/35*b^2*c^2)*arctanh((d*x^2+c)^(1/2)/c^(1/2)))*d)/x^6

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 447, normalized size of antiderivative = 2.35 \[ \int \frac {\left (a+b x^2\right )^2}{x^7 \left (c+d x^2\right )^{3/2}} \, dx=\left [\frac {3 \, {\left ({\left (24 \, b^{2} c^{2} d^{2} - 60 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{8} + {\left (24 \, b^{2} c^{3} d - 60 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{6}\right )} \sqrt {c} \log \left (-\frac {d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) - 2 \, {\left (3 \, {\left (24 \, b^{2} c^{3} d - 60 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{6} + 8 \, a^{2} c^{4} + {\left (24 \, b^{2} c^{4} - 60 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} x^{4} + 2 \, {\left (12 \, a b c^{4} - 7 \, a^{2} c^{3} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{96 \, {\left (c^{5} d x^{8} + c^{6} x^{6}\right )}}, -\frac {3 \, {\left ({\left (24 \, b^{2} c^{2} d^{2} - 60 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{8} + {\left (24 \, b^{2} c^{3} d - 60 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{6}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + {\left (3 \, {\left (24 \, b^{2} c^{3} d - 60 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{6} + 8 \, a^{2} c^{4} + {\left (24 \, b^{2} c^{4} - 60 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} x^{4} + 2 \, {\left (12 \, a b c^{4} - 7 \, a^{2} c^{3} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{48 \, {\left (c^{5} d x^{8} + c^{6} x^{6}\right )}}\right ] \]

[In]

integrate((b*x^2+a)^2/x^7/(d*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

[1/96*(3*((24*b^2*c^2*d^2 - 60*a*b*c*d^3 + 35*a^2*d^4)*x^8 + (24*b^2*c^3*d - 60*a*b*c^2*d^2 + 35*a^2*c*d^3)*x^
6)*sqrt(c)*log(-(d*x^2 + 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/x^2) - 2*(3*(24*b^2*c^3*d - 60*a*b*c^2*d^2 + 35*a^2*
c*d^3)*x^6 + 8*a^2*c^4 + (24*b^2*c^4 - 60*a*b*c^3*d + 35*a^2*c^2*d^2)*x^4 + 2*(12*a*b*c^4 - 7*a^2*c^3*d)*x^2)*
sqrt(d*x^2 + c))/(c^5*d*x^8 + c^6*x^6), -1/48*(3*((24*b^2*c^2*d^2 - 60*a*b*c*d^3 + 35*a^2*d^4)*x^8 + (24*b^2*c
^3*d - 60*a*b*c^2*d^2 + 35*a^2*c*d^3)*x^6)*sqrt(-c)*arctan(sqrt(-c)/sqrt(d*x^2 + c)) + (3*(24*b^2*c^3*d - 60*a
*b*c^2*d^2 + 35*a^2*c*d^3)*x^6 + 8*a^2*c^4 + (24*b^2*c^4 - 60*a*b*c^3*d + 35*a^2*c^2*d^2)*x^4 + 2*(12*a*b*c^4
- 7*a^2*c^3*d)*x^2)*sqrt(d*x^2 + c))/(c^5*d*x^8 + c^6*x^6)]

Sympy [F]

\[ \int \frac {\left (a+b x^2\right )^2}{x^7 \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{2}}{x^{7} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate((b*x**2+a)**2/x**7/(d*x**2+c)**(3/2),x)

[Out]

Integral((a + b*x**2)**2/(x**7*(c + d*x**2)**(3/2)), x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.30 \[ \int \frac {\left (a+b x^2\right )^2}{x^7 \left (c+d x^2\right )^{3/2}} \, dx=\frac {3 \, b^{2} d \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right )}{2 \, c^{\frac {5}{2}}} - \frac {15 \, a b d^{2} \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right )}{4 \, c^{\frac {7}{2}}} + \frac {35 \, a^{2} d^{3} \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right )}{16 \, c^{\frac {9}{2}}} - \frac {3 \, b^{2} d}{2 \, \sqrt {d x^{2} + c} c^{2}} + \frac {15 \, a b d^{2}}{4 \, \sqrt {d x^{2} + c} c^{3}} - \frac {35 \, a^{2} d^{3}}{16 \, \sqrt {d x^{2} + c} c^{4}} - \frac {b^{2}}{2 \, \sqrt {d x^{2} + c} c x^{2}} + \frac {5 \, a b d}{4 \, \sqrt {d x^{2} + c} c^{2} x^{2}} - \frac {35 \, a^{2} d^{2}}{48 \, \sqrt {d x^{2} + c} c^{3} x^{2}} - \frac {a b}{2 \, \sqrt {d x^{2} + c} c x^{4}} + \frac {7 \, a^{2} d}{24 \, \sqrt {d x^{2} + c} c^{2} x^{4}} - \frac {a^{2}}{6 \, \sqrt {d x^{2} + c} c x^{6}} \]

[In]

integrate((b*x^2+a)^2/x^7/(d*x^2+c)^(3/2),x, algorithm="maxima")

[Out]

3/2*b^2*d*arcsinh(c/(sqrt(c*d)*abs(x)))/c^(5/2) - 15/4*a*b*d^2*arcsinh(c/(sqrt(c*d)*abs(x)))/c^(7/2) + 35/16*a
^2*d^3*arcsinh(c/(sqrt(c*d)*abs(x)))/c^(9/2) - 3/2*b^2*d/(sqrt(d*x^2 + c)*c^2) + 15/4*a*b*d^2/(sqrt(d*x^2 + c)
*c^3) - 35/16*a^2*d^3/(sqrt(d*x^2 + c)*c^4) - 1/2*b^2/(sqrt(d*x^2 + c)*c*x^2) + 5/4*a*b*d/(sqrt(d*x^2 + c)*c^2
*x^2) - 35/48*a^2*d^2/(sqrt(d*x^2 + c)*c^3*x^2) - 1/2*a*b/(sqrt(d*x^2 + c)*c*x^4) + 7/24*a^2*d/(sqrt(d*x^2 + c
)*c^2*x^4) - 1/6*a^2/(sqrt(d*x^2 + c)*c*x^6)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.41 \[ \int \frac {\left (a+b x^2\right )^2}{x^7 \left (c+d x^2\right )^{3/2}} \, dx=-\frac {{\left (24 \, b^{2} c^{2} d - 60 \, a b c d^{2} + 35 \, a^{2} d^{3}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{16 \, \sqrt {-c} c^{4}} - \frac {b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}}{\sqrt {d x^{2} + c} c^{4}} - \frac {24 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c^{2} d - 48 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c^{3} d + 24 \, \sqrt {d x^{2} + c} b^{2} c^{4} d - 84 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b c d^{2} + 192 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c^{2} d^{2} - 108 \, \sqrt {d x^{2} + c} a b c^{3} d^{2} + 57 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} d^{3} - 136 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} c d^{3} + 87 \, \sqrt {d x^{2} + c} a^{2} c^{2} d^{3}}{48 \, c^{4} d^{3} x^{6}} \]

[In]

integrate((b*x^2+a)^2/x^7/(d*x^2+c)^(3/2),x, algorithm="giac")

[Out]

-1/16*(24*b^2*c^2*d - 60*a*b*c*d^2 + 35*a^2*d^3)*arctan(sqrt(d*x^2 + c)/sqrt(-c))/(sqrt(-c)*c^4) - (b^2*c^2*d
- 2*a*b*c*d^2 + a^2*d^3)/(sqrt(d*x^2 + c)*c^4) - 1/48*(24*(d*x^2 + c)^(5/2)*b^2*c^2*d - 48*(d*x^2 + c)^(3/2)*b
^2*c^3*d + 24*sqrt(d*x^2 + c)*b^2*c^4*d - 84*(d*x^2 + c)^(5/2)*a*b*c*d^2 + 192*(d*x^2 + c)^(3/2)*a*b*c^2*d^2 -
108*sqrt(d*x^2 + c)*a*b*c^3*d^2 + 57*(d*x^2 + c)^(5/2)*a^2*d^3 - 136*(d*x^2 + c)^(3/2)*a^2*c*d^3 + 87*sqrt(d*
x^2 + c)*a^2*c^2*d^3)/(c^4*d^3*x^6)

Mupad [B] (verification not implemented)

Time = 6.33 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.29 \[ \int \frac {\left (a+b x^2\right )^2}{x^7 \left (c+d x^2\right )^{3/2}} \, dx=\frac {d\,\mathrm {atanh}\left (\frac {\sqrt {d\,x^2+c}}{\sqrt {c}}\right )\,\left (35\,a^2\,d^2-60\,a\,b\,c\,d+24\,b^2\,c^2\right )}{16\,c^{9/2}}-\frac {\frac {a^2\,d^3-2\,a\,b\,c\,d^2+b^2\,c^2\,d}{c}-\frac {\left (d\,x^2+c\right )\,\left (77\,a^2\,d^3-132\,a\,b\,c\,d^2+56\,b^2\,c^2\,d\right )}{16\,c^2}+\frac {{\left (d\,x^2+c\right )}^2\,\left (35\,a^2\,d^3-60\,a\,b\,c\,d^2+24\,b^2\,c^2\,d\right )}{6\,c^3}-\frac {{\left (d\,x^2+c\right )}^3\,\left (35\,a^2\,d^3-60\,a\,b\,c\,d^2+24\,b^2\,c^2\,d\right )}{16\,c^4}}{3\,c\,{\left (d\,x^2+c\right )}^{5/2}-{\left (d\,x^2+c\right )}^{7/2}+c^3\,\sqrt {d\,x^2+c}-3\,c^2\,{\left (d\,x^2+c\right )}^{3/2}} \]

[In]

int((a + b*x^2)^2/(x^7*(c + d*x^2)^(3/2)),x)

[Out]

(d*atanh((c + d*x^2)^(1/2)/c^(1/2))*(35*a^2*d^2 + 24*b^2*c^2 - 60*a*b*c*d))/(16*c^(9/2)) - ((a^2*d^3 + b^2*c^2
*d - 2*a*b*c*d^2)/c - ((c + d*x^2)*(77*a^2*d^3 + 56*b^2*c^2*d - 132*a*b*c*d^2))/(16*c^2) + ((c + d*x^2)^2*(35*
a^2*d^3 + 24*b^2*c^2*d - 60*a*b*c*d^2))/(6*c^3) - ((c + d*x^2)^3*(35*a^2*d^3 + 24*b^2*c^2*d - 60*a*b*c*d^2))/(
16*c^4))/(3*c*(c + d*x^2)^(5/2) - (c + d*x^2)^(7/2) + c^3*(c + d*x^2)^(1/2) - 3*c^2*(c + d*x^2)^(3/2))