\(\int \frac {(a+b x^2)^2 (c+d x^2)^{3/2}}{x^7} \, dx\) [624]

Optimal result

Integrand size = 24, antiderivative size = 187 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^7} \, dx=\frac {d \left (24 b^2 c^2+a d (12 b c-a d)\right ) \sqrt {c+d x^2}}{16 c^2}-\frac {\left (24 b^2 c^2+a d (12 b c-a d)\right ) \left (c+d x^2\right )^{3/2}}{48 c^2 x^2}-\frac {a^2 \left (c+d x^2\right )^{5/2}}{6 c x^6}-\frac {a (12 b c-a d) \left (c+d x^2\right )^{5/2}}{24 c^2 x^4}-\frac {d \left (24 b^2 c^2+a d (12 b c-a d)\right ) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{16 c^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.98, 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, 43, 52, 65, 214} \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^7} \, dx=-\frac {a^2 \left (c+d x^2\right )^{5/2}}{6 c x^6}-\frac {d \left (a d (12 b c-a d)+24 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{16 c^{3/2}}-\frac {\left (c+d x^2\right )^{3/2} \left (\frac {a d (12 b c-a d)}{c^2}+24 b^2\right )}{48 x^2}+\frac {d \sqrt {c+d x^2} \left (a d (12 b c-a d)+24 b^2 c^2\right )}{16 c^2}-\frac {a \left (c+d x^2\right )^{5/2} (12 b c-a d)}{24 c^2 x^4} \]

[In]

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

[Out]

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

Rule 43

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

Rule 52

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] + Dist[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] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && 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 (c+d x)^{3/2}}{x^4} \, dx,x,x^2\right ) \\ & = -\frac {a^2 \left (c+d x^2\right )^{5/2}}{6 c x^6}+\frac {\text {Subst}\left (\int \frac {\left (\frac {1}{2} a (12 b c-a d)+3 b^2 c x\right ) (c+d x)^{3/2}}{x^3} \, dx,x,x^2\right )}{6 c} \\ & = -\frac {a^2 \left (c+d x^2\right )^{5/2}}{6 c x^6}-\frac {a (12 b c-a d) \left (c+d x^2\right )^{5/2}}{24 c^2 x^4}+\frac {1}{48} \left (24 b^2+\frac {a d (12 b c-a d)}{c^2}\right ) \text {Subst}\left (\int \frac {(c+d x)^{3/2}}{x^2} \, dx,x,x^2\right ) \\ & = -\frac {\left (24 b^2+\frac {a d (12 b c-a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}}{48 x^2}-\frac {a^2 \left (c+d x^2\right )^{5/2}}{6 c x^6}-\frac {a (12 b c-a d) \left (c+d x^2\right )^{5/2}}{24 c^2 x^4}+\frac {1}{32} \left (d \left (24 b^2+\frac {a d (12 b c-a d)}{c^2}\right )\right ) \text {Subst}\left (\int \frac {\sqrt {c+d x}}{x} \, dx,x,x^2\right ) \\ & = \frac {1}{16} d \left (24 b^2+\frac {a d (12 b c-a d)}{c^2}\right ) \sqrt {c+d x^2}-\frac {\left (24 b^2+\frac {a d (12 b c-a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}}{48 x^2}-\frac {a^2 \left (c+d x^2\right )^{5/2}}{6 c x^6}-\frac {a (12 b c-a d) \left (c+d x^2\right )^{5/2}}{24 c^2 x^4}+\frac {1}{32} \left (c d \left (24 b^2+\frac {a d (12 b c-a d)}{c^2}\right )\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right ) \\ & = \frac {1}{16} d \left (24 b^2+\frac {a d (12 b c-a d)}{c^2}\right ) \sqrt {c+d x^2}-\frac {\left (24 b^2+\frac {a d (12 b c-a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}}{48 x^2}-\frac {a^2 \left (c+d x^2\right )^{5/2}}{6 c x^6}-\frac {a (12 b c-a d) \left (c+d x^2\right )^{5/2}}{24 c^2 x^4}+\frac {1}{16} \left (c \left (24 b^2+\frac {a d (12 b c-a d)}{c^2}\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right ) \\ & = \frac {1}{16} d \left (24 b^2+\frac {a d (12 b c-a d)}{c^2}\right ) \sqrt {c+d x^2}-\frac {\left (24 b^2+\frac {a d (12 b c-a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}}{48 x^2}-\frac {a^2 \left (c+d x^2\right )^{5/2}}{6 c x^6}-\frac {a (12 b c-a d) \left (c+d x^2\right )^{5/2}}{24 c^2 x^4}-\frac {1}{16} \sqrt {c} d \left (24 b^2+\frac {a d (12 b c-a d)}{c^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.71 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^7} \, dx=-\frac {\sqrt {c+d x^2} \left (24 b^2 c x^4 \left (c-2 d x^2\right )+12 a b c x^2 \left (2 c+5 d x^2\right )+a^2 \left (8 c^2+14 c d x^2+3 d^2 x^4\right )\right )}{48 c x^6}+\frac {d \left (-24 b^2 c^2-12 a b c d+a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{16 c^{3/2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.94 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.68

method	result	size
pseudoelliptic	\(-\frac {-\frac {3 d \,x^{6} \left (a^{2} d^{2}-12 a b c d -24 b^{2} c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{\sqrt {c}}\right )}{8}+\sqrt {d \,x^{2}+c}\, \left (\frac {7 x^{2} \left (-\frac {24}{7} b^{2} x^{4}+\frac {30}{7} a b \,x^{2}+a^{2}\right ) d \,c^{\frac {3}{2}}}{4}+\left (3 b^{2} x^{4}+3 a b \,x^{2}+a^{2}\right ) c^{\frac {5}{2}}+\frac {3 \sqrt {c}\, a^{2} d^{2} x^{4}}{8}\right )}{6 c^{\frac {3}{2}} x^{6}}\)	\(128\)
risch	\(-\frac {\sqrt {d \,x^{2}+c}\, \left (3 a^{2} d^{2} x^{4}+60 x^{4} a b c d +24 b^{2} c^{2} x^{4}+14 a^{2} c d \,x^{2}+24 a b \,c^{2} x^{2}+8 a^{2} c^{2}\right )}{48 x^{6} c}-\frac {d \left (-16 b^{2} c \sqrt {d \,x^{2}+c}+\frac {\left (-a^{2} d^{2}+12 a b c d +24 b^{2} c^{2}\right ) \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{\sqrt {c}}\right )}{16 c}\)	\(151\)
default	\(a^{2} \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}}}{6 c \,x^{6}}-\frac {d \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}}}{4 c \,x^{4}}+\frac {d \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}}}{2 c \,x^{2}}+\frac {3 d \left (\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {d \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )\right )\right )}{2 c}\right )}{4 c}\right )}{6 c}\right )+b^{2} \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}}}{2 c \,x^{2}}+\frac {3 d \left (\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {d \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )\right )\right )}{2 c}\right )+2 a b \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}}}{4 c \,x^{4}}+\frac {d \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}}}{2 c \,x^{2}}+\frac {3 d \left (\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {d \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )\right )\right )}{2 c}\right )}{4 c}\right )\)	\(314\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.61 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^7} \, dx=\left [-\frac {3 \, {\left (24 \, b^{2} c^{2} d + 12 \, a b c d^{2} - a^{2} d^{3}\right )} \sqrt {c} x^{6} \log \left (-\frac {d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) - 2 \, {\left (48 \, b^{2} c^{2} d x^{6} - 8 \, a^{2} c^{3} - 3 \, {\left (8 \, b^{2} c^{3} + 20 \, a b c^{2} d + a^{2} c d^{2}\right )} x^{4} - 2 \, {\left (12 \, a b c^{3} + 7 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{96 \, c^{2} x^{6}}, \frac {3 \, {\left (24 \, b^{2} c^{2} d + 12 \, a b c d^{2} - a^{2} d^{3}\right )} \sqrt {-c} x^{6} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + {\left (48 \, b^{2} c^{2} d x^{6} - 8 \, a^{2} c^{3} - 3 \, {\left (8 \, b^{2} c^{3} + 20 \, a b c^{2} d + a^{2} c d^{2}\right )} x^{4} - 2 \, {\left (12 \, a b c^{3} + 7 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{48 \, c^{2} x^{6}}\right ] \]

[In]

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

[Out]

[-1/96*(3*(24*b^2*c^2*d + 12*a*b*c*d^2 - a^2*d^3)*sqrt(c)*x^6*log(-(d*x^2 + 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/x
^2) - 2*(48*b^2*c^2*d*x^6 - 8*a^2*c^3 - 3*(8*b^2*c^3 + 20*a*b*c^2*d + a^2*c*d^2)*x^4 - 2*(12*a*b*c^3 + 7*a^2*c
^2*d)*x^2)*sqrt(d*x^2 + c))/(c^2*x^6), 1/48*(3*(24*b^2*c^2*d + 12*a*b*c*d^2 - a^2*d^3)*sqrt(-c)*x^6*arctan(sqr
t(-c)/sqrt(d*x^2 + c)) + (48*b^2*c^2*d*x^6 - 8*a^2*c^3 - 3*(8*b^2*c^3 + 20*a*b*c^2*d + a^2*c*d^2)*x^4 - 2*(12*
a*b*c^3 + 7*a^2*c^2*d)*x^2)*sqrt(d*x^2 + c))/(c^2*x^6)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 367 vs. \(2 (172) = 344\).

Time = 84.18 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.96 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^7} \, dx=- \frac {a^{2} c^{2}}{6 \sqrt {d} x^{7} \sqrt {\frac {c}{d x^{2}} + 1}} - \frac {11 a^{2} c \sqrt {d}}{24 x^{5} \sqrt {\frac {c}{d x^{2}} + 1}} - \frac {17 a^{2} d^{\frac {3}{2}}}{48 x^{3} \sqrt {\frac {c}{d x^{2}} + 1}} - \frac {a^{2} d^{\frac {5}{2}}}{16 c x \sqrt {\frac {c}{d x^{2}} + 1}} + \frac {a^{2} d^{3} \operatorname {asinh}{\left (\frac {\sqrt {c}}{\sqrt {d} x} \right )}}{16 c^{\frac {3}{2}}} - \frac {a b c^{2}}{2 \sqrt {d} x^{5} \sqrt {\frac {c}{d x^{2}} + 1}} - \frac {3 a b c \sqrt {d}}{4 x^{3} \sqrt {\frac {c}{d x^{2}} + 1}} - \frac {a b d^{\frac {3}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{x} - \frac {a b d^{\frac {3}{2}}}{4 x \sqrt {\frac {c}{d x^{2}} + 1}} - \frac {3 a b d^{2} \operatorname {asinh}{\left (\frac {\sqrt {c}}{\sqrt {d} x} \right )}}{4 \sqrt {c}} - \frac {3 b^{2} \sqrt {c} d \operatorname {asinh}{\left (\frac {\sqrt {c}}{\sqrt {d} x} \right )}}{2} - \frac {b^{2} c \sqrt {d} \sqrt {\frac {c}{d x^{2}} + 1}}{2 x} + \frac {b^{2} c \sqrt {d}}{x \sqrt {\frac {c}{d x^{2}} + 1}} + \frac {b^{2} d^{\frac {3}{2}} x}{\sqrt {\frac {c}{d x^{2}} + 1}} \]

[In]

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

[Out]

-a**2*c**2/(6*sqrt(d)*x**7*sqrt(c/(d*x**2) + 1)) - 11*a**2*c*sqrt(d)/(24*x**5*sqrt(c/(d*x**2) + 1)) - 17*a**2*
d**(3/2)/(48*x**3*sqrt(c/(d*x**2) + 1)) - a**2*d**(5/2)/(16*c*x*sqrt(c/(d*x**2) + 1)) + a**2*d**3*asinh(sqrt(c
)/(sqrt(d)*x))/(16*c**(3/2)) - a*b*c**2/(2*sqrt(d)*x**5*sqrt(c/(d*x**2) + 1)) - 3*a*b*c*sqrt(d)/(4*x**3*sqrt(c
/(d*x**2) + 1)) - a*b*d**(3/2)*sqrt(c/(d*x**2) + 1)/x - a*b*d**(3/2)/(4*x*sqrt(c/(d*x**2) + 1)) - 3*a*b*d**2*a
sinh(sqrt(c)/(sqrt(d)*x))/(4*sqrt(c)) - 3*b**2*sqrt(c)*d*asinh(sqrt(c)/(sqrt(d)*x))/2 - b**2*c*sqrt(d)*sqrt(c/
(d*x**2) + 1)/(2*x) + b**2*c*sqrt(d)/(x*sqrt(c/(d*x**2) + 1)) + b**2*d**(3/2)*x/sqrt(c/(d*x**2) + 1)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.61 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^7} \, dx=-\frac {3}{2} \, b^{2} \sqrt {c} d \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right ) - \frac {3 \, a b d^{2} \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right )}{4 \, \sqrt {c}} + \frac {a^{2} d^{3} \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right )}{16 \, c^{\frac {3}{2}}} + \frac {3}{2} \, \sqrt {d x^{2} + c} b^{2} d + \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} d}{2 \, c} + \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d^{2}}{4 \, c^{2}} + \frac {3 \, \sqrt {d x^{2} + c} a b d^{2}}{4 \, c} - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d^{3}}{48 \, c^{3}} - \frac {\sqrt {d x^{2} + c} a^{2} d^{3}}{16 \, c^{2}} - \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2}}{2 \, c x^{2}} - \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a b d}{4 \, c^{2} x^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} d^{2}}{48 \, c^{3} x^{2}} - \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a b}{2 \, c x^{4}} + \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} d}{24 \, c^{2} x^{4}} - \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2}}{6 \, c x^{6}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.39 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^7} \, dx=\frac {48 \, \sqrt {d x^{2} + c} b^{2} d^{2} + \frac {3 \, {\left (24 \, b^{2} c^{2} d^{2} + 12 \, a b c d^{3} - a^{2} d^{4}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{\sqrt {-c} c} - \frac {24 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c^{2} d^{2} - 48 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c^{3} d^{2} + 24 \, \sqrt {d x^{2} + c} b^{2} c^{4} d^{2} + 60 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b c d^{3} - 96 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c^{2} d^{3} + 36 \, \sqrt {d x^{2} + c} a b c^{3} d^{3} + 3 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} d^{4} + 8 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} c d^{4} - 3 \, \sqrt {d x^{2} + c} a^{2} c^{2} d^{4}}{c d^{3} x^{6}}}{48 \, d} \]

[In]

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

[Out]

1/48*(48*sqrt(d*x^2 + c)*b^2*d^2 + 3*(24*b^2*c^2*d^2 + 12*a*b*c*d^3 - a^2*d^4)*arctan(sqrt(d*x^2 + c)/sqrt(-c)
)/(sqrt(-c)*c) - (24*(d*x^2 + c)^(5/2)*b^2*c^2*d^2 - 48*(d*x^2 + c)^(3/2)*b^2*c^3*d^2 + 24*sqrt(d*x^2 + c)*b^2
*c^4*d^2 + 60*(d*x^2 + c)^(5/2)*a*b*c*d^3 - 96*(d*x^2 + c)^(3/2)*a*b*c^2*d^3 + 36*sqrt(d*x^2 + c)*a*b*c^3*d^3
+ 3*(d*x^2 + c)^(5/2)*a^2*d^4 + 8*(d*x^2 + c)^(3/2)*a^2*c*d^4 - 3*sqrt(d*x^2 + c)*a^2*c^2*d^4)/(c*d^3*x^6))/d

Mupad [B] (verification not implemented)

Time = 6.84 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.15 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^7} \, dx=\frac {\sqrt {d\,x^2+c}\,\left (-\frac {a^2\,c\,d^3}{16}+\frac {3\,a\,b\,c^2\,d^2}{4}+\frac {b^2\,c^3\,d}{2}\right )-{\left (d\,x^2+c\right )}^{3/2}\,\left (-\frac {a^2\,d^3}{6}+2\,a\,b\,c\,d^2+b^2\,c^2\,d\right )+\frac {{\left (d\,x^2+c\right )}^{5/2}\,\left (a^2\,d^3+20\,a\,b\,c\,d^2+8\,b^2\,c^2\,d\right )}{16\,c}}{3\,c\,{\left (d\,x^2+c\right )}^2-3\,c^2\,\left (d\,x^2+c\right )-{\left (d\,x^2+c\right )}^3+c^3}+b^2\,d\,\sqrt {d\,x^2+c}-\frac {d\,\mathrm {atanh}\left (\frac {\sqrt {d\,x^2+c}}{\sqrt {c}}\right )\,\left (-a^2\,d^2+12\,a\,b\,c\,d+24\,b^2\,c^2\right )}{16\,c^{3/2}} \]

[In]

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

[Out]

((c + d*x^2)^(1/2)*((b^2*c^3*d)/2 - (a^2*c*d^3)/16 + (3*a*b*c^2*d^2)/4) - (c + d*x^2)^(3/2)*(b^2*c^2*d - (a^2*
d^3)/6 + 2*a*b*c*d^2) + ((c + d*x^2)^(5/2)*(a^2*d^3 + 8*b^2*c^2*d + 20*a*b*c*d^2))/(16*c))/(3*c*(c + d*x^2)^2
- 3*c^2*(c + d*x^2) - (c + d*x^2)^3 + c^3) + b^2*d*(c + d*x^2)^(1/2) - (d*atanh((c + d*x^2)^(1/2)/c^(1/2))*(24
*b^2*c^2 - a^2*d^2 + 12*a*b*c*d))/(16*c^(3/2))