3.4.7 \(\int \frac {x^5}{(\frac {e (a+b x^2)}{c+d x^2})^{3/2}} \, dx\) [307]

Optimal. Leaf size=354 \[ -\frac {\left (b^2 c^2+5 a d (2 b c-7 a d)\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{16 b^4 d e^2}-\frac {\left (b^2 c^2+5 a d (2 b c-7 a d)\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^2}{24 b^3 d (b c-a d) e^2}-\frac {a^2 \left (c+d x^2\right )^3}{b (b c-a d)^2 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {\left (b^2 c^2-2 a b c d+7 a^2 d^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^3}{6 b^2 d (b c-a d)^2 e^2}-\frac {(b c-a d) \left (b^2 c^2+5 a d (2 b c-7 a d)\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {b} \sqrt {e}}\right )}{16 b^{9/2} d^{3/2} e^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.36, antiderivative size = 353, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1981, 1980, 473, 393, 205, 214} \begin {gather*} \frac {\left (c+d x^2\right )^3 \left (7 a^2 d^2-2 a b c d+b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{6 b^2 d e^2 (b c-a d)^2}-\frac {a^2 \left (c+d x^2\right )^3}{b e (b c-a d)^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {\left (c+d x^2\right )^2 \left (\frac {5 a (2 b c-7 a d)}{b^2}+\frac {c^2}{d}\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{24 b e^2 (b c-a d)}-\frac {(b c-a d) \left (5 a d (2 b c-7 a d)+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {b} \sqrt {e}}\right )}{16 b^{9/2} d^{3/2} e^{3/2}}-\frac {\left (c+d x^2\right ) \left (5 a d (2 b c-7 a d)+b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{16 b^4 d e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/((e*(a + b*x^2))/(c + d*x^2))^(3/2),x]

[Out]

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

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

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 393

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

Rule 473

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

Rule 1980

Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)))/((c_) + (d_.)*(x_)))^(p_), x_Symbol] :> With[{q = Denominator[p]}
, Dist[q*e*(b*c - a*d), Subst[Int[x^(q*(p + 1) - 1)*(((-a)*e + c*x^q)^m/(b*e - d*x^q)^(m + 2)), x], x, (e*((a
+ b*x)/(c + d*x)))^(1/q)], x]] /; FreeQ[{a, b, c, d, e, m}, x] && FractionQ[p] && IntegerQ[m]

Rule 1981

Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> Dist[1/n, Sub
st[Int[x^(Simplify[(m + 1)/n] - 1)*(e*((a + b*x)/(c + d*x)))^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n,
p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {x^5}{\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx &=((b c-a d) e) \text {Subst}\left (\int \frac {\left (-a e+c x^2\right )^2}{x^2 \left (b e-d x^2\right )^4} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )\\ &=-\frac {a^2 \left (c+d x^2\right )^3}{b (b c-a d)^2 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {(b c-a d) \text {Subst}\left (\int \frac {-a (2 b c-7 a d) e^2+b c^2 e x^2}{\left (b e-d x^2\right )^4} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )}{b}\\ &=-\frac {a^2 \left (c+d x^2\right )^3}{b (b c-a d)^2 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {\left (b^2 c^2-2 a b c d+7 a^2 d^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^3}{6 b^2 d (b c-a d)^2 e^2}-\frac {\left ((b c-a d) \left (b^2 c^2+5 a d (2 b c-7 a d)\right ) e\right ) \text {Subst}\left (\int \frac {1}{\left (b e-d x^2\right )^3} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )}{6 b^2 d}\\ &=-\frac {\left (b^2 c^2+5 a d (2 b c-7 a d)\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^2}{24 b^3 d (b c-a d) e^2}-\frac {a^2 \left (c+d x^2\right )^3}{b (b c-a d)^2 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {\left (b^2 c^2-2 a b c d+7 a^2 d^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^3}{6 b^2 d (b c-a d)^2 e^2}-\frac {\left ((b c-a d) \left (b^2 c^2+5 a d (2 b c-7 a d)\right )\right ) \text {Subst}\left (\int \frac {1}{\left (b e-d x^2\right )^2} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )}{8 b^3 d}\\ &=-\frac {\left (b^2 c^2+5 a d (2 b c-7 a d)\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{16 b^4 d e^2}-\frac {\left (b^2 c^2+5 a d (2 b c-7 a d)\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^2}{24 b^3 d (b c-a d) e^2}-\frac {a^2 \left (c+d x^2\right )^3}{b (b c-a d)^2 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {\left (b^2 c^2-2 a b c d+7 a^2 d^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^3}{6 b^2 d (b c-a d)^2 e^2}-\frac {\left ((b c-a d) \left (b^2 c^2+5 a d (2 b c-7 a d)\right )\right ) \text {Subst}\left (\int \frac {1}{b e-d x^2} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )}{16 b^4 d e}\\ &=-\frac {\left (b^2 c^2+5 a d (2 b c-7 a d)\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{16 b^4 d e^2}-\frac {\left (b^2 c^2+5 a d (2 b c-7 a d)\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^2}{24 b^3 d (b c-a d) e^2}-\frac {a^2 \left (c+d x^2\right )^3}{b (b c-a d)^2 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {\left (b^2 c^2-2 a b c d+7 a^2 d^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^3}{6 b^2 d (b c-a d)^2 e^2}-\frac {(b c-a d) \left (b^2 c^2+5 a d (2 b c-7 a d)\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {b} \sqrt {e}}\right )}{16 b^{9/2} d^{3/2} e^{3/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 4.62, size = 247, normalized size = 0.70 \begin {gather*} \frac {\sqrt {d} \sqrt {\frac {b \left (c+d x^2\right )}{b c-a d}} \left (105 a^3 d^2+5 a^2 b d \left (-20 c+7 d x^2\right )+a b^2 \left (3 c^2-38 c d x^2-14 d^2 x^4\right )+b^3 x^2 \left (3 c^2+14 c d x^2+8 d^2 x^4\right )\right )-3 \sqrt {b c-a d} \left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \sqrt {a+b x^2} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b c-a d}}\right )}{48 b^4 d^{3/2} e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {\frac {b \left (c+d x^2\right )}{b c-a d}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5/((e*(a + b*x^2))/(c + d*x^2))^(3/2),x]

[Out]

(Sqrt[d]*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*(105*a^3*d^2 + 5*a^2*b*d*(-20*c + 7*d*x^2) + a*b^2*(3*c^2 - 38*c*d*
x^2 - 14*d^2*x^4) + b^3*x^2*(3*c^2 + 14*c*d*x^2 + 8*d^2*x^4)) - 3*Sqrt[b*c - a*d]*(b^2*c^2 + 10*a*b*c*d - 35*a
^2*d^2)*Sqrt[a + b*x^2]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/(48*b^4*d^(3/2)*e*Sqrt[(e*(a + b*x
^2))/(c + d*x^2)]*Sqrt[(b*(c + d*x^2))/(b*c - a*d)])

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1026\) vs. \(2(324)=648\).
time = 0.15, size = 1027, normalized size = 2.90

method	result	size
risch	\(\frac {\left (8 b^{2} d^{2} x^{4}-22 a b \,d^{2} x^{2}+14 b^{2} c d \,x^{2}+57 a^{2} d^{2}-52 a b c d +3 b^{2} c^{2}\right ) \left (b \,x^{2}+a \right )}{48 d \,b^{4} e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}+\frac {\left (-\frac {35 d^{2} \ln \left (\frac {\frac {1}{2} a d e +\frac {1}{2} b c e +d e b \,x^{2}}{\sqrt {d e b}}+\sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}\right ) a^{3}}{32 b^{4} \sqrt {d e b}}+\frac {45 d \ln \left (\frac {\frac {1}{2} a d e +\frac {1}{2} b c e +d e b \,x^{2}}{\sqrt {d e b}}+\sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}\right ) a^{2} c}{32 b^{3} \sqrt {d e b}}-\frac {9 \ln \left (\frac {\frac {1}{2} a d e +\frac {1}{2} b c e +d e b \,x^{2}}{\sqrt {d e b}}+\sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}\right ) c^{2} a}{32 b^{2} \sqrt {d e b}}-\frac {\ln \left (\frac {\frac {1}{2} a d e +\frac {1}{2} b c e +d e b \,x^{2}}{\sqrt {d e b}}+\sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}\right ) c^{3}}{32 b d \sqrt {d e b}}+\frac {d^{3} a^{4} x^{2}}{b^{4} \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {2 d^{2} a^{3} x^{2} c}{b^{3} \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}+\frac {d \,a^{2} x^{2} c^{2}}{b^{2} \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}+\frac {d^{2} a^{4} c}{b^{4} \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {2 d \,a^{3} c^{2}}{b^{3} \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}+\frac {a^{2} c^{3}}{b^{2} \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}\right ) \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\)	\(730\)
default	\(\frac {\left (-60 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, a \,b^{2} d^{2} x^{4}+12 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, b^{3} c d \,x^{4}-105 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} b \,d^{3} x^{2}+135 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b^{2} c \,d^{2} x^{2}-27 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{3} c^{2} d \,x^{2}-3 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{4} c^{3} x^{2}+16 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {b d}\, b^{2} d \,x^{2}+54 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, a^{2} b \,d^{2} x^{2}-108 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, a \,b^{2} c d \,x^{2}+6 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, b^{3} c^{2} x^{2}-105 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{4} d^{3}+135 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} b c \,d^{2}-27 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b^{2} c^{2} d -3 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{3} c^{3}+96 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {b d}\, a^{3} d^{2}-96 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {b d}\, a^{2} b c d +16 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {b d}\, a b d +114 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, a^{3} d^{2}-120 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, a^{2} b c d +6 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, a \,b^{2} c^{2}\right ) \left (b \,x^{2}+a \right )}{96 d \,b^{4} \sqrt {b d}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \left (d \,x^{2}+c \right ) \left (\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}\right )^{\frac {3}{2}}}\)	\(1027\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(e*(b*x^2+a)/(d*x^2+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/96*(-60*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)*a*b^2*d^2*x^4+12*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)
*(b*d)^(1/2)*b^3*c*d*x^4-105*ln(1/2*(2*b*d*x^2+2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d
)^(1/2))*a^3*b*d^3*x^2+135*ln(1/2*(2*b*d*x^2+2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^
(1/2))*a^2*b^2*c*d^2*x^2-27*ln(1/2*(2*b*d*x^2+2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)
^(1/2))*a*b^3*c^2*d*x^2-3*ln(1/2*(2*b*d*x^2+2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(
1/2))*b^4*c^3*x^2+16*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(3/2)*(b*d)^(1/2)*b^2*d*x^2+54*(b*d*x^4+a*d*x^2+b*c*x^2+a*c
)^(1/2)*(b*d)^(1/2)*a^2*b*d^2*x^2-108*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)*a*b^2*c*d*x^2+6*(b*d*x^4
+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)*b^3*c^2*x^2-105*ln(1/2*(2*b*d*x^2+2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2
)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*d^3+135*ln(1/2*(2*b*d*x^2+2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^
(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*b*c*d^2-27*ln(1/2*(2*b*d*x^2+2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)
+a*d+b*c)/(b*d)^(1/2))*a^2*b^2*c^2*d-3*ln(1/2*(2*b*d*x^2+2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)+a*d
+b*c)/(b*d)^(1/2))*a*b^3*c^3+96*((d*x^2+c)*(b*x^2+a))^(1/2)*(b*d)^(1/2)*a^3*d^2-96*((d*x^2+c)*(b*x^2+a))^(1/2)
*(b*d)^(1/2)*a^2*b*c*d+16*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(3/2)*(b*d)^(1/2)*a*b*d+114*(b*d*x^4+a*d*x^2+b*c*x^2+a
*c)^(1/2)*(b*d)^(1/2)*a^3*d^2-120*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)*a^2*b*c*d+6*(b*d*x^4+a*d*x^2
+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)*a*b^2*c^2)/d/b^4*(b*x^2+a)/(b*d)^(1/2)/((d*x^2+c)*(b*x^2+a))^(1/2)/(d*x^2+c)/(
e*(b*x^2+a)/(d*x^2+c))^(3/2)

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Maxima [A]
time = 0.52, size = 428, normalized size = 1.21 \begin {gather*} \frac {1}{96} \, {\left (\frac {2 \, {\left (48 \, a^{2} b^{4} c d - 48 \, a^{3} b^{3} d^{2} + \frac {3 \, {\left (b^{3} c^{3} d^{2} + 9 \, a b^{2} c^{2} d^{3} - 45 \, a^{2} b c d^{4} + 35 \, a^{3} d^{5}\right )} {\left (b x^{2} + a\right )}^{3}}{{\left (d x^{2} + c\right )}^{3}} - \frac {8 \, {\left (b^{4} c^{3} d + 9 \, a b^{3} c^{2} d^{2} - 45 \, a^{2} b^{2} c d^{3} + 35 \, a^{3} b d^{4}\right )} {\left (b x^{2} + a\right )}^{2}}{{\left (d x^{2} + c\right )}^{2}} - \frac {3 \, {\left (b^{5} c^{3} - 23 \, a b^{4} c^{2} d + 99 \, a^{2} b^{3} c d^{2} - 77 \, a^{3} b^{2} d^{3}\right )} {\left (b x^{2} + a\right )}}{d x^{2} + c}\right )}}{b^{4} d^{4} \left (\frac {b x^{2} + a}{d x^{2} + c}\right )^{\frac {7}{2}} - 3 \, b^{5} d^{3} \left (\frac {b x^{2} + a}{d x^{2} + c}\right )^{\frac {5}{2}} + 3 \, b^{6} d^{2} \left (\frac {b x^{2} + a}{d x^{2} + c}\right )^{\frac {3}{2}} - b^{7} d \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}} + \frac {3 \, {\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d - 45 \, a^{2} b c d^{2} + 35 \, a^{3} d^{3}\right )} \log \left (\frac {d \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} - \sqrt {b d}}{d \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} + \sqrt {b d}}\right )}{\sqrt {b d} b^{4} d}\right )} e^{\left (-\frac {3}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(e*(b*x^2+a)/(d*x^2+c))^(3/2),x, algorithm="maxima")

[Out]

1/96*(2*(48*a^2*b^4*c*d - 48*a^3*b^3*d^2 + 3*(b^3*c^3*d^2 + 9*a*b^2*c^2*d^3 - 45*a^2*b*c*d^4 + 35*a^3*d^5)*(b*
x^2 + a)^3/(d*x^2 + c)^3 - 8*(b^4*c^3*d + 9*a*b^3*c^2*d^2 - 45*a^2*b^2*c*d^3 + 35*a^3*b*d^4)*(b*x^2 + a)^2/(d*
x^2 + c)^2 - 3*(b^5*c^3 - 23*a*b^4*c^2*d + 99*a^2*b^3*c*d^2 - 77*a^3*b^2*d^3)*(b*x^2 + a)/(d*x^2 + c))/(b^4*d^
4*((b*x^2 + a)/(d*x^2 + c))^(7/2) - 3*b^5*d^3*((b*x^2 + a)/(d*x^2 + c))^(5/2) + 3*b^6*d^2*((b*x^2 + a)/(d*x^2
+ c))^(3/2) - b^7*d*sqrt((b*x^2 + a)/(d*x^2 + c))) + 3*(b^3*c^3 + 9*a*b^2*c^2*d - 45*a^2*b*c*d^2 + 35*a^3*d^3)
*log((d*sqrt((b*x^2 + a)/(d*x^2 + c)) - sqrt(b*d))/(d*sqrt((b*x^2 + a)/(d*x^2 + c)) + sqrt(b*d)))/(sqrt(b*d)*b
^4*d))*e^(-3/2)

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Fricas [A]
time = 0.64, size = 750, normalized size = 2.12 \begin {gather*} \left [\frac {{\left (3 \, {\left (a b^{3} c^{3} + 9 \, a^{2} b^{2} c^{2} d - 45 \, a^{3} b c d^{2} + 35 \, a^{4} d^{3} + {\left (b^{4} c^{3} + 9 \, a b^{3} c^{2} d - 45 \, a^{2} b^{2} c d^{2} + 35 \, a^{3} b d^{3}\right )} x^{2}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} - 4 \, {\left (2 \, b d^{2} x^{4} + b c^{2} + a c d + {\left (3 \, b c d + a d^{2}\right )} x^{2}\right )} \sqrt {b d} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}\right ) + 4 \, {\left (8 \, b^{4} d^{4} x^{8} + 3 \, a b^{3} c^{3} d - 100 \, a^{2} b^{2} c^{2} d^{2} + 105 \, a^{3} b c d^{3} + 2 \, {\left (11 \, b^{4} c d^{3} - 7 \, a b^{3} d^{4}\right )} x^{6} + {\left (17 \, b^{4} c^{2} d^{2} - 52 \, a b^{3} c d^{3} + 35 \, a^{2} b^{2} d^{4}\right )} x^{4} + {\left (3 \, b^{4} c^{3} d - 35 \, a b^{3} c^{2} d^{2} - 65 \, a^{2} b^{2} c d^{3} + 105 \, a^{3} b d^{4}\right )} x^{2}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}\right )} e^{\left (-\frac {3}{2}\right )}}{192 \, {\left (b^{6} d^{2} x^{2} + a b^{5} d^{2}\right )}}, \frac {{\left (3 \, {\left (a b^{3} c^{3} + 9 \, a^{2} b^{2} c^{2} d - 45 \, a^{3} b c d^{2} + 35 \, a^{4} d^{3} + {\left (b^{4} c^{3} + 9 \, a b^{3} c^{2} d - 45 \, a^{2} b^{2} c d^{2} + 35 \, a^{3} b d^{3}\right )} x^{2}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {-b d} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}}{2 \, {\left (b^{2} d x^{2} + a b d\right )}}\right ) + 2 \, {\left (8 \, b^{4} d^{4} x^{8} + 3 \, a b^{3} c^{3} d - 100 \, a^{2} b^{2} c^{2} d^{2} + 105 \, a^{3} b c d^{3} + 2 \, {\left (11 \, b^{4} c d^{3} - 7 \, a b^{3} d^{4}\right )} x^{6} + {\left (17 \, b^{4} c^{2} d^{2} - 52 \, a b^{3} c d^{3} + 35 \, a^{2} b^{2} d^{4}\right )} x^{4} + {\left (3 \, b^{4} c^{3} d - 35 \, a b^{3} c^{2} d^{2} - 65 \, a^{2} b^{2} c d^{3} + 105 \, a^{3} b d^{4}\right )} x^{2}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}\right )} e^{\left (-\frac {3}{2}\right )}}{96 \, {\left (b^{6} d^{2} x^{2} + a b^{5} d^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(e*(b*x^2+a)/(d*x^2+c))^(3/2),x, algorithm="fricas")

[Out]

[1/192*(3*(a*b^3*c^3 + 9*a^2*b^2*c^2*d - 45*a^3*b*c*d^2 + 35*a^4*d^3 + (b^4*c^3 + 9*a*b^3*c^2*d - 45*a^2*b^2*c
*d^2 + 35*a^3*b*d^3)*x^2)*sqrt(b*d)*log(8*b^2*d^2*x^4 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 8*(b^2*c*d + a*b*d^2)*
x^2 - 4*(2*b*d^2*x^4 + b*c^2 + a*c*d + (3*b*c*d + a*d^2)*x^2)*sqrt(b*d)*sqrt((b*x^2 + a)/(d*x^2 + c))) + 4*(8*
b^4*d^4*x^8 + 3*a*b^3*c^3*d - 100*a^2*b^2*c^2*d^2 + 105*a^3*b*c*d^3 + 2*(11*b^4*c*d^3 - 7*a*b^3*d^4)*x^6 + (17
*b^4*c^2*d^2 - 52*a*b^3*c*d^3 + 35*a^2*b^2*d^4)*x^4 + (3*b^4*c^3*d - 35*a*b^3*c^2*d^2 - 65*a^2*b^2*c*d^3 + 105
*a^3*b*d^4)*x^2)*sqrt((b*x^2 + a)/(d*x^2 + c)))*e^(-3/2)/(b^6*d^2*x^2 + a*b^5*d^2), 1/96*(3*(a*b^3*c^3 + 9*a^2
*b^2*c^2*d - 45*a^3*b*c*d^2 + 35*a^4*d^3 + (b^4*c^3 + 9*a*b^3*c^2*d - 45*a^2*b^2*c*d^2 + 35*a^3*b*d^3)*x^2)*sq
rt(-b*d)*arctan(1/2*(2*b*d*x^2 + b*c + a*d)*sqrt(-b*d)*sqrt((b*x^2 + a)/(d*x^2 + c))/(b^2*d*x^2 + a*b*d)) + 2*
(8*b^4*d^4*x^8 + 3*a*b^3*c^3*d - 100*a^2*b^2*c^2*d^2 + 105*a^3*b*c*d^3 + 2*(11*b^4*c*d^3 - 7*a*b^3*d^4)*x^6 +
(17*b^4*c^2*d^2 - 52*a*b^3*c*d^3 + 35*a^2*b^2*d^4)*x^4 + (3*b^4*c^3*d - 35*a*b^3*c^2*d^2 - 65*a^2*b^2*c*d^3 +
105*a^3*b*d^4)*x^2)*sqrt((b*x^2 + a)/(d*x^2 + c)))*e^(-3/2)/(b^6*d^2*x^2 + a*b^5*d^2)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(e*(b*x**2+a)/(d*x**2+c))**(3/2),x)

[Out]

Timed out

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(e*(b*x^2+a)/(d*x^2+c))^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(sa

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^5}{{\left (\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/((e*(a + b*x^2))/(c + d*x^2))^(3/2),x)

[Out]

int(x^5/((e*(a + b*x^2))/(c + d*x^2))^(3/2), x)

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