3.4.44 \(\int \frac {x^5}{\sqrt {a+\frac {b}{c+d x^2}}} \, dx\) [344]

Optimal. Leaf size=225 \[ \frac {\left (5 b^2+12 a b c+8 a^2 c^2\right ) \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{16 a^3 d^3}-\frac {(5 b+8 a c) \left (c+d x^2\right )^2 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{24 a^2 d^3}+\frac {x^2 \left (c+d x^2\right )^2 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{6 a d^2}-\frac {b \left (5 b^2+12 a b c+8 a^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {a}}\right )}{16 a^{7/2} d^3} \]

[Out]

-1/16*b*(8*a^2*c^2+12*a*b*c+5*b^2)*arctanh(((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/a^(1/2))/a^(7/2)/d^3+1/16*(8*a^2*
c^2+12*a*b*c+5*b^2)*(d*x^2+c)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/a^3/d^3-1/24*(8*a*c+5*b)*(d*x^2+c)^2*((a*d*x^2
+a*c+b)/(d*x^2+c))^(1/2)/a^2/d^3+1/6*x^2*(d*x^2+c)^2*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/a/d^2

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Rubi [A]
time = 0.29, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1985, 1981, 1980, 424, 393, 205, 214} \begin {gather*} -\frac {(8 a c+5 b) \left (c+d x^2\right )^2 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{24 a^2 d^3}-\frac {b \left (8 a^2 c^2+12 a b c+5 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{\sqrt {a}}\right )}{16 a^{7/2} d^3}+\frac {\left (8 a^2 c^2+12 a b c+5 b^2\right ) \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{16 a^3 d^3}+\frac {x^2 \left (c+d x^2\right )^2 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{6 a d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/Sqrt[a + b/(c + d*x^2)],x]

[Out]

((5*b^2 + 12*a*b*c + 8*a^2*c^2)*(c + d*x^2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(16*a^3*d^3) - ((5*b + 8*a*
c)*(c + d*x^2)^2*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(24*a^2*d^3) + (x^2*(c + d*x^2)^2*Sqrt[(b + a*c + a*d*
x^2)/(c + d*x^2)])/(6*a*d^2) - (b*(5*b^2 + 12*a*b*c + 8*a^2*c^2)*ArcTanh[Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]
/Sqrt[a]])/(16*a^(7/2)*d^3)

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 424

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a*d - c*b)*x*(a + b*x^n)^(
p + 1)*((c + d*x^n)^(q - 1)/(a*b*n*(p + 1))), x] - Dist[1/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)
^(q - 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*(p + q) + 1))*x^n, x], x], x] /;
FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q
, x]

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]]

Rule 1985

Int[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u*((b + a*c + a*d*x^n)/(c + d*x^n))^p
, x] /; FreeQ[{a, b, c, d, n, p}, x]

Rubi steps

\begin {align*} \int \frac {x^5}{\sqrt {a+\frac {b}{c+d x^2}}} \, dx &=\frac {\sqrt {b+a \left (c+d x^2\right )} \int \frac {x^5 \sqrt {c+d x^2}}{\sqrt {b+a \left (c+d x^2\right )}} \, dx}{\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=\frac {\sqrt {b+a \left (c+d x^2\right )} \int \frac {x^5 \sqrt {c+d x^2}}{\sqrt {b+a c+a d x^2}} \, dx}{\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=\frac {\sqrt {b+a \left (c+d x^2\right )} \text {Subst}\left (\int \frac {x^2 \sqrt {c+d x}}{\sqrt {b+a c+a d x}} \, dx,x,x^2\right )}{2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=\frac {x^2 \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{6 a d^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\sqrt {b+a \left (c+d x^2\right )} \text {Subst}\left (\int \frac {\sqrt {c+d x} \left (-c (b+a c)-\frac {1}{2} (5 b+8 a c) d x\right )}{\sqrt {b+a c+a d x}} \, dx,x,x^2\right )}{6 a d^2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {(5 b+8 a c) \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{24 a^2 d^3 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {x^2 \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{6 a d^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (\left (-2 a c (b+a c) d^2+\frac {1}{2} (5 b+8 a c) d \left (\frac {a c d}{2}+\frac {3}{2} (b+a c) d\right )\right ) \sqrt {b+a \left (c+d x^2\right )}\right ) \text {Subst}\left (\int \frac {\sqrt {c+d x}}{\sqrt {b+a c+a d x}} \, dx,x,x^2\right )}{12 a^2 d^4 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=\frac {\left (5 b^2+12 a b c+8 a^2 c^2\right ) \left (b+a \left (c+d x^2\right )\right )}{16 a^3 d^3 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {(5 b+8 a c) \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{24 a^2 d^3 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {x^2 \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{6 a d^2 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\left (b \left (-2 a c (b+a c) d^2+\frac {1}{2} (5 b+8 a c) d \left (\frac {a c d}{2}+\frac {3}{2} (b+a c) d\right )\right ) \sqrt {b+a \left (c+d x^2\right )}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+d x} \sqrt {b+a c+a d x}} \, dx,x,x^2\right )}{24 a^3 d^4 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=\frac {\left (5 b^2+12 a b c+8 a^2 c^2\right ) \left (b+a \left (c+d x^2\right )\right )}{16 a^3 d^3 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {(5 b+8 a c) \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{24 a^2 d^3 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {x^2 \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{6 a d^2 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\left (b \left (-2 a c (b+a c) d^2+\frac {1}{2} (5 b+8 a c) d \left (\frac {a c d}{2}+\frac {3}{2} (b+a c) d\right )\right ) \sqrt {b+a \left (c+d x^2\right )}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+a x^2}} \, dx,x,\sqrt {c+d x^2}\right )}{12 a^3 d^5 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=\frac {\left (5 b^2+12 a b c+8 a^2 c^2\right ) \left (b+a \left (c+d x^2\right )\right )}{16 a^3 d^3 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {(5 b+8 a c) \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{24 a^2 d^3 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {x^2 \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{6 a d^2 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\left (b \left (-2 a c (b+a c) d^2+\frac {1}{2} (5 b+8 a c) d \left (\frac {a c d}{2}+\frac {3}{2} (b+a c) d\right )\right ) \sqrt {b+a \left (c+d x^2\right )}\right ) \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\sqrt {c+d x^2}}{\sqrt {b+a \left (c+d x^2\right )}}\right )}{12 a^3 d^5 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=\frac {\left (5 b^2+12 a b c+8 a^2 c^2\right ) \left (b+a \left (c+d x^2\right )\right )}{16 a^3 d^3 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {(5 b+8 a c) \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{24 a^2 d^3 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {x^2 \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{6 a d^2 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {b \left (5 b^2+12 a b c+8 a^2 c^2\right ) \sqrt {b+a \left (c+d x^2\right )} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x^2}}{\sqrt {b+a \left (c+d x^2\right )}}\right )}{16 a^{7/2} d^3 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ \end {align*}

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Mathematica [A]
time = 0.24, size = 148, normalized size = 0.66 \begin {gather*} \frac {\sqrt {a} \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (15 b^2+2 a b \left (13 c-5 d x^2\right )+8 a^2 \left (c^2-c d x^2+d^2 x^4\right )\right )-3 b \left (5 b^2+12 a b c+8 a^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {a}}\right )}{48 a^{7/2} d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a]*(c + d*x^2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(15*b^2 + 2*a*b*(13*c - 5*d*x^2) + 8*a^2*(c^2 - c*d
*x^2 + d^2*x^4)) - 3*b*(5*b^2 + 12*a*b*c + 8*a^2*c^2)*ArcTanh[Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]/Sqrt[a]])/
(48*a^(7/2)*d^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(532\) vs. \(2(205)=410\).
time = 0.08, size = 533, normalized size = 2.37

method	result	size
risch	\(\frac {\left (8 d^{2} a^{2} x^{4}-8 a^{2} c d \,x^{2}-10 a b d \,x^{2}+8 a^{2} c^{2}+26 a b c +15 b^{2}\right ) \left (a d \,x^{2}+a c +b \right )}{48 d^{3} a^{3} \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}+\frac {\left (-\frac {b \ln \left (\frac {a c d +\frac {1}{2} b d +a \,d^{2} x^{2}}{\sqrt {a \,d^{2}}}+\sqrt {c^{2} a +b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}\right ) c^{2}}{4 d^{2} a \sqrt {a \,d^{2}}}-\frac {3 b^{2} \ln \left (\frac {a c d +\frac {1}{2} b d +a \,d^{2} x^{2}}{\sqrt {a \,d^{2}}}+\sqrt {c^{2} a +b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}\right ) c}{8 d^{2} a^{2} \sqrt {a \,d^{2}}}-\frac {5 b^{3} \ln \left (\frac {a c d +\frac {1}{2} b d +a \,d^{2} x^{2}}{\sqrt {a \,d^{2}}}+\sqrt {c^{2} a +b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}\right )}{32 d^{2} a^{3} \sqrt {a \,d^{2}}}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}}{\sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\)	\(380\)
default	\(\frac {\sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (-48 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, x^{2} c \,a^{2} d \sqrt {a \,d^{2}}-36 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, x^{2} b a d \sqrt {a \,d^{2}}-24 \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) a^{2} b \,c^{2} d -36 \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) b^{2} c a d +16 \left (a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c \right )^{\frac {3}{2}} a \sqrt {a \,d^{2}}+36 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, c b a \sqrt {a \,d^{2}}-15 \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) b^{3} d +30 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, b^{2} \sqrt {a \,d^{2}}\right )}{96 a^{3} d^{3} \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, \sqrt {a \,d^{2}}}\)	\(533\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a+b/(d*x^2+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/96*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)*(d*x^2+c)/a^3/d^3*(-48*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*
x^2*c*a^2*d*(a*d^2)^(1/2)-36*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*x^2*b*a*d*(a*d^2)^(1/2)-24*ln(1/2
*(2*a*d^2*x^2+2*a*c*d+2*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*d^2)^(1/2)+b*d)/(a*d^2)^(1/2))*a^2*
b*c^2*d-36*ln(1/2*(2*a*d^2*x^2+2*a*c*d+2*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*d^2)^(1/2)+b*d)/(a
*d^2)^(1/2))*b^2*c*a*d+16*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(3/2)*a*(a*d^2)^(1/2)+36*(a*d^2*x^4+2*a*c*
d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*c*b*a*(a*d^2)^(1/2)-15*ln(1/2*(2*a*d^2*x^2+2*a*c*d+2*(a*d^2*x^4+2*a*c*d*x^2+b*d
*x^2+a*c^2+b*c)^(1/2)*(a*d^2)^(1/2)+b*d)/(a*d^2)^(1/2))*b^3*d+30*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/
2)*b^2*(a*d^2)^(1/2))/((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)/(a*d^2)^(1/2)

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Maxima [A]
time = 0.51, size = 340, normalized size = 1.51 \begin {gather*} -\frac {3 \, {\left (8 \, a^{2} b c^{2} + 12 \, a b^{2} c + 5 \, b^{3}\right )} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {5}{2}} - 8 \, {\left (6 \, a^{3} b c^{2} + 12 \, a^{2} b^{2} c + 5 \, a b^{3}\right )} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {3}{2}} + 3 \, {\left (8 \, a^{4} b c^{2} + 20 \, a^{3} b^{2} c + 11 \, a^{2} b^{3}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{48 \, {\left (a^{6} d^{3} - \frac {3 \, {\left (a d x^{2} + a c + b\right )} a^{5} d^{3}}{d x^{2} + c} + \frac {3 \, {\left (a d x^{2} + a c + b\right )}^{2} a^{4} d^{3}}{{\left (d x^{2} + c\right )}^{2}} - \frac {{\left (a d x^{2} + a c + b\right )}^{3} a^{3} d^{3}}{{\left (d x^{2} + c\right )}^{3}}\right )}} + \frac {{\left (8 \, a^{2} c^{2} + 12 \, a b c + 5 \, b^{2}\right )} b \log \left (-\frac {\sqrt {a} - \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{\sqrt {a} + \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}\right )}{32 \, a^{\frac {7}{2}} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a+b/(d*x^2+c))^(1/2),x, algorithm="maxima")

[Out]

-1/48*(3*(8*a^2*b*c^2 + 12*a*b^2*c + 5*b^3)*((a*d*x^2 + a*c + b)/(d*x^2 + c))^(5/2) - 8*(6*a^3*b*c^2 + 12*a^2*
b^2*c + 5*a*b^3)*((a*d*x^2 + a*c + b)/(d*x^2 + c))^(3/2) + 3*(8*a^4*b*c^2 + 20*a^3*b^2*c + 11*a^2*b^3)*sqrt((a
*d*x^2 + a*c + b)/(d*x^2 + c)))/(a^6*d^3 - 3*(a*d*x^2 + a*c + b)*a^5*d^3/(d*x^2 + c) + 3*(a*d*x^2 + a*c + b)^2
*a^4*d^3/(d*x^2 + c)^2 - (a*d*x^2 + a*c + b)^3*a^3*d^3/(d*x^2 + c)^3) + 1/32*(8*a^2*c^2 + 12*a*b*c + 5*b^2)*b*
log(-(sqrt(a) - sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/(sqrt(a) + sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))))/(a^(
7/2)*d^3)

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Fricas [A]
time = 0.37, size = 425, normalized size = 1.89 \begin {gather*} \left [\frac {3 \, {\left (8 \, a^{2} b c^{2} + 12 \, a b^{2} c + 5 \, b^{3}\right )} \sqrt {a} \log \left (8 \, a^{2} d^{2} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (2 \, a^{2} c + a b\right )} d x^{2} + 8 \, a b c + b^{2} - 4 \, {\left (2 \, a d^{2} x^{4} + {\left (4 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + b c\right )} \sqrt {a} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}\right ) + 4 \, {\left (8 \, a^{3} d^{3} x^{6} - 10 \, a^{2} b d^{2} x^{4} + 8 \, a^{3} c^{3} + 26 \, a^{2} b c^{2} + 15 \, a b^{2} c + {\left (16 \, a^{2} b c + 15 \, a b^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{192 \, a^{4} d^{3}}, \frac {3 \, {\left (8 \, a^{2} b c^{2} + 12 \, a b^{2} c + 5 \, b^{3}\right )} \sqrt {-a} \arctan \left (\frac {{\left (2 \, a d x^{2} + 2 \, a c + b\right )} \sqrt {-a} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{2 \, {\left (a^{2} d x^{2} + a^{2} c + a b\right )}}\right ) + 2 \, {\left (8 \, a^{3} d^{3} x^{6} - 10 \, a^{2} b d^{2} x^{4} + 8 \, a^{3} c^{3} + 26 \, a^{2} b c^{2} + 15 \, a b^{2} c + {\left (16 \, a^{2} b c + 15 \, a b^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{96 \, a^{4} d^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a+b/(d*x^2+c))^(1/2),x, algorithm="fricas")

[Out]

[1/192*(3*(8*a^2*b*c^2 + 12*a*b^2*c + 5*b^3)*sqrt(a)*log(8*a^2*d^2*x^4 + 8*a^2*c^2 + 8*(2*a^2*c + a*b)*d*x^2 +
8*a*b*c + b^2 - 4*(2*a*d^2*x^4 + (4*a*c + b)*d*x^2 + 2*a*c^2 + b*c)*sqrt(a)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 +
c))) + 4*(8*a^3*d^3*x^6 - 10*a^2*b*d^2*x^4 + 8*a^3*c^3 + 26*a^2*b*c^2 + 15*a*b^2*c + (16*a^2*b*c + 15*a*b^2)*
d*x^2)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/(a^4*d^3), 1/96*(3*(8*a^2*b*c^2 + 12*a*b^2*c + 5*b^3)*sqrt(-a)*a
rctan(1/2*(2*a*d*x^2 + 2*a*c + b)*sqrt(-a)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))/(a^2*d*x^2 + a^2*c + a*b)) +
2*(8*a^3*d^3*x^6 - 10*a^2*b*d^2*x^4 + 8*a^3*c^3 + 26*a^2*b*c^2 + 15*a*b^2*c + (16*a^2*b*c + 15*a*b^2)*d*x^2)*s
qrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/(a^4*d^3)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{5}}{\sqrt {\frac {a c + a d x^{2} + b}{c + d x^{2}}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(a+b/(d*x**2+c))**(1/2),x)

[Out]

Integral(x**5/sqrt((a*c + a*d*x**2 + b)/(c + d*x**2)), x)

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Giac [A]
time = 3.41, size = 226, normalized size = 1.00 \begin {gather*} \frac {2 \, \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c} {\left (2 \, x^{2} {\left (\frac {4 \, x^{2}}{a d} - \frac {4 \, a^{2} c d^{3} + 5 \, a b d^{3}}{a^{3} d^{5}}\right )} + \frac {8 \, a^{2} c^{2} d^{2} + 26 \, a b c d^{2} + 15 \, b^{2} d^{2}}{a^{3} d^{5}}\right )} + \frac {3 \, {\left (8 \, a^{2} b c^{2} + 12 \, a b^{2} c + 5 \, b^{3}\right )} \log \left ({\left | -2 \, a c d - 2 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} \sqrt {a} {\left | d \right |} - b d \right |}\right )}{a^{\frac {7}{2}} d^{2} {\left | d \right |}}}{96 \, \mathrm {sgn}\left (d x^{2} + c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a+b/(d*x^2+c))^(1/2),x, algorithm="giac")

[Out]

1/96*(2*sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c)*(2*x^2*(4*x^2/(a*d) - (4*a^2*c*d^3 + 5*a*b*d^3)/
(a^3*d^5)) + (8*a^2*c^2*d^2 + 26*a*b*c*d^2 + 15*b^2*d^2)/(a^3*d^5)) + 3*(8*a^2*b*c^2 + 12*a*b^2*c + 5*b^3)*log
(abs(-2*a*c*d - 2*(sqrt(a*d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))*sqrt(a)*abs(d) - b
*d))/(a^(7/2)*d^2*abs(d)))/sgn(d*x^2 + c)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^5}{\sqrt {a+\frac {b}{d\,x^2+c}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a + b/(c + d*x^2))^(1/2),x)

[Out]

int(x^5/(a + b/(c + d*x^2))^(1/2), x)

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