3.4.31 \(\int x^5 (a+\frac {b}{c+d x^2})^{3/2} \, dx\) [331]

Optimal. Leaf size=249 \[ -\frac {b c^2 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{d^3}-\frac {\left (5 b^2+60 a b c-24 a^2 c^2\right ) \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{48 a d^3}-\frac {(b+12 a c) \left (c+d x^2\right )^2 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{24 d^3}+\frac {\left (c+d x^2\right )^3 \left (\frac {b+a c+a d x^2}{c+d x^2}\right )^{5/2}}{6 a d^3}-\frac {b \left (b^2+12 a b c-24 a^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {a}}\right )}{16 a^{3/2} d^3} \]

[Out]

1/6*(d*x^2+c)^3*((a*d*x^2+a*c+b)/(d*x^2+c))^(5/2)/a/d^3-1/16*b*(-24*a^2*c^2+12*a*b*c+b^2)*arctanh(((a*d*x^2+a*
c+b)/(d*x^2+c))^(1/2)/a^(1/2))/a^(3/2)/d^3-b*c^2*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/d^3-1/48*(-24*a^2*c^2+60*a*
b*c+5*b^2)*(d*x^2+c)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/a/d^3-1/24*(12*a*c+b)*(d*x^2+c)^2*((a*d*x^2+a*c+b)/(d*x
^2+c))^(1/2)/d^3

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Rubi [A]
time = 0.34, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {1985, 1981, 1980, 474, 466, 1171, 396, 214} \begin {gather*} -\frac {\left (-24 a^2 c^2+60 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}}}{48 a d^3}-\frac {b \left (-24 a^2 c^2+12 a b c+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^{3/2} d^3}-\frac {b c^2 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{d^3}+\frac {\left (c+d x^2\right )^3 \left (\frac {a c+a d x^2+b}{c+d x^2}\right )^{5/2}}{6 a d^3}-\frac {(12 a c+b) \left (c+d x^2\right )^2 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{24 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b/(c + d*x^2))^(3/2),x]

[Out]

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

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 396

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

Rule 466

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

Rule 474

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

Rule 1171

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1
)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

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

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Mathematica [A]
time = 0.25, size = 150, normalized size = 0.60 \begin {gather*} \frac {\sqrt {a} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (3 b^2 \left (c+d x^2\right )-2 a b \left (47 c^2+16 c d x^2-7 d^2 x^4\right )+8 a^2 \left (c^3+d^3 x^6\right )\right )-3 b \left (b^2+12 a b c-24 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^{3/2} d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(3*b^2*(c + d*x^2) - 2*a*b*(47*c^2 + 16*c*d*x^2 - 7*d^2*x^4) +
8*a^2*(c^3 + d^3*x^6)) - 3*b*(b^2 + 12*a*b*c - 24*a^2*c^2)*ArcTanh[Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]/Sqrt[
a]])/(48*a^(3/2)*d^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1017\) vs. \(2(227)=454\).
time = 0.13, size = 1018, normalized size = 4.09

method	result	size
risch	\(\frac {\left (8 d^{2} a^{2} x^{4}-8 a^{2} c d \,x^{2}+14 a b d \,x^{2}+8 a^{2} c^{2}-46 a b c +3 b^{2}\right ) \left (d \,x^{2}+c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{48 d^{3} a}+\frac {\left (\frac {3 b a \ln \left (\frac {a c d +\frac {1}{2} b d +a \,d^{2} x^{2}}{\sqrt {a \,d^{2}}}+\sqrt {c^{2} a +b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}\right ) c^{2}}{4 d^{2} \sqrt {a \,d^{2}}}-\frac {3 b^{2} \ln \left (\frac {a c d +\frac {1}{2} b d +a \,d^{2} x^{2}}{\sqrt {a \,d^{2}}}+\sqrt {c^{2} a +b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}\right ) c}{8 d^{2} \sqrt {a \,d^{2}}}-\frac {b^{3} \ln \left (\frac {a c d +\frac {1}{2} b d +a \,d^{2} x^{2}}{\sqrt {a \,d^{2}}}+\sqrt {c^{2} a +b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}\right )}{32 a \,d^{2} \sqrt {a \,d^{2}}}-\frac {b a \,c^{2} x^{2}}{d^{2} \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}}-\frac {b a \,c^{3}}{d^{3} \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}}-\frac {b^{2} c^{2}}{d^{3} \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}}\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}}{a d \,x^{2}+a c +b}\)	\(508\)
default	\(-\frac {\left (48 \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}}\, a^{2} c \,d^{2} x^{4}-12 \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}}\, a b \,d^{2} x^{4}-72 \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^{2} x^{2}+48 \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}}\, a^{2} c^{2} d \,x^{2}+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 ) a \,b^{2} c \,d^{2} x^{2}-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}} \sqrt {a \,d^{2}}\, a d \,x^{2}+96 \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}}\, a b c d \,x^{2}-72 \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^{3} d +3 \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^{2} x^{2}-6 \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^{2} d \,x^{2}+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 ) a \,b^{2} c^{2} d +96 \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, \sqrt {a \,d^{2}}\, a b \,c^{2}-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}} \sqrt {a \,d^{2}}\, a c +108 \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}}\, a b \,c^{2}+3 \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} c d -6 \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^{2} c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{96 d^{3} a \sqrt {a \,d^{2}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}}\)	\(1018\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/96*(48*(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)*a^2*c*d^2*x^4-12*(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)*a*b*d^2*x^4-72*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^2*x^2+48*(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)*a^2*c^2*d*x^2+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))*a*b^2*c*d^2*x^2-16*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)
^(3/2)*(a*d^2)^(1/2)*a*d*x^2+96*(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)*a*b*c*d*x^2-72*l
n(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^3*d+3*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^2*x^2-6*(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^2*d*x^2+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))*a*
b^2*c^2*d+96*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*(a*d^2)^(1/2)*a*b*c^2-16*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b
*c)^(3/2)*(a*d^2)^(1/2)*a*c+108*(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)*a*b*c^2+3*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*
c*d-6*(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^2*c)/d^3/a*((a*d*x^2+a*c+b)/(d*x^2+c))^(
1/2)/(a*d^2)^(1/2)/((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)

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Maxima [A]
time = 0.52, size = 368, normalized size = 1.48 \begin {gather*} -\frac {b c^{2} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{d^{3}} - \frac {3 \, {\left (8 \, a^{2} b c^{2} - 20 \, a b^{2} c + 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 - 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} - 12 \, a^{3} b^{2} c - a^{2} b^{3}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{48 \, {\left (a^{4} d^{3} - \frac {3 \, {\left (a d x^{2} + a c + b\right )} a^{3} d^{3}}{d x^{2} + c} + \frac {3 \, {\left (a d x^{2} + a c + b\right )}^{2} a^{2} d^{3}}{{\left (d x^{2} + c\right )}^{2}} - \frac {{\left (a d x^{2} + a c + b\right )}^{3} a d^{3}}{{\left (d x^{2} + c\right )}^{3}}\right )}} - \frac {{\left (24 \, a^{2} c^{2} - 12 \, a b c - 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 {3}{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))^(3/2),x, algorithm="maxima")

[Out]

-b*c^2*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))/d^3 - 1/48*(3*(8*a^2*b*c^2 - 20*a*b^2*c + 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 - a*b^3)*((a*d*x^2 + a*c + b)/(d*x^2 + c))^(3/2) + 3*(8*
a^4*b*c^2 - 12*a^3*b^2*c - a^2*b^3)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/(a^4*d^3 - 3*(a*d*x^2 + a*c + b)*a^
3*d^3/(d*x^2 + c) + 3*(a*d*x^2 + a*c + b)^2*a^2*d^3/(d*x^2 + c)^2 - (a*d*x^2 + a*c + b)^3*a*d^3/(d*x^2 + c)^3)
- 1/32*(24*a^2*c^2 - 12*a*b*c - 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^(3/2)*d^3)

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Fricas [A]
time = 0.42, size = 427, normalized size = 1.71 \begin {gather*} \left [\frac {3 \, {\left (24 \, a^{2} b c^{2} - 12 \, a b^{2} c - 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} + 14 \, a^{2} b d^{2} x^{4} + 8 \, a^{3} c^{3} - 94 \, a^{2} b c^{2} + 3 \, a b^{2} c - {\left (32 \, a^{2} b c - 3 \, a b^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{192 \, a^{2} d^{3}}, -\frac {3 \, {\left (24 \, a^{2} b c^{2} - 12 \, a b^{2} c - 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} + 14 \, a^{2} b d^{2} x^{4} + 8 \, a^{3} c^{3} - 94 \, a^{2} b c^{2} + 3 \, a b^{2} c - {\left (32 \, a^{2} b c - 3 \, a b^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{96 \, a^{2} 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))^(3/2),x, algorithm="fricas")

[Out]

[1/192*(3*(24*a^2*b*c^2 - 12*a*b^2*c - 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 + 14*a^2*b*d^2*x^4 + 8*a^3*c^3 - 94*a^2*b*c^2 + 3*a*b^2*c - (32*a^2*b*c - 3*a*b^2)*d*x
^2)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/(a^2*d^3), -1/96*(3*(24*a^2*b*c^2 - 12*a*b^2*c - b^3)*sqrt(-a)*arct
an(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 + 14*a^2*b*d^2*x^4 + 8*a^3*c^3 - 94*a^2*b*c^2 + 3*a*b^2*c - (32*a^2*b*c - 3*a*b^2)*d*x^2)*sqrt((
a*d*x^2 + a*c + b)/(d*x^2 + c)))/(a^2*d^3)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{5} \left (\frac {a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 527 vs. \(2 (229) = 458\).
time = 3.67, size = 527, normalized size = 2.12 \begin {gather*} \frac {1}{48} \, \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c} {\left (2 \, {\left (\frac {4 \, a x^{2} \mathrm {sgn}\left (d x^{2} + c\right )}{d} - \frac {4 \, a^{3} c d^{6} \mathrm {sgn}\left (d x^{2} + c\right ) - 7 \, a^{2} b d^{6} \mathrm {sgn}\left (d x^{2} + c\right )}{a^{2} d^{8}}\right )} x^{2} + \frac {8 \, a^{3} c^{2} d^{5} \mathrm {sgn}\left (d x^{2} + c\right ) - 46 \, a^{2} b c d^{5} \mathrm {sgn}\left (d x^{2} + c\right ) + 3 \, a b^{2} d^{5} \mathrm {sgn}\left (d x^{2} + c\right )}{a^{2} d^{8}}\right )} - \frac {{\left (24 \, a^{\frac {5}{2}} b c^{2} \mathrm {sgn}\left (d x^{2} + c\right ) - 12 \, a^{\frac {3}{2}} b^{2} c \mathrm {sgn}\left (d x^{2} + c\right ) - \sqrt {a} b^{3} \mathrm {sgn}\left (d x^{2} + c\right )\right )} \log \left ({\left | -2 \, a^{\frac {5}{2}} c^{3} d - 6 \, {\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 )} a^{2} c^{2} {\left | d \right |} - 6 \, {\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 )}^{2} a^{\frac {3}{2}} c d - a^{\frac {3}{2}} b c^{2} 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 )}^{3} a {\left | d \right |} - 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 )} a b c {\left | d \right |} - {\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 )}^{2} \sqrt {a} b d \right |}\right )}{96 \, a^{2} d^{2} {\left | d \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c)*(2*(4*a*x^2*sgn(d*x^2 + c)/d - (4*a^3*c*d^6*sgn(d*x
^2 + c) - 7*a^2*b*d^6*sgn(d*x^2 + c))/(a^2*d^8))*x^2 + (8*a^3*c^2*d^5*sgn(d*x^2 + c) - 46*a^2*b*c*d^5*sgn(d*x^
2 + c) + 3*a*b^2*d^5*sgn(d*x^2 + c))/(a^2*d^8)) - 1/96*(24*a^(5/2)*b*c^2*sgn(d*x^2 + c) - 12*a^(3/2)*b^2*c*sgn
(d*x^2 + c) - sqrt(a)*b^3*sgn(d*x^2 + c))*log(abs(-2*a^(5/2)*c^3*d - 6*(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))*a^2*c^2*abs(d) - 6*(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))^2*a^(3/2)*c*d - a^(3/2)*b*c^2*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))^3*a*abs(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))*a*b
*c*abs(d) - (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))^2*sqrt(a)*b*d))/(a^2*d^2
*abs(d))

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

[Out]

int(x^5*(a + b/(c + d*x^2))^(3/2), x)

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