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

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

[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 = 0.73, antiderivative size = 311, normalized size of antiderivative = 1.25, number of steps used = 10, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6722, 1975, 446, 89, 80, 50, 63, 217, 206} \[ -\frac {\left (-24 a^2 c^2+12 a b c+b^2\right ) \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}} \left (a \left (c+d x^2\right )+b\right )}{24 a b d^3}-\frac {\left (-24 a^2 c^2+12 a b c+b^2\right ) \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{16 a d^3}-\frac {b \left (-24 a^2 c^2+12 a b c+b^2\right ) \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x^2}}{\sqrt {a \left (c+d x^2\right )+b}}\right )}{16 a^{3/2} d^3 \sqrt {a \left (c+d x^2\right )+b}}-\frac {c^2 \sqrt {a+\frac {b}{c+d x^2}} \left (a \left (c+d x^2\right )+b\right )^2}{b d^3}+\frac {\left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}} \left (a \left (c+d x^2\right )+b\right )^2}{6 a d^3} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b/(c + d*x^2))^(3/2),x]

[Out]

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

Rule 50

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 63

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 80

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

Rule 89

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 446

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]]

Rule 1975

Int[(u_)^(p_.)*(v_)^(q_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*ExpandToSum[u, x]^p*ExpandToSum[v, x]^q
, x] /; FreeQ[{e, m, p, q}, x] && BinomialQ[{u, v}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0]
&&  !BinomialMatchQ[{u, v}, x]

Rule 6722

Int[(u_.)*((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[(a + b*v^n)^FracPart[p]/(v^(n*FracPart[p])*(b + a/
v^n)^FracPart[p]), Int[u*v^(n*p)*(b + a/v^n)^p, x], x] /; FreeQ[{a, b, p}, x] &&  !IntegerQ[p] && ILtQ[n, 0] &
& BinomialQ[v, x] &&  !LinearQ[v, 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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 0.41, size = 142, normalized size = 0.57 \[ \frac {\sqrt {a} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (8 a^2 \left (c^3+d^3 x^6\right )-2 a b \left (47 c^2+16 c d x^2-7 d^2 x^4\right )+3 b^2 \left (c+d x^2\right )\right )-3 b \left (-24 a^2 c^2+12 a b c+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{c+d x^2}}}{\sqrt {a}}\right )}{48 a^{3/2} d^3} \]

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[a + b/(c + d*x^2)]/Sqrt[a]])/(48*a^(3/
2)*d^3)

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fricas [A]  time = 0.99, size = 427, normalized size = 1.71 \[ \left [\frac {3 \, {\left (24 \, a^{2} b c^{2} - 12 \, a b^{2} c - b^{3}\right )} \sqrt {a} \log \left (8 \, a^{2} d^{2} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (2 \, a^{2} c + a b\right )} d x^{2} + 8 \, a b c + b^{2} + 4 \, {\left (2 \, a d^{2} x^{4} + {\left (4 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + b c\right )} \sqrt {a} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}\right ) + 4 \, {\left (8 \, a^{3} d^{3} x^{6} + 14 \, a^{2} b d^{2} x^{4} + 8 \, a^{3} c^{3} - 94 \, a^{2} b c^{2} + 3 \, a b^{2} c - {\left (32 \, a^{2} b c - 3 \, a b^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{192 \, a^{2} d^{3}}, -\frac {3 \, {\left (24 \, a^{2} b c^{2} - 12 \, a b^{2} c - b^{3}\right )} \sqrt {-a} \arctan \left (\frac {{\left (2 \, a d x^{2} + 2 \, a c + b\right )} \sqrt {-a} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{2 \, {\left (a^{2} d x^{2} + a^{2} c + a b\right )}}\right ) - 2 \, {\left (8 \, a^{3} d^{3} x^{6} + 14 \, a^{2} b d^{2} x^{4} + 8 \, a^{3} c^{3} - 94 \, a^{2} b c^{2} + 3 \, a b^{2} c - {\left (32 \, a^{2} b c - 3 \, a b^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{96 \, a^{2} d^{3}}\right ] \]

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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giac [B]  time = 2.10, size = 527, normalized size = 2.12 \[ \frac {1}{48} \, \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c} {\left (2 \, {\left (\frac {4 \, a x^{2} \mathrm {sgn}\left (d x^{2} + c\right )}{d} - \frac {4 \, a^{3} c d^{6} \mathrm {sgn}\left (d x^{2} + c\right ) - 7 \, a^{2} b d^{6} \mathrm {sgn}\left (d x^{2} + c\right )}{a^{2} d^{8}}\right )} x^{2} + \frac {8 \, a^{3} c^{2} d^{5} \mathrm {sgn}\left (d x^{2} + c\right ) - 46 \, a^{2} b c d^{5} \mathrm {sgn}\left (d x^{2} + c\right ) + 3 \, a b^{2} d^{5} \mathrm {sgn}\left (d x^{2} + c\right )}{a^{2} d^{8}}\right )} - \frac {{\left (24 \, a^{\frac {5}{2}} b c^{2} \mathrm {sgn}\left (d x^{2} + c\right ) - 12 \, a^{\frac {3}{2}} b^{2} c \mathrm {sgn}\left (d x^{2} + c\right ) - \sqrt {a} b^{3} \mathrm {sgn}\left (d x^{2} + c\right )\right )} \log \left ({\left | -2 \, a^{\frac {5}{2}} c^{3} d - 6 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} a^{2} c^{2} {\left | d \right |} - 6 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{2} a^{\frac {3}{2}} c d - a^{\frac {3}{2}} b c^{2} d - 2 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{3} a {\left | d \right |} - 2 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} a b c {\left | d \right |} - {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{2} \sqrt {a} b d \right |}\right )}{96 \, a^{2} d^{2} {\left | d \right |}} \]

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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maple [B]  time = 0.07, size = 1018, normalized size = 4.09 \[ -\frac {\left (-72 a^{2} b \,c^{2} d^{2} x^{2} \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +b d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}}{2 \sqrt {a \,d^{2}}}\right )+48 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, a^{2} c \,d^{2} x^{4}+36 a \,b^{2} c \,d^{2} x^{2} \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +b d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}}{2 \sqrt {a \,d^{2}}}\right )-72 a^{2} b \,c^{3} d \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +b d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}}{2 \sqrt {a \,d^{2}}}\right )-12 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, a b \,d^{2} x^{4}+3 b^{3} d^{2} x^{2} \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +b d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}}{2 \sqrt {a \,d^{2}}}\right )+48 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, a^{2} c^{2} d \,x^{2}+36 a \,b^{2} c^{2} d \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +b d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}}{2 \sqrt {a \,d^{2}}}\right )+96 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, a b c d \,x^{2}+3 b^{3} c d \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +b d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}}{2 \sqrt {a \,d^{2}}}\right )-6 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, b^{2} d \,x^{2}+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}+108 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \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}+a \,c^{2}+b c \right )^{\frac {3}{2}} \sqrt {a \,d^{2}}\, a d \,x^{2}-6 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, b^{2} c -16 \left (a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c \right )^{\frac {3}{2}} \sqrt {a \,d^{2}}\, a c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{96 \sqrt {a \,d^{2}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, a \,d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)*x^4*a^2*c*d^2-12*(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)*x^4*a*b*d^2-72*ln(1/2*(2*a*d^2*x^2+2*a*c*d+b*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))/(a*d^2)^(1/2))*x^2*a^2*b*c^2*d^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)*x^2*a^2*c^2*d+36*ln(1/2*(2*a*d^2*x^2+2*a*c*d+b*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))/(a*d^2)^(1/2))*x^2*a*b^2*c*d^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)*x^2*a*d+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)*x^2*a*b*c*d+3*ln
(1/2*(2*a*d^2*x^2+2*a*c*d+b*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))/(a*d^2)^(1/2))*
x^2*b^3*d^2-72*ln(1/2*(2*a*d^2*x^2+2*a*c*d+b*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)
)/(a*d^2)^(1/2))*a^2*b*c^3*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)*x^2*b^2*d+36*ln(1
/2*(2*a*d^2*x^2+2*a*c*d+b*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))/(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+b*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))/(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 = 1.71, size = 368, normalized size = 1.48 \[ -\frac {b c^{2} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{d^{3}} - \frac {3 \, {\left (8 \, a^{2} b c^{2} - 20 \, a b^{2} c + b^{3}\right )} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {5}{2}} - 8 \, {\left (6 \, a^{3} b c^{2} - 12 \, a^{2} b^{2} c - a b^{3}\right )} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {3}{2}} + 3 \, {\left (8 \, a^{4} b c^{2} - 12 \, a^{3} b^{2} c - a^{2} b^{3}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{48 \, {\left (a^{4} d^{3} - \frac {3 \, {\left (a d x^{2} + a c + b\right )} a^{3} d^{3}}{d x^{2} + c} + \frac {3 \, {\left (a d x^{2} + a c + b\right )}^{2} a^{2} d^{3}}{{\left (d x^{2} + c\right )}^{2}} - \frac {{\left (a d x^{2} + a c + b\right )}^{3} a d^{3}}{{\left (d x^{2} + c\right )}^{3}}\right )}} - \frac {{\left (24 \, a^{2} c^{2} - 12 \, a b c - b^{2}\right )} b \log \left (-\frac {\sqrt {a} - \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{\sqrt {a} + \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}\right )}{32 \, a^{\frac {3}{2}} d^{3}} \]

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

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

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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