\(\int \frac {x^5 (a+b x^2)^{3/2}}{\sqrt {c+d x^2}} \, dx\) [944]

Optimal result

Integrand size = 26, antiderivative size = 276 \[ \int \frac {x^5 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=-\frac {(b c-a d) \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{128 b^2 d^4}+\frac {\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{192 b^2 d^3}-\frac {(7 b c+3 a d) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{48 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{8 b d}+\frac {(b c-a d)^2 \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{128 b^{5/2} d^{9/2}} \]

[Out]

1/128*(-a*d+b*c)^2*(3*a^2*d^2+10*a*b*c*d+35*b^2*c^2)*arctanh(d^(1/2)*(b*x^2+a)^(1/2)/b^(1/2)/(d*x^2+c)^(1/2))/
b^(5/2)/d^(9/2)+1/192*(3*a^2*d^2+10*a*b*c*d+35*b^2*c^2)*(b*x^2+a)^(3/2)*(d*x^2+c)^(1/2)/b^2/d^3-1/48*(3*a*d+7*
b*c)*(b*x^2+a)^(5/2)*(d*x^2+c)^(1/2)/b^2/d^2+1/8*x^2*(b*x^2+a)^(5/2)*(d*x^2+c)^(1/2)/b/d-1/128*(-a*d+b*c)*(3*a
^2*d^2+10*a*b*c*d+35*b^2*c^2)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b^2/d^4

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {457, 92, 81, 52, 65, 223, 212} \[ \int \frac {x^5 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\frac {(b c-a d)^2 \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{128 b^{5/2} d^{9/2}}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d) \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right )}{128 b^2 d^4}+\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right )}{192 b^2 d^3}-\frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} (3 a d+7 b c)}{48 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{8 b d} \]

[In]

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

[Out]

-1/128*((b*c - a*d)*(35*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(b^2*d^4) + ((35*b^
2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(192*b^2*d^3) - ((7*b*c + 3*a*d)*(a + b*x^2
)^(5/2)*Sqrt[c + d*x^2])/(48*b^2*d^2) + (x^2*(a + b*x^2)^(5/2)*Sqrt[c + d*x^2])/(8*b*d) + ((b*c - a*d)^2*(35*b
^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x^2])/(Sqrt[b]*Sqrt[c + d*x^2])])/(128*b^(5/2)*d^
(9/2))

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 81

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 92

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a + b*x
)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 212

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

Rule 223

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 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 {x^2 (a+b x)^{3/2}}{\sqrt {c+d x}} \, dx,x,x^2\right ) \\ & = \frac {x^2 \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{8 b d}+\frac {\text {Subst}\left (\int \frac {(a+b x)^{3/2} \left (-a c-\frac {1}{2} (7 b c+3 a d) x\right )}{\sqrt {c+d x}} \, dx,x,x^2\right )}{8 b d} \\ & = -\frac {(7 b c+3 a d) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{48 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{8 b d}+\frac {\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \text {Subst}\left (\int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx,x,x^2\right )}{96 b^2 d^2} \\ & = \frac {\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{192 b^2 d^3}-\frac {(7 b c+3 a d) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{48 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{8 b d}-\frac {\left ((b c-a d) \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx,x,x^2\right )}{128 b^2 d^3} \\ & = -\frac {(b c-a d) \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{128 b^2 d^4}+\frac {\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{192 b^2 d^3}-\frac {(7 b c+3 a d) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{48 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{8 b d}+\frac {\left ((b c-a d)^2 \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )}{256 b^2 d^4} \\ & = -\frac {(b c-a d) \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{128 b^2 d^4}+\frac {\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{192 b^2 d^3}-\frac {(7 b c+3 a d) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{48 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{8 b d}+\frac {\left ((b c-a d)^2 \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x^2}\right )}{128 b^3 d^4} \\ & = -\frac {(b c-a d) \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{128 b^2 d^4}+\frac {\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{192 b^2 d^3}-\frac {(7 b c+3 a d) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{48 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{8 b d}+\frac {\left ((b c-a d)^2 \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2}}\right )}{128 b^3 d^4} \\ & = -\frac {(b c-a d) \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{128 b^2 d^4}+\frac {\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{192 b^2 d^3}-\frac {(7 b c+3 a d) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{48 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{8 b d}+\frac {(b c-a d)^2 \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{128 b^{5/2} d^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.90 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.84 \[ \int \frac {x^5 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\frac {-b \sqrt {d} \sqrt {a+b x^2} \left (c+d x^2\right ) \left (9 a^3 d^3+3 a^2 b d^2 \left (5 c-2 d x^2\right )+a b^2 d \left (-145 c^2+92 c d x^2-72 d^2 x^4\right )+b^3 \left (105 c^3-70 c^2 d x^2+56 c d^2 x^4-48 d^3 x^6\right )\right )+3 (b c-a d)^{5/2} \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \sqrt {\frac {b \left (c+d x^2\right )}{b c-a d}} \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b c-a d}}\right )}{384 b^3 d^{9/2} \sqrt {c+d x^2}} \]

[In]

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

[Out]

(-(b*Sqrt[d]*Sqrt[a + b*x^2]*(c + d*x^2)*(9*a^3*d^3 + 3*a^2*b*d^2*(5*c - 2*d*x^2) + a*b^2*d*(-145*c^2 + 92*c*d
*x^2 - 72*d^2*x^4) + b^3*(105*c^3 - 70*c^2*d*x^2 + 56*c*d^2*x^4 - 48*d^3*x^6))) + 3*(b*c - a*d)^(5/2)*(35*b^2*
c^2 + 10*a*b*c*d + 3*a^2*d^2)*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a
*d]])/(384*b^3*d^(9/2)*Sqrt[c + d*x^2])

Maple [A] (verified)

Time = 3.17 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.02

method	result	size
risch	\(-\frac {\left (-48 b^{3} d^{3} x^{6}-72 a \,b^{2} d^{3} x^{4}+56 b^{3} c \,d^{2} x^{4}-6 x^{2} a^{2} b \,d^{3}+92 x^{2} a \,b^{2} c \,d^{2}-70 x^{2} b^{3} c^{2} d +9 a^{3} d^{3}+15 a^{2} b c \,d^{2}-145 a \,b^{2} c^{2} d +105 b^{3} c^{3}\right ) \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{384 b^{2} d^{4}}+\frac {\left (3 a^{4} d^{4}+4 a^{3} b c \,d^{3}+18 a^{2} b^{2} c^{2} d^{2}-60 a \,b^{3} c^{3} d +35 b^{4} c^{4}\right ) \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{256 b^{2} d^{4} \sqrt {b d}\, \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\)	\(282\)
default	\(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (96 b^{3} d^{3} x^{6} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+144 a \,b^{2} d^{3} x^{4} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}-112 b^{3} c \,d^{2} x^{4} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+12 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a^{2} b \,d^{3} x^{2}-184 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a \,b^{2} c \,d^{2} x^{2}+140 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, b^{3} c^{2} d \,x^{2}+9 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{4} d^{4}+12 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} b c \,d^{3}+54 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b^{2} c^{2} d^{2}-180 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{3} c^{3} d +105 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{4} c^{4}-18 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a^{3} d^{3}-30 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a^{2} b c \,d^{2}+290 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a \,b^{2} c^{2} d -210 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, b^{3} c^{3}\right )}{768 b^{2} d^{4} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}}\)	\(658\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {b \,x^{6} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{8 d}+\frac {3 x^{4} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a}{16 d}-\frac {3 \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a^{3}}{128 b^{2} d}+\frac {145 \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a \,c^{2}}{384 d^{3}}-\frac {35 b \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, c^{3}}{128 d^{4}}+\frac {3 \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) a^{4}}{256 b^{2} \sqrt {b d}}-\frac {7 b \,x^{4} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, c}{48 d^{2}}+\frac {\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, x^{2} a^{2}}{64 b d}-\frac {23 \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, x^{2} a c}{96 d^{2}}+\frac {35 b \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, x^{2} c^{2}}{192 d^{3}}-\frac {5 \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a^{2} c}{128 b \,d^{2}}+\frac {9 \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) a^{2} c^{2}}{128 d^{2} \sqrt {b d}}+\frac {35 b^{2} \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) c^{4}}{256 d^{4} \sqrt {b d}}+\frac {a^{3} c \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right )}{64 b d \sqrt {b d}}-\frac {15 b a \,c^{3} \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right )}{64 d^{3} \sqrt {b d}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\)	\(680\)

[In]

int(x^5*(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/384/b^2*(-48*b^3*d^3*x^6-72*a*b^2*d^3*x^4+56*b^3*c*d^2*x^4-6*a^2*b*d^3*x^2+92*a*b^2*c*d^2*x^2-70*b^3*c^2*d*
x^2+9*a^3*d^3+15*a^2*b*c*d^2-145*a*b^2*c^2*d+105*b^3*c^3)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d^4+1/256/b^2*(3*a^4
*d^4+4*a^3*b*c*d^3+18*a^2*b^2*c^2*d^2-60*a*b^3*c^3*d+35*b^4*c^4)/d^4*ln((1/2*a*d+1/2*b*c+b*d*x^2)/(b*d)^(1/2)+
(b*d*x^4+(a*d+b*c)*x^2+a*c)^(1/2))/(b*d)^(1/2)*((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 574, normalized size of antiderivative = 2.08 \[ \int \frac {x^5 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\left [\frac {3 \, {\left (35 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 3 \, a^{4} d^{4}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {b d}\right ) + 4 \, {\left (48 \, b^{4} d^{4} x^{6} - 105 \, b^{4} c^{3} d + 145 \, a b^{3} c^{2} d^{2} - 15 \, a^{2} b^{2} c d^{3} - 9 \, a^{3} b d^{4} - 8 \, {\left (7 \, b^{4} c d^{3} - 9 \, a b^{3} d^{4}\right )} x^{4} + 2 \, {\left (35 \, b^{4} c^{2} d^{2} - 46 \, a b^{3} c d^{3} + 3 \, a^{2} b^{2} d^{4}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{1536 \, b^{3} d^{5}}, -\frac {3 \, {\left (35 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 3 \, a^{4} d^{4}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-b d}}{2 \, {\left (b^{2} d^{2} x^{4} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )}}\right ) - 2 \, {\left (48 \, b^{4} d^{4} x^{6} - 105 \, b^{4} c^{3} d + 145 \, a b^{3} c^{2} d^{2} - 15 \, a^{2} b^{2} c d^{3} - 9 \, a^{3} b d^{4} - 8 \, {\left (7 \, b^{4} c d^{3} - 9 \, a b^{3} d^{4}\right )} x^{4} + 2 \, {\left (35 \, b^{4} c^{2} d^{2} - 46 \, a b^{3} c d^{3} + 3 \, a^{2} b^{2} d^{4}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{768 \, b^{3} d^{5}}\right ] \]

[In]

integrate(x^5*(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

[1/1536*(3*(35*b^4*c^4 - 60*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 + 3*a^4*d^4)*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*x^2 + b*c + a*d)*sqrt(b*x^2 + a
)*sqrt(d*x^2 + c)*sqrt(b*d)) + 4*(48*b^4*d^4*x^6 - 105*b^4*c^3*d + 145*a*b^3*c^2*d^2 - 15*a^2*b^2*c*d^3 - 9*a^
3*b*d^4 - 8*(7*b^4*c*d^3 - 9*a*b^3*d^4)*x^4 + 2*(35*b^4*c^2*d^2 - 46*a*b^3*c*d^3 + 3*a^2*b^2*d^4)*x^2)*sqrt(b*
x^2 + a)*sqrt(d*x^2 + c))/(b^3*d^5), -1/768*(3*(35*b^4*c^4 - 60*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d
^3 + 3*a^4*d^4)*sqrt(-b*d)*arctan(1/2*(2*b*d*x^2 + b*c + a*d)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(-b*d)/(b^2*
d^2*x^4 + a*b*c*d + (b^2*c*d + a*b*d^2)*x^2)) - 2*(48*b^4*d^4*x^6 - 105*b^4*c^3*d + 145*a*b^3*c^2*d^2 - 15*a^2
*b^2*c*d^3 - 9*a^3*b*d^4 - 8*(7*b^4*c*d^3 - 9*a*b^3*d^4)*x^4 + 2*(35*b^4*c^2*d^2 - 46*a*b^3*c*d^3 + 3*a^2*b^2*
d^4)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(b^3*d^5)]

Sympy [F]

\[ \int \frac {x^5 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\int \frac {x^{5} \left (a + b x^{2}\right )^{\frac {3}{2}}}{\sqrt {c + d x^{2}}}\, dx \]

[In]

integrate(x**5*(b*x**2+a)**(3/2)/(d*x**2+c)**(1/2),x)

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^5 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(x^5*(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more detail

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.11 \[ \int \frac {x^5 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\frac {{\left (\sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d} \sqrt {b x^{2} + a} {\left (2 \, {\left (b x^{2} + a\right )} {\left (4 \, {\left (b x^{2} + a\right )} {\left (\frac {6 \, {\left (b x^{2} + a\right )}}{b^{3} d} - \frac {7 \, b^{7} c d^{5} + 9 \, a b^{6} d^{6}}{b^{9} d^{7}}\right )} + \frac {35 \, b^{8} c^{2} d^{4} + 10 \, a b^{7} c d^{5} + 3 \, a^{2} b^{6} d^{6}}{b^{9} d^{7}}\right )} - \frac {3 \, {\left (35 \, b^{9} c^{3} d^{3} - 25 \, a b^{8} c^{2} d^{4} - 7 \, a^{2} b^{7} c d^{5} - 3 \, a^{3} b^{6} d^{6}\right )}}{b^{9} d^{7}}\right )} - \frac {3 \, {\left (35 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 3 \, a^{4} d^{4}\right )} \log \left ({\left | -\sqrt {b x^{2} + a} \sqrt {b d} + \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b^{2} d^{4}}\right )} b}{384 \, {\left | b \right |}} \]

[In]

integrate(x^5*(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="giac")

[Out]

1/384*(sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d)*sqrt(b*x^2 + a)*(2*(b*x^2 + a)*(4*(b*x^2 + a)*(6*(b*x^2 + a)/(b^3
*d) - (7*b^7*c*d^5 + 9*a*b^6*d^6)/(b^9*d^7)) + (35*b^8*c^2*d^4 + 10*a*b^7*c*d^5 + 3*a^2*b^6*d^6)/(b^9*d^7)) -
3*(35*b^9*c^3*d^3 - 25*a*b^8*c^2*d^4 - 7*a^2*b^7*c*d^5 - 3*a^3*b^6*d^6)/(b^9*d^7)) - 3*(35*b^4*c^4 - 60*a*b^3*
c^3*d + 18*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 + 3*a^4*d^4)*log(abs(-sqrt(b*x^2 + a)*sqrt(b*d) + sqrt(b^2*c + (b*x
^2 + a)*b*d - a*b*d)))/(sqrt(b*d)*b^2*d^4))*b/abs(b)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\int \frac {x^5\,{\left (b\,x^2+a\right )}^{3/2}}{\sqrt {d\,x^2+c}} \,d x \]

[In]

int((x^5*(a + b*x^2)^(3/2))/(c + d*x^2)^(1/2),x)

[Out]

int((x^5*(a + b*x^2)^(3/2))/(c + d*x^2)^(1/2), x)