\(\int \frac {x^3 (a+b x^2)^{5/2}}{\sqrt {c+d x^2}} \, dx\) [955]

Optimal result

Integrand size = 26, antiderivative size = 237 \[ \int \frac {x^3 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=-\frac {5 (b c-a d)^2 (7 b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{128 b d^4}+\frac {5 (b c-a d) (7 b c+a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{192 b d^3}-\frac {(7 b c+a d) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{48 b d^2}+\frac {\left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{8 b d}+\frac {5 (b c-a d)^3 (7 b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{128 b^{3/2} d^{9/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {457, 81, 52, 65, 223, 212} \[ \int \frac {x^3 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\frac {5 (b c-a d)^3 (a d+7 b c) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{128 b^{3/2} d^{9/2}}-\frac {5 \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)^2 (a d+7 b c)}{128 b d^4}+\frac {5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d) (a d+7 b c)}{192 b d^3}-\frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} (a d+7 b c)}{48 b d^2}+\frac {\left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{8 b d} \]

[In]

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

[Out]

(-5*(b*c - a*d)^2*(7*b*c + a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(128*b*d^4) + (5*(b*c - a*d)*(7*b*c + a*d)*(a
+ b*x^2)^(3/2)*Sqrt[c + d*x^2])/(192*b*d^3) - ((7*b*c + a*d)*(a + b*x^2)^(5/2)*Sqrt[c + d*x^2])/(48*b*d^2) +
((a + b*x^2)^(7/2)*Sqrt[c + d*x^2])/(8*b*d) + (5*(b*c - a*d)^3*(7*b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x^2])
/(Sqrt[b]*Sqrt[c + d*x^2])])/(128*b^(3/2)*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 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 (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx,x,x^2\right ) \\ & = \frac {\left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{8 b d}-\frac {(7 b c+a d) \text {Subst}\left (\int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}} \, dx,x,x^2\right )}{16 b d} \\ & = -\frac {(7 b c+a d) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{48 b d^2}+\frac {\left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{8 b d}+\frac {(5 (b c-a d) (7 b c+a d)) \text {Subst}\left (\int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx,x,x^2\right )}{96 b d^2} \\ & = \frac {5 (b c-a d) (7 b c+a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{192 b d^3}-\frac {(7 b c+a d) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{48 b d^2}+\frac {\left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{8 b d}-\frac {\left (5 (b c-a d)^2 (7 b c+a d)\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx,x,x^2\right )}{128 b d^3} \\ & = -\frac {5 (b c-a d)^2 (7 b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{128 b d^4}+\frac {5 (b c-a d) (7 b c+a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{192 b d^3}-\frac {(7 b c+a d) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{48 b d^2}+\frac {\left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{8 b d}+\frac {\left (5 (b c-a d)^3 (7 b c+a d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )}{256 b d^4} \\ & = -\frac {5 (b c-a d)^2 (7 b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{128 b d^4}+\frac {5 (b c-a d) (7 b c+a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{192 b d^3}-\frac {(7 b c+a d) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{48 b d^2}+\frac {\left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{8 b d}+\frac {\left (5 (b c-a d)^3 (7 b c+a d)\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^2 d^4} \\ & = -\frac {5 (b c-a d)^2 (7 b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{128 b d^4}+\frac {5 (b c-a d) (7 b c+a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{192 b d^3}-\frac {(7 b c+a d) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{48 b d^2}+\frac {\left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{8 b d}+\frac {\left (5 (b c-a d)^3 (7 b c+a d)\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^2 d^4} \\ & = -\frac {5 (b c-a d)^2 (7 b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{128 b d^4}+\frac {5 (b c-a d) (7 b c+a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{192 b d^3}-\frac {(7 b c+a d) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{48 b d^2}+\frac {\left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{8 b d}+\frac {5 (b c-a d)^3 (7 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{128 b^{3/2} d^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.22 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.90 \[ \int \frac {x^3 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\frac {b \sqrt {d} \sqrt {a+b x^2} \left (c+d x^2\right ) \left (15 a^3 d^3+a^2 b d^2 \left (-191 c+118 d x^2\right )+a b^2 d \left (265 c^2-172 c d x^2+136 d^2 x^4\right )+b^3 \left (-105 c^3+70 c^2 d x^2-56 c d^2 x^4+48 d^3 x^6\right )\right )+15 (b c-a d)^{7/2} (7 b c+a d) \sqrt {\frac {b \left (c+d x^2\right )}{b c-a d}} \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b c-a d}}\right )}{384 b^2 d^{9/2} \sqrt {c+d x^2}} \]

[In]

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

[Out]

(b*Sqrt[d]*Sqrt[a + b*x^2]*(c + d*x^2)*(15*a^3*d^3 + a^2*b*d^2*(-191*c + 118*d*x^2) + a*b^2*d*(265*c^2 - 172*c
*d*x^2 + 136*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)) + 15*(b*c - a*d)^(7/2)*(7*b
*c + a*d)*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^2*d^(9/
2)*Sqrt[c + d*x^2])

Maple [A] (verified)

Time = 3.15 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.19

method	result	size
risch	\(\frac {\left (48 b^{3} d^{3} x^{6}+136 a \,b^{2} d^{3} x^{4}-56 b^{3} c \,d^{2} x^{4}+118 x^{2} a^{2} b \,d^{3}-172 x^{2} a \,b^{2} c \,d^{2}+70 x^{2} b^{3} c^{2} d +15 a^{3} d^{3}-191 a^{2} b c \,d^{2}+265 a \,b^{2} c^{2} d -105 b^{3} c^{3}\right ) \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{384 b \,d^{4}}-\frac {5 \left (a^{4} d^{4}+4 a^{3} b c \,d^{3}-18 a^{2} b^{2} c^{2} d^{2}+20 a \,b^{3} c^{3} d -7 b^{4} c^{4}\right ) \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{256 b \,d^{4} \sqrt {b d}\, \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\)	\(281\)
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}-272 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}-236 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a^{2} b \,d^{3} x^{2}+344 \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}+15 \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}+60 \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}-270 \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}+300 \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}-30 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a^{3} d^{3}+382 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a^{2} b c \,d^{2}-530 \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 \,d^{4} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}}\)	\(658\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {59 \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, x^{2} a^{2}}{192 d}+\frac {5 \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a^{3}}{128 b d}-\frac {191 \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a^{2} c}{384 d^{2}}-\frac {35 b^{2} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, c^{3}}{128 d^{4}}-\frac {5 \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) a^{4}}{256 b \sqrt {b d}}+\frac {b^{2} x^{6} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{8 d}+\frac {17 b \,x^{4} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a}{48 d}-\frac {7 b^{2} x^{4} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, c}{48 d^{2}}+\frac {35 b^{2} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, x^{2} c^{2}}{192 d^{3}}+\frac {265 b \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a \,c^{2}}{384 d^{3}}-\frac {5 a^{3} c \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right )}{64 d \sqrt {b d}}+\frac {35 b^{3} \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) c^{4}}{256 d^{4} \sqrt {b d}}-\frac {43 b \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, x^{2} a c}{96 d^{2}}+\frac {45 b \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) a^{2} c^{2}}{128 d^{2} \sqrt {b d}}-\frac {25 b^{2} a \,c^{3} \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right )}{64 d^{3} \sqrt {b d}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\)	\(685\)

[In]

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

[Out]

1/384/b*(48*b^3*d^3*x^6+136*a*b^2*d^3*x^4-56*b^3*c*d^2*x^4+118*a^2*b*d^3*x^2-172*a*b^2*c*d^2*x^2+70*b^3*c^2*d*
x^2+15*a^3*d^3-191*a^2*b*c*d^2+265*a*b^2*c^2*d-105*b^3*c^3)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d^4-5/256/b*(a^4*d
^4+4*a^3*b*c*d^3-18*a^2*b^2*c^2*d^2+20*a*b^3*c^3*d-7*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 = 0.31 (sec) , antiderivative size = 574, normalized size of antiderivative = 2.42 \[ \int \frac {x^3 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\left [-\frac {15 \, {\left (7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} - 4 \, {\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {b d}\right ) - 4 \, {\left (48 \, b^{4} d^{4} x^{6} - 105 \, b^{4} c^{3} d + 265 \, a b^{3} c^{2} d^{2} - 191 \, a^{2} b^{2} c d^{3} + 15 \, a^{3} b d^{4} - 8 \, {\left (7 \, b^{4} c d^{3} - 17 \, a b^{3} d^{4}\right )} x^{4} + 2 \, {\left (35 \, b^{4} c^{2} d^{2} - 86 \, a b^{3} c d^{3} + 59 \, a^{2} b^{2} d^{4}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{1536 \, b^{2} d^{5}}, -\frac {15 \, {\left (7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-b d}}{2 \, {\left (b^{2} d^{2} x^{4} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )}}\right ) - 2 \, {\left (48 \, b^{4} d^{4} x^{6} - 105 \, b^{4} c^{3} d + 265 \, a b^{3} c^{2} d^{2} - 191 \, a^{2} b^{2} c d^{3} + 15 \, a^{3} b d^{4} - 8 \, {\left (7 \, b^{4} c d^{3} - 17 \, a b^{3} d^{4}\right )} x^{4} + 2 \, {\left (35 \, b^{4} c^{2} d^{2} - 86 \, a b^{3} c d^{3} + 59 \, a^{2} b^{2} d^{4}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{768 \, b^{2} d^{5}}\right ] \]

[In]

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

[Out]

[-1/1536*(15*(7*b^4*c^4 - 20*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^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 + 265*a*b^3*c^2*d^2 - 191*a^2*b^2*c*d^3 + 15*a
^3*b*d^4 - 8*(7*b^4*c*d^3 - 17*a*b^3*d^4)*x^4 + 2*(35*b^4*c^2*d^2 - 86*a*b^3*c*d^3 + 59*a^2*b^2*d^4)*x^2)*sqrt
(b*x^2 + a)*sqrt(d*x^2 + c))/(b^2*d^5), -1/768*(15*(7*b^4*c^4 - 20*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 4*a^3*b*
c*d^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 + 265*a*b^3*c^2*d^2 - 191*a
^2*b^2*c*d^3 + 15*a^3*b*d^4 - 8*(7*b^4*c*d^3 - 17*a*b^3*d^4)*x^4 + 2*(35*b^4*c^2*d^2 - 86*a*b^3*c*d^3 + 59*a^2
*b^2*d^4)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(b^2*d^5)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate(x^3*(b*x^2+a)^(5/2)/(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 = 0.32 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.28 \[ \int \frac {x^3 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\frac {{\left (\sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d} \sqrt {b x^{2} + a} {\left (2 \, {\left (b x^{2} + a\right )} {\left (4 \, {\left (b x^{2} + a\right )} {\left (\frac {6 \, {\left (b x^{2} + a\right )}}{b^{2} d} - \frac {7 \, b^{3} c d^{5} + a b^{2} d^{6}}{b^{4} d^{7}}\right )} + \frac {5 \, {\left (7 \, b^{4} c^{2} d^{4} - 6 \, a b^{3} c d^{5} - a^{2} b^{2} d^{6}\right )}}{b^{4} d^{7}}\right )} - \frac {15 \, {\left (7 \, b^{5} c^{3} d^{3} - 13 \, a b^{4} c^{2} d^{4} + 5 \, a^{2} b^{3} c d^{5} + a^{3} b^{2} d^{6}\right )}}{b^{4} d^{7}}\right )} - \frac {15 \, {\left (7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} \log \left ({\left | -\sqrt {b x^{2} + a} \sqrt {b d} + \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b d^{4}}\right )} b}{384 \, {\left | b \right |}} \]

[In]

integrate(x^3*(b*x^2+a)^(5/2)/(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^2
*d) - (7*b^3*c*d^5 + a*b^2*d^6)/(b^4*d^7)) + 5*(7*b^4*c^2*d^4 - 6*a*b^3*c*d^5 - a^2*b^2*d^6)/(b^4*d^7)) - 15*(
7*b^5*c^3*d^3 - 13*a*b^4*c^2*d^4 + 5*a^2*b^3*c*d^5 + a^3*b^2*d^6)/(b^4*d^7)) - 15*(7*b^4*c^4 - 20*a*b^3*c^3*d
+ 18*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^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*d^4))*b/abs(b)

Mupad [F(-1)]

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

[In]

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

[Out]

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