\(\int \frac {x^5}{(a+b x^2)^{3/2} \sqrt {c+d x^2}} \, dx\) [980]

Optimal result

Integrand size = 26, antiderivative size = 129 \[ \int \frac {x^5}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=-\frac {a^2 \sqrt {c+d x^2}}{b^2 (b c-a d) \sqrt {a+b x^2}}+\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{2 b^2 d}-\frac {(b c+3 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{2 b^{5/2} d^{3/2}} \]

[Out]

-1/2*(3*a*d+b*c)*arctanh(d^(1/2)*(b*x^2+a)^(1/2)/b^(1/2)/(d*x^2+c)^(1/2))/b^(5/2)/d^(3/2)-a^2*(d*x^2+c)^(1/2)/
b^2/(-a*d+b*c)/(b*x^2+a)^(1/2)+1/2*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b^2/d

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {457, 91, 81, 65, 223, 212} \[ \int \frac {x^5}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=-\frac {a^2 \sqrt {c+d x^2}}{b^2 \sqrt {a+b x^2} (b c-a d)}-\frac {(3 a d+b c) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{2 b^{5/2} d^{3/2}}+\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{2 b^2 d} \]

[In]

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

[Out]

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

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 91

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

Mathematica [A] (verified)

Time = 1.77 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.19 \[ \int \frac {x^5}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {b} \sqrt {d} \sqrt {c+d x^2} \left (-3 a^2 d+b^2 c x^2+a b \left (c-d x^2\right )\right )-\left (b^2 c^2+2 a b c d-3 a^2 d^2\right ) \sqrt {a+b x^2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {d} \sqrt {a+b x^2}}\right )}{2 b^{5/2} d^{3/2} (b c-a d) \sqrt {a+b x^2}} \]

[In]

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

[Out]

(Sqrt[b]*Sqrt[d]*Sqrt[c + d*x^2]*(-3*a^2*d + b^2*c*x^2 + a*b*(c - d*x^2)) - (b^2*c^2 + 2*a*b*c*d - 3*a^2*d^2)*
Sqrt[a + b*x^2]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/(Sqrt[d]*Sqrt[a + b*x^2])])/(2*b^(5/2)*d^(3/2)*(b*c - a*d)*S
qrt[a + b*x^2])

Maple [A] (verified)

Time = 3.14 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.40

method	result	size
risch	\(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{2 b^{2} d}-\frac {\left (\frac {\left (3 a d +b c \right ) \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right )}{2 \sqrt {b d}}-\frac {2 a^{2} d \left (d \,x^{2}+c \right )}{\left (a d -b c \right ) \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{2 b^{2} d \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\)	\(180\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (-\frac {3 a \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right )}{4 b^{2} \sqrt {b d}}+\frac {\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{2 b^{2} d}-\frac {\ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) c}{4 b d \sqrt {b d}}-\frac {a^{2} \sqrt {b d \left (x^{2}+\frac {a}{b}\right )^{2}+\left (-a d +b c \right ) \left (x^{2}+\frac {a}{b}\right )}}{b^{3} \left (-a d +b c \right ) \left (x^{2}+\frac {a}{b}\right )}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\)	\(250\)
default	\(-\frac {\left (3 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b \,d^{2} x^{2}-2 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{2} c d \,x^{2}-\ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{3} c^{2} x^{2}-2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, b^{2} c \,x^{2}+3 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} d^{2}-2 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b c d -\ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{2} c^{2}-6 \sqrt {b d}\, \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, a^{2} d +2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a b c \right ) \sqrt {d \,x^{2}+c}}{4 b^{2} \sqrt {b \,x^{2}+a}\, \sqrt {b d}\, d \left (a d -b c \right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}\)	\(511\)

[In]

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

[Out]

1/2*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b^2/d-1/2/b^2/d*(1/2*(3*a*d+b*c)*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)-2*a^2*d*(d*x^2+c)/(a*d-b*c)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (105) = 210\).

Time = 0.32 (sec) , antiderivative size = 498, normalized size of antiderivative = 3.86 \[ \int \frac {x^5}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\left [\frac {{\left (a b^{2} c^{2} + 2 \, a^{2} b c d - 3 \, a^{3} d^{2} + {\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} - 4 \, {\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {b d}\right ) + 4 \, {\left (a b^{2} c d - 3 \, a^{2} b d^{2} + {\left (b^{3} c d - a b^{2} d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{8 \, {\left (a b^{4} c d^{2} - a^{2} b^{3} d^{3} + {\left (b^{5} c d^{2} - a b^{4} d^{3}\right )} x^{2}\right )}}, \frac {{\left (a b^{2} c^{2} + 2 \, a^{2} b c d - 3 \, a^{3} d^{2} + {\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-b d}}{2 \, {\left (b^{2} d^{2} x^{4} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )}}\right ) + 2 \, {\left (a b^{2} c d - 3 \, a^{2} b d^{2} + {\left (b^{3} c d - a b^{2} d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{4 \, {\left (a b^{4} c d^{2} - a^{2} b^{3} d^{3} + {\left (b^{5} c d^{2} - a b^{4} d^{3}\right )} x^{2}\right )}}\right ] \]

[In]

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

[Out]

[1/8*((a*b^2*c^2 + 2*a^2*b*c*d - 3*a^3*d^2 + (b^3*c^2 + 2*a*b^2*c*d - 3*a^2*b*d^2)*x^2)*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*(a*b^2*c*d - 3*a^2*b*d^2 + (b^3*c*d - a*b^2*d^2)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^
2 + c))/(a*b^4*c*d^2 - a^2*b^3*d^3 + (b^5*c*d^2 - a*b^4*d^3)*x^2), 1/4*((a*b^2*c^2 + 2*a^2*b*c*d - 3*a^3*d^2 +
(b^3*c^2 + 2*a*b^2*c*d - 3*a^2*b*d^2)*x^2)*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*(a*b^2*c*d - 3*a^2*b*d^2 + (b^3*
c*d - a*b^2*d^2)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(a*b^4*c*d^2 - a^2*b^3*d^3 + (b^5*c*d^2 - a*b^4*d^3)*x^
2)]

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 = 0.34 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.43 \[ \int \frac {x^5}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=-\frac {2 \, a^{2} d}{{\left (b^{2} c - a b d - {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}\right )} \sqrt {b d} {\left | b \right |}} + \frac {{\left (b c + 3 \, a d\right )} \log \left ({\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}\right )}{4 \, \sqrt {b d} b d {\left | b \right |}} + \frac {\sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d} \sqrt {b x^{2} + a} {\left | b \right |}}{2 \, b^{4} d} \]

[In]

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

[Out]

-2*a^2*d/((b^2*c - a*b*d - (sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^2)*sqrt(b*d)*ab
s(b)) + 1/4*(b*c + 3*a*d)*log((sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^2)/(sqrt(b*d
)*b*d*abs(b)) + 1/2*sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d)*sqrt(b*x^2 + a)*abs(b)/(b^4*d)

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)