3.934 \(\int \frac {x^5 \sqrt {a+b x^2}}{\sqrt {c+d x^2}} \, dx\)

Optimal. Leaf size=209 \[ \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (a^2 d^2+2 a b c d+5 b^2 c^2\right )}{16 b^2 d^3}-\frac {(b c-a d) \left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{16 b^{5/2} d^{7/2}}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (3 a d+5 b c)}{24 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{6 b d} \]

[Out]

-1/16*(-a*d+b*c)*(a^2*d^2+2*a*b*c*d+5*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^(7/2)-1/24*(3*a*d+5*b*c)*(b*x^2+a)^(3/2)*(d*x^2+c)^(1/2)/b^2/d^2+1/6*x^2*(b*x^2+a)^(3/2)*(d*x^2+c)^(1/2)/b
/d+1/16*(a^2*d^2+2*a*b*c*d+5*b^2*c^2)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b^2/d^3

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Rubi [A]  time = 0.25, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {446, 90, 80, 50, 63, 217, 206} \[ \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (a^2 d^2+2 a b c d+5 b^2 c^2\right )}{16 b^2 d^3}-\frac {(b c-a d) \left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{16 b^{5/2} d^{7/2}}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (3 a d+5 b c)}{24 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{6 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((5*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(16*b^2*d^3) - ((5*b*c + 3*a*d)*(a + b*x^2
)^(3/2)*Sqrt[c + d*x^2])/(24*b^2*d^2) + (x^2*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(6*b*d) - ((b*c - a*d)*(5*b^2*
c^2 + 2*a*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x^2])/(Sqrt[b]*Sqrt[c + d*x^2])])/(16*b^(5/2)*d^(7/2))

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 90

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 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]]

Rubi steps

\begin {align*} \int \frac {x^5 \sqrt {a+b x^2}}{\sqrt {c+d x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2 \sqrt {a+b x}}{\sqrt {c+d x}} \, dx,x,x^2\right )\\ &=\frac {x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{6 b d}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a+b x} \left (-a c-\frac {1}{2} (5 b c+3 a d) x\right )}{\sqrt {c+d x}} \, dx,x,x^2\right )}{6 b d}\\ &=-\frac {(5 b c+3 a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{24 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{6 b d}+\frac {\left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx,x,x^2\right )}{16 b^2 d^2}\\ &=\frac {\left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{16 b^2 d^3}-\frac {(5 b c+3 a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{24 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{6 b d}-\frac {\left ((b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )}{32 b^2 d^3}\\ &=\frac {\left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{16 b^2 d^3}-\frac {(5 b c+3 a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{24 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{6 b d}-\frac {\left ((b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x^2}\right )}{16 b^3 d^3}\\ &=\frac {\left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{16 b^2 d^3}-\frac {(5 b c+3 a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{24 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{6 b d}-\frac {\left ((b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right )\right ) \operatorname {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 )}{16 b^3 d^3}\\ &=\frac {\left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{16 b^2 d^3}-\frac {(5 b c+3 a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{24 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{6 b d}-\frac {(b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{16 b^{5/2} d^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.45, size = 187, normalized size = 0.89 \[ \frac {-3 \left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) (b c-a d)^{3/2} \sqrt {\frac {b \left (c+d x^2\right )}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b c-a d}}\right )-b \sqrt {d} \sqrt {a+b x^2} \left (c+d x^2\right ) \left (3 a^2 d^2-2 a b d \left (d x^2-2 c\right )+b^2 \left (-15 c^2+10 c d x^2-8 d^2 x^4\right )\right )}{48 b^3 d^{7/2} \sqrt {c+d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(b*Sqrt[d]*Sqrt[a + b*x^2]*(c + d*x^2)*(3*a^2*d^2 - 2*a*b*d*(-2*c + d*x^2) + b^2*(-15*c^2 + 10*c*d*x^2 - 8*d
^2*x^4))) - 3*(b*c - a*d)^(3/2)*(5*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*ArcSinh[(S
qrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/(48*b^3*d^(7/2)*Sqrt[c + d*x^2])

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fricas [A]  time = 1.22, size = 442, normalized size = 2.11 \[ \left [-\frac {3 \, {\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {b d}\right ) - 4 \, {\left (8 \, b^{3} d^{3} x^{4} + 15 \, b^{3} c^{2} d - 4 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3} - 2 \, {\left (5 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{192 \, b^{3} d^{4}}, \frac {3 \, {\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-b d}}{2 \, {\left (b^{2} d^{2} x^{4} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )}}\right ) + 2 \, {\left (8 \, b^{3} d^{3} x^{4} + 15 \, b^{3} c^{2} d - 4 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3} - 2 \, {\left (5 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{96 \, b^{3} d^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/192*(3*(5*b^3*c^3 - 3*a*b^2*c^2*d - a^2*b*c*d^2 - a^3*d^3)*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*(8*b^3*d^3*x^4 + 15*b^3*c^2*d - 4*a*b^2*c*d^2 - 3*a^2*b*d^3 - 2*(5*b^3*c*d^2 - a*b^2*d^3)*x^2)*sqrt(b*x^
2 + a)*sqrt(d*x^2 + c))/(b^3*d^4), 1/96*(3*(5*b^3*c^3 - 3*a*b^2*c^2*d - a^2*b*c*d^2 - a^3*d^3)*sqrt(-b*d)*arct
an(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*(8*b^3*d^3*x^4 + 15*b^3*c^2*d - 4*a*b^2*c*d^2 - 3*a^2*b*d^3 - 2*(5*b^3*c*d^2 - a*b^2*d^3)*x
^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(b^3*d^4)]

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giac [A]  time = 0.48, size = 226, normalized size = 1.08 \[ \frac {{\left (\sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d} \sqrt {b x^{2} + a} {\left (2 \, {\left (b x^{2} + a\right )} {\left (\frac {4 \, {\left (b x^{2} + a\right )}}{b^{3} d} - \frac {5 \, b^{7} c d^{3} + 7 \, a b^{6} d^{4}}{b^{9} d^{5}}\right )} + \frac {3 \, {\left (5 \, b^{8} c^{2} d^{2} + 2 \, a b^{7} c d^{3} + a^{2} b^{6} d^{4}\right )}}{b^{9} d^{5}}\right )} + \frac {3 \, {\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | -\sqrt {b x^{2} + a} \sqrt {b d} + \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b^{2} d^{3}}\right )} b}{48 \, {\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(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)/(b^3*d) - (5*b^7*c*d
^3 + 7*a*b^6*d^4)/(b^9*d^5)) + 3*(5*b^8*c^2*d^2 + 2*a*b^7*c*d^3 + a^2*b^6*d^4)/(b^9*d^5)) + 3*(5*b^3*c^3 - 3*a
*b^2*c^2*d - a^2*b*c*d^2 - a^3*d^3)*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^3))*b/abs(b)

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maple [B]  time = 0.08, size = 532, normalized size = 2.55 \[ \frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (16 \sqrt {x^{4} b d +x^{2} a d +b c \,x^{2}+a c}\, \sqrt {b d}\, b^{2} d^{2} x^{4}+3 a^{3} d^{3} \ln \left (\frac {2 b d \,x^{2}+a d +b c +2 \sqrt {x^{4} b d +x^{2} a d +b c \,x^{2}+a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+3 a^{2} b c \,d^{2} \ln \left (\frac {2 b d \,x^{2}+a d +b c +2 \sqrt {x^{4} b d +x^{2} a d +b c \,x^{2}+a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+9 a \,b^{2} c^{2} d \ln \left (\frac {2 b d \,x^{2}+a d +b c +2 \sqrt {x^{4} b d +x^{2} a d +b c \,x^{2}+a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-15 b^{3} c^{3} \ln \left (\frac {2 b d \,x^{2}+a d +b c +2 \sqrt {x^{4} b d +x^{2} a d +b c \,x^{2}+a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+4 \sqrt {x^{4} b d +x^{2} a d +b c \,x^{2}+a c}\, \sqrt {b d}\, a b \,d^{2} x^{2}-20 \sqrt {x^{4} b d +x^{2} a d +b c \,x^{2}+a c}\, \sqrt {b d}\, b^{2} c d \,x^{2}-6 \sqrt {x^{4} b d +x^{2} a d +b c \,x^{2}+a c}\, \sqrt {b d}\, a^{2} d^{2}-8 \sqrt {x^{4} b d +x^{2} a d +b c \,x^{2}+a c}\, \sqrt {b d}\, a b c d +30 \sqrt {x^{4} b d +x^{2} a d +b c \,x^{2}+a c}\, \sqrt {b d}\, b^{2} c^{2}\right )}{96 \sqrt {x^{4} b d +x^{2} a d +b c \,x^{2}+a c}\, \sqrt {b d}\, b^{2} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(16*x^4*b^2*d^2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)+4*(b*d*x^
4+a*d*x^2+b*c*x^2+a*c)^(1/2)*x^2*a*b*d^2*(b*d)^(1/2)-20*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*x^2*c*b^2*d*(b*d)^
(1/2)+3*d^3*ln(1/2*(2*x^2*b*d+2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3+3*ln
(1/2*(2*x^2*b*d+2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*c*b*d^2+9*ln(1/2*(
2*x^2*b*d+2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a*c^2*b^2*d-15*b^3*ln(1/2*(2
*x^2*b*d+2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*c^3-6*(b*d*x^4+a*d*x^2+b*c*x^
2+a*c)^(1/2)*a^2*d^2*(b*d)^(1/2)-8*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*a*c*b*d*(b*d)^(1/2)+30*(b*d*x^4+a*d*x^2
+b*c*x^2+a*c)^(1/2)*c^2*b^2*(b*d)^(1/2))/d^3/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/b^2/(b*d)^(1/2)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^2+a)^(1/2)/(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 details)Is a*d-b*c zero or nonzero?

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mupad [B]  time = 54.78, size = 993, normalized size = 4.75 \[ \frac {\mathrm {atanh}\left (\frac {\sqrt {d}\,\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}{\sqrt {b}\,\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}\right )\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2+2\,a\,b\,c\,d+5\,b^2\,c^2\right )}{8\,b^{5/2}\,d^{7/2}}-\frac {\frac {\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )\,\left (\frac {a^3\,b^3\,d^3}{8}+\frac {a^2\,b^4\,c\,d^2}{8}+\frac {3\,a\,b^5\,c^2\,d}{8}-\frac {5\,b^6\,c^3}{8}\right )}{d^9\,\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}-\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^5\,\left (\frac {19\,a^3\,b\,d^3}{4}+\frac {275\,a^2\,b^2\,c\,d^2}{4}+\frac {313\,a\,b^3\,c^2\,d}{4}+\frac {33\,b^4\,c^3}{4}\right )}{d^7\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^5}-\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^7\,\left (\frac {19\,a^3\,d^3}{4}+\frac {275\,a^2\,b\,c\,d^2}{4}+\frac {313\,a\,b^2\,c^2\,d}{4}+\frac {33\,b^3\,c^3}{4}\right )}{d^6\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^7}-\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^3\,\left (\frac {17\,a^3\,b^2\,d^3}{24}+\frac {91\,a^2\,b^3\,c\,d^2}{8}+\frac {17\,a\,b^4\,c^2\,d}{8}-\frac {85\,b^5\,c^3}{24}\right )}{d^8\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^3}+\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^{11}\,\left (\frac {a^3\,d^3}{8}+\frac {a^2\,b\,c\,d^2}{8}+\frac {3\,a\,b^2\,c^2\,d}{8}-\frac {5\,b^3\,c^3}{8}\right )}{b^2\,d^4\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^{11}}-\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^9\,\left (\frac {17\,a^3\,d^3}{24}+\frac {91\,a^2\,b\,c\,d^2}{8}+\frac {17\,a\,b^2\,c^2\,d}{8}-\frac {85\,b^3\,c^3}{24}\right )}{b\,d^5\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^9}+\frac {\sqrt {a}\,\sqrt {c}\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^8\,\left (16\,d\,a^2+48\,b\,c\,a\right )}{d^4\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^8}+\frac {\sqrt {a}\,\sqrt {c}\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^4\,\left (16\,d\,a^2\,b^2+48\,c\,a\,b^3\right )}{d^6\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^4}+\frac {\sqrt {a}\,\sqrt {c}\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^6\,\left (32\,a^2\,b\,d^2+\frac {352\,a\,b^2\,c\,d}{3}+64\,b^3\,c^2\right )}{d^6\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^6}}{\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^{12}}{{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^{12}}+\frac {b^6}{d^6}-\frac {6\,b^5\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^2}{d^5\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^2}+\frac {15\,b^4\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^4}{d^4\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^4}-\frac {20\,b^3\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^6}{d^3\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^6}+\frac {15\,b^2\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^8}{d^2\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^8}-\frac {6\,b\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^{10}}{d\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^{10}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atanh((d^(1/2)*((a + b*x^2)^(1/2) - a^(1/2)))/(b^(1/2)*((c + d*x^2)^(1/2) - c^(1/2))))*(a*d - b*c)*(a^2*d^2 +
5*b^2*c^2 + 2*a*b*c*d))/(8*b^(5/2)*d^(7/2)) - ((((a + b*x^2)^(1/2) - a^(1/2))*((a^3*b^3*d^3)/8 - (5*b^6*c^3)/
8 + (a^2*b^4*c*d^2)/8 + (3*a*b^5*c^2*d)/8))/(d^9*((c + d*x^2)^(1/2) - c^(1/2))) - (((a + b*x^2)^(1/2) - a^(1/2
))^5*((33*b^4*c^3)/4 + (19*a^3*b*d^3)/4 + (275*a^2*b^2*c*d^2)/4 + (313*a*b^3*c^2*d)/4))/(d^7*((c + d*x^2)^(1/2
) - c^(1/2))^5) - (((a + b*x^2)^(1/2) - a^(1/2))^7*((19*a^3*d^3)/4 + (33*b^3*c^3)/4 + (313*a*b^2*c^2*d)/4 + (2
75*a^2*b*c*d^2)/4))/(d^6*((c + d*x^2)^(1/2) - c^(1/2))^7) - (((a + b*x^2)^(1/2) - a^(1/2))^3*((17*a^3*b^2*d^3)
/24 - (85*b^5*c^3)/24 + (91*a^2*b^3*c*d^2)/8 + (17*a*b^4*c^2*d)/8))/(d^8*((c + d*x^2)^(1/2) - c^(1/2))^3) + ((
(a + b*x^2)^(1/2) - a^(1/2))^11*((a^3*d^3)/8 - (5*b^3*c^3)/8 + (3*a*b^2*c^2*d)/8 + (a^2*b*c*d^2)/8))/(b^2*d^4*
((c + d*x^2)^(1/2) - c^(1/2))^11) - (((a + b*x^2)^(1/2) - a^(1/2))^9*((17*a^3*d^3)/24 - (85*b^3*c^3)/24 + (17*
a*b^2*c^2*d)/8 + (91*a^2*b*c*d^2)/8))/(b*d^5*((c + d*x^2)^(1/2) - c^(1/2))^9) + (a^(1/2)*c^(1/2)*((a + b*x^2)^
(1/2) - a^(1/2))^8*(16*a^2*d + 48*a*b*c))/(d^4*((c + d*x^2)^(1/2) - c^(1/2))^8) + (a^(1/2)*c^(1/2)*((a + b*x^2
)^(1/2) - a^(1/2))^4*(16*a^2*b^2*d + 48*a*b^3*c))/(d^6*((c + d*x^2)^(1/2) - c^(1/2))^4) + (a^(1/2)*c^(1/2)*((a
+ b*x^2)^(1/2) - a^(1/2))^6*(64*b^3*c^2 + 32*a^2*b*d^2 + (352*a*b^2*c*d)/3))/(d^6*((c + d*x^2)^(1/2) - c^(1/2
))^6))/(((a + b*x^2)^(1/2) - a^(1/2))^12/((c + d*x^2)^(1/2) - c^(1/2))^12 + b^6/d^6 - (6*b^5*((a + b*x^2)^(1/2
) - a^(1/2))^2)/(d^5*((c + d*x^2)^(1/2) - c^(1/2))^2) + (15*b^4*((a + b*x^2)^(1/2) - a^(1/2))^4)/(d^4*((c + d*
x^2)^(1/2) - c^(1/2))^4) - (20*b^3*((a + b*x^2)^(1/2) - a^(1/2))^6)/(d^3*((c + d*x^2)^(1/2) - c^(1/2))^6) + (1
5*b^2*((a + b*x^2)^(1/2) - a^(1/2))^8)/(d^2*((c + d*x^2)^(1/2) - c^(1/2))^8) - (6*b*((a + b*x^2)^(1/2) - a^(1/
2))^10)/(d*((c + d*x^2)^(1/2) - c^(1/2))^10))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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