\(\int \frac {x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx\) [975]

Optimal result

Integrand size = 26, antiderivative size = 342 \[ \int \frac {x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {\left (8 b^2 c^2+7 a b c d+8 a^2 d^2\right ) x \sqrt {a+b x^2}}{15 b^3 d^2 \sqrt {c+d x^2}}-\frac {4 (b c+a d) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 b^2 d^2}+\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b d}-\frac {\sqrt {c} \left (8 b^2 c^2+7 a b c d+8 a^2 d^2\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b^3 d^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {4 c^{3/2} (b c+a d) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 b^2 d^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]

[Out]

1/15*(8*a^2*d^2+7*a*b*c*d+8*b^2*c^2)*x*(b*x^2+a)^(1/2)/b^3/d^2/(d*x^2+c)^(1/2)+4/15*c^(3/2)*(a*d+b*c)*(1/(1+d*
x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*EllipticF(x*d^(1/2)/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2))*(b*x^2+a)^(1/
2)/b^2/d^(5/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-1/15*(8*a^2*d^2+7*a*b*c*d+8*b^2*c^2)*(1/(1+d*x^
2/c))^(1/2)*(1+d*x^2/c)^(1/2)*EllipticE(x*d^(1/2)/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2))*c^(1/2)*(b*x^2+
a)^(1/2)/b^3/d^(5/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-4/15*(a*d+b*c)*x*(b*x^2+a)^(1/2)*(d*x^2+c
)^(1/2)/b^2/d^2+1/5*x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b/d

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 342, 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 = {490, 596, 545, 429, 506, 422} \[ \int \frac {x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=-\frac {\sqrt {c} \sqrt {a+b x^2} \left (8 a^2 d^2+7 a b c d+8 b^2 c^2\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b^3 d^{5/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {x \sqrt {a+b x^2} \left (8 a^2 d^2+7 a b c d+8 b^2 c^2\right )}{15 b^3 d^2 \sqrt {c+d x^2}}+\frac {4 c^{3/2} \sqrt {a+b x^2} (a d+b c) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 b^2 d^{5/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {4 x \sqrt {a+b x^2} \sqrt {c+d x^2} (a d+b c)}{15 b^2 d^2}+\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b d} \]

[In]

Int[x^6/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]

[Out]

((8*b^2*c^2 + 7*a*b*c*d + 8*a^2*d^2)*x*Sqrt[a + b*x^2])/(15*b^3*d^2*Sqrt[c + d*x^2]) - (4*(b*c + a*d)*x*Sqrt[a
+ b*x^2]*Sqrt[c + d*x^2])/(15*b^2*d^2) + (x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(5*b*d) - (Sqrt[c]*(8*b^2*c^2
+ 7*a*b*c*d + 8*a^2*d^2)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*b^3*d^(5
/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (4*c^(3/2)*(b*c + a*d)*Sqrt[a + b*x^2]*EllipticF[
ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*b^2*d^(5/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c +
d*x^2])

Rule 422

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sq
rt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

Rule 429

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*
Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rule 490

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

Rule 506

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt
[c + d*x^2])), x] - Dist[c/b, Int[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]

Rule 545

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

Rule 596

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

Rubi steps \begin{align*} \text {integral}& = \frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b d}-\frac {\int \frac {x^2 \left (3 a c+4 (b c+a d) x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{5 b d} \\ & = -\frac {4 (b c+a d) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 b^2 d^2}+\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b d}+\frac {\int \frac {4 a c (b c+a d)+\left (8 b^2 c^2+7 a b c d+8 a^2 d^2\right ) x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{15 b^2 d^2} \\ & = -\frac {4 (b c+a d) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 b^2 d^2}+\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b d}+\frac {(4 a c (b c+a d)) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{15 b^2 d^2}+\frac {\left (8 b^2 c^2+7 a b c d+8 a^2 d^2\right ) \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{15 b^2 d^2} \\ & = \frac {\left (8 b^2 c^2+7 a b c d+8 a^2 d^2\right ) x \sqrt {a+b x^2}}{15 b^3 d^2 \sqrt {c+d x^2}}-\frac {4 (b c+a d) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 b^2 d^2}+\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b d}+\frac {4 c^{3/2} (b c+a d) \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b^2 d^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {\left (c \left (8 b^2 c^2+7 a b c d+8 a^2 d^2\right )\right ) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{15 b^3 d^2} \\ & = \frac {\left (8 b^2 c^2+7 a b c d+8 a^2 d^2\right ) x \sqrt {a+b x^2}}{15 b^3 d^2 \sqrt {c+d x^2}}-\frac {4 (b c+a d) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 b^2 d^2}+\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b d}-\frac {\sqrt {c} \left (8 b^2 c^2+7 a b c d+8 a^2 d^2\right ) \sqrt {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b^3 d^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {4 c^{3/2} (b c+a d) \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b^2 d^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.00 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.73 \[ \int \frac {x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {-\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (4 b c+4 a d-3 b d x^2\right )-i c \left (8 b^2 c^2+7 a b c d+8 a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i c \left (8 b^2 c^2+3 a b c d+4 a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{15 a^2 \left (\frac {b}{a}\right )^{5/2} d^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \]

[In]

Integrate[x^6/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]

[Out]

(-(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(4*b*c + 4*a*d - 3*b*d*x^2)) - I*c*(8*b^2*c^2 + 7*a*b*c*d + 8*a^2*d^2
)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + I*c*(8*b^2*c^2 + 3*
a*b*c*d + 4*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(
15*a^2*(b/a)^(5/2)*d^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])

Maple [A] (verified)

Time = 7.16 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.11

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{5 b d}-\frac {\left (4 a d +4 b c \right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{15 b^{2} d^{2}}+\frac {\left (4 a d +4 b c \right ) a c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{15 b^{2} d^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {\left (-\frac {3 a c}{5 b d}+\frac {\left (4 a d +4 b c \right ) \left (2 a d +2 b c \right )}{15 b^{2} d^{2}}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\)	\(379\)
risch	\(-\frac {x \left (-3 b d \,x^{2}+4 a d +4 b c \right ) \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{15 b^{2} d^{2}}+\frac {\left (\frac {4 a^{2} c d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}+\frac {4 b \,c^{2} a \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {\left (8 a^{2} d^{2}+7 a b c d +8 b^{2} c^{2}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{15 b^{2} d^{2} \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\)	\(419\)
default	\(-\frac {\left (-3 \sqrt {-\frac {b}{a}}\, b^{2} d^{3} x^{7}+\sqrt {-\frac {b}{a}}\, a b \,d^{3} x^{5}+\sqrt {-\frac {b}{a}}\, b^{2} c \,d^{2} x^{5}+4 \sqrt {-\frac {b}{a}}\, a^{2} d^{3} x^{3}+5 \sqrt {-\frac {b}{a}}\, a b c \,d^{2} x^{3}+4 \sqrt {-\frac {b}{a}}\, b^{2} c^{2} d \,x^{3}+4 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c \,d^{2}+3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} d +8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{3}-8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c \,d^{2}-7 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} d -8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{3}+4 \sqrt {-\frac {b}{a}}\, a^{2} c \,d^{2} x +4 \sqrt {-\frac {b}{a}}\, a b \,c^{2} d x \right ) \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{15 d^{3} b^{2} \sqrt {-\frac {b}{a}}\, \left (b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c \right )}\)	\(546\)

[In]

int(x^6/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(1/5/b/d*x^3*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)-1
/15/b^2/d^2*(4*a*d+4*b*c)*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/15/b^2/d^2*(4*a*d+4*b*c)*a*c/(-b/a)^(1/2)*(1
+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c
/b)^(1/2))-(-3/5*a*c/b/d+1/15/b^2/d^2*(4*a*d+4*b*c)*(2*a*d+2*b*c))*c/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c
)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/d*(EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-EllipticE(x*
(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))))

Fricas [A] (verification not implemented)

none

Time = 0.08 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.68 \[ \int \frac {x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=-\frac {{\left (8 \, b^{2} c^{3} + 7 \, a b c^{2} d + 8 \, a^{2} c d^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (8 \, b^{2} c^{3} + 7 \, a b c^{2} d + 4 \, a^{2} d^{3} + 4 \, {\left (2 \, a^{2} + a b\right )} c d^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (3 \, b^{2} d^{3} x^{4} + 8 \, b^{2} c^{2} d + 7 \, a b c d^{2} + 8 \, a^{2} d^{3} - 4 \, {\left (b^{2} c d^{2} + a b d^{3}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{15 \, b^{3} d^{4} x} \]

[In]

integrate(x^6/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

-1/15*((8*b^2*c^3 + 7*a*b*c^2*d + 8*a^2*c*d^2)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*
c)) - (8*b^2*c^3 + 7*a*b*c^2*d + 4*a^2*d^3 + 4*(2*a^2 + a*b)*c*d^2)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_f(arcsin(s
qrt(-c/d)/x), a*d/(b*c)) - (3*b^2*d^3*x^4 + 8*b^2*c^2*d + 7*a*b*c*d^2 + 8*a^2*d^3 - 4*(b^2*c*d^2 + a*b*d^3)*x^
2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(b^3*d^4*x)

Sympy [F]

\[ \int \frac {x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {x^{6}}{\sqrt {a + b x^{2}} \sqrt {c + d x^{2}}}\, dx \]

[In]

integrate(x**6/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2),x)

[Out]

Integral(x**6/(sqrt(a + b*x**2)*sqrt(c + d*x**2)), x)

Maxima [F]

\[ \int \frac {x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {x^{6}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}} \,d x } \]

[In]

integrate(x^6/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^6/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)), x)

Giac [F]

\[ \int \frac {x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {x^{6}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}} \,d x } \]

[In]

integrate(x^6/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="giac")

[Out]

integrate(x^6/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {x^6}{\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}} \,d x \]

[In]

int(x^6/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)),x)

[Out]

int(x^6/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)), x)