\(\int (a+b x^2)^{3/2} (c+d x^2)^{3/2} \, dx\) [172]

Optimal result

Integrand size = 23, antiderivative size = 410 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \, dx=-\frac {2 (b c+a d) \left (b^2 c^2-6 a b c d+a^2 d^2\right ) x \sqrt {a+b x^2}}{35 b^2 d \sqrt {c+d x^2}}+\frac {1}{35} \left (9 a c+\frac {b c^2}{d}-\frac {2 a^2 d}{b}\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}+\frac {2 (4 b c-a d) x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{35 b}+\frac {d x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{7 b}+\frac {2 \sqrt {c} (b c+a d) \left (b^2 c^2-6 a b c d+a^2 d^2\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{35 b^2 d^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {c^{3/2} \left (b^2 c^2-18 a b c d+a^2 d^2\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{35 b d^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]

[Out]

-2/35*(a*d+b*c)*(a^2*d^2-6*a*b*c*d+b^2*c^2)*x*(b*x^2+a)^(1/2)/b^2/d/(d*x^2+c)^(1/2)-1/35*c^(3/2)*(a^2*d^2-18*a
*b*c*d+b^2*c^2)*(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/d^(3/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)+2/35*(a*d+b*c)*(a^2*d^2-6
*a*b*c*d+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^2/d^(3/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)+2/35*(-a*d+4*
b*c)*x*(b*x^2+a)^(3/2)*(d*x^2+c)^(1/2)/b+1/7*d*x*(b*x^2+a)^(5/2)*(d*x^2+c)^(1/2)/b+1/35*(9*a*c+b*c^2/d-2*a^2*d
/b)*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {427, 542, 545, 429, 506, 422} \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \, dx=\frac {2 \sqrt {c} \sqrt {a+b x^2} (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{35 b^2 d^{3/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {c^{3/2} \sqrt {a+b x^2} \left (a^2 d^2-18 a b c d+b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{35 b d^{3/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {2 x \sqrt {a+b x^2} (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right )}{35 b^2 d \sqrt {c+d x^2}}+\frac {1}{35} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-\frac {2 a^2 d}{b}+9 a c+\frac {b c^2}{d}\right )+\frac {d x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{7 b}+\frac {2 x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (4 b c-a d)}{35 b} \]

[In]

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

[Out]

(-2*(b*c + a*d)*(b^2*c^2 - 6*a*b*c*d + a^2*d^2)*x*Sqrt[a + b*x^2])/(35*b^2*d*Sqrt[c + d*x^2]) + ((9*a*c + (b*c
^2)/d - (2*a^2*d)/b)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/35 + (2*(4*b*c - a*d)*x*(a + b*x^2)^(3/2)*Sqrt[c + d*x
^2])/(35*b) + (d*x*(a + b*x^2)^(5/2)*Sqrt[c + d*x^2])/(7*b) + (2*Sqrt[c]*(b*c + a*d)*(b^2*c^2 - 6*a*b*c*d + a^
2*d^2)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(35*b^2*d^(3/2)*Sqrt[(c*(a + b
*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (c^(3/2)*(b^2*c^2 - 18*a*b*c*d + a^2*d^2)*Sqrt[a + b*x^2]*EllipticF
[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(35*b*d^(3/2)*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 427

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*x*(a + b*x^n)^(p + 1)*((c
+ d*x^n)^(q - 1)/(b*(n*(p + q) + 1))), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]

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 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 542

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

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]

Rubi steps \begin{align*} \text {integral}& = \frac {d x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{7 b}+\frac {\int \frac {\left (a+b x^2\right )^{3/2} \left (c (7 b c-a d)+2 d (4 b c-a d) x^2\right )}{\sqrt {c+d x^2}} \, dx}{7 b} \\ & = \frac {2 (4 b c-a d) x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{35 b}+\frac {d x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{7 b}+\frac {\int \frac {\sqrt {a+b x^2} \left (3 a c d (9 b c-a d)+3 d \left (b^2 c^2+9 a b c d-2 a^2 d^2\right ) x^2\right )}{\sqrt {c+d x^2}} \, dx}{35 b d} \\ & = \frac {1}{35} \left (9 a c+\frac {b c^2}{d}-\frac {2 a^2 d}{b}\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}+\frac {2 (4 b c-a d) x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{35 b}+\frac {d x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{7 b}+\frac {\int \frac {-3 a c d \left (b^2 c^2-18 a b c d+a^2 d^2\right )-6 d (b c+a d) \left (b^2 c^2-6 a b c d+a^2 d^2\right ) x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{105 b d^2} \\ & = \frac {1}{35} \left (9 a c+\frac {b c^2}{d}-\frac {2 a^2 d}{b}\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}+\frac {2 (4 b c-a d) x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{35 b}+\frac {d x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{7 b}-\frac {\left (a c \left (b^2 c^2-18 a b c d+a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{35 b d}-\frac {\left (2 (b c+a d) \left (b^2 c^2-6 a b c d+a^2 d^2\right )\right ) \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{35 b d} \\ & = -\frac {2 (b c+a d) \left (b^2 c^2-6 a b c d+a^2 d^2\right ) x \sqrt {a+b x^2}}{35 b^2 d \sqrt {c+d x^2}}+\frac {1}{35} \left (9 a c+\frac {b c^2}{d}-\frac {2 a^2 d}{b}\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}+\frac {2 (4 b c-a d) x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{35 b}+\frac {d x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{7 b}-\frac {c^{3/2} \left (b^2 c^2-18 a b c d+a^2 d^2\right ) \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{35 b d^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {\left (2 c (b c+a d) \left (b^2 c^2-6 a b c d+a^2 d^2\right )\right ) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{35 b^2 d} \\ & = -\frac {2 (b c+a d) \left (b^2 c^2-6 a b c d+a^2 d^2\right ) x \sqrt {a+b x^2}}{35 b^2 d \sqrt {c+d x^2}}+\frac {1}{35} \left (9 a c+\frac {b c^2}{d}-\frac {2 a^2 d}{b}\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}+\frac {2 (4 b c-a d) x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{35 b}+\frac {d x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{7 b}+\frac {2 \sqrt {c} (b c+a d) \left (b^2 c^2-6 a b c d+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 )}{35 b^2 d^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {c^{3/2} \left (b^2 c^2-18 a b c d+a^2 d^2\right ) \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{35 b d^{3/2} \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 = 4.18 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.74 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (a^2 d^2+a b d \left (17 c+8 d x^2\right )+b^2 \left (c^2+8 c d x^2+5 d^2 x^4\right )\right )+2 i c \left (b^3 c^3-5 a b^2 c^2 d-5 a^2 b c d^2+a^3 d^3\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c \left (2 b^3 c^3-11 a b^2 c^2 d+8 a^2 b c d^2+a^3 d^3\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{35 b \sqrt {\frac {b}{a}} d^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \]

[In]

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

[Out]

(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(a^2*d^2 + a*b*d*(17*c + 8*d*x^2) + b^2*(c^2 + 8*c*d*x^2 + 5*d^2*x^4))
+ (2*I)*c*(b^3*c^3 - 5*a*b^2*c^2*d - 5*a^2*b*c*d^2 + a^3*d^3)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Elliptic
E[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*c*(2*b^3*c^3 - 11*a*b^2*c^2*d + 8*a^2*b*c*d^2 + a^3*d^3)*Sqrt[1 + (
b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(35*b*Sqrt[b/a]*d^2*Sqrt[a + b*x
^2]*Sqrt[c + d*x^2])

Maple [A] (verified)

Time = 4.94 (sec) , antiderivative size = 566, normalized size of antiderivative = 1.38

method	result	size
risch	\(\frac {x \left (5 b^{2} d^{2} x^{4}+8 x^{2} a b \,d^{2}+8 x^{2} b^{2} c d +a^{2} d^{2}+17 a b c d +b^{2} c^{2}\right ) \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{35 b d}-\frac {\left (\frac {a^{3} c \,d^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}+\frac {b^{2} c^{3} a \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {18 a^{2} b \,c^{2} d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {\left (2 a^{3} d^{3}-10 a^{2} b c \,d^{2}-10 a \,b^{2} c^{2} d +2 b^{3} c^{3}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{35 b d \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\)	\(566\)
elliptic	\(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {b d \,x^{5} \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{7}+\frac {\left (2 a b \,d^{2}+2 b^{2} c d -\frac {b d \left (6 a d +6 b c \right )}{7}\right ) x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{5 b d}+\frac {\left (a^{2} d^{2}+\frac {23 a b c d}{7}+b^{2} c^{2}-\frac {\left (2 a b \,d^{2}+2 b^{2} c d -\frac {b d \left (6 a d +6 b c \right )}{7}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{3 b d}+\frac {\left (a^{2} c^{2}-\frac {\left (a^{2} d^{2}+\frac {23 a b c d}{7}+b^{2} c^{2}-\frac {\left (2 a b \,d^{2}+2 b^{2} c d -\frac {b d \left (6 a d +6 b c \right )}{7}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) a c}{3 b d}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {\left (2 a^{2} c d +2 b \,c^{2} a -\frac {3 \left (2 a b \,d^{2}+2 b^{2} c d -\frac {b d \left (6 a d +6 b c \right )}{7}\right ) a c}{5 b d}-\frac {\left (a^{2} d^{2}+\frac {23 a b c d}{7}+b^{2} c^{2}-\frac {\left (2 a b \,d^{2}+2 b^{2} c d -\frac {b d \left (6 a d +6 b c \right )}{7}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right )}{b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\)	\(684\)
default	\(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (5 \sqrt {-\frac {b}{a}}\, b^{3} d^{4} x^{9}+13 \sqrt {-\frac {b}{a}}\, a \,b^{2} d^{4} x^{7}+13 \sqrt {-\frac {b}{a}}\, b^{3} c \,d^{3} x^{7}+9 \sqrt {-\frac {b}{a}}\, a^{2} b \,d^{4} x^{5}+38 \sqrt {-\frac {b}{a}}\, a \,b^{2} c \,d^{3} x^{5}+9 \sqrt {-\frac {b}{a}}\, b^{3} c^{2} d^{2} x^{5}+\sqrt {-\frac {b}{a}}\, a^{3} d^{4} x^{3}+26 \sqrt {-\frac {b}{a}}\, a^{2} b c \,d^{3} x^{3}+26 \sqrt {-\frac {b}{a}}\, a \,b^{2} c^{2} d^{2} x^{3}+\sqrt {-\frac {b}{a}}\, b^{3} c^{3} d \,x^{3}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{3} c \,d^{3}+8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} b \,c^{2} d^{2}-11 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{2} c^{3} d +2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{3} c^{4}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{3} c \,d^{3}+10 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} b \,c^{2} d^{2}+10 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{2} c^{3} d -2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{3} c^{4}+\sqrt {-\frac {b}{a}}\, a^{3} c \,d^{3} x +17 \sqrt {-\frac {b}{a}}\, a^{2} b \,c^{2} d^{2} x +\sqrt {-\frac {b}{a}}\, a \,b^{2} c^{3} d x \right )}{35 b \,d^{2} \left (b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c \right ) \sqrt {-\frac {b}{a}}}\)	\(780\)

[In]

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

[Out]

1/35/b/d*x*(5*b^2*d^2*x^4+8*a*b*d^2*x^2+8*b^2*c*d*x^2+a^2*d^2+17*a*b*c*d+b^2*c^2)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1
/2)-1/35/b/d*(a^3*c*d^2/(-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)*E
llipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))+b^2*c^3*a/(-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))-18*a^2*b*c^2*d/(-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))-(2*a^3*d^3-10*a^2*b*c*d^2-10*a*b^2*c^2*d+2*b^3*c^3)*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))))*((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 = 0.10 (sec) , antiderivative size = 316, normalized size of antiderivative = 0.77 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \, dx=\frac {2 \, {\left (b^{3} c^{4} - 5 \, a b^{2} c^{3} d - 5 \, a^{2} b c^{2} d^{2} + a^{3} c d^{3}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (2 \, b^{3} c^{4} - 10 \, a b^{2} c^{3} d + a^{3} d^{4} - {\left (10 \, a^{2} b - a b^{2}\right )} c^{2} d^{2} + 2 \, {\left (a^{3} - 9 \, a^{2} b\right )} c d^{3}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) + {\left (5 \, b^{3} d^{4} x^{6} - 2 \, b^{3} c^{3} d + 10 \, a b^{2} c^{2} d^{2} + 10 \, a^{2} b c d^{3} - 2 \, a^{3} d^{4} + 8 \, {\left (b^{3} c d^{3} + a b^{2} d^{4}\right )} x^{4} + {\left (b^{3} c^{2} d^{2} + 17 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{35 \, b^{2} d^{3} x} \]

[In]

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

[Out]

1/35*(2*(b^3*c^4 - 5*a*b^2*c^3*d - 5*a^2*b*c^2*d^2 + a^3*c*d^3)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(
-c/d)/x), a*d/(b*c)) - (2*b^3*c^4 - 10*a*b^2*c^3*d + a^3*d^4 - (10*a^2*b - a*b^2)*c^2*d^2 + 2*(a^3 - 9*a^2*b)*
c*d^3)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) + (5*b^3*d^4*x^6 - 2*b^3*c^3*d + 10*
a*b^2*c^2*d^2 + 10*a^2*b*c*d^3 - 2*a^3*d^4 + 8*(b^3*c*d^3 + a*b^2*d^4)*x^4 + (b^3*c^2*d^2 + 17*a*b^2*c*d^3 + a
^2*b*d^4)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(b^2*d^3*x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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