\(\int \frac {(a+b x^2)^{7/2}}{(c+d x^2)^{3/2}} \, dx\) [207]

Optimal result

Integrand size = 23, antiderivative size = 445 \[ \int \frac {\left (a+b x^2\right )^{7/2}}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {\left (48 b^3 c^3-128 a b^2 c^2 d+103 a^2 b c d^2-15 a^3 d^3\right ) x \sqrt {a+b x^2}}{15 c d^3 \sqrt {c+d x^2}}-\frac {(b c-a d) x \left (a+b x^2\right )^{5/2}}{c d \sqrt {c+d x^2}}-\frac {b \left (24 b^2 c^2-43 a b c d+15 a^2 d^2\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 c d^3}+\frac {b (6 b c-5 a d) x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 c d^2}-\frac {\left (48 b^3 c^3-128 a b^2 c^2 d+103 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 \sqrt {c} d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {b \sqrt {c} \left (24 b^2 c^2-61 a b c d+45 a^2 d^2\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]

[Out]

-(-a*d+b*c)*x*(b*x^2+a)^(5/2)/c/d/(d*x^2+c)^(1/2)+1/15*(-15*a^3*d^3+103*a^2*b*c*d^2-128*a*b^2*c^2*d+48*b^3*c^3
)*x*(b*x^2+a)^(1/2)/c/d^3/(d*x^2+c)^(1/2)-1/15*(-15*a^3*d^3+103*a^2*b*c*d^2-128*a*b^2*c^2*d+48*b^3*c^3)*(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))*(b*x^2+a)^(
1/2)/d^(7/2)/c^(1/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)+1/15*b*(45*a^2*d^2-61*a*b*c*d+24*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))*c^(1
/2)*(b*x^2+a)^(1/2)/d^(7/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)+1/5*b*(-5*a*d+6*b*c)*x*(b*x^2+a)^(
3/2)*(d*x^2+c)^(1/2)/c/d^2-1/15*b*(15*a^2*d^2-43*a*b*c*d+24*b^2*c^2)*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/c/d^3

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 445, 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 = {424, 542, 545, 429, 506, 422} \[ \int \frac {\left (a+b x^2\right )^{7/2}}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {b \sqrt {c} \sqrt {a+b x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 d^{7/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (15 a^2 d^2-43 a b c d+24 b^2 c^2\right )}{15 c d^3}-\frac {\sqrt {a+b x^2} \left (-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 \sqrt {c} d^{7/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {x \sqrt {a+b x^2} \left (-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right )}{15 c d^3 \sqrt {c+d x^2}}+\frac {b x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (6 b c-5 a d)}{5 c d^2}-\frac {x \left (a+b x^2\right )^{5/2} (b c-a d)}{c d \sqrt {c+d x^2}} \]

[In]

Int[(a + b*x^2)^(7/2)/(c + d*x^2)^(3/2),x]

[Out]

((48*b^3*c^3 - 128*a*b^2*c^2*d + 103*a^2*b*c*d^2 - 15*a^3*d^3)*x*Sqrt[a + b*x^2])/(15*c*d^3*Sqrt[c + d*x^2]) -
((b*c - a*d)*x*(a + b*x^2)^(5/2))/(c*d*Sqrt[c + d*x^2]) - (b*(24*b^2*c^2 - 43*a*b*c*d + 15*a^2*d^2)*x*Sqrt[a
+ b*x^2]*Sqrt[c + d*x^2])/(15*c*d^3) + (b*(6*b*c - 5*a*d)*x*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(5*c*d^2) - ((4
8*b^3*c^3 - 128*a*b^2*c^2*d + 103*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[
c]], 1 - (b*c)/(a*d)])/(15*Sqrt[c]*d^(7/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (b*Sqrt[c]
*(24*b^2*c^2 - 61*a*b*c*d + 45*a^2*d^2)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)
])/(15*d^(7/2)*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 424

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a*d - c*b)*x*(a + b*x^n)^(
p + 1)*((c + d*x^n)^(q - 1)/(a*b*n*(p + 1))), x] - Dist[1/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)
^(q - 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*(p + q) + 1))*x^n, x], x], x] /;
FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[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 {(b c-a d) x \left (a+b x^2\right )^{5/2}}{c d \sqrt {c+d x^2}}+\frac {\int \frac {\left (a+b x^2\right )^{3/2} \left (a b c+b (6 b c-5 a d) x^2\right )}{\sqrt {c+d x^2}} \, dx}{c d} \\ & = -\frac {(b c-a d) x \left (a+b x^2\right )^{5/2}}{c d \sqrt {c+d x^2}}+\frac {b (6 b c-5 a d) x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 c d^2}+\frac {\int \frac {\sqrt {a+b x^2} \left (-2 a b c (3 b c-5 a d)-b \left (24 b^2 c^2-43 a b c d+15 a^2 d^2\right ) x^2\right )}{\sqrt {c+d x^2}} \, dx}{5 c d^2} \\ & = -\frac {(b c-a d) x \left (a+b x^2\right )^{5/2}}{c d \sqrt {c+d x^2}}-\frac {b \left (24 b^2 c^2-43 a b c d+15 a^2 d^2\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 c d^3}+\frac {b (6 b c-5 a d) x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 c d^2}+\frac {\int \frac {a b c \left (24 b^2 c^2-61 a b c d+45 a^2 d^2\right )+b \left (48 b^3 c^3-128 a b^2 c^2 d+103 a^2 b c d^2-15 a^3 d^3\right ) x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{15 c d^3} \\ & = -\frac {(b c-a d) x \left (a+b x^2\right )^{5/2}}{c d \sqrt {c+d x^2}}-\frac {b \left (24 b^2 c^2-43 a b c d+15 a^2 d^2\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 c d^3}+\frac {b (6 b c-5 a d) x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 c d^2}+\frac {\left (a b \left (24 b^2 c^2-61 a b c d+45 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{15 d^3}+\frac {\left (b \left (48 b^3 c^3-128 a b^2 c^2 d+103 a^2 b c d^2-15 a^3 d^3\right )\right ) \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{15 c d^3} \\ & = \frac {\left (48 b^3 c^3-128 a b^2 c^2 d+103 a^2 b c d^2-15 a^3 d^3\right ) x \sqrt {a+b x^2}}{15 c d^3 \sqrt {c+d x^2}}-\frac {(b c-a d) x \left (a+b x^2\right )^{5/2}}{c d \sqrt {c+d x^2}}-\frac {b \left (24 b^2 c^2-43 a b c d+15 a^2 d^2\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 c d^3}+\frac {b (6 b c-5 a d) x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 c d^2}+\frac {b \sqrt {c} \left (24 b^2 c^2-61 a b c d+45 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 )}{15 d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {\left (48 b^3 c^3-128 a b^2 c^2 d+103 a^2 b c d^2-15 a^3 d^3\right ) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{15 d^3} \\ & = \frac {\left (48 b^3 c^3-128 a b^2 c^2 d+103 a^2 b c d^2-15 a^3 d^3\right ) x \sqrt {a+b x^2}}{15 c d^3 \sqrt {c+d x^2}}-\frac {(b c-a d) x \left (a+b x^2\right )^{5/2}}{c d \sqrt {c+d x^2}}-\frac {b \left (24 b^2 c^2-43 a b c d+15 a^2 d^2\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 c d^3}+\frac {b (6 b c-5 a d) x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 c d^2}-\frac {\left (48 b^3 c^3-128 a b^2 c^2 d+103 a^2 b c d^2-15 a^3 d^3\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 \sqrt {c} d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {b \sqrt {c} \left (24 b^2 c^2-61 a b c d+45 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 )}{15 d^{7/2} \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 = 5.17 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.71 \[ \int \frac {\left (a+b x^2\right )^{7/2}}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (-45 a^2 b c d^2+15 a^3 d^3+a b^2 c d \left (61 c+16 d x^2\right )-3 b^3 c \left (8 c^2+2 c d x^2-d^2 x^4\right )\right )+i b c \left (-48 b^3 c^3+128 a b^2 c^2 d-103 a^2 b c d^2+15 a^3 d^3\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+4 i b c \left (12 b^3 c^3-38 a b^2 c^2 d+41 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{15 \sqrt {\frac {b}{a}} c d^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \]

[In]

Integrate[(a + b*x^2)^(7/2)/(c + d*x^2)^(3/2),x]

[Out]

(Sqrt[b/a]*d*x*(a + b*x^2)*(-45*a^2*b*c*d^2 + 15*a^3*d^3 + a*b^2*c*d*(61*c + 16*d*x^2) - 3*b^3*c*(8*c^2 + 2*c*
d*x^2 - d^2*x^4)) + I*b*c*(-48*b^3*c^3 + 128*a*b^2*c^2*d - 103*a^2*b*c*d^2 + 15*a^3*d^3)*Sqrt[1 + (b*x^2)/a]*S
qrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (4*I)*b*c*(12*b^3*c^3 - 38*a*b^2*c^2*d + 4
1*a^2*b*c*d^2 - 15*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)])/(15*Sqrt[b/a]*c*d^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])

Maple [A] (verified)

Time = 9.53 (sec) , antiderivative size = 701, normalized size of antiderivative = 1.58

method	result	size
risch	\(\frac {b^{2} x \left (3 b d \,x^{2}+16 a d -9 b c \right ) \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{15 d^{3}}+\frac {\left (-\frac {b^{2} \left (58 a^{2} d^{2}-83 a b c d +33 b^{2} c^{2}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}+\frac {b \left (60 a^{3} d^{3}-106 a^{2} b c \,d^{2}+69 a \,b^{2} c^{2} d -15 b^{3} c^{3}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{d \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}+\frac {\left (15 a^{4} d^{4}-60 a^{3} b c \,d^{3}+90 a^{2} b^{2} c^{2} d^{2}-60 a \,b^{3} c^{3} d +15 b^{4} c^{4}\right ) \left (\frac {\left (b d \,x^{2}+a d \right ) x}{c \left (a d -b c \right ) \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}+\frac {\left (\frac {1}{c}-\frac {a d}{c \left (a d -b c \right )}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}+\frac {b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\left (a d -b c \right ) \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}\right )}{d}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{15 d^{3} \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\)	\(701\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {\left (b d \,x^{2}+a d \right ) \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x}{c \,d^{4} \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}+\frac {b^{3} x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{5 d^{2}}+\frac {\left (\frac {b^{3} \left (4 a d -b c \right )}{d^{2}}-\frac {b^{3} \left (4 a d +4 b c \right )}{5 d^{2}}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{3 b d}+\frac {\left (\frac {b \left (4 a^{3} d^{3}-6 a^{2} b c \,d^{2}+4 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{d^{4}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (a d -b c \right )}{d^{4} c}-\frac {a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{d^{3} c}-\frac {\left (\frac {b^{3} \left (4 a d -b c \right )}{d^{2}}-\frac {b^{3} \left (4 a d +4 b c \right )}{5 d^{2}}\right ) a c}{3 b d}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {\left (\frac {b^{2} \left (6 a^{2} d^{2}-4 a b c d +b^{2} c^{2}\right )}{d^{3}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b}{d^{3} c}-\frac {3 b^{3} a c}{5 d^{2}}-\frac {\left (\frac {b^{3} \left (4 a d -b c \right )}{d^{2}}-\frac {b^{3} \left (4 a d +4 b c \right )}{5 d^{2}}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\)	\(744\)
default	\(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (3 \sqrt {-\frac {b}{a}}\, b^{4} c \,d^{3} x^{7}+19 \sqrt {-\frac {b}{a}}\, a \,b^{3} c \,d^{3} x^{5}-6 \sqrt {-\frac {b}{a}}\, b^{4} c^{2} d^{2} x^{5}+15 \sqrt {-\frac {b}{a}}\, a^{3} b \,d^{4} x^{3}-29 \sqrt {-\frac {b}{a}}\, a^{2} b^{2} c \,d^{3} x^{3}+55 \sqrt {-\frac {b}{a}}\, a \,b^{3} c^{2} d^{2} x^{3}-24 \sqrt {-\frac {b}{a}}\, b^{4} c^{3} d \,x^{3}+60 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{3} b c \,d^{3}-164 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} b^{2} c^{2} d^{2}+152 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{3} c^{3} d -48 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{4} c^{4}-15 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{3} b c \,d^{3}+103 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} b^{2} c^{2} d^{2}-128 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{3} c^{3} d +48 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{4} c^{4}+15 \sqrt {-\frac {b}{a}}\, a^{4} d^{4} x -45 \sqrt {-\frac {b}{a}}\, a^{3} b c \,d^{3} x +61 \sqrt {-\frac {b}{a}}\, a^{2} b^{2} c^{2} d^{2} x -24 \sqrt {-\frac {b}{a}}\, a \,b^{3} c^{3} d x \right )}{15 d^{4} \left (b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c \right ) \sqrt {-\frac {b}{a}}\, c}\)	\(755\)

[In]

int((b*x^2+a)^(7/2)/(d*x^2+c)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/15*b^2*x*(3*b*d*x^2+16*a*d-9*b*c)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d^3+1/15/d^3*(-b^2*(58*a^2*d^2-83*a*b*c*d+
33*b^2*c^2)*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*(Elliptic
F(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*(60*a^3*d^3-1
06*a^2*b*c*d^2+69*a*b^2*c^2*d-15*b^3*c^3)/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))+(15*a^4*d^4-60*a^3*b*c*d^3+90*a^2*b^2*c^
2*d^2-60*a*b^3*c^3*d+15*b^4*c^4)/d*((b*d*x^2+a*d)/c/(a*d-b*c)*x/((x^2+c/d)*(b*d*x^2+a*d))^(1/2)+(1/c-1/c/(a*d-
b*c)*a*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))+b/(a*d-b*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))-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 = 472, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b x^2\right )^{7/2}}{\left (c+d x^2\right )^{3/2}} \, dx=-\frac {{\left ({\left (48 \, b^{3} c^{4} d - 128 \, a b^{2} c^{3} d^{2} + 103 \, a^{2} b c^{2} d^{3} - 15 \, a^{3} c d^{4}\right )} x^{3} + {\left (48 \, b^{3} c^{5} - 128 \, a b^{2} c^{4} d + 103 \, a^{2} b c^{3} d^{2} - 15 \, a^{3} c^{2} d^{3}\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (48 \, b^{3} c^{4} d - 128 \, a b^{2} c^{3} d^{2} + 45 \, a^{3} d^{5} + {\left (103 \, a^{2} b + 24 \, a b^{2}\right )} c^{2} d^{3} - {\left (15 \, a^{3} + 61 \, a^{2} b\right )} c d^{4}\right )} x^{3} + {\left (48 \, b^{3} c^{5} - 128 \, a b^{2} c^{4} d + 45 \, a^{3} c d^{4} + {\left (103 \, a^{2} b + 24 \, a b^{2}\right )} c^{3} d^{2} - {\left (15 \, a^{3} + 61 \, a^{2} b\right )} c^{2} d^{3}\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (3 \, b^{3} c d^{4} x^{6} + 48 \, b^{3} c^{4} d - 128 \, a b^{2} c^{3} d^{2} + 103 \, a^{2} b c^{2} d^{3} - 15 \, a^{3} c d^{4} - 2 \, {\left (3 \, b^{3} c^{2} d^{3} - 8 \, a b^{2} c d^{4}\right )} x^{4} + {\left (24 \, b^{3} c^{3} d^{2} - 67 \, a b^{2} c^{2} d^{3} + 58 \, a^{2} b c d^{4}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{15 \, {\left (c d^{6} x^{3} + c^{2} d^{5} x\right )}} \]

[In]

integrate((b*x^2+a)^(7/2)/(d*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

-1/15*(((48*b^3*c^4*d - 128*a*b^2*c^3*d^2 + 103*a^2*b*c^2*d^3 - 15*a^3*c*d^4)*x^3 + (48*b^3*c^5 - 128*a*b^2*c^
4*d + 103*a^2*b*c^3*d^2 - 15*a^3*c^2*d^3)*x)*sqrt(b*d)*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c))
- ((48*b^3*c^4*d - 128*a*b^2*c^3*d^2 + 45*a^3*d^5 + (103*a^2*b + 24*a*b^2)*c^2*d^3 - (15*a^3 + 61*a^2*b)*c*d^4
)*x^3 + (48*b^3*c^5 - 128*a*b^2*c^4*d + 45*a^3*c*d^4 + (103*a^2*b + 24*a*b^2)*c^3*d^2 - (15*a^3 + 61*a^2*b)*c^
2*d^3)*x)*sqrt(b*d)*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (3*b^3*c*d^4*x^6 + 48*b^3*c^4*d -
128*a*b^2*c^3*d^2 + 103*a^2*b*c^2*d^3 - 15*a^3*c*d^4 - 2*(3*b^3*c^2*d^3 - 8*a*b^2*c*d^4)*x^4 + (24*b^3*c^3*d^
2 - 67*a*b^2*c^2*d^3 + 58*a^2*b*c*d^4)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(c*d^6*x^3 + c^2*d^5*x)

Sympy [F]

\[ \int \frac {\left (a+b x^2\right )^{7/2}}{\left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {7}{2}}}{\left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate((b*x**2+a)**(7/2)/(d*x**2+c)**(3/2),x)

[Out]

Integral((a + b*x**2)**(7/2)/(c + d*x**2)**(3/2), x)

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{7/2}}{\left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((b*x^2+a)^(7/2)/(d*x^2+c)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(7/2)/(d*x^2 + c)^(3/2), x)

Giac [F]

\[ \int \frac {\left (a+b x^2\right )^{7/2}}{\left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((b*x^2+a)^(7/2)/(d*x^2+c)^(3/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(7/2)/(d*x^2 + c)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{7/2}}{\left (c+d x^2\right )^{3/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{7/2}}{{\left (d\,x^2+c\right )}^{3/2}} \,d x \]

[In]

int((a + b*x^2)^(7/2)/(c + d*x^2)^(3/2),x)

[Out]

int((a + b*x^2)^(7/2)/(c + d*x^2)^(3/2), x)