\(\int \frac {x^2 (a+b x^2)^{5/2}}{\sqrt {c+d x^2}} \, dx\) [961]

Optimal result

Integrand size = 26, antiderivative size = 436 \[ \int \frac {x^2 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=-\frac {\left (48 b^3 c^3-128 a b^2 c^2 d+103 a^2 b c d^2-15 a^3 d^3\right ) x \sqrt {a+b x^2}}{105 b d^3 \sqrt {c+d x^2}}+\frac {\left (24 b^2 c^2-61 a b c d+45 a^2 d^2\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{105 d^3}-\frac {2 b (3 b c-5 a d) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{35 d^2}+\frac {b x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 d}+\frac {\sqrt {c} \left (48 b^3 c^3-128 a b^2 c^2 d+103 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{105 b d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {c^{3/2} \left (24 b^2 c^2-61 a b c d+45 a^2 d^2\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{105 d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]

[Out]

-1/105*(-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)/b/d^3/(d*x^2+c)^(1/2)-1/105*
c^(3/2)*(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))*(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)+
1/105*(-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)*Ellipti
cE(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/d^(7/2)/(c*(b*x^2+a)/a/(d*
x^2+c))^(1/2)/(d*x^2+c)^(1/2)+1/7*b*x^3*(b*x^2+a)^(3/2)*(d*x^2+c)^(1/2)/d+1/105*(45*a^2*d^2-61*a*b*c*d+24*b^2*
c^2)*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d^3-2/35*b*(-5*a*d+3*b*c)*x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d^2

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {488, 595, 596, 545, 429, 506, 422} \[ \int \frac {x^2 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=-\frac {c^{3/2} \sqrt {a+b x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{105 d^{7/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right )}{105 d^3}+\frac {\sqrt {c} \sqrt {a+b x^2} \left (-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{105 b d^{7/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {x \sqrt {a+b x^2} \left (-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right )}{105 b d^3 \sqrt {c+d x^2}}-\frac {2 b x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-5 a d)}{35 d^2}+\frac {b x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 d} \]

[In]

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

[Out]

-1/105*((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])/(b*d^3*Sqrt[c + d*x^2
]) + ((24*b^2*c^2 - 61*a*b*c*d + 45*a^2*d^2)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(105*d^3) - (2*b*(3*b*c - 5*a*
d)*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(35*d^2) + (b*x^3*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(7*d) + (Sqrt[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[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqr
t[c]], 1 - (b*c)/(a*d)])/(105*b*d^(7/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (c^(3/2)*(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)])/(1
05*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 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 488

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*(e*x)^
(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(b*e*(m + n*(p + q) + 1))), x] + Dist[1/(b*(m + n*(p + q) + 1
)), Int[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*((c*b - a*d)*(m + 1) + c*b*n*(p + q)) + (d*(c*b - a*d
)*(m + 1) + d*n*(q - 1)*(b*c - a*d) + c*b*d*n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && N
eQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 1] && 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 595

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

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 {b x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 d}+\frac {\int \frac {x^2 \sqrt {a+b x^2} \left (-a (3 b c-7 a d)-2 b (3 b c-5 a d) x^2\right )}{\sqrt {c+d x^2}} \, dx}{7 d} \\ & = -\frac {2 b (3 b c-5 a d) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{35 d^2}+\frac {b x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 d}+\frac {\int \frac {x^2 \left (a \left (18 b^2 c^2-45 a b c d+35 a^2 d^2\right )+b \left (24 b^2 c^2-61 a b c d+45 a^2 d^2\right ) x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{35 d^2} \\ & = \frac {\left (24 b^2 c^2-61 a b c d+45 a^2 d^2\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{105 d^3}-\frac {2 b (3 b c-5 a d) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{35 d^2}+\frac {b x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 d}-\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}{105 b d^3} \\ & = \frac {\left (24 b^2 c^2-61 a b c d+45 a^2 d^2\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{105 d^3}-\frac {2 b (3 b c-5 a d) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{35 d^2}+\frac {b x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 d}-\frac {\left (a c \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}{105 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 ) \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{105 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}}{105 b d^3 \sqrt {c+d x^2}}+\frac {\left (24 b^2 c^2-61 a b c d+45 a^2 d^2\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{105 d^3}-\frac {2 b (3 b c-5 a d) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{35 d^2}+\frac {b x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 d}-\frac {c^{3/2} \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 )}{105 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 (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 )\right ) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{105 b 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}}{105 b d^3 \sqrt {c+d x^2}}+\frac {\left (24 b^2 c^2-61 a b c d+45 a^2 d^2\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{105 d^3}-\frac {2 b (3 b c-5 a d) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{35 d^2}+\frac {b x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 d}+\frac {\sqrt {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 {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{105 b 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 {c^{3/2} \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 )}{105 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 = 2.97 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.70 \[ \int \frac {x^2 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (45 a^2 d^2+a b d \left (-61 c+45 d x^2\right )+3 b^2 \left (8 c^2-6 c d x^2+5 d^2 x^4\right )\right )-i c \left (-48 b^3 c^3+128 a b^2 c^2 d-103 a^2 b c d^2+15 a^3 d^3\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+4 i c \left (-12 b^3 c^3+38 a b^2 c^2 d-41 a^2 b c d^2+15 a^3 d^3\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{105 \sqrt {\frac {b}{a}} d^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \]

[In]

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

[Out]

(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(45*a^2*d^2 + a*b*d*(-61*c + 45*d*x^2) + 3*b^2*(8*c^2 - 6*c*d*x^2 + 5*d
^2*x^4)) - I*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]*Sqrt[1 + (d*
x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (4*I)*c*(-12*b^3*c^3 + 38*a*b^2*c^2*d - 41*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)])/(105*S
qrt[b/a]*d^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])

Maple [A] (verified)

Time = 8.79 (sec) , antiderivative size = 563, normalized size of antiderivative = 1.29

method	result	size
risch	\(\frac {x \left (15 b^{2} d^{2} x^{4}+45 x^{2} a b \,d^{2}-18 x^{2} b^{2} c d +45 a^{2} d^{2}-61 a b c d +24 b^{2} c^{2}\right ) \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{105 d^{3}}-\frac {\left (\frac {45 a^{3} c \,d^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}+\frac {24 b^{2} c^{3} a \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {61 a^{2} b \,c^{2} d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}+\frac {\left (15 a^{3} d^{3}-103 a^{2} b c \,d^{2}+128 a \,b^{2} c^{2} d -48 b^{3} c^{3}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{105 d^{3} \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\)	\(563\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {b^{2} x^{5} \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{7 d}+\frac {\left (3 a \,b^{2}-\frac {b^{2} \left (6 a d +6 b c \right )}{7 d}\right ) x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{5 b d}+\frac {\left (3 a^{2} b -\frac {5 a \,b^{2} c}{7 d}-\frac {\left (3 a \,b^{2}-\frac {b^{2} \left (6 a d +6 b c \right )}{7 d}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{3 b d}-\frac {\left (3 a^{2} b -\frac {5 a \,b^{2} c}{7 d}-\frac {\left (3 a \,b^{2}-\frac {b^{2} \left (6 a d +6 b c \right )}{7 d}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) a c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{3 b d \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {\left (a^{3}-\frac {3 \left (3 a \,b^{2}-\frac {b^{2} \left (6 a d +6 b c \right )}{7 d}\right ) a c}{5 b d}-\frac {\left (3 a^{2} b -\frac {5 a \,b^{2} c}{7 d}-\frac {\left (3 a \,b^{2}-\frac {b^{2} \left (6 a d +6 b c \right )}{7 d}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\)	\(612\)
default	\(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (-15 \sqrt {-\frac {b}{a}}\, b^{3} d^{4} x^{9}-60 \sqrt {-\frac {b}{a}}\, a \,b^{2} d^{4} x^{7}+3 \sqrt {-\frac {b}{a}}\, b^{3} c \,d^{3} x^{7}-90 \sqrt {-\frac {b}{a}}\, a^{2} b \,d^{4} x^{5}+19 \sqrt {-\frac {b}{a}}\, a \,b^{2} c \,d^{3} x^{5}-6 \sqrt {-\frac {b}{a}}\, b^{3} c^{2} d^{2} x^{5}-45 \sqrt {-\frac {b}{a}}\, a^{3} d^{4} x^{3}-29 \sqrt {-\frac {b}{a}}\, a^{2} b c \,d^{3} x^{3}+55 \sqrt {-\frac {b}{a}}\, a \,b^{2} c^{2} d^{2} x^{3}-24 \sqrt {-\frac {b}{a}}\, b^{3} 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} 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 \,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^{2} 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^{3} 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} 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 \,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^{2} 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^{3} c^{4}-45 \sqrt {-\frac {b}{a}}\, a^{3} c \,d^{3} x +61 \sqrt {-\frac {b}{a}}\, a^{2} b \,c^{2} d^{2} x -24 \sqrt {-\frac {b}{a}}\, a \,b^{2} c^{3} d x \right )}{105 d^{4} \left (b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c \right ) \sqrt {-\frac {b}{a}}}\)	\(782\)

[In]

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

[Out]

1/105*x*(15*b^2*d^2*x^4+45*a*b*d^2*x^2-18*b^2*c*d*x^2+45*a^2*d^2-61*a*b*c*d+24*b^2*c^2)*(b*x^2+a)^(1/2)*(d*x^2
+c)^(1/2)/d^3-1/105/d^3*(45*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)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))+24*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))-61*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))+(15*a^3*d^3-103*a^2*b*c*d^2+128*a*b^2*c^2*d-48*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 = 0.09 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.74 \[ \int \frac {x^2 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\frac {{\left (48 \, b^{3} c^{4} - 128 \, a b^{2} c^{3} d + 103 \, a^{2} b c^{2} d^{2} - 15 \, a^{3} c d^{3}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (48 \, b^{3} c^{4} - 128 \, a b^{2} c^{3} d + 45 \, a^{3} d^{4} + {\left (103 \, a^{2} b + 24 \, a b^{2}\right )} c^{2} d^{2} - {\left (15 \, a^{3} + 61 \, a^{2} b\right )} c d^{3}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) + {\left (15 \, b^{3} d^{4} x^{6} - 48 \, b^{3} c^{3} d + 128 \, a b^{2} c^{2} d^{2} - 103 \, a^{2} b c d^{3} + 15 \, a^{3} d^{4} - 9 \, {\left (2 \, b^{3} c d^{3} - 5 \, a b^{2} d^{4}\right )} x^{4} + {\left (24 \, b^{3} c^{2} d^{2} - 61 \, a b^{2} c d^{3} + 45 \, a^{2} b d^{4}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{105 \, b d^{5} x} \]

[In]

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

[Out]

1/105*((48*b^3*c^4 - 128*a*b^2*c^3*d + 103*a^2*b*c^2*d^2 - 15*a^3*c*d^3)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_e(arc
sin(sqrt(-c/d)/x), a*d/(b*c)) - (48*b^3*c^4 - 128*a*b^2*c^3*d + 45*a^3*d^4 + (103*a^2*b + 24*a*b^2)*c^2*d^2 -
(15*a^3 + 61*a^2*b)*c*d^3)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) + (15*b^3*d^4*x^
6 - 48*b^3*c^3*d + 128*a*b^2*c^2*d^2 - 103*a^2*b*c*d^3 + 15*a^3*d^4 - 9*(2*b^3*c*d^3 - 5*a*b^2*d^4)*x^4 + (24*
b^3*c^2*d^2 - 61*a*b^2*c*d^3 + 45*a^2*b*d^4)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(b*d^5*x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\int \frac {x^2\,{\left (b\,x^2+a\right )}^{5/2}}{\sqrt {d\,x^2+c}} \,d x \]

[In]

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

[Out]

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