\(\int \sqrt {e x} (a+b x^2)^2 \sqrt {c+d x^2} \, dx\) [823]

Optimal result

Integrand size = 28, antiderivative size = 425 \[ \int \sqrt {e x} \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=\frac {2 \left (39 a^2 d^2+b c (7 b c-26 a d)\right ) (e x)^{3/2} \sqrt {c+d x^2}}{195 d^2 e}+\frac {4 c \left (39 a^2 d^2+b c (7 b c-26 a d)\right ) \sqrt {e x} \sqrt {c+d x^2}}{195 d^{5/2} \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 b (7 b c-26 a d) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{117 d^2 e}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{13 d e^3}-\frac {4 c^{5/4} \left (39 a^2 d^2+b c (7 b c-26 a d)\right ) \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{195 d^{11/4} \sqrt {c+d x^2}}+\frac {2 c^{5/4} \left (39 a^2 d^2+b c (7 b c-26 a d)\right ) \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{195 d^{11/4} \sqrt {c+d x^2}} \]

[Out]

-2/117*b*(-26*a*d+7*b*c)*(e*x)^(3/2)*(d*x^2+c)^(3/2)/d^2/e+2/13*b^2*(e*x)^(7/2)*(d*x^2+c)^(3/2)/d/e^3+2/195*(3
9*a^2*d^2+b*c*(-26*a*d+7*b*c))*(e*x)^(3/2)*(d*x^2+c)^(1/2)/d^2/e+4/195*c*(39*a^2*d^2+b*c*(-26*a*d+7*b*c))*(e*x
)^(1/2)*(d*x^2+c)^(1/2)/d^(5/2)/(c^(1/2)+x*d^(1/2))-4/195*c^(5/4)*(39*a^2*d^2+b*c*(-26*a*d+7*b*c))*(cos(2*arct
an(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2)))^2)^(1/2)/cos(2*arctan(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2)))*EllipticE
(sin(2*arctan(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2))),1/2*2^(1/2))*(c^(1/2)+x*d^(1/2))*e^(1/2)*((d*x^2+c)/(c^(1/
2)+x*d^(1/2))^2)^(1/2)/d^(11/4)/(d*x^2+c)^(1/2)+2/195*c^(5/4)*(39*a^2*d^2+b*c*(-26*a*d+7*b*c))*(cos(2*arctan(d
^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2)))^2)^(1/2)/cos(2*arctan(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2)))*EllipticF(sin
(2*arctan(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2))),1/2*2^(1/2))*(c^(1/2)+x*d^(1/2))*e^(1/2)*((d*x^2+c)/(c^(1/2)+x
*d^(1/2))^2)^(1/2)/d^(11/4)/(d*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {475, 470, 285, 335, 311, 226, 1210} \[ \int \sqrt {e x} \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=\frac {2 c^{5/4} \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (39 a^2 d^2+b c (7 b c-26 a d)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{195 d^{11/4} \sqrt {c+d x^2}}-\frac {4 c^{5/4} \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (39 a^2 d^2+b c (7 b c-26 a d)\right ) E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{195 d^{11/4} \sqrt {c+d x^2}}+\frac {2 (e x)^{3/2} \sqrt {c+d x^2} \left (39 a^2 d^2+b c (7 b c-26 a d)\right )}{195 d^2 e}+\frac {4 c \sqrt {e x} \sqrt {c+d x^2} \left (39 a^2 d^2+b c (7 b c-26 a d)\right )}{195 d^{5/2} \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 b (e x)^{3/2} \left (c+d x^2\right )^{3/2} (7 b c-26 a d)}{117 d^2 e}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{13 d e^3} \]

[In]

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

[Out]

(2*(39*a^2*d^2 + b*c*(7*b*c - 26*a*d))*(e*x)^(3/2)*Sqrt[c + d*x^2])/(195*d^2*e) + (4*c*(39*a^2*d^2 + b*c*(7*b*
c - 26*a*d))*Sqrt[e*x]*Sqrt[c + d*x^2])/(195*d^(5/2)*(Sqrt[c] + Sqrt[d]*x)) - (2*b*(7*b*c - 26*a*d)*(e*x)^(3/2
)*(c + d*x^2)^(3/2))/(117*d^2*e) + (2*b^2*(e*x)^(7/2)*(c + d*x^2)^(3/2))/(13*d*e^3) - (4*c^(5/4)*(39*a^2*d^2 +
b*c*(7*b*c - 26*a*d))*Sqrt[e]*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticE[2*Arc
Tan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(195*d^(11/4)*Sqrt[c + d*x^2]) + (2*c^(5/4)*(39*a^2*d^2 + b*
c*(7*b*c - 26*a*d))*Sqrt[e]*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticF[2*ArcTan
[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(195*d^(11/4)*Sqrt[c + d*x^2])

Rule 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 285

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

Rule 311

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, Dist[1/q, Int[1/Sqrt[a + b*x^4], x],
x] - Dist[1/q, Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 470

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

Rule 475

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

Rule 1210

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a +
c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4
]))*EllipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rubi steps \begin{align*} \text {integral}& = \frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{13 d e^3}+\frac {2 \int \sqrt {e x} \sqrt {c+d x^2} \left (\frac {13 a^2 d}{2}-\frac {1}{2} b (7 b c-26 a d) x^2\right ) \, dx}{13 d} \\ & = -\frac {2 b (7 b c-26 a d) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{117 d^2 e}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{13 d e^3}+\frac {1}{39} \left (39 a^2+\frac {b c (7 b c-26 a d)}{d^2}\right ) \int \sqrt {e x} \sqrt {c+d x^2} \, dx \\ & = \frac {2 \left (39 a^2+\frac {b c (7 b c-26 a d)}{d^2}\right ) (e x)^{3/2} \sqrt {c+d x^2}}{195 e}-\frac {2 b (7 b c-26 a d) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{117 d^2 e}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{13 d e^3}+\frac {1}{195} \left (2 c \left (39 a^2+\frac {b c (7 b c-26 a d)}{d^2}\right )\right ) \int \frac {\sqrt {e x}}{\sqrt {c+d x^2}} \, dx \\ & = \frac {2 \left (39 a^2+\frac {b c (7 b c-26 a d)}{d^2}\right ) (e x)^{3/2} \sqrt {c+d x^2}}{195 e}-\frac {2 b (7 b c-26 a d) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{117 d^2 e}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{13 d e^3}+\frac {\left (4 c \left (39 a^2+\frac {b c (7 b c-26 a d)}{d^2}\right )\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{195 e} \\ & = \frac {2 \left (39 a^2+\frac {b c (7 b c-26 a d)}{d^2}\right ) (e x)^{3/2} \sqrt {c+d x^2}}{195 e}-\frac {2 b (7 b c-26 a d) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{117 d^2 e}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{13 d e^3}+\frac {\left (4 c^{3/2} \left (39 a^2+\frac {b c (7 b c-26 a d)}{d^2}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{195 \sqrt {d}}-\frac {\left (4 c^{3/2} \left (39 a^2+\frac {b c (7 b c-26 a d)}{d^2}\right )\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {d} x^2}{\sqrt {c} e}}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{195 \sqrt {d}} \\ & = \frac {2 \left (39 a^2+\frac {b c (7 b c-26 a d)}{d^2}\right ) (e x)^{3/2} \sqrt {c+d x^2}}{195 e}+\frac {4 c \left (39 a^2+\frac {b c (7 b c-26 a d)}{d^2}\right ) \sqrt {e x} \sqrt {c+d x^2}}{195 \sqrt {d} \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 b (7 b c-26 a d) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{117 d^2 e}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{13 d e^3}-\frac {4 c^{5/4} \left (39 a^2+\frac {b c (7 b c-26 a d)}{d^2}\right ) \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{195 d^{3/4} \sqrt {c+d x^2}}+\frac {2 c^{5/4} \left (39 a^2+\frac {b c (7 b c-26 a d)}{d^2}\right ) \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{195 d^{3/4} \sqrt {c+d x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 21.24 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.34 \[ \int \sqrt {e x} \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=\frac {2 \sqrt {e x} \left (-x \left (c+d x^2\right ) \left (-117 a^2 d^2-26 a b d \left (2 c+5 d x^2\right )+b^2 \left (14 c^2-10 c d x^2-45 d^2 x^4\right )\right )+6 c \left (7 b^2 c^2-26 a b c d+39 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-\frac {c}{d x^2}\right )\right )}{585 d^2 \sqrt {c+d x^2}} \]

[In]

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

[Out]

(2*Sqrt[e*x]*(-(x*(c + d*x^2)*(-117*a^2*d^2 - 26*a*b*d*(2*c + 5*d*x^2) + b^2*(14*c^2 - 10*c*d*x^2 - 45*d^2*x^4
))) + 6*c*(7*b^2*c^2 - 26*a*b*c*d + 39*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*Hypergeometric2F1[-1/4, 1/2, 3/4, -(c/(d
*x^2))]))/(585*d^2*Sqrt[c + d*x^2])

Maple [A] (verified)

Time = 3.12 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.68

method	result	size
risch	\(\frac {2 x^{2} \left (45 b^{2} d^{2} x^{4}+130 x^{2} a b \,d^{2}+10 x^{2} b^{2} c d +117 a^{2} d^{2}+52 a b c d -14 b^{2} c^{2}\right ) \sqrt {d \,x^{2}+c}\, e}{585 d^{2} \sqrt {e x}}+\frac {2 c \left (39 a^{2} d^{2}-26 a b c d +7 b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right ) e \sqrt {e x \left (d \,x^{2}+c \right )}}{195 d^{3} \sqrt {d e \,x^{3}+c e x}\, \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\)	\(290\)
elliptic	\(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \sqrt {e x}\, \left (\frac {2 b^{2} x^{5} \sqrt {d e \,x^{3}+c e x}}{13}+\frac {2 \left (b \left (2 a d +b c \right ) e -\frac {11 b^{2} c e}{13}\right ) x^{3} \sqrt {d e \,x^{3}+c e x}}{9 d e}+\frac {2 \left (a \left (a d +2 b c \right ) e -\frac {7 \left (b \left (2 a d +b c \right ) e -\frac {11 b^{2} c e}{13}\right ) c}{9 d}\right ) x \sqrt {d e \,x^{3}+c e x}}{5 d e}+\frac {\left (a^{2} c e -\frac {3 \left (a \left (a d +2 b c \right ) e -\frac {7 \left (b \left (2 a d +b c \right ) e -\frac {11 b^{2} c e}{13}\right ) c}{9 d}\right ) c}{5 d}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{e x \sqrt {d \,x^{2}+c}}\)	\(367\)
default	\(\frac {2 \sqrt {e x}\, \left (45 b^{2} d^{4} x^{8}+130 a b \,d^{4} x^{6}+55 b^{2} c \,d^{3} x^{6}+234 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c^{2} d^{2}-156 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{3} d +42 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{4}-117 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c^{2} d^{2}+78 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{3} d -21 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{4}+117 a^{2} d^{4} x^{4}+182 c a b \,x^{4} d^{3}-4 b^{2} c^{2} d^{2} x^{4}+117 a^{2} c \,d^{3} x^{2}+52 a b \,c^{2} d^{2} x^{2}-14 b^{2} c^{3} d \,x^{2}\right )}{585 \sqrt {d \,x^{2}+c}\, d^{3} x}\)	\(658\)

[In]

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

[Out]

2/585*x^2*(45*b^2*d^2*x^4+130*a*b*d^2*x^2+10*b^2*c*d*x^2+117*a^2*d^2+52*a*b*c*d-14*b^2*c^2)*(d*x^2+c)^(1/2)/d^
2*e/(e*x)^(1/2)+2/195*c*(39*a^2*d^2-26*a*b*c*d+7*b^2*c^2)/d^3*(-c*d)^(1/2)*((x+(-c*d)^(1/2)/d)/(-c*d)^(1/2)*d)
^(1/2)*(-2*(x-(-c*d)^(1/2)/d)/(-c*d)^(1/2)*d)^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)/(d*e*x^3+c*e*x)^(1/2)*(-2*(-c*d)
^(1/2)/d*EllipticE(((x+(-c*d)^(1/2)/d)/(-c*d)^(1/2)*d)^(1/2),1/2*2^(1/2))+(-c*d)^(1/2)/d*EllipticF(((x+(-c*d)^
(1/2)/d)/(-c*d)^(1/2)*d)^(1/2),1/2*2^(1/2)))*e*(e*x*(d*x^2+c))^(1/2)/(e*x)^(1/2)/(d*x^2+c)^(1/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.10 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.32 \[ \int \sqrt {e x} \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=-\frac {2 \, {\left (6 \, {\left (7 \, b^{2} c^{3} - 26 \, a b c^{2} d + 39 \, a^{2} c d^{2}\right )} \sqrt {d e} {\rm weierstrassZeta}\left (-\frac {4 \, c}{d}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right )\right ) - {\left (45 \, b^{2} d^{3} x^{5} + 10 \, {\left (b^{2} c d^{2} + 13 \, a b d^{3}\right )} x^{3} - {\left (14 \, b^{2} c^{2} d - 52 \, a b c d^{2} - 117 \, a^{2} d^{3}\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {e x}\right )}}{585 \, d^{3}} \]

[In]

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

[Out]

-2/585*(6*(7*b^2*c^3 - 26*a*b*c^2*d + 39*a^2*c*d^2)*sqrt(d*e)*weierstrassZeta(-4*c/d, 0, weierstrassPInverse(-
4*c/d, 0, x)) - (45*b^2*d^3*x^5 + 10*(b^2*c*d^2 + 13*a*b*d^3)*x^3 - (14*b^2*c^2*d - 52*a*b*c*d^2 - 117*a^2*d^3
)*x)*sqrt(d*x^2 + c)*sqrt(e*x))/d^3

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.24 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.35 \[ \int \sqrt {e x} \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=\frac {a^{2} \sqrt {c} \sqrt {e} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \Gamma \left (\frac {7}{4}\right )} + \frac {a b \sqrt {c} \sqrt {e} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{\Gamma \left (\frac {11}{4}\right )} + \frac {b^{2} \sqrt {c} \sqrt {e} x^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \Gamma \left (\frac {15}{4}\right )} \]

[In]

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

[Out]

a**2*sqrt(c)*sqrt(e)*x**(3/2)*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), d*x**2*exp_polar(I*pi)/c)/(2*gamma(7/4)) +
a*b*sqrt(c)*sqrt(e)*x**(7/2)*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), d*x**2*exp_polar(I*pi)/c)/gamma(11/4) + b
**2*sqrt(c)*sqrt(e)*x**(11/2)*gamma(11/4)*hyper((-1/2, 11/4), (15/4,), d*x**2*exp_polar(I*pi)/c)/(2*gamma(15/4
))

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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