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

Optimal. Leaf size=425 \[ \frac{2 c^{5/4} \sqrt{e} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (39 a^2 d^2+b c (7 b c-26 a d)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{195 d^{11/4} \sqrt{c+d x^2}}-\frac{4 c^{5/4} \sqrt{e} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (39 a^2 d^2+b c (7 b c-26 a d)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{195 d^{11/4} \sqrt{c+d x^2}}+\frac{2 (e x)^{3/2} \sqrt{c+d x^2} \left (39 a^2 d^2+b c (7 b c-26 a d)\right )}{195 d^2 e}+\frac{4 c \sqrt{e x} \sqrt{c+d x^2} \left (39 a^2 d^2+b c (7 b c-26 a d)\right )}{195 d^{5/2} \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 b (e x)^{3/2} \left (c+d x^2\right )^{3/2} (7 b c-26 a d)}{117 d^2 e}+\frac{2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{13 d e^3} \]

[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*ArcTan[(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])

_______________________________________________________________________________________

Rubi [A]  time = 0.98075, antiderivative size = 425, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{2 c^{5/4} \sqrt{e} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (39 a^2 d^2+b c (7 b c-26 a d)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{195 d^{11/4} \sqrt{c+d x^2}}-\frac{4 c^{5/4} \sqrt{e} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (39 a^2 d^2+b c (7 b c-26 a d)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{195 d^{11/4} \sqrt{c+d x^2}}+\frac{2 (e x)^{3/2} \sqrt{c+d x^2} \left (39 a^2 d^2+b c (7 b c-26 a d)\right )}{195 d^2 e}+\frac{4 c \sqrt{e x} \sqrt{c+d x^2} \left (39 a^2 d^2+b c (7 b c-26 a d)\right )}{195 d^{5/2} \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 b (e x)^{3/2} \left (c+d x^2\right )^{3/2} (7 b c-26 a d)}{117 d^2 e}+\frac{2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{13 d e^3} \]

Antiderivative was successfully verified.

[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*ArcTan[(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])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 89.4098, size = 405, normalized size = 0.95 \[ \frac{2 b^{2} \left (e x\right )^{\frac{7}{2}} \left (c + d x^{2}\right )^{\frac{3}{2}}}{13 d e^{3}} + \frac{2 b \left (e x\right )^{\frac{3}{2}} \left (c + d x^{2}\right )^{\frac{3}{2}} \left (26 a d - 7 b c\right )}{117 d^{2} e} - \frac{4 c^{\frac{5}{4}} \sqrt{e} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (39 a^{2} d^{2} - b c \left (26 a d - 7 b c\right )\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{195 d^{\frac{11}{4}} \sqrt{c + d x^{2}}} + \frac{2 c^{\frac{5}{4}} \sqrt{e} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (39 a^{2} d^{2} - b c \left (26 a d - 7 b c\right )\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{195 d^{\frac{11}{4}} \sqrt{c + d x^{2}}} + \frac{4 c \sqrt{e x} \sqrt{c + d x^{2}} \left (39 a^{2} d^{2} - b c \left (26 a d - 7 b c\right )\right )}{195 d^{\frac{5}{2}} \left (\sqrt{c} + \sqrt{d} x\right )} + \frac{2 \left (e x\right )^{\frac{3}{2}} \sqrt{c + d x^{2}} \left (39 a^{2} d^{2} - b c \left (26 a d - 7 b c\right )\right )}{195 d^{2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2*(e*x)**(1/2)*(d*x**2+c)**(1/2),x)

[Out]

2*b**2*(e*x)**(7/2)*(c + d*x**2)**(3/2)/(13*d*e**3) + 2*b*(e*x)**(3/2)*(c + d*x*
*2)**(3/2)*(26*a*d - 7*b*c)/(117*d**2*e) - 4*c**(5/4)*sqrt(e)*sqrt((c + d*x**2)/
(sqrt(c) + sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(39*a**2*d**2 - b*c*(26*a*d - 7*
b*c))*elliptic_e(2*atan(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)*sqrt(e)*sqrt((c + d*x**2)/(sqrt(c) + sqrt(d)*
x)**2)*(sqrt(c) + sqrt(d)*x)*(39*a**2*d**2 - b*c*(26*a*d - 7*b*c))*elliptic_f(2*
atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(195*d**(11/4)*sqrt(c + d*x**2
)) + 4*c*sqrt(e*x)*sqrt(c + d*x**2)*(39*a**2*d**2 - b*c*(26*a*d - 7*b*c))/(195*d
**(5/2)*(sqrt(c) + sqrt(d)*x)) + 2*(e*x)**(3/2)*sqrt(c + d*x**2)*(39*a**2*d**2 -
b*c*(26*a*d - 7*b*c))/(195*d**2*e)

_______________________________________________________________________________________

Mathematica [C]  time = 1.24173, size = 282, normalized size = 0.66 \[ \frac{2 e \left (d x^2 \left (c+d x^2\right ) \left (117 a^2 d^2+26 a b d \left (2 c+5 d x^2\right )+b^2 \left (-14 c^2+10 c d x^2+45 d^2 x^4\right )\right )+\frac{6 c \left (39 a^2 d^2-26 a b c d+7 b^2 c^2\right ) \left (\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \left (c+d x^2\right )+\sqrt{c} \sqrt{d} x^{3/2} \sqrt{\frac{c}{d x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )-\sqrt{c} \sqrt{d} x^{3/2} \sqrt{\frac{c}{d x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right )}{585 d^3 \sqrt{e x} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*e*(d*x^2*(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[(I
*Sqrt[c])/Sqrt[d]]*(c + d*x^2) - Sqrt[c]*Sqrt[d]*Sqrt[1 + c/(d*x^2)]*x^(3/2)*Ell
ipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1] + Sqrt[c]*Sqrt[d]*Sqrt[
1 + c/(d*x^2)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -
1]))/Sqrt[(I*Sqrt[c])/Sqrt[d]]))/(585*d^3*Sqrt[e*x]*Sqrt[c + d*x^2])

_______________________________________________________________________________________

Maple [A]  time = 0.065, size = 658, normalized size = 1.6 \[{\frac{2}{585\,{d}^{3}x}\sqrt{ex} \left ( 45\,{x}^{8}{b}^{2}{d}^{4}+130\,{x}^{6}ab{d}^{4}+55\,{x}^{6}{b}^{2}c{d}^{3}+234\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}{c}^{2}{d}^{2}-156\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) ab{c}^{3}d+42\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{4}-117\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}{c}^{2}{d}^{2}+78\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) ab{c}^{3}d-21\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{4}+117\,{x}^{4}{a}^{2}{d}^{4}+182\,{x}^{4}abc{d}^{3}-4\,{x}^{4}{b}^{2}{c}^{2}{d}^{2}+117\,{x}^{2}{a}^{2}c{d}^{3}+52\,{x}^{2}ab{c}^{2}{d}^{2}-14\,{x}^{2}{b}^{2}{c}^{3}d \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/585*(e*x)^(1/2)/(d*x^2+c)^(1/2)/d^3*(45*x^8*b^2*d^4+130*x^6*a*b*d^4+55*x^6*b^2
*c*d^3+234*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/
(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticE(((d*x+(-c*d)^(1/2))/(-c*
d)^(1/2))^(1/2),1/2*2^(1/2))*a^2*c^2*d^2-156*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(
1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*
EllipticE(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*b*c^3*d+42*((d*
x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(
1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticE(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)
,1/2*2^(1/2))*b^2*c^4-117*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x
+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticF(((d*x+(-c
*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*a^2*c^2*d^2+78*((d*x+(-c*d)^(1/2))/(
-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(
1/2)*d)^(1/2)*EllipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*b
*c^3*d-21*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(
-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticF(((d*x+(-c*d)^(1/2))/(-c*d
)^(1/2))^(1/2),1/2*2^(1/2))*b^2*c^4+117*x^4*a^2*d^4+182*x^4*a*b*c*d^3-4*x^4*b^2*
c^2*d^2+117*x^2*a^2*c*d^3+52*x^2*a*b*c^2*d^2-14*x^2*b^2*c^3*d)/x

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{2} \sqrt{d x^{2} + c} \sqrt{e x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)*sqrt(e*x),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt{d x^{2} + c} \sqrt{e x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)*sqrt(e*x),x, algorithm="fricas")

[Out]

integral((b^2*x^4 + 2*a*b*x^2 + a^2)*sqrt(d*x^2 + c)*sqrt(e*x), x)

_______________________________________________________________________________________

Sympy [A]  time = 23.6811, size = 148, normalized size = 0.35 \[ \frac{a^{2} \sqrt{c} \left (e x\right )^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 e \Gamma \left (\frac{7}{4}\right )} + \frac{a b \sqrt{c} \left (e x\right )^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{e^{3} \Gamma \left (\frac{11}{4}\right )} + \frac{b^{2} \sqrt{c} \left (e x\right )^{\frac{11}{2}} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{5} \Gamma \left (\frac{15}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{2} \sqrt{d x^{2} + c} \sqrt{e x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)*sqrt(e*x),x, algorithm="giac")

[Out]

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