3.907 \(\int \frac{\left (c-d x^2\right )^{3/2}}{\sqrt{e x} \left (a-b x^2\right )^2} \, dx\)
Optimal. Leaf size=366 \[ \frac{3 \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (a d+b c) (b c-a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^2 b^2 \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2}}+\frac{3 \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (a d+b c) (b c-a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^2 b^2 \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} d^{3/4} \sqrt{1-\frac{d x^2}{c}} (3 a d+b c) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 a b^2 \sqrt{e} \sqrt{c-d x^2}}+\frac{\sqrt{e x} \sqrt{c-d x^2} (b c-a d)}{2 a b e \left (a-b x^2\right )} \]
[Out]
((b*c - a*d)*Sqrt[e*x]*Sqrt[c - d*x^2])/(2*a*b*e*(a - b*x^2)) + (c^(1/4)*d^(3/4)
*(b*c + 3*a*d)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)
*Sqrt[e])], -1])/(2*a*b^2*Sqrt[e]*Sqrt[c - d*x^2]) + (3*c^(1/4)*(b*c - a*d)*(b*c
+ a*d)*Sqrt[1 - (d*x^2)/c]*EllipticPi[-((Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d])), A
rcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(4*a^2*b^2*d^(1/4)*Sqrt[e]*Sq
rt[c - d*x^2]) + (3*c^(1/4)*(b*c - a*d)*(b*c + a*d)*Sqrt[1 - (d*x^2)/c]*Elliptic
Pi[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt
[e])], -1])/(4*a^2*b^2*d^(1/4)*Sqrt[e]*Sqrt[c - d*x^2])
_______________________________________________________________________________________
Rubi [A] time = 1.48969, antiderivative size = 366, normalized size of antiderivative = 1.,
number of steps used = 10, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267
\[ \frac{3 \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (a d+b c) (b c-a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^2 b^2 \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2}}+\frac{3 \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (a d+b c) (b c-a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^2 b^2 \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} d^{3/4} \sqrt{1-\frac{d x^2}{c}} (3 a d+b c) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 a b^2 \sqrt{e} \sqrt{c-d x^2}}+\frac{\sqrt{e x} \sqrt{c-d x^2} (b c-a d)}{2 a b e \left (a-b x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(c - d*x^2)^(3/2)/(Sqrt[e*x]*(a - b*x^2)^2),x]
[Out]
((b*c - a*d)*Sqrt[e*x]*Sqrt[c - d*x^2])/(2*a*b*e*(a - b*x^2)) + (c^(1/4)*d^(3/4)
*(b*c + 3*a*d)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)
*Sqrt[e])], -1])/(2*a*b^2*Sqrt[e]*Sqrt[c - d*x^2]) + (3*c^(1/4)*(b*c - a*d)*(b*c
+ a*d)*Sqrt[1 - (d*x^2)/c]*EllipticPi[-((Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d])), A
rcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(4*a^2*b^2*d^(1/4)*Sqrt[e]*Sq
rt[c - d*x^2]) + (3*c^(1/4)*(b*c - a*d)*(b*c + a*d)*Sqrt[1 - (d*x^2)/c]*Elliptic
Pi[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt
[e])], -1])/(4*a^2*b^2*d^(1/4)*Sqrt[e]*Sqrt[c - d*x^2])
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-d*x**2+c)**(3/2)/(e*x)**(1/2)/(-b*x**2+a)**2,x)
[Out]
Timed out
_______________________________________________________________________________________
Mathematica [C] time = 0.647606, size = 428, normalized size = 1.17 \[ \frac{x \left (\frac{10 x^2 \left (d x^2-c\right ) (b c-a d) \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+9 a c \left (a d \left (5 c-2 d x^2\right )+b c \left (6 d x^2-5 c\right )\right ) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{a \left (2 x^2 \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}-\frac{25 c^2 (a d+3 b c) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{2 x^2 \left (2 b c F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}\right )}{10 b \sqrt{e x} \left (b x^2-a\right ) \sqrt{c-d x^2}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(c - d*x^2)^(3/2)/(Sqrt[e*x]*(a - b*x^2)^2),x]
[Out]
(x*((-25*c^2*(3*b*c + a*d)*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a])/(5*
a*c*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + 2*x^2*(2*b*c*AppellF1[5/4
, 1/2, 2, 9/4, (d*x^2)/c, (b*x^2)/a] + a*d*AppellF1[5/4, 3/2, 1, 9/4, (d*x^2)/c,
(b*x^2)/a])) + (9*a*c*(a*d*(5*c - 2*d*x^2) + b*c*(-5*c + 6*d*x^2))*AppellF1[5/4
, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a] + 10*(b*c - a*d)*x^2*(-c + d*x^2)*(2*b*c*Ap
pellF1[9/4, 1/2, 2, 13/4, (d*x^2)/c, (b*x^2)/a] + a*d*AppellF1[9/4, 3/2, 1, 13/4
, (d*x^2)/c, (b*x^2)/a]))/(a*(9*a*c*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2
)/a] + 2*x^2*(2*b*c*AppellF1[9/4, 1/2, 2, 13/4, (d*x^2)/c, (b*x^2)/a] + a*d*Appe
llF1[9/4, 3/2, 1, 13/4, (d*x^2)/c, (b*x^2)/a])))))/(10*b*Sqrt[e*x]*(-a + b*x^2)*
Sqrt[c - d*x^2])
_______________________________________________________________________________________
Maple [B] time = 0.04, size = 2531, normalized size = 6.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-d*x^2+c)^(3/2)/(e*x)^(1/2)/(-b*x^2+a)^2,x)
[Out]
1/8*(-d*x^2+c)^(1/2)/b*d*(-2*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2
*2^(1/2))*2^(1/2)*x^2*b^3*c^2*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)
^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)+2*El
lipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*a*b^2*c^2*((d
*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/
(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)+3*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)
^(1/2))^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2^(1/2))*2^(1/2)*x
^2*a^2*b^2*c*d^2*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)
^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)-3*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/
2))^(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2))*2^(1/2)*x^2*a
^2*b^2*c*d^2*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/
2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)+4*x*a^2*b*c*d^2*(a*b)^(1/2)-8*x*a*b^2*c^2*d*(
a*b)^(1/2)-4*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2
)*x^2*a*b^2*c*d*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^
(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)+3*(c*d)^(1/2)*((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*d/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))
^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2^(1/2))*x^2*b^3*c^2+3*(c
*d)^(1/2)*((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*d/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)*EllipticPi(((d*x+(c*d)^(1/2)
)/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2))*x^
2*b^3*c^2-3*(c*d)^(1/2)*((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*d/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)*EllipticPi(((d
*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1
/2*2^(1/2))*a*b^2*c^2-3*(c*d)^(1/2)*((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*d/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)*El
lipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b-(a*b
)^(1/2)*d),1/2*2^(1/2))*a*b^2*c^2-3*((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*d/(c*d)^(1/2))^(1/2)*EllipticPi(((d
*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1
/2*2^(1/2))*x^2*b^4*c^3+3*((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*d/(c*d)^(1/2))^(1/2)*EllipticPi(((d*x+(c*d)^(
1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2)
)*x^2*b^4*c^3+3*((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*d/(c*d)^(1/2))^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d
)^(1/2))^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2^(1/2))*a*b^3*c^
3-3*((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*d/(c*d)^(1/2))^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/
2),(c*d)^(1/2)*b/((c*d)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2))*a*b^3*c^3-4*x^3*a^2*
b*d^3*(a*b)^(1/2)-4*x^3*b^3*c^2*d*(a*b)^(1/2)-3*EllipticPi(((d*x+(c*d)^(1/2))/(c
*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2^(1/2))*2^(1/2
)*x^2*a^2*b*d^2*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^
(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)-3*EllipticPi(((d*x
+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b-(a*b)^(1/2)*d),1/2
*2^(1/2))*2^(1/2)*x^2*a^2*b*d^2*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*
d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)+6*
EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*x^2*a^2*b*d
^2*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*
(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)+4*EllipticF(((d*x+(c*d)^(1/2))/
(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*a^2*b*c*d*((d*x+(c*d)^(1/2))/(c*d)^(1/2)
)^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1
/2)*(a*b)^(1/2)+4*x*b^3*c^3*(a*b)^(1/2)+8*x^3*a*b^2*c*d^2*(a*b)^(1/2)-6*Elliptic
F(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*a^3*d^2*((d*x+(c*d)
^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1
/2))^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)-3*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^
(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2^(1/2))*2^(1/2)*a^3*b*c*d
^2*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*
(-x*d/(c*d)^(1/2))^(1/2)+3*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d
)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2^(1/2))*2^(1/2)*a^3*d^2*((d*x+(c*d)
^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1
/2))^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)+3*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^
(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2))*2^(1/2)*a^3*b*c*d
^2*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*
(-x*d/(c*d)^(1/2))^(1/2)+3*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d
)^(1/2)*b/((c*d)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2))*2^(1/2)*a^3*d^2*((d*x+(c*d)
^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1
/2))^(1/2)*(c*d)^(1/2)*(a*b)^(1/2))/a/(e*x)^(1/2)/(d*x^2-c)/(b*x^2-a)/(a*b)^(1/2
)/((a*b)^(1/2)*d+(c*d)^(1/2)*b)/((c*d)^(1/2)*b-(a*b)^(1/2)*d)
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-d x^{2} + c\right )}^{\frac{3}{2}}}{{\left (b x^{2} - a\right )}^{2} \sqrt{e x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-d*x^2 + c)^(3/2)/((b*x^2 - a)^2*sqrt(e*x)),x, algorithm="maxima")
[Out]
integrate((-d*x^2 + c)^(3/2)/((b*x^2 - a)^2*sqrt(e*x)), x)
_______________________________________________________________________________________
Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-d*x^2 + c)^(3/2)/((b*x^2 - a)^2*sqrt(e*x)),x, algorithm="fricas")
[Out]
Exception raised: TypeError
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-d*x**2+c)**(3/2)/(e*x)**(1/2)/(-b*x**2+a)**2,x)
[Out]
Timed out
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-d x^{2} + c\right )}^{\frac{3}{2}}}{{\left (b x^{2} - a\right )}^{2} \sqrt{e x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-d*x^2 + c)^(3/2)/((b*x^2 - a)^2*sqrt(e*x)),x, algorithm="giac")
[Out]
integrate((-d*x^2 + c)^(3/2)/((b*x^2 - a)^2*sqrt(e*x)), x)