3.903 \(\int \frac{(e x)^{7/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx\)

Optimal. Leaf size=429 \[ \frac{\sqrt [4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} \left (231 a^2 d^2-259 a b c d+48 b^2 c^2\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{42 b^4 \sqrt [4]{d} \sqrt{c-d x^2}}-\frac{\sqrt [4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (5 b c-11 a d) (b c-a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 b^4 \sqrt [4]{d} \sqrt{c-d x^2}}-\frac{\sqrt [4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (5 b c-11 a d) (b c-a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 b^4 \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{e^3 \sqrt{e x} \sqrt{c-d x^2} (57 b c-77 a d)}{42 b^3}+\frac{e (e x)^{5/2} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}-\frac{11 d e (e x)^{5/2} \sqrt{c-d x^2}}{14 b^2} \]

[Out]

((57*b*c - 77*a*d)*e^3*Sqrt[e*x]*Sqrt[c - d*x^2])/(42*b^3) - (11*d*e*(e*x)^(5/2)
*Sqrt[c - d*x^2])/(14*b^2) + (e*(e*x)^(5/2)*(c - d*x^2)^(3/2))/(2*b*(a - b*x^2))
+ (c^(1/4)*(48*b^2*c^2 - 259*a*b*c*d + 231*a^2*d^2)*e^(7/2)*Sqrt[1 - (d*x^2)/c]
*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(42*b^4*d^(1/4)*S
qrt[c - d*x^2]) - (c^(1/4)*(5*b*c - 11*a*d)*(b*c - a*d)*e^(7/2)*Sqrt[1 - (d*x^2)
/c]*EllipticPi[-((Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d])), ArcSin[(d^(1/4)*Sqrt[e*x]
)/(c^(1/4)*Sqrt[e])], -1])/(4*b^4*d^(1/4)*Sqrt[c - d*x^2]) - (c^(1/4)*(5*b*c - 1
1*a*d)*(b*c - a*d)*e^(7/2)*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqrt[b]*Sqrt[c])/(Sqr
t[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(4*b^4*d^(1/4
)*Sqrt[c - d*x^2])

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Rubi [A]  time = 2.54322, antiderivative size = 429, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\sqrt [4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} \left (231 a^2 d^2-259 a b c d+48 b^2 c^2\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{42 b^4 \sqrt [4]{d} \sqrt{c-d x^2}}-\frac{\sqrt [4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (5 b c-11 a d) (b c-a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 b^4 \sqrt [4]{d} \sqrt{c-d x^2}}-\frac{\sqrt [4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (5 b c-11 a d) (b c-a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 b^4 \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{e^3 \sqrt{e x} \sqrt{c-d x^2} (57 b c-77 a d)}{42 b^3}+\frac{e (e x)^{5/2} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}-\frac{11 d e (e x)^{5/2} \sqrt{c-d x^2}}{14 b^2} \]

Antiderivative was successfully verified.

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

[Out]

((57*b*c - 77*a*d)*e^3*Sqrt[e*x]*Sqrt[c - d*x^2])/(42*b^3) - (11*d*e*(e*x)^(5/2)
*Sqrt[c - d*x^2])/(14*b^2) + (e*(e*x)^(5/2)*(c - d*x^2)^(3/2))/(2*b*(a - b*x^2))
+ (c^(1/4)*(48*b^2*c^2 - 259*a*b*c*d + 231*a^2*d^2)*e^(7/2)*Sqrt[1 - (d*x^2)/c]
*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(42*b^4*d^(1/4)*S
qrt[c - d*x^2]) - (c^(1/4)*(5*b*c - 11*a*d)*(b*c - a*d)*e^(7/2)*Sqrt[1 - (d*x^2)
/c]*EllipticPi[-((Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d])), ArcSin[(d^(1/4)*Sqrt[e*x]
)/(c^(1/4)*Sqrt[e])], -1])/(4*b^4*d^(1/4)*Sqrt[c - d*x^2]) - (c^(1/4)*(5*b*c - 1
1*a*d)*(b*c - a*d)*e^(7/2)*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqrt[b]*Sqrt[c])/(Sqr
t[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(4*b^4*d^(1/4
)*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((e*x)**(7/2)*(-d*x**2+c)**(3/2)/(-b*x**2+a)**2,x)

[Out]

Timed out

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Mathematica [C]  time = 1.06209, size = 392, normalized size = 0.91 \[ \frac{(e x)^{7/2} \left (\frac{9 a c x^2 \left (231 a^2 d^2-259 a b c d+48 b^2 c^2\right ) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{2 x^2 \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}-\frac{25 a^2 c^2 (77 a d-57 b c) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{2 x^2 \left (2 b c F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}+5 \left (c-d x^2\right ) \left (77 a^2 d-a b \left (57 c+44 d x^2\right )-12 b^2 x^2 \left (d x^2-3 c\right )\right )\right )}{210 b^3 x^3 \left (b x^2-a\right ) \sqrt{c-d x^2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((e*x)^(7/2)*(5*(c - d*x^2)*(77*a^2*d - 12*b^2*x^2*(-3*c + d*x^2) - a*b*(57*c +
44*d*x^2)) - (25*a^2*c^2*(-57*b*c + 77*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*(48*b^2*c^2 - 259*a*b*c*d + 231*a^2*d^2)
*x^2*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/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*AppellF1[9/4, 3/2, 1, 13/4, (d*x^2)/c, (b*x^2)/a]))))/(210
*b^3*x^3*(-a + b*x^2)*Sqrt[c - d*x^2])

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Maple [B]  time = 0.042, size = 3790, normalized size = 8.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(7/2)*(-d*x^2+c)^(3/2)/(-b*x^2+a)^2,x)

[Out]

1/168*e^3*(e*x)^(1/2)*(-d*x^2+c)^(1/2)/b^3*(-48*x^7*b^4*c*d^3*(a*b)^(1/2)+176*x^
5*a^2*b^2*d^4*(a*b)^(1/2)+192*x^5*b^4*c^2*d^2*(a*b)^(1/2)-308*x^3*a^3*b*d^4*(a*b
)^(1/2)-144*x^3*b^4*c^3*d*(a*b)^(1/2)+360*x^3*a^2*b^2*c*d^3*(a*b)^(1/2)+92*x^3*a
*b^3*c^2*d^2*(a*b)^(1/2)+308*x*a^3*b*c*d^3*(a*b)^(1/2)-536*x*a^2*b^2*c^2*d^2*(a*
b)^(1/2)+228*x*a*b^3*c^3*d*(a*b)^(1/2)+336*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)*(-x*d/(c*d)^(1/2))^(1/2)*(
c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)-980*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^2*c*d^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)*((d*x+(c*d)^
(1/2))/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)+614*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^3*c^2*d*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^
(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)
*(a*b)^(1/2)-105*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*b^3*c^2*d*((-d*x+(c*d)^
(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*((d*x+(c*d)^(1/2)
)/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)-105*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*b^3
*c^2*d*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/
2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)-105*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*b^4*c^3*d*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1
/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)+96*EllipticF(((d*x+(c*d)^(1/2))
/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*a*b^3*c^3*((-d*x+(c*d)^(1/2))/(c*d)^(1/
2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(
1/2)*(a*b)^(1/2)-96*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))
*2^(1/2)*x^2*b^4*c^3*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(
1/2)*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)-368*x^5*a*b^3
*c*d^3*(a*b)^(1/2)+48*x^7*a*b^3*d^4*(a*b)^(1/2)+105*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^2*b^2*c^2*d*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1
/2)*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)+462*EllipticF(
((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*x^2*a^3*b*d^3*((-d*x+
(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*((d*x+(c*d)
^(1/2))/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)-231*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^3*b*d^3*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)
^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)+980*EllipticF(((d*x+(c*
d)^(1/2))/(c*d)^(1/2))^(1/2),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)*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*
d)^(1/2))^(1/2)*(a*b)^(1/2)-614*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),
1/2*2^(1/2))*2^(1/2)*a^2*b^2*c^2*d*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/
(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)
-336*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)*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(
1/2)*(a*b)^(1/2)+105*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^2*b^2*c^2*d*((-d*x+(c*d
)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*((d*x+(c*d)^(1/
2))/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)-336*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)*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)
*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)-231*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^3*b*d^3*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))
^(1/2)*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)+336*Ellipti
cPi(((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^3*c^2*d^2*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^
(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)+336*Ellipti
cPi(((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)*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(
a*b)^(1/2)+231*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^4*d^3*((-d*x+(c*d)^(1/2))/(c*
d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1
/2))^(1/2)*(a*b)^(1/2)+336*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^2*c^2*d^2*((-
d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/
(c*d)^(1/2))^(1/2)-105*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^2*b^3*c^3*d*((-d*x+(c
*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^
(1/2))^(1/2)+231*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^4*b*c*d^3*((-d*x+(c*d)^(1/2
))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(
1/2)+231*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^4*d^3*((-d*x+(c*d)^(1/2))/(c*d)^(1/
2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(
1/2)*(a*b)^(1/2)-336*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^2*c^2*d^2*((-d*x+(c
*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^
(1/2))^(1/2)+105*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^2*b^3*c^3*d*((-d*x+(c*d)^(1
/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))
^(1/2)-462*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*
a^4*d^3*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1
/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)-231*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^4*b*c*d^3*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2)
)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)+231*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^3*b^2*c*d^3*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^
(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)-336*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^3*c^2*d^2*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^
(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)+105*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*b^4*c^3*d*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2
)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)-231*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^3*b^2*c*d^3*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*
((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2))/x/(d*x^2-c)/(a*b)^(1/2)/((a*b)^(1/2)*d+(c
*d)^(1/2)*b)/((c*d)^(1/2)*b-(a*b)^(1/2)*d)/(b*x^2-a)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-d x^{2} + c\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{7}{2}}}{{\left (b x^{2} - a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (d e^{3} x^{5} - c e^{3} x^{3}\right )} \sqrt{-d x^{2} + c} \sqrt{e x}}{b^{2} x^{4} - 2 \, a b x^{2} + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(d*e^3*x^5 - c*e^3*x^3)*sqrt(-d*x^2 + c)*sqrt(e*x)/(b^2*x^4 - 2*a*b*x^
2 + a^2), x)

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-d x^{2} + c\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{7}{2}}}{{\left (b x^{2} - a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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