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

Optimal. Leaf size=421 \[ -\frac{\sqrt [4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (b c-a d)^2 \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt{a} b^{5/2} \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (b c-a d)^2 \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt{a} b^{5/2} \sqrt [4]{d} \sqrt{c-d x^2}}-\frac{2 c^{3/4} \sqrt [4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (7 b c-5 a d) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{5 b^2 \sqrt{c-d x^2}}+\frac{2 c^{3/4} \sqrt [4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (7 b c-5 a d) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{5 b^2 \sqrt{c-d x^2}}+\frac{2 d (e x)^{3/2} \sqrt{c-d x^2}}{5 b e} \]

[Out]

(2*d*(e*x)^(3/2)*Sqrt[c - d*x^2])/(5*b*e) + (2*c^(3/4)*d^(1/4)*(7*b*c - 5*a*d)*S
qrt[e]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e]
)], -1])/(5*b^2*Sqrt[c - d*x^2]) - (2*c^(3/4)*d^(1/4)*(7*b*c - 5*a*d)*Sqrt[e]*Sq
rt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/
(5*b^2*Sqrt[c - d*x^2]) - (c^(1/4)*(b*c - a*d)^2*Sqrt[e]*Sqrt[1 - (d*x^2)/c]*Ell
ipticPi[-((Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d])), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1
/4)*Sqrt[e])], -1])/(Sqrt[a]*b^(5/2)*d^(1/4)*Sqrt[c - d*x^2]) + (c^(1/4)*(b*c -
a*d)^2*Sqrt[e]*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])/(Sqrt[a]*b^(5/2)*d^(1/4)*
Sqrt[c - d*x^2])

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Rubi [A]  time = 2.25314, antiderivative size = 421, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{\sqrt [4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (b c-a d)^2 \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt{a} b^{5/2} \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (b c-a d)^2 \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt{a} b^{5/2} \sqrt [4]{d} \sqrt{c-d x^2}}-\frac{2 c^{3/4} \sqrt [4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (7 b c-5 a d) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{5 b^2 \sqrt{c-d x^2}}+\frac{2 c^{3/4} \sqrt [4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (7 b c-5 a d) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{5 b^2 \sqrt{c-d x^2}}+\frac{2 d (e x)^{3/2} \sqrt{c-d x^2}}{5 b e} \]

Antiderivative was successfully verified.

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

[Out]

(2*d*(e*x)^(3/2)*Sqrt[c - d*x^2])/(5*b*e) + (2*c^(3/4)*d^(1/4)*(7*b*c - 5*a*d)*S
qrt[e]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e]
)], -1])/(5*b^2*Sqrt[c - d*x^2]) - (2*c^(3/4)*d^(1/4)*(7*b*c - 5*a*d)*Sqrt[e]*Sq
rt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/
(5*b^2*Sqrt[c - d*x^2]) - (c^(1/4)*(b*c - a*d)^2*Sqrt[e]*Sqrt[1 - (d*x^2)/c]*Ell
ipticPi[-((Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d])), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1
/4)*Sqrt[e])], -1])/(Sqrt[a]*b^(5/2)*d^(1/4)*Sqrt[c - d*x^2]) + (c^(1/4)*(b*c -
a*d)^2*Sqrt[e]*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])/(Sqrt[a]*b^(5/2)*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((-d*x**2+c)**(3/2)*(e*x)**(1/2)/(-b*x**2+a),x)

[Out]

Timed out

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Mathematica [C]  time = 0.810788, size = 427, normalized size = 1.01 \[ \frac{2 x \sqrt{e x} \left (\frac{49 a c^2 (3 a d-5 b c) F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{2 x^2 \left (2 b c F_1\left (\frac{7}{4};\frac{1}{2},2;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{7}{4};\frac{3}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+7 a c F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}+\frac{-42 d x^2 \left (a-b x^2\right ) \left (c-d x^2\right ) \left (2 b c F_1\left (\frac{11}{4};\frac{1}{2},2;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{11}{4};\frac{3}{2},1;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )-33 a c d \left (7 a c-2 a d x^2-14 b c x^2+7 b d x^4\right ) F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{2 x^2 \left (2 b c F_1\left (\frac{11}{4};\frac{1}{2},2;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{11}{4};\frac{3}{2},1;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+11 a c F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}\right )}{105 b \left (b x^2-a\right ) \sqrt{c-d x^2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*x*Sqrt[e*x]*((49*a*c^2*(-5*b*c + 3*a*d)*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c,
(b*x^2)/a])/(7*a*c*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 2*x^2*(2*
b*c*AppellF1[7/4, 1/2, 2, 11/4, (d*x^2)/c, (b*x^2)/a] + a*d*AppellF1[7/4, 3/2, 1
, 11/4, (d*x^2)/c, (b*x^2)/a])) + (-33*a*c*d*(7*a*c - 14*b*c*x^2 - 2*a*d*x^2 + 7
*b*d*x^4)*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a] - 42*d*x^2*(a - b*x^
2)*(c - d*x^2)*(2*b*c*AppellF1[11/4, 1/2, 2, 15/4, (d*x^2)/c, (b*x^2)/a] + a*d*A
ppellF1[11/4, 3/2, 1, 15/4, (d*x^2)/c, (b*x^2)/a]))/(11*a*c*AppellF1[7/4, 1/2, 1
, 11/4, (d*x^2)/c, (b*x^2)/a] + 2*x^2*(2*b*c*AppellF1[11/4, 1/2, 2, 15/4, (d*x^2
)/c, (b*x^2)/a] + a*d*AppellF1[11/4, 3/2, 1, 15/4, (d*x^2)/c, (b*x^2)/a]))))/(10
5*b*(-a + b*x^2)*Sqrt[c - d*x^2])

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Maple [B]  time = 0.03, size = 1927, normalized size = 4.6 \[ \text{result too large to display} \]

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),x)

[Out]

-1/10*(-d*x^2+c)^(1/2)*(e*x)^(1/2)*d*(5*((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^2*b*c*d^2-5*((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)^(1/2)*(c*d)^(1/2)*a^2*d^2-10*((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)*EllipticP
i(((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*d+10*((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)^(1/2)*(c*d)^(1/2)*a*b*c*d+5*((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)*Elliptic
Pi(((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))*b^3*c^3-5*(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))*b^2*c^2+5*((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^2*b*c*d^2+5*((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)^(1/2)*(c*d)^(1/2)*a^2*d^2-10*((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^2*c^2*d-10*((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)^(1/2)*(c*d)^(1/2)*a*b*c*d+5*((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))*b^3*c^3+5*(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))*b^2*c^2-20*((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)*EllipticE(((d
*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a^2*b*c*d^2+48*((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)*EllipticE(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*b^2*
c^2*d-28*((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)*EllipticE(((d*x+(c*d)^(1/2))/(c*d)^(1/2))
^(1/2),1/2*2^(1/2))*b^3*c^3+10*((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)*EllipticF(((d*x+(c*
d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a^2*b*c*d^2-24*((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)*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*b^2*c^2*d+
14*((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)*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)
,1/2*2^(1/2))*b^3*c^3+4*x^4*a*b^2*d^3-4*x^4*b^3*c*d^2-4*x^2*a*b^2*c*d^2+4*x^2*b^
3*c^2*d)/b^2/x/(d*x^2-c)/((a*b)^(1/2)*d+(c*d)^(1/2)*b)/((c*d)^(1/2)*b-(a*b)^(1/2
)*d)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [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)*sqrt(e*x)/(b*x^2 - a),x, algorithm="fricas")

[Out]

Timed out

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

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),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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