3.866 \(\int \frac{(e x)^{5/2} \sqrt{c-d x^2}}{a-b x^2} \, dx\)

Optimal. Leaf size=414 \[ -\frac{\sqrt{a} \sqrt [4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^{5/2} \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{\sqrt{a} \sqrt [4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^{5/2} \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{2 c^{3/4} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (2 b c-5 a d) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{5 b^2 d^{3/4} \sqrt{c-d x^2}}-\frac{2 c^{3/4} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (2 b c-5 a d) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{5 b^2 d^{3/4} \sqrt{c-d x^2}}-\frac{2 e (e x)^{3/2} \sqrt{c-d x^2}}{5 b} \]

[Out]

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

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Rubi [A]  time = 2.19546, antiderivative size = 414, 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 \[ -\frac{\sqrt{a} \sqrt [4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^{5/2} \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{\sqrt{a} \sqrt [4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^{5/2} \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{2 c^{3/4} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (2 b c-5 a d) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{5 b^2 d^{3/4} \sqrt{c-d x^2}}-\frac{2 c^{3/4} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (2 b c-5 a d) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{5 b^2 d^{3/4} \sqrt{c-d x^2}}-\frac{2 e (e x)^{3/2} \sqrt{c-d x^2}}{5 b} \]

Antiderivative was successfully verified.

[In]  Int[((e*x)^(5/2)*Sqrt[c - d*x^2])/(a - b*x^2),x]

[Out]

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

[Out]

Timed out

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Mathematica [C]  time = 1.20133, size = 418, normalized size = 1.01 \[ \frac{2 e (e x)^{3/2} \left (\frac{14 x^2 \left (a-b x^2\right ) \left (c-d x^2\right ) \left (2 b c F_1\left (\frac{11}{4};\frac{1}{2},2;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{11}{4};\frac{3}{2},1;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+11 a c \left (7 a c-2 a d x^2-9 b c x^2+7 b d x^4\right ) F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{2 x^2 \left (2 b c F_1\left (\frac{11}{4};\frac{1}{2},2;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{11}{4};\frac{3}{2},1;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+11 a c F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}-\frac{49 a^2 c^2 F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{2 x^2 \left (2 b c F_1\left (\frac{7}{4};\frac{1}{2},2;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{7}{4};\frac{3}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+7 a c F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}\right )}{35 b \left (b x^2-a\right ) \sqrt{c-d x^2}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((e*x)^(5/2)*Sqrt[c - d*x^2])/(a - b*x^2),x]

[Out]

(2*e*(e*x)^(3/2)*((-49*a^2*c^2*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])) + (11*a*c*(7*a*c - 9*b*c*x^2 - 2*a*d*x^2 + 7*b*d*x^4)*Appell
F1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a] + 14*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*AppellF1[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]))))/(35*b*(-a + b*x^2)*Sq
rt[c - d*x^2])

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Maple [B]  time = 0.065, size = 1491, normalized size = 3.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(5/2)*(-d*x^2+c)^(1/2)/(-b*x^2+a),x)

[Out]

1/10*e^2*(e*x)^(1/2)*(-d*x^2+c)^(1/2)*(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)*EllipticP
i(((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^2-5*((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)*EllipticP
i(((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-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*b*c*d-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)*Elliptic
E(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a^2*b*c*d^2+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))*a
*b^2*c^2*d-8*((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^2-14*((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))*a*b^2*c^
2*d+4*((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+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^2-5*((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^2*c^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*b)^(
1/2)*(c*d)^(1/2)*a*b*c*d+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)/x/b^2/(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. \[ -\int \frac{\sqrt{-d x^{2} + c} \left (e x\right )^{\frac{5}{2}}}{b x^{2} - a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(-d*x^2 + c)*(e*x)^(5/2)/(b*x^2 - a),x, algorithm="maxima")

[Out]

-integrate(sqrt(-d*x^2 + c)*(e*x)^(5/2)/(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(-sqrt(-d*x^2 + c)*(e*x)^(5/2)/(b*x^2 - a),x, algorithm="fricas")

[Out]

Timed out

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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)**(5/2)*(-d*x**2+c)**(1/2)/(-b*x**2+a),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{\sqrt{-d x^{2} + c} \left (e x\right )^{\frac{5}{2}}}{b x^{2} - a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(-d*x^2 + c)*(e*x)^(5/2)/(b*x^2 - a),x, algorithm="giac")

[Out]

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