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

Optimal. Leaf size=305 \[ -\frac{2 \sqrt [4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (3 a d+b c) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{3 b^2 d^{5/4} \sqrt{c-d x^2}}+\frac{a \sqrt [4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^2 \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{a \sqrt [4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^2 \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{2 e^3 \sqrt{e x} \sqrt{c-d x^2}}{3 b d} \]

[Out]

(2*e^3*Sqrt[e*x]*Sqrt[c - d*x^2])/(3*b*d) - (2*c^(1/4)*(b*c + 3*a*d)*e^(7/2)*Sqr
t[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(
3*b^2*d^(5/4)*Sqrt[c - d*x^2]) + (a*c^(1/4)*e^(7/2)*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)*S
qrt[e])], -1])/(b^2*d^(1/4)*Sqrt[c - d*x^2]) + (a*c^(1/4)*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])/(b^2*d^(1/4)*Sqrt[c - d*x^2])

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Rubi [A]  time = 1.32492, antiderivative size = 305, 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 \[ -\frac{2 \sqrt [4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (3 a d+b c) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{3 b^2 d^{5/4} \sqrt{c-d x^2}}+\frac{a \sqrt [4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^2 \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{a \sqrt [4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^2 \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{2 e^3 \sqrt{e x} \sqrt{c-d x^2}}{3 b d} \]

Antiderivative was successfully verified.

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

[Out]

(2*e^3*Sqrt[e*x]*Sqrt[c - d*x^2])/(3*b*d) - (2*c^(1/4)*(b*c + 3*a*d)*e^(7/2)*Sqr
t[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(
3*b^2*d^(5/4)*Sqrt[c - d*x^2]) + (a*c^(1/4)*e^(7/2)*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)*S
qrt[e])], -1])/(b^2*d^(1/4)*Sqrt[c - d*x^2]) + (a*c^(1/4)*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])/(b^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)**(7/2)/(-b*x**2+a)/(-d*x**2+c)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 0.731736, size = 423, normalized size = 1.39 \[ \frac{2 (e x)^{7/2} \left (\frac{25 a^2 c^2 F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{2 x^2 \left (2 b c F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}+\frac{-10 x^2 \left (a-b x^2\right ) \left (c-d x^2\right ) \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )-9 a c \left (5 a c-2 a d x^2-4 b c x^2+5 b d x^4\right ) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{2 x^2 \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}\right )}{15 b d x^3 \left (b x^2-a\right ) \sqrt{c-d x^2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*(e*x)^(7/2)*((25*a^2*c^2*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*(5*a*c - 4*b*c*x^2 - 2*a*d*x^2 + 5*b*d*x^4)*AppellF1[5/
4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a] - 10*x^2*(a - b*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]))/(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*AppellF
1[9/4, 3/2, 1, 13/4, (d*x^2)/c, (b*x^2)/a]))))/(15*b*d*x^3*(-a + b*x^2)*Sqrt[c -
d*x^2])

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Maple [B]  time = 0.052, size = 853, normalized size = 2.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(7/2)/(-b*x^2+a)/(-d*x^2+c)^(1/2),x)

[Out]

-1/6/b/d*(6*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)
*a^2*d^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)*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(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*b*c*d*((-d*x+(c*d)^(1/2))/(c*d)^
(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)*(c*d)^(1/2)*((d*x+(c*d)^(1/2))
/(c*d)^(1/2))^(1/2)-2*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2
))*2^(1/2)*b^2*c^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)*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)+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))*a^2*b*c*d^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))*(a*b)^(1/2)*(c*d)^(1/2)*a^2*d^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/
((c*d)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2))*a^2*b*c*d^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/((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+4*x^3*a*b*d^3*(
a*b)^(1/2)-4*x^3*b^2*c*d^2*(a*b)^(1/2)-4*x*a*b*c*d^2*(a*b)^(1/2)+4*x*b^2*c^2*d*(
a*b)^(1/2))*(-d*x^2+c)^(1/2)*e^3*(e*x)^(1/2)/x/((c*d)^(1/2)*b-(a*b)^(1/2)*d)/((a
*b)^(1/2)*d+(c*d)^(1/2)*b)/(a*b)^(1/2)/(d*x^2-c)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-integrate((e*x)^(7/2)/((b*x^2 - a)*sqrt(-d*x^2 + c)), 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(-(e*x)^(7/2)/((b*x^2 - a)*sqrt(-d*x^2 + c)),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)**(7/2)/(-b*x**2+a)/(-d*x**2+c)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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