3.86 \(\int \frac{\left (a-b x^4\right )^{5/2}}{\left (c-d x^4\right )^2} \, dx\)

Optimal. Leaf size=365 \[ -\frac{\sqrt [4]{a} b^{3/4} \sqrt{1-\frac{b x^4}{a}} \left (-3 a^2 d^2-26 a b c d+21 b^2 c^2\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{12 c d^3 \sqrt{a-b x^4}}+\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (3 a d+7 b c) (b c-a d)^2 \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^3 \sqrt{a-b x^4}}+\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (3 a d+7 b c) (b c-a d)^2 \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^3 \sqrt{a-b x^4}}+\frac{b x \sqrt{a-b x^4} (7 b c-3 a d)}{12 c d^2}-\frac{x \left (a-b x^4\right )^{3/2} (b c-a d)}{4 c d \left (c-d x^4\right )} \]

[Out]

(b*(7*b*c - 3*a*d)*x*Sqrt[a - b*x^4])/(12*c*d^2) - ((b*c - a*d)*x*(a - b*x^4)^(3
/2))/(4*c*d*(c - d*x^4)) - (a^(1/4)*b^(3/4)*(21*b^2*c^2 - 26*a*b*c*d - 3*a^2*d^2
)*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(12*c*d^3*Sqrt
[a - b*x^4]) + (a^(1/4)*(b*c - a*d)^2*(7*b*c + 3*a*d)*Sqrt[1 - (b*x^4)/a]*Ellipt
icPi[-((Sqrt[a]*Sqrt[d])/(Sqrt[b]*Sqrt[c])), ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(
8*b^(1/4)*c^2*d^3*Sqrt[a - b*x^4]) + (a^(1/4)*(b*c - a*d)^2*(7*b*c + 3*a*d)*Sqrt
[1 - (b*x^4)/a]*EllipticPi[(Sqrt[a]*Sqrt[d])/(Sqrt[b]*Sqrt[c]), ArcSin[(b^(1/4)*
x)/a^(1/4)], -1])/(8*b^(1/4)*c^2*d^3*Sqrt[a - b*x^4])

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Rubi [A]  time = 1.04876, antiderivative size = 365, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348 \[ -\frac{\sqrt [4]{a} b^{3/4} \sqrt{1-\frac{b x^4}{a}} \left (-3 a^2 d^2-26 a b c d+21 b^2 c^2\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{12 c d^3 \sqrt{a-b x^4}}+\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (3 a d+7 b c) (b c-a d)^2 \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^3 \sqrt{a-b x^4}}+\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (3 a d+7 b c) (b c-a d)^2 \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^3 \sqrt{a-b x^4}}+\frac{b x \sqrt{a-b x^4} (7 b c-3 a d)}{12 c d^2}-\frac{x \left (a-b x^4\right )^{3/2} (b c-a d)}{4 c d \left (c-d x^4\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a - b*x^4)^(5/2)/(c - d*x^4)^2,x]

[Out]

(b*(7*b*c - 3*a*d)*x*Sqrt[a - b*x^4])/(12*c*d^2) - ((b*c - a*d)*x*(a - b*x^4)^(3
/2))/(4*c*d*(c - d*x^4)) - (a^(1/4)*b^(3/4)*(21*b^2*c^2 - 26*a*b*c*d - 3*a^2*d^2
)*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(12*c*d^3*Sqrt
[a - b*x^4]) + (a^(1/4)*(b*c - a*d)^2*(7*b*c + 3*a*d)*Sqrt[1 - (b*x^4)/a]*Ellipt
icPi[-((Sqrt[a]*Sqrt[d])/(Sqrt[b]*Sqrt[c])), ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(
8*b^(1/4)*c^2*d^3*Sqrt[a - b*x^4]) + (a^(1/4)*(b*c - a*d)^2*(7*b*c + 3*a*d)*Sqrt
[1 - (b*x^4)/a]*EllipticPi[(Sqrt[a]*Sqrt[d])/(Sqrt[b]*Sqrt[c]), ArcSin[(b^(1/4)*
x)/a^(1/4)], -1])/(8*b^(1/4)*c^2*d^3*Sqrt[a - b*x^4])

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Rubi in Sympy [A]  time = 168.627, size = 333, normalized size = 0.91 \[ \frac{\sqrt [4]{a} b^{\frac{3}{4}} \sqrt{1 - \frac{b x^{4}}{a}} \left (3 a^{2} d^{2} + 26 a b c d - 21 b^{2} c^{2}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{12 c d^{3} \sqrt{a - b x^{4}}} + \frac{\sqrt [4]{a} \sqrt{1 - \frac{b x^{4}}{a}} \left (a d - b c\right )^{2} \left (3 a d + 7 b c\right ) \Pi \left (- \frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{8 \sqrt [4]{b} c^{2} d^{3} \sqrt{a - b x^{4}}} + \frac{\sqrt [4]{a} \sqrt{1 - \frac{b x^{4}}{a}} \left (a d - b c\right )^{2} \left (3 a d + 7 b c\right ) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{8 \sqrt [4]{b} c^{2} d^{3} \sqrt{a - b x^{4}}} - \frac{b x \sqrt{a - b x^{4}} \left (3 a d - 7 b c\right )}{12 c d^{2}} + \frac{x \left (a - b x^{4}\right )^{\frac{3}{2}} \left (a d - b c\right )}{4 c d \left (c - d x^{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b*x**4+a)**(5/2)/(-d*x**4+c)**2,x)

[Out]

a**(1/4)*b**(3/4)*sqrt(1 - b*x**4/a)*(3*a**2*d**2 + 26*a*b*c*d - 21*b**2*c**2)*e
lliptic_f(asin(b**(1/4)*x/a**(1/4)), -1)/(12*c*d**3*sqrt(a - b*x**4)) + a**(1/4)
*sqrt(1 - b*x**4/a)*(a*d - b*c)**2*(3*a*d + 7*b*c)*elliptic_pi(-sqrt(a)*sqrt(d)/
(sqrt(b)*sqrt(c)), asin(b**(1/4)*x/a**(1/4)), -1)/(8*b**(1/4)*c**2*d**3*sqrt(a -
b*x**4)) + a**(1/4)*sqrt(1 - b*x**4/a)*(a*d - b*c)**2*(3*a*d + 7*b*c)*elliptic_
pi(sqrt(a)*sqrt(d)/(sqrt(b)*sqrt(c)), asin(b**(1/4)*x/a**(1/4)), -1)/(8*b**(1/4)
*c**2*d**3*sqrt(a - b*x**4)) - b*x*sqrt(a - b*x**4)*(3*a*d - 7*b*c)/(12*c*d**2)
+ x*(a - b*x**4)**(3/2)*(a*d - b*c)/(4*c*d*(c - d*x**4))

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Mathematica [C]  time = 1.3619, size = 491, normalized size = 1.35 \[ \frac{x \left (\frac{-10 x^4 \left (a-b x^4\right ) \left (3 a^2 d^2-6 a b c d+b^2 c \left (7 c-4 d x^4\right )\right ) \left (2 a d F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )-9 a c \left (15 a^3 d^2-6 a^2 b d \left (5 c+3 d x^4\right )+a b^2 c \left (35 c-16 d x^4\right )+2 b^3 c x^4 \left (10 d x^4-7 c\right )\right ) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )}{c \left (2 x^4 \left (2 a d F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )+9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )}-\frac{25 a^2 \left (9 a^2 d^2+6 a b c d-7 b^2 c^2\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )}{2 x^4 \left (2 a d F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )+5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )}\right )}{60 d^2 \sqrt{a-b x^4} \left (d x^4-c\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a - b*x^4)^(5/2)/(c - d*x^4)^2,x]

[Out]

(x*((-25*a^2*(-7*b^2*c^2 + 6*a*b*c*d + 9*a^2*d^2)*AppellF1[1/4, 1/2, 1, 5/4, (b*
x^4)/a, (d*x^4)/c])/(5*a*c*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c] + 2*
x^4*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, (b*x^4)/a, (d*x^4)/c] + b*c*AppellF1[5/4,
3/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c])) + (-9*a*c*(15*a^3*d^2 + a*b^2*c*(35*c - 16*
d*x^4) - 6*a^2*b*d*(5*c + 3*d*x^4) + 2*b^3*c*x^4*(-7*c + 10*d*x^4))*AppellF1[5/4
, 1/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c] - 10*x^4*(a - b*x^4)*(-6*a*b*c*d + 3*a^2*d^
2 + b^2*c*(7*c - 4*d*x^4))*(2*a*d*AppellF1[9/4, 1/2, 2, 13/4, (b*x^4)/a, (d*x^4)
/c] + b*c*AppellF1[9/4, 3/2, 1, 13/4, (b*x^4)/a, (d*x^4)/c]))/(c*(9*a*c*AppellF1
[5/4, 1/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c] + 2*x^4*(2*a*d*AppellF1[9/4, 1/2, 2, 13
/4, (b*x^4)/a, (d*x^4)/c] + b*c*AppellF1[9/4, 3/2, 1, 13/4, (b*x^4)/a, (d*x^4)/c
])))))/(60*d^2*Sqrt[a - b*x^4]*(-c + d*x^4))

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Maple [C]  time = 0.04, size = 412, normalized size = 1.1 \[ -{\frac{ \left ({a}^{2}{d}^{2}-2\,cabd+{b}^{2}{c}^{2} \right ) x}{4\,c{d}^{2} \left ( d{x}^{4}-c \right ) }\sqrt{-b{x}^{4}+a}}+{\frac{{b}^{2}x}{3\,{d}^{2}}\sqrt{-b{x}^{4}+a}}+{1 \left ({\frac{{b}^{2} \left ( 3\,ad-2\,bc \right ) }{{d}^{3}}}+{\frac{b \left ({a}^{2}{d}^{2}-2\,cabd+{b}^{2}{c}^{2} \right ) }{4\,c{d}^{3}}}-{\frac{a{b}^{2}}{3\,{d}^{2}}} \right ) \sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}}-{\frac{1}{32\,c{d}^{4}}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}d-c \right ) }{\frac{3\,{a}^{3}{d}^{3}+{a}^{2}c{d}^{2}b-11\,a{c}^{2}d{b}^{2}+7\,{c}^{3}{b}^{3}}{{{\it \_alpha}}^{3}} \left ( -{1{\it Artanh} \left ({\frac{-2\,{{\it \_alpha}}^{2}b{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \right ){\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}}-2\,{\frac{{{\it \_alpha}}^{3}d}{c\sqrt{-b{x}^{4}+a}}\sqrt{1-{\frac{\sqrt{b}{x}^{2}}{\sqrt{a}}}}\sqrt{1+{\frac{\sqrt{b}{x}^{2}}{\sqrt{a}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}},{\frac{\sqrt{a}{{\it \_alpha}}^{2}d}{c\sqrt{b}}},{1\sqrt{-{\frac{\sqrt{b}}{\sqrt{a}}}}{\frac{1}{\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}}}}} \right ){\frac{1}{\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}}}}} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x^4+a)^(5/2)/(-d*x^4+c)^2,x)

[Out]

-1/4*(a^2*d^2-2*a*b*c*d+b^2*c^2)/c/d^2*x*(-b*x^4+a)^(1/2)/(d*x^4-c)+1/3*b^2/d^2*
x*(-b*x^4+a)^(1/2)+(b^2*(3*a*d-2*b*c)/d^3+1/4*b/d^3*(a^2*d^2-2*a*b*c*d+b^2*c^2)/
c-1/3*b^2/d^2*a)/(1/a^(1/2)*b^(1/2))^(1/2)*(1-b^(1/2)*x^2/a^(1/2))^(1/2)*(1+b^(1
/2)*x^2/a^(1/2))^(1/2)/(-b*x^4+a)^(1/2)*EllipticF(x*(1/a^(1/2)*b^(1/2))^(1/2),I)
-1/32/c/d^4*sum((3*a^3*d^3+a^2*b*c*d^2-11*a*b^2*c^2*d+7*b^3*c^3)/_alpha^3*(-1/((
a*d-b*c)/d)^(1/2)*arctanh(1/2*(-2*_alpha^2*b*x^2+2*a)/((a*d-b*c)/d)^(1/2)/(-b*x^
4+a)^(1/2))-2/(1/a^(1/2)*b^(1/2))^(1/2)*_alpha^3*d/c*(1-b^(1/2)*x^2/a^(1/2))^(1/
2)*(1+b^(1/2)*x^2/a^(1/2))^(1/2)/(-b*x^4+a)^(1/2)*EllipticPi(x*(1/a^(1/2)*b^(1/2
))^(1/2),a^(1/2)/b^(1/2)*_alpha^2/c*d,(-1/a^(1/2)*b^(1/2))^(1/2)/(1/a^(1/2)*b^(1
/2))^(1/2))),_alpha=RootOf(_Z^4*d-c))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-b x^{4} + a\right )}^{\frac{5}{2}}}{{\left (d x^{4} - c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(5/2)/(d*x^4 - c)^2,x, algorithm="maxima")

[Out]

integrate((-b*x^4 + a)^(5/2)/(d*x^4 - c)^2, 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((-b*x^4 + a)^(5/2)/(d*x^4 - c)^2,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((-b*x**4+a)**(5/2)/(-d*x**4+c)**2,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-b x^{4} + a\right )}^{\frac{5}{2}}}{{\left (d x^{4} - c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(5/2)/(d*x^4 - c)^2,x, algorithm="giac")

[Out]

integrate((-b*x^4 + a)^(5/2)/(d*x^4 - c)^2, x)