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

Optimal. Leaf size=321 \[ \frac{\sqrt [4]{a} b^{3/4} \sqrt{1-\frac{b x^4}{a}} \left (47 a^2 d^2-56 a b c d+21 b^2 c^2\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{21 d^3 \sqrt{a-b x^4}}-\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (b c-a d)^3 \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^3 \sqrt{a-b x^4}}-\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (b c-a d)^3 \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^3 \sqrt{a-b x^4}}-\frac{b x \sqrt{a-b x^4} (7 b c-13 a d)}{21 d^2}+\frac{b x \left (a-b x^4\right )^{3/2}}{7 d} \]

[Out]

-(b*(7*b*c - 13*a*d)*x*Sqrt[a - b*x^4])/(21*d^2) + (b*x*(a - b*x^4)^(3/2))/(7*d)
+ (a^(1/4)*b^(3/4)*(21*b^2*c^2 - 56*a*b*c*d + 47*a^2*d^2)*Sqrt[1 - (b*x^4)/a]*E
llipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(21*d^3*Sqrt[a - b*x^4]) - (a^(1/4)*(
b*c - a*d)^3*Sqrt[1 - (b*x^4)/a]*EllipticPi[-((Sqrt[a]*Sqrt[d])/(Sqrt[b]*Sqrt[c]
)), ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(2*b^(1/4)*c*d^3*Sqrt[a - b*x^4]) - (a^(1/
4)*(b*c - a*d)^3*Sqrt[1 - (b*x^4)/a]*EllipticPi[(Sqrt[a]*Sqrt[d])/(Sqrt[b]*Sqrt[
c]), ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(2*b^(1/4)*c*d^3*Sqrt[a - b*x^4])

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Rubi [A]  time = 0.914184, antiderivative size = 321, 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 \[ \frac{\sqrt [4]{a} b^{3/4} \sqrt{1-\frac{b x^4}{a}} \left (47 a^2 d^2-56 a b c d+21 b^2 c^2\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{21 d^3 \sqrt{a-b x^4}}-\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (b c-a d)^3 \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^3 \sqrt{a-b x^4}}-\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (b c-a d)^3 \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^3 \sqrt{a-b x^4}}-\frac{b x \sqrt{a-b x^4} (7 b c-13 a d)}{21 d^2}+\frac{b x \left (a-b x^4\right )^{3/2}}{7 d} \]

Antiderivative was successfully verified.

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

[Out]

-(b*(7*b*c - 13*a*d)*x*Sqrt[a - b*x^4])/(21*d^2) + (b*x*(a - b*x^4)^(3/2))/(7*d)
+ (a^(1/4)*b^(3/4)*(21*b^2*c^2 - 56*a*b*c*d + 47*a^2*d^2)*Sqrt[1 - (b*x^4)/a]*E
llipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(21*d^3*Sqrt[a - b*x^4]) - (a^(1/4)*(
b*c - a*d)^3*Sqrt[1 - (b*x^4)/a]*EllipticPi[-((Sqrt[a]*Sqrt[d])/(Sqrt[b]*Sqrt[c]
)), ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(2*b^(1/4)*c*d^3*Sqrt[a - b*x^4]) - (a^(1/
4)*(b*c - a*d)^3*Sqrt[1 - (b*x^4)/a]*EllipticPi[(Sqrt[a]*Sqrt[d])/(Sqrt[b]*Sqrt[
c]), ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(2*b^(1/4)*c*d^3*Sqrt[a - b*x^4])

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Rubi in Sympy [A]  time = 151.844, size = 292, normalized size = 0.91 \[ \frac{\sqrt [4]{a} b^{\frac{3}{4}} \sqrt{1 - \frac{b x^{4}}{a}} \left (47 a^{2} d^{2} - 56 a b c d + 21 b^{2} c^{2}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{21 d^{3} \sqrt{a - b x^{4}}} + \frac{\sqrt [4]{a} \sqrt{1 - \frac{b x^{4}}{a}} \left (a d - b c\right )^{3} \Pi \left (- \frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{2 \sqrt [4]{b} c d^{3} \sqrt{a - b x^{4}}} + \frac{\sqrt [4]{a} \sqrt{1 - \frac{b x^{4}}{a}} \left (a d - b c\right )^{3} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{2 \sqrt [4]{b} c d^{3} \sqrt{a - b x^{4}}} + \frac{b x \left (a - b x^{4}\right )^{\frac{3}{2}}}{7 d} + \frac{b x \sqrt{a - b x^{4}} \left (13 a d - 7 b c\right )}{21 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(1/4)*b**(3/4)*sqrt(1 - b*x**4/a)*(47*a**2*d**2 - 56*a*b*c*d + 21*b**2*c**2)*
elliptic_f(asin(b**(1/4)*x/a**(1/4)), -1)/(21*d**3*sqrt(a - b*x**4)) + a**(1/4)*
sqrt(1 - b*x**4/a)*(a*d - b*c)**3*elliptic_pi(-sqrt(a)*sqrt(d)/(sqrt(b)*sqrt(c))
, asin(b**(1/4)*x/a**(1/4)), -1)/(2*b**(1/4)*c*d**3*sqrt(a - b*x**4)) + a**(1/4)
*sqrt(1 - b*x**4/a)*(a*d - b*c)**3*elliptic_pi(sqrt(a)*sqrt(d)/(sqrt(b)*sqrt(c))
, asin(b**(1/4)*x/a**(1/4)), -1)/(2*b**(1/4)*c*d**3*sqrt(a - b*x**4)) + b*x*(a -
b*x**4)**(3/2)/(7*d) + b*x*sqrt(a - b*x**4)*(13*a*d - 7*b*c)/(21*d**2)

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Mathematica [C]  time = 1.72882, size = 385, normalized size = 1.2 \[ \frac{x \left (-\frac{9 a b c x^4 \left (47 a^2 d^2-56 a b c d+21 b^2 c^2\right ) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )}{\left (c-d x^4\right ) \left (2 x^4 \left (2 a d F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )+9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )}+\frac{25 a^2 c \left (21 a^2 d^2-16 a b c d+7 b^2 c^2\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )}{\left (c-d x^4\right ) \left (2 x^4 \left (2 a d F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )+5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )}+5 b \left (b x^4-a\right ) \left (-16 a d+7 b c+3 b d x^4\right )\right )}{105 d^2 \sqrt{a-b x^4}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x*(5*b*(-a + b*x^4)*(7*b*c - 16*a*d + 3*b*d*x^4) + (25*a^2*c*(7*b^2*c^2 - 16*a*
b*c*d + 21*a^2*d^2)*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c])/((c - d*x^
4)*(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*Appell
F1[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*b*c*(21*b^2*c^2 - 56*a*b*c*d + 47*a^2*d^2)*x^4*Appel
lF1[5/4, 1/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c])/((c - d*x^4)*(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])))))/
(105*d^2*Sqrt[a - b*x^4])

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Maple [C]  time = 0.069, size = 408, normalized size = 1.3 \[ -{\frac{{b}^{2}{x}^{5}}{7\,d}\sqrt{-b{x}^{4}+a}}+{\frac{x}{3\,b} \left ({\frac{{b}^{2} \left ( 3\,ad-bc \right ) }{{d}^{2}}}-{\frac{5\,a{b}^{2}}{7\,d}} \right ) \sqrt{-b{x}^{4}+a}}-{1 \left ( -{\frac{b \left ( 3\,{a}^{2}{d}^{2}-3\,cabd+{b}^{2}{c}^{2} \right ) }{{d}^{3}}}+{\frac{a}{3\,b} \left ({\frac{{b}^{2} \left ( 3\,ad-bc \right ) }{{d}^{2}}}-{\frac{5\,a{b}^{2}}{7\,d}} \right ) } \right ) \sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}}+{\frac{1}{8\,{d}^{4}}\sum _{{\it \_alpha}={\it RootOf} \left ( d{{\it \_Z}}^{4}-c \right ) }{\frac{-{a}^{3}{d}^{3}+3\,{a}^{2}c{d}^{2}b-3\,a{c}^{2}d{b}^{2}+{c}^{3}{b}^{3}}{{{\it \_alpha}}^{3}} \left ( -{1{\it Artanh} \left ({\frac{-2\,{{\it \_alpha}}^{2}b{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \right ){\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}}-2\,{\frac{{{\it \_alpha}}^{3}d}{c\sqrt{-b{x}^{4}+a}}\sqrt{1-{\frac{\sqrt{b}{x}^{2}}{\sqrt{a}}}}\sqrt{1+{\frac{\sqrt{b}{x}^{2}}{\sqrt{a}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}},{\frac{\sqrt{a}{{\it \_alpha}}^{2}d}{c\sqrt{b}}},{1\sqrt{-{\frac{\sqrt{b}}{\sqrt{a}}}}{\frac{1}{\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}}}}} \right ){\frac{1}{\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}}}}} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/7*b^2/d*x^5*(-b*x^4+a)^(1/2)+1/3*(b^2/d^2*(3*a*d-b*c)-5/7*b^2/d*a)/b*x*(-b*x^
4+a)^(1/2)-(-b*(3*a^2*d^2-3*a*b*c*d+b^2*c^2)/d^3+1/3*(b^2/d^2*(3*a*d-b*c)-5/7*b^
2/d*a)/b*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)+1/8/
d^4*sum((-a^3*d^3+3*a^2*b*c*d^2-3*a*b^2*c^2*d+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. \[ -\int \frac{{\left (-b x^{4} + a\right )}^{\frac{5}{2}}}{d x^{4} - c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{a^{2} \sqrt{a - b x^{4}}}{- c + d x^{4}}\, dx - \int \frac{b^{2} x^{8} \sqrt{a - b x^{4}}}{- c + d x^{4}}\, dx - \int \left (- \frac{2 a b x^{4} \sqrt{a - b x^{4}}}{- c + d x^{4}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(a**2*sqrt(a - b*x**4)/(-c + d*x**4), x) - Integral(b**2*x**8*sqrt(a -
b*x**4)/(-c + d*x**4), x) - Integral(-2*a*b*x**4*sqrt(a - b*x**4)/(-c + d*x**4),
x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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