3.85 \(\int \frac{\left (a-b x^4\right )^{7/2}}{\left (c-d x^4\right )^2} \, dx\)

Optimal. Leaf size=426 \[ -\frac{b x \sqrt{a-b x^4} \left (21 a^2 d^2-122 a b c d+77 b^2 c^2\right )}{84 c d^3}+\frac{\sqrt [4]{a} b^{3/4} \sqrt{1-\frac{b x^4}{a}} \left (21 a^3 d^3+349 a^2 b c d^2-553 a b^2 c^2 d+231 b^3 c^3\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{84 c d^4 \sqrt{a-b x^4}}-\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (3 a d+11 b c) (b c-a d)^3 \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^4 \sqrt{a-b x^4}}-\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (3 a d+11 b c) (b c-a d)^3 \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^4 \sqrt{a-b x^4}}+\frac{b x \left (a-b x^4\right )^{3/2} (11 b c-7 a d)}{28 c d^2}-\frac{x \left (a-b x^4\right )^{5/2} (b c-a d)}{4 c d \left (c-d x^4\right )} \]

[Out]

-(b*(77*b^2*c^2 - 122*a*b*c*d + 21*a^2*d^2)*x*Sqrt[a - b*x^4])/(84*c*d^3) + (b*(
11*b*c - 7*a*d)*x*(a - b*x^4)^(3/2))/(28*c*d^2) - ((b*c - a*d)*x*(a - b*x^4)^(5/
2))/(4*c*d*(c - d*x^4)) + (a^(1/4)*b^(3/4)*(231*b^3*c^3 - 553*a*b^2*c^2*d + 349*
a^2*b*c*d^2 + 21*a^3*d^3)*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/
4)], -1])/(84*c*d^4*Sqrt[a - b*x^4]) - (a^(1/4)*(b*c - a*d)^3*(11*b*c + 3*a*d)*S
qrt[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^4*Sqrt[a - b*x^4]) - (a^(1/4)*(b*c - a*
d)^3*(11*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^4*Sqrt[a - b*x^4])

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Rubi [A]  time = 1.39091, antiderivative size = 426, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348 \[ -\frac{b x \sqrt{a-b x^4} \left (21 a^2 d^2-122 a b c d+77 b^2 c^2\right )}{84 c d^3}+\frac{\sqrt [4]{a} b^{3/4} \sqrt{1-\frac{b x^4}{a}} \left (21 a^3 d^3+349 a^2 b c d^2-553 a b^2 c^2 d+231 b^3 c^3\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{84 c d^4 \sqrt{a-b x^4}}-\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (3 a d+11 b c) (b c-a d)^3 \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^4 \sqrt{a-b x^4}}-\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (3 a d+11 b c) (b c-a d)^3 \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^4 \sqrt{a-b x^4}}+\frac{b x \left (a-b x^4\right )^{3/2} (11 b c-7 a d)}{28 c d^2}-\frac{x \left (a-b x^4\right )^{5/2} (b c-a d)}{4 c d \left (c-d x^4\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a - b*x^4)^(7/2)/(c - d*x^4)^2,x]

[Out]

-(b*(77*b^2*c^2 - 122*a*b*c*d + 21*a^2*d^2)*x*Sqrt[a - b*x^4])/(84*c*d^3) + (b*(
11*b*c - 7*a*d)*x*(a - b*x^4)^(3/2))/(28*c*d^2) - ((b*c - a*d)*x*(a - b*x^4)^(5/
2))/(4*c*d*(c - d*x^4)) + (a^(1/4)*b^(3/4)*(231*b^3*c^3 - 553*a*b^2*c^2*d + 349*
a^2*b*c*d^2 + 21*a^3*d^3)*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/
4)], -1])/(84*c*d^4*Sqrt[a - b*x^4]) - (a^(1/4)*(b*c - a*d)^3*(11*b*c + 3*a*d)*S
qrt[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^4*Sqrt[a - b*x^4]) - (a^(1/4)*(b*c - a*
d)^3*(11*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^4*Sqrt[a - b*x^4])

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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((-b*x**4+a)**(7/2)/(-d*x**4+c)**2,x)

[Out]

Timed out

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Mathematica [C]  time = 1.82599, size = 580, normalized size = 1.36 \[ \frac{x \left (\frac{25 a^2 \left (63 a^3 d^3+63 a^2 b c d^2-155 a b^2 c^2 d+77 b^3 c^3\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )}{2 x^4 \left (2 a d F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )+5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )}+\frac{9 a c \left (105 a^4 d^3-63 a^3 b d^2 \left (5 c+2 d x^4\right )+a^2 b^2 c d \left (775 c-494 d x^4\right )+a b^3 c \left (-385 c^2-2 c d x^4+520 d^2 x^8\right )+2 b^4 c x^4 \left (77 c^2-110 c d x^4-30 d^2 x^8\right )\right ) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )-10 x^4 \left (b x^4-a\right ) \left (21 a^3 d^3-63 a^2 b c d^2+a b^2 c d \left (155 c-92 d x^4\right )+b^3 c \left (-77 c^2+44 c d x^4+12 d^2 x^8\right )\right ) \left (2 a d F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )}{c \left (2 x^4 \left (2 a d F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )+9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )}\right )}{420 d^3 \sqrt{a-b x^4} \left (c-d x^4\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a - b*x^4)^(7/2)/(c - d*x^4)^2,x]

[Out]

(x*((25*a^2*(77*b^3*c^3 - 155*a*b^2*c^2*d + 63*a^2*b*c*d^2 + 63*a^3*d^3)*AppellF
1[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*(105*a^4*d^
3 + a^2*b^2*c*d*(775*c - 494*d*x^4) - 63*a^3*b*d^2*(5*c + 2*d*x^4) + 2*b^4*c*x^4
*(77*c^2 - 110*c*d*x^4 - 30*d^2*x^8) + a*b^3*c*(-385*c^2 - 2*c*d*x^4 + 520*d^2*x
^8))*AppellF1[5/4, 1/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c] - 10*x^4*(-a + b*x^4)*(-63
*a^2*b*c*d^2 + 21*a^3*d^3 + a*b^2*c*d*(155*c - 92*d*x^4) + b^3*c*(-77*c^2 + 44*c
*d*x^4 + 12*d^2*x^8))*(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]))))
)/(420*d^3*Sqrt[a - b*x^4]*(c - d*x^4))

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Maple [C]  time = 0.041, size = 540, normalized size = 1.3 \[ -{\frac{ \left ({a}^{3}{d}^{3}-3\,{a}^{2}c{d}^{2}b+3\,a{c}^{2}d{b}^{2}-{c}^{3}{b}^{3} \right ) x}{4\,c{d}^{3} \left ( d{x}^{4}-c \right ) }\sqrt{-b{x}^{4}+a}}-{\frac{{b}^{3}{x}^{5}}{7\,{d}^{2}}\sqrt{-b{x}^{4}+a}}-{\frac{x}{3\,b} \left ( -2\,{\frac{{b}^{3} \left ( 2\,ad-bc \right ) }{{d}^{3}}}+{\frac{5\,a{b}^{3}}{7\,{d}^{2}}} \right ) \sqrt{-b{x}^{4}+a}}+{1 \left ({\frac{{b}^{2} \left ( 6\,{a}^{2}{d}^{2}-8\,cabd+3\,{b}^{2}{c}^{2} \right ) }{{d}^{4}}}+{\frac{b \left ({a}^{3}{d}^{3}-3\,{a}^{2}c{d}^{2}b+3\,a{c}^{2}d{b}^{2}-{c}^{3}{b}^{3} \right ) }{4\,{d}^{4}c}}+{\frac{a}{3\,b} \left ( -2\,{\frac{{b}^{3} \left ( 2\,ad-bc \right ) }{{d}^{3}}}+{\frac{5\,a{b}^{3}}{7\,{d}^{2}}} \right ) } \right ) \sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}}-{\frac{1}{32\,{d}^{5}c}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}d-c \right ) }{\frac{3\,{a}^{4}{d}^{4}+2\,{a}^{3}b{d}^{3}c-24\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}+30\,a{b}^{3}{c}^{3}d-11\,{b}^{4}{c}^{4}}{{{\it \_alpha}}^{3}} \left ( -{1{\it Artanh} \left ({\frac{-2\,{{\it \_alpha}}^{2}b{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \right ){\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}}-2\,{\frac{{{\it \_alpha}}^{3}d}{c\sqrt{-b{x}^{4}+a}}\sqrt{1-{\frac{\sqrt{b}{x}^{2}}{\sqrt{a}}}}\sqrt{1+{\frac{\sqrt{b}{x}^{2}}{\sqrt{a}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}},{\frac{\sqrt{a}{{\it \_alpha}}^{2}d}{c\sqrt{b}}},{1\sqrt{-{\frac{\sqrt{b}}{\sqrt{a}}}}{\frac{1}{\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}}}}} \right ){\frac{1}{\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}}}}} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x^4+a)^(7/2)/(-d*x^4+c)^2,x)

[Out]

-1/4*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/c/d^3*x*(-b*x^4+a)^(1/2)/(d*x
^4-c)-1/7*b^3/d^2*x^5*(-b*x^4+a)^(1/2)-1/3*(-2*b^3/d^3*(2*a*d-b*c)+5/7*b^3/d^2*a
)/b*x*(-b*x^4+a)^(1/2)+(b^2*(6*a^2*d^2-8*a*b*c*d+3*b^2*c^2)/d^4+1/4*b/d^4*(a^3*d
^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/c+1/3*(-2*b^3/d^3*(2*a*d-b*c)+5/7*b^3/d^
2*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)-1/32/d
^5/c*sum((3*a^4*d^4+2*a^3*b*c*d^3-24*a^2*b^2*c^2*d^2+30*a*b^3*c^3*d-11*b^4*c^4)/
_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. \[ \int \frac{{\left (-b x^{4} + a\right )}^{\frac{7}{2}}}{{\left (d x^{4} - c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(7/2)/(d*x^4 - c)^2,x, algorithm="maxima")

[Out]

integrate((-b*x^4 + a)^(7/2)/(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)^(7/2)/(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)**(7/2)/(-d*x**4+c)**2,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-b x^{4} + a\right )}^{\frac{7}{2}}}{{\left (d x^{4} - c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(7/2)/(d*x^4 - c)^2,x, algorithm="giac")

[Out]

integrate((-b*x^4 + a)^(7/2)/(d*x^4 - c)^2, x)