3.91 \(\int \frac{1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )^2} \, dx\)

Optimal. Leaf size=439 \[ \frac{b x \left (-3 a^2 d^2-17 a b c d+5 b^2 c^2\right )}{12 a^2 c \sqrt{a-b x^4} (b c-a d)^3}+\frac{b^{3/4} \sqrt{1-\frac{b x^4}{a}} \left (-3 a^2 d^2-17 a b c d+5 b^2 c^2\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{12 a^{7/4} c \sqrt{a-b x^4} (b c-a d)^3}+\frac{\sqrt [4]{a} d^2 \sqrt{1-\frac{b x^4}{a}} (13 b c-3 a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt{a-b x^4} (b c-a d)^3}+\frac{\sqrt [4]{a} d^2 \sqrt{1-\frac{b x^4}{a}} (13 b c-3 a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt{a-b x^4} (b c-a d)^3}-\frac{d x}{4 c \left (a-b x^4\right )^{3/2} \left (c-d x^4\right ) (b c-a d)}+\frac{b x (3 a d+2 b c)}{12 a c \left (a-b x^4\right )^{3/2} (b c-a d)^2} \]

[Out]

(b*(2*b*c + 3*a*d)*x)/(12*a*c*(b*c - a*d)^2*(a - b*x^4)^(3/2)) + (b*(5*b^2*c^2 -
17*a*b*c*d - 3*a^2*d^2)*x)/(12*a^2*c*(b*c - a*d)^3*Sqrt[a - b*x^4]) - (d*x)/(4*
c*(b*c - a*d)*(a - b*x^4)^(3/2)*(c - d*x^4)) + (b^(3/4)*(5*b^2*c^2 - 17*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
*a^(7/4)*c*(b*c - a*d)^3*Sqrt[a - b*x^4]) + (a^(1/4)*d^2*(13*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*(b*c - a*d)^3*Sqrt[a - b*x^4]) + (a^(1/4)*d^2*
(13*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*(b*c - a*d)^3*Sqrt[a - b*x
^4])

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Rubi [A]  time = 1.43859, antiderivative size = 439, 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 \[ \frac{b x \left (-3 a^2 d^2-17 a b c d+5 b^2 c^2\right )}{12 a^2 c \sqrt{a-b x^4} (b c-a d)^3}+\frac{b^{3/4} \sqrt{1-\frac{b x^4}{a}} \left (-3 a^2 d^2-17 a b c d+5 b^2 c^2\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{12 a^{7/4} c \sqrt{a-b x^4} (b c-a d)^3}+\frac{\sqrt [4]{a} d^2 \sqrt{1-\frac{b x^4}{a}} (13 b c-3 a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt{a-b x^4} (b c-a d)^3}+\frac{\sqrt [4]{a} d^2 \sqrt{1-\frac{b x^4}{a}} (13 b c-3 a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt{a-b x^4} (b c-a d)^3}-\frac{d x}{4 c \left (a-b x^4\right )^{3/2} \left (c-d x^4\right ) (b c-a d)}+\frac{b x (3 a d+2 b c)}{12 a c \left (a-b x^4\right )^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a - b*x^4)^(5/2)*(c - d*x^4)^2),x]

[Out]

(b*(2*b*c + 3*a*d)*x)/(12*a*c*(b*c - a*d)^2*(a - b*x^4)^(3/2)) + (b*(5*b^2*c^2 -
17*a*b*c*d - 3*a^2*d^2)*x)/(12*a^2*c*(b*c - a*d)^3*Sqrt[a - b*x^4]) - (d*x)/(4*
c*(b*c - a*d)*(a - b*x^4)^(3/2)*(c - d*x^4)) + (b^(3/4)*(5*b^2*c^2 - 17*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
*a^(7/4)*c*(b*c - a*d)^3*Sqrt[a - b*x^4]) + (a^(1/4)*d^2*(13*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*(b*c - a*d)^3*Sqrt[a - b*x^4]) + (a^(1/4)*d^2*
(13*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*(b*c - a*d)^3*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(1/(-b*x**4+a)**(5/2)/(-d*x**4+c)**2,x)

[Out]

Timed out

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Mathematica [C]  time = 2.18446, size = 617, normalized size = 1.41 \[ \frac{x \left (\frac{25 a \left (9 a^3 d^3-36 a^2 b c d^2+17 a b^2 c^2 d-5 b^3 c^3\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )}{2 x^4 \left (2 a d F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )+5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )}+\frac{10 x^4 \left (3 a^4 d^3-6 a^3 b d^3 x^4+a^2 b^2 d \left (19 c^2-19 c d x^4+3 d^2 x^8\right )+a b^3 c \left (-7 c^2-10 c d x^4+17 d^2 x^8\right )+5 b^4 c^2 x^4 \left (c-d x^4\right )\right ) \left (2 a d F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )+9 a c \left (15 a^4 d^3-33 a^3 b d^3 x^4+a^2 b^2 d \left (95 c^2-112 c d x^4+18 d^2 x^8\right )+a b^3 c \left (-35 c^2-45 c d x^4+102 d^2 x^8\right )+5 b^4 c^2 x^4 \left (5 c-6 d x^4\right )\right ) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )}{c \left (a-b x^4\right ) \left (2 x^4 \left (2 a d F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )+9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )}\right )}{60 a^2 \sqrt{a-b x^4} \left (c-d x^4\right ) (a d-b c)^3} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((a - b*x^4)^(5/2)*(c - d*x^4)^2),x]

[Out]

(x*((25*a*(-5*b^3*c^3 + 17*a*b^2*c^2*d - 36*a^2*b*c*d^2 + 9*a^3*d^3)*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^4*d^3 - 3
3*a^3*b*d^3*x^4 + 5*b^4*c^2*x^4*(5*c - 6*d*x^4) + a^2*b^2*d*(95*c^2 - 112*c*d*x^
4 + 18*d^2*x^8) + a*b^3*c*(-35*c^2 - 45*c*d*x^4 + 102*d^2*x^8))*AppellF1[5/4, 1/
2, 1, 9/4, (b*x^4)/a, (d*x^4)/c] + 10*x^4*(3*a^4*d^3 - 6*a^3*b*d^3*x^4 + 5*b^4*c
^2*x^4*(c - d*x^4) + a^2*b^2*d*(19*c^2 - 19*c*d*x^4 + 3*d^2*x^8) + a*b^3*c*(-7*c
^2 - 10*c*d*x^4 + 17*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*(a - b*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])))))/(60*a^2*(-(b*c) + a*d)^3*Sqrt[a - b*x^4]*(c - d*x^4))

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Maple [C]  time = 0.057, size = 484, normalized size = 1.1 \[ -{\frac{b{d}^{3}x}{ \left ( 4\,{a}^{2}{d}^{2}-8\,cabd+4\,{b}^{2}{c}^{2} \right ) \left ( ad-bc \right ) c \left ( bd{x}^{4}-bc \right ) }\sqrt{-b{x}^{4}+a}}+{\frac{x}{6\, \left ( ad-bc \right ) ^{2}a}\sqrt{-b{x}^{4}+a} \left ({x}^{4}-{\frac{a}{b}} \right ) ^{-2}}+{\frac{{b}^{2}x \left ( 17\,ad-5\,bc \right ) }{12\,{a}^{2} \left ( ad-bc \right ) ^{3}}{\frac{1}{\sqrt{- \left ({x}^{4}-{\frac{a}{b}} \right ) b}}}}+{1 \left ({\frac{{d}^{2}b}{ \left ( 4\,{a}^{2}{d}^{2}-8\,cabd+4\,{b}^{2}{c}^{2} \right ) \left ( ad-bc \right ) c}}+{\frac{{b}^{2} \left ( 17\,ad-5\,bc \right ) }{12\,{a}^{2} \left ( ad-bc \right ) ^{3}}} \right ) \sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}}-{\frac{d}{32\,c}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}d-c \right ) }{\frac{3\,ad-13\,bc}{ \left ( ad-bc \right ) ^{3}{{\it \_alpha}}^{3}} \left ( -{1{\it Artanh} \left ({\frac{-2\,{{\it \_alpha}}^{2}b{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \right ){\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}}-2\,{\frac{{{\it \_alpha}}^{3}d}{c\sqrt{-b{x}^{4}+a}}\sqrt{1-{\frac{\sqrt{b}{x}^{2}}{\sqrt{a}}}}\sqrt{1+{\frac{\sqrt{b}{x}^{2}}{\sqrt{a}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}},{\frac{\sqrt{a}{{\it \_alpha}}^{2}d}{c\sqrt{b}}},{1\sqrt{-{\frac{\sqrt{b}}{\sqrt{a}}}}{\frac{1}{\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}}}}} \right ){\frac{1}{\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}}}}} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-b*x^4+a)^(5/2)/(-d*x^4+c)^2,x)

[Out]

-1/4*b*d^3/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)/c*x*(-b*x^4+a)^(1/2)/(b*d*x^4-b
*c)+1/6/(a*d-b*c)^2/a*x*(-b*x^4+a)^(1/2)/(x^4-a/b)^2+1/12*b^2/a^2*x*(17*a*d-5*b*
c)/(a*d-b*c)^3/(-(x^4-a/b)*b)^(1/2)+(1/4*b*d^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-
b*c)/c+1/12*b^2/a^2*(17*a*d-5*b*c)/(a*d-b*c)^3)/(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/c*sum((3*a*d-13*b*c)/(a*d-b*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. \[ \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{5}{2}}{\left (d x^{4} - c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(5/2)*(d*x^4 - c)^2),x, algorithm="maxima")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(5/2)*(d*x^4 - c)^2),x, algorithm="giac")

[Out]

integrate(1/((-b*x^4 + a)^(5/2)*(d*x^4 - c)^2), x)