3.116 \(\int \frac{1}{\left (a+b x^4\right )^{7/4} \left (c+d x^4\right )^2} \, dx\)

Optimal. Leaf size=390 \[ -\frac{b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} \left (3 a^2 d^2-32 a b c d+8 b^2 c^2\right ) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{12 a^{3/2} c \left (a+b x^4\right )^{3/4} (b c-a d)^3}+\frac{d^2 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (10 b c-3 a d) \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)^3}+\frac{d^2 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (10 b c-3 a d) \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)^3}+\frac{b x (3 a d+4 b c)}{12 a c \left (a+b x^4\right )^{3/4} (b c-a d)^2}-\frac{d x}{4 c \left (a+b x^4\right )^{3/4} \left (c+d x^4\right ) (b c-a d)} \]

[Out]

(b*(4*b*c + 3*a*d)*x)/(12*a*c*(b*c - a*d)^2*(a + b*x^4)^(3/4)) - (d*x)/(4*c*(b*c
- a*d)*(a + b*x^4)^(3/4)*(c + d*x^4)) - (b^(3/2)*(8*b^2*c^2 - 32*a*b*c*d + 3*a^
2*d^2)*(1 + a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/
(12*a^(3/2)*c*(b*c - a*d)^3*(a + b*x^4)^(3/4)) + (d^2*(10*b*c - 3*a*d)*Sqrt[a/(a
+ b*x^4)]*Sqrt[a + b*x^4]*EllipticPi[-(Sqrt[b*c - a*d]/(Sqrt[b]*Sqrt[c])), ArcS
in[(b^(1/4)*x)/(a + b*x^4)^(1/4)], -1])/(8*b^(1/4)*c^2*(b*c - a*d)^3) + (d^2*(10
*b*c - 3*a*d)*Sqrt[a/(a + b*x^4)]*Sqrt[a + b*x^4]*EllipticPi[Sqrt[b*c - a*d]/(Sq
rt[b]*Sqrt[c]), ArcSin[(b^(1/4)*x)/(a + b*x^4)^(1/4)], -1])/(8*b^(1/4)*c^2*(b*c
- a*d)^3)

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Rubi [A]  time = 1.16421, antiderivative size = 390, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476 \[ -\frac{b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} \left (3 a^2 d^2-32 a b c d+8 b^2 c^2\right ) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{12 a^{3/2} c \left (a+b x^4\right )^{3/4} (b c-a d)^3}+\frac{d^2 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (10 b c-3 a d) \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)^3}+\frac{d^2 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (10 b c-3 a d) \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)^3}+\frac{b x (3 a d+4 b c)}{12 a c \left (a+b x^4\right )^{3/4} (b c-a d)^2}-\frac{d x}{4 c \left (a+b x^4\right )^{3/4} \left (c+d x^4\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^4)^(7/4)*(c + d*x^4)^2),x]

[Out]

(b*(4*b*c + 3*a*d)*x)/(12*a*c*(b*c - a*d)^2*(a + b*x^4)^(3/4)) - (d*x)/(4*c*(b*c
- a*d)*(a + b*x^4)^(3/4)*(c + d*x^4)) - (b^(3/2)*(8*b^2*c^2 - 32*a*b*c*d + 3*a^
2*d^2)*(1 + a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/
(12*a^(3/2)*c*(b*c - a*d)^3*(a + b*x^4)^(3/4)) + (d^2*(10*b*c - 3*a*d)*Sqrt[a/(a
+ b*x^4)]*Sqrt[a + b*x^4]*EllipticPi[-(Sqrt[b*c - a*d]/(Sqrt[b]*Sqrt[c])), ArcS
in[(b^(1/4)*x)/(a + b*x^4)^(1/4)], -1])/(8*b^(1/4)*c^2*(b*c - a*d)^3) + (d^2*(10
*b*c - 3*a*d)*Sqrt[a/(a + b*x^4)]*Sqrt[a + b*x^4]*EllipticPi[Sqrt[b*c - a*d]/(Sq
rt[b]*Sqrt[c]), ArcSin[(b^(1/4)*x)/(a + b*x^4)^(1/4)], -1])/(8*b^(1/4)*c^2*(b*c
- a*d)^3)

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Rubi in Sympy [A]  time = 157.932, size = 350, normalized size = 0.9 \[ \frac{d x}{4 c \left (a + b x^{4}\right )^{\frac{3}{4}} \left (c + d x^{4}\right ) \left (a d - b c\right )} + \frac{d^{2} \sqrt{\frac{a}{a + b x^{4}}} \sqrt{a + b x^{4}} \left (3 a d - 10 b c\right ) \Pi \left (- \frac{\sqrt{- a d + b c}}{\sqrt{b} \sqrt{c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}\middle | -1\right )}{8 \sqrt [4]{b} c^{2} \left (a d - b c\right )^{3}} + \frac{d^{2} \sqrt{\frac{a}{a + b x^{4}}} \sqrt{a + b x^{4}} \left (3 a d - 10 b c\right ) \Pi \left (\frac{\sqrt{- a d + b c}}{\sqrt{b} \sqrt{c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}\middle | -1\right )}{8 \sqrt [4]{b} c^{2} \left (a d - b c\right )^{3}} + \frac{b x \left (3 a d + 4 b c\right )}{12 a c \left (a + b x^{4}\right )^{\frac{3}{4}} \left (a d - b c\right )^{2}} + \frac{b^{\frac{3}{2}} x^{3} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} \left (3 a^{2} d^{2} - 32 a b c d + 8 b^{2} c^{2}\right ) F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{12 a^{\frac{3}{2}} c \left (a + b x^{4}\right )^{\frac{3}{4}} \left (a d - b c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**4+a)**(7/4)/(d*x**4+c)**2,x)

[Out]

d*x/(4*c*(a + b*x**4)**(3/4)*(c + d*x**4)*(a*d - b*c)) + d**2*sqrt(a/(a + b*x**4
))*sqrt(a + b*x**4)*(3*a*d - 10*b*c)*elliptic_pi(-sqrt(-a*d + b*c)/(sqrt(b)*sqrt
(c)), asin(b**(1/4)*x/(a + b*x**4)**(1/4)), -1)/(8*b**(1/4)*c**2*(a*d - b*c)**3)
+ d**2*sqrt(a/(a + b*x**4))*sqrt(a + b*x**4)*(3*a*d - 10*b*c)*elliptic_pi(sqrt(
-a*d + b*c)/(sqrt(b)*sqrt(c)), asin(b**(1/4)*x/(a + b*x**4)**(1/4)), -1)/(8*b**(
1/4)*c**2*(a*d - b*c)**3) + b*x*(3*a*d + 4*b*c)/(12*a*c*(a + b*x**4)**(3/4)*(a*d
- b*c)**2) + b**(3/2)*x**3*(a/(b*x**4) + 1)**(3/4)*(3*a**2*d**2 - 32*a*b*c*d +
8*b**2*c**2)*elliptic_f(atan(sqrt(a)/(sqrt(b)*x**2))/2, 2)/(12*a**(3/2)*c*(a + b
*x**4)**(3/4)*(a*d - b*c)**3)

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Mathematica [C]  time = 1.03826, size = 485, normalized size = 1.24 \[ \frac{x \left (\frac{9 a c \left (15 a^2 d^2+21 a b d^2 x^4+4 b^2 c \left (5 c+7 d x^4\right )\right ) F_1\left (\frac{5}{4};\frac{3}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )-5 x^4 \left (3 a^2 d^2+3 a b d^2 x^4+4 b^2 c \left (c+d x^4\right )\right ) \left (4 a d F_1\left (\frac{9}{4};\frac{3}{4},2;\frac{13}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+3 b c F_1\left (\frac{9}{4};\frac{7}{4},1;\frac{13}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )}{a c \left (9 a c F_1\left (\frac{5}{4};\frac{3}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )-x^4 \left (4 a d F_1\left (\frac{9}{4};\frac{3}{4},2;\frac{13}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+3 b c F_1\left (\frac{9}{4};\frac{7}{4},1;\frac{13}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )\right )}-\frac{25 \left (9 a^2 d^2-24 a b c d+8 b^2 c^2\right ) F_1\left (\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{x^4 \left (4 a d F_1\left (\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+3 b c F_1\left (\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )-5 a c F_1\left (\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}\right )}{60 \left (a+b x^4\right )^{3/4} \left (c+d x^4\right ) (b c-a d)^2} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((a + b*x^4)^(7/4)*(c + d*x^4)^2),x]

[Out]

(x*((-25*(8*b^2*c^2 - 24*a*b*c*d + 9*a^2*d^2)*AppellF1[1/4, 3/4, 1, 5/4, -((b*x^
4)/a), -((d*x^4)/c)])/(-5*a*c*AppellF1[1/4, 3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)
/c)] + x^4*(4*a*d*AppellF1[5/4, 3/4, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + 3*b*c
*AppellF1[5/4, 7/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])) + (9*a*c*(15*a^2*d^2 +
21*a*b*d^2*x^4 + 4*b^2*c*(5*c + 7*d*x^4))*AppellF1[5/4, 3/4, 1, 9/4, -((b*x^4)/
a), -((d*x^4)/c)] - 5*x^4*(3*a^2*d^2 + 3*a*b*d^2*x^4 + 4*b^2*c*(c + d*x^4))*(4*a
*d*AppellF1[9/4, 3/4, 2, 13/4, -((b*x^4)/a), -((d*x^4)/c)] + 3*b*c*AppellF1[9/4,
7/4, 1, 13/4, -((b*x^4)/a), -((d*x^4)/c)]))/(a*c*(9*a*c*AppellF1[5/4, 3/4, 1, 9
/4, -((b*x^4)/a), -((d*x^4)/c)] - x^4*(4*a*d*AppellF1[9/4, 3/4, 2, 13/4, -((b*x^
4)/a), -((d*x^4)/c)] + 3*b*c*AppellF1[9/4, 7/4, 1, 13/4, -((b*x^4)/a), -((d*x^4)
/c)])))))/(60*(b*c - a*d)^2*(a + b*x^4)^(3/4)*(c + d*x^4))

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Maple [F]  time = 0.06, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( d{x}^{4}+c \right ) ^{2}} \left ( b{x}^{4}+a \right ) ^{-{\frac{7}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^4+a)^(7/4)/(d*x^4+c)^2,x)

[Out]

int(1/(b*x^4+a)^(7/4)/(d*x^4+c)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{7}{4}}{\left (d x^{4} + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(7/4)*(d*x^4 + c)^2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(7/4)*(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)^(7/4)*(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)**(7/4)/(d*x**4+c)**2,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{7}{4}}{\left (d x^{4} + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(7/4)*(d*x^4 + c)^2),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(7/4)*(d*x^4 + c)^2), x)