3.105 \(\int \frac{1}{\left (a+b x^4\right )^{11/4} \left (c+d x^4\right )} \, dx\)

Optimal. Leaf size=357 \[ \frac{b x (6 b c-13 a d)}{21 a^2 \left (a+b x^4\right )^{3/4} (b c-a d)^2}-\frac{b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} \left (47 a^2 d^2-38 a b c d+12 b^2 c^2\right ) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{21 a^{5/2} \left (a+b x^4\right )^{3/4} (b c-a d)^3}-\frac{d^3 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c (b c-a d)^3}-\frac{d^3 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c (b c-a d)^3}+\frac{b x}{7 a \left (a+b x^4\right )^{7/4} (b c-a d)} \]

[Out]

(b*x)/(7*a*(b*c - a*d)*(a + b*x^4)^(7/4)) + (b*(6*b*c - 13*a*d)*x)/(21*a^2*(b*c
- a*d)^2*(a + b*x^4)^(3/4)) - (b^(3/2)*(12*b^2*c^2 - 38*a*b*c*d + 47*a^2*d^2)*(1
+ a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(21*a^(5/
2)*(b*c - a*d)^3*(a + b*x^4)^(3/4)) - (d^3*Sqrt[a/(a + b*x^4)]*Sqrt[a + b*x^4]*E
llipticPi[-(Sqrt[b*c - a*d]/(Sqrt[b]*Sqrt[c])), ArcSin[(b^(1/4)*x)/(a + b*x^4)^(
1/4)], -1])/(2*b^(1/4)*c*(b*c - a*d)^3) - (d^3*Sqrt[a/(a + b*x^4)]*Sqrt[a + b*x^
4]*EllipticPi[Sqrt[b*c - a*d]/(Sqrt[b]*Sqrt[c]), ArcSin[(b^(1/4)*x)/(a + b*x^4)^
(1/4)], -1])/(2*b^(1/4)*c*(b*c - a*d)^3)

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Rubi [A]  time = 1.07988, antiderivative size = 357, 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 \[ \frac{b x (6 b c-13 a d)}{21 a^2 \left (a+b x^4\right )^{3/4} (b c-a d)^2}-\frac{b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} \left (47 a^2 d^2-38 a b c d+12 b^2 c^2\right ) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{21 a^{5/2} \left (a+b x^4\right )^{3/4} (b c-a d)^3}-\frac{d^3 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c (b c-a d)^3}-\frac{d^3 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c (b c-a d)^3}+\frac{b x}{7 a \left (a+b x^4\right )^{7/4} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^4)^(11/4)*(c + d*x^4)),x]

[Out]

(b*x)/(7*a*(b*c - a*d)*(a + b*x^4)^(7/4)) + (b*(6*b*c - 13*a*d)*x)/(21*a^2*(b*c
- a*d)^2*(a + b*x^4)^(3/4)) - (b^(3/2)*(12*b^2*c^2 - 38*a*b*c*d + 47*a^2*d^2)*(1
+ a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(21*a^(5/
2)*(b*c - a*d)^3*(a + b*x^4)^(3/4)) - (d^3*Sqrt[a/(a + b*x^4)]*Sqrt[a + b*x^4]*E
llipticPi[-(Sqrt[b*c - a*d]/(Sqrt[b]*Sqrt[c])), ArcSin[(b^(1/4)*x)/(a + b*x^4)^(
1/4)], -1])/(2*b^(1/4)*c*(b*c - a*d)^3) - (d^3*Sqrt[a/(a + b*x^4)]*Sqrt[a + b*x^
4]*EllipticPi[Sqrt[b*c - a*d]/(Sqrt[b]*Sqrt[c]), ArcSin[(b^(1/4)*x)/(a + b*x^4)^
(1/4)], -1])/(2*b^(1/4)*c*(b*c - a*d)^3)

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Rubi in Sympy [A]  time = 161.555, size = 318, normalized size = 0.89 \[ \frac{d^{3} \sqrt{\frac{a}{a + b x^{4}}} \sqrt{a + b x^{4}} \Pi \left (- \frac{\sqrt{- a d + b c}}{\sqrt{b} \sqrt{c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}\middle | -1\right )}{2 \sqrt [4]{b} c \left (a d - b c\right )^{3}} + \frac{d^{3} \sqrt{\frac{a}{a + b x^{4}}} \sqrt{a + b x^{4}} \Pi \left (\frac{\sqrt{- a d + b c}}{\sqrt{b} \sqrt{c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}\middle | -1\right )}{2 \sqrt [4]{b} c \left (a d - b c\right )^{3}} - \frac{b x}{7 a \left (a + b x^{4}\right )^{\frac{7}{4}} \left (a d - b c\right )} - \frac{b x \left (13 a d - 6 b c\right )}{21 a^{2} \left (a + b x^{4}\right )^{\frac{3}{4}} \left (a d - b c\right )^{2}} + \frac{b^{\frac{3}{2}} x^{3} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} \left (47 a^{2} d^{2} - 38 a b c d + 12 b^{2} c^{2}\right ) F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{21 a^{\frac{5}{2}} \left (a + b x^{4}\right )^{\frac{3}{4}} \left (a d - b c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**4+a)**(11/4)/(d*x**4+c),x)

[Out]

d**3*sqrt(a/(a + b*x**4))*sqrt(a + b*x**4)*elliptic_pi(-sqrt(-a*d + b*c)/(sqrt(b
)*sqrt(c)), asin(b**(1/4)*x/(a + b*x**4)**(1/4)), -1)/(2*b**(1/4)*c*(a*d - b*c)*
*3) + d**3*sqrt(a/(a + b*x**4))*sqrt(a + b*x**4)*elliptic_pi(sqrt(-a*d + b*c)/(s
qrt(b)*sqrt(c)), asin(b**(1/4)*x/(a + b*x**4)**(1/4)), -1)/(2*b**(1/4)*c*(a*d -
b*c)**3) - b*x/(7*a*(a + b*x**4)**(7/4)*(a*d - b*c)) - b*x*(13*a*d - 6*b*c)/(21*
a**2*(a + b*x**4)**(3/4)*(a*d - b*c)**2) + b**(3/2)*x**3*(a/(b*x**4) + 1)**(3/4)
*(47*a**2*d**2 - 38*a*b*c*d + 12*b**2*c**2)*elliptic_f(atan(sqrt(a)/(sqrt(b)*x**
2))/2, 2)/(21*a**(5/2)*(a + b*x**4)**(3/4)*(a*d - b*c)**3)

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Mathematica [C]  time = 1.56098, size = 407, normalized size = 1.14 \[ \frac{x \left (\frac{25 a c \left (21 a^2 d^2-26 a b c d+12 b^2 c^2\right ) F_1\left (\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{\left (c+d x^4\right ) \left (5 a c F_1\left (\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )-x^4 \left (4 a d F_1\left (\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+3 b c F_1\left (\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )\right )}+\frac{5 b \left (-16 a^2 d+a b \left (9 c-13 d x^4\right )+6 b^2 c x^4\right )}{a+b x^4}+\frac{18 a b c d x^4 (13 a d-6 b c) F_1\left (\frac{5}{4};\frac{3}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{\left (c+d x^4\right ) \left (x^4 \left (4 a d F_1\left (\frac{9}{4};\frac{3}{4},2;\frac{13}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+3 b c F_1\left (\frac{9}{4};\frac{7}{4},1;\frac{13}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )-9 a c F_1\left (\frac{5}{4};\frac{3}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )}\right )}{105 a^2 \left (a+b x^4\right )^{3/4} (b c-a d)^2} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((a + b*x^4)^(11/4)*(c + d*x^4)),x]

[Out]

(x*((5*b*(-16*a^2*d + 6*b^2*c*x^4 + a*b*(9*c - 13*d*x^4)))/(a + b*x^4) + (25*a*c
*(12*b^2*c^2 - 26*a*b*c*d + 21*a^2*d^2)*AppellF1[1/4, 3/4, 1, 5/4, -((b*x^4)/a),
-((d*x^4)/c)])/((c + d*x^4)*(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)]))) + (18*a*b*c*d*(
-6*b*c + 13*a*d)*x^4*AppellF1[5/4, 3/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])/((c
+ d*x^4)*(-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)])))))/(105*a^2*(b*c - a*d)^2*(a + b
*x^4)^(3/4))

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Maple [F]  time = 0.058, size = 0, normalized size = 0. \[ \int{\frac{1}{d{x}^{4}+c} \left ( b{x}^{4}+a \right ) ^{-{\frac{11}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^4+a)^(11/4)/(d*x^4+c),x)

[Out]

int(1/(b*x^4+a)^(11/4)/(d*x^4+c),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{11}{4}}{\left (d x^{4} + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(11/4)*(d*x^4 + c)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(11/4)*(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(1/((b*x^4 + a)^(11/4)*(d*x^4 + c)),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)**(11/4)/(d*x**4+c),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{11}{4}}{\left (d x^{4} + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(11/4)*(d*x^4 + c)),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(11/4)*(d*x^4 + c)), x)