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

Optimal. Leaf size=298 \[ \frac{\sqrt{a} b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 c d \left (a+b x^4\right )^{3/4}}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (3 a d+2 b c) \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (3 a d+2 b c) \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d}-\frac{x \sqrt [4]{a+b x^4} (b c-a d)}{4 c d \left (c+d x^4\right )} \]

[Out]

-((b*c - a*d)*x*(a + b*x^4)^(1/4))/(4*c*d*(c + d*x^4)) + (Sqrt[a]*b^(3/2)*(1 + a
/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(4*c*d*(a + b
*x^4)^(3/4)) + ((2*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])), ArcSin[(b^(1/4)*x)/(a + b*x^4)^(1/4)], -1])
/(8*b^(1/4)*c^2*d) + ((2*b*c + 3*a*d)*Sqrt[a/(a + b*x^4)]*Sqrt[a + b*x^4]*Ellipt
icPi[Sqrt[b*c - a*d]/(Sqrt[b]*Sqrt[c]), ArcSin[(b^(1/4)*x)/(a + b*x^4)^(1/4)], -
1])/(8*b^(1/4)*c^2*d)

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Rubi [A]  time = 0.703559, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429 \[ \frac{\sqrt{a} b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 c d \left (a+b x^4\right )^{3/4}}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (3 a d+2 b c) \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (3 a d+2 b c) \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d}-\frac{x \sqrt [4]{a+b x^4} (b c-a d)}{4 c d \left (c+d x^4\right )} \]

Antiderivative was successfully verified.

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

[Out]

-((b*c - a*d)*x*(a + b*x^4)^(1/4))/(4*c*d*(c + d*x^4)) + (Sqrt[a]*b^(3/2)*(1 + a
/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(4*c*d*(a + b
*x^4)^(3/4)) + ((2*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])), ArcSin[(b^(1/4)*x)/(a + b*x^4)^(1/4)], -1])
/(8*b^(1/4)*c^2*d) + ((2*b*c + 3*a*d)*Sqrt[a/(a + b*x^4)]*Sqrt[a + b*x^4]*Ellipt
icPi[Sqrt[b*c - a*d]/(Sqrt[b]*Sqrt[c]), ArcSin[(b^(1/4)*x)/(a + b*x^4)^(1/4)], -
1])/(8*b^(1/4)*c^2*d)

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Rubi in Sympy [A]  time = 83.37, size = 260, normalized size = 0.87 \[ \frac{\sqrt{a} b^{\frac{3}{2}} x^{3} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{4 c d \left (a + b x^{4}\right )^{\frac{3}{4}}} + \frac{x \sqrt [4]{a + b x^{4}} \left (a d - b c\right )}{4 c d \left (c + d x^{4}\right )} + \frac{\sqrt{\frac{a}{a + b x^{4}}} \sqrt{a + b x^{4}} \left (3 a d + 2 b c\right ) \Pi \left (- \frac{\sqrt{- a d + b c}}{\sqrt{b} \sqrt{c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}\middle | -1\right )}{8 \sqrt [4]{b} c^{2} d} + \frac{\sqrt{\frac{a}{a + b x^{4}}} \sqrt{a + b x^{4}} \left (3 a d + 2 b c\right ) \Pi \left (\frac{\sqrt{- a d + b c}}{\sqrt{b} \sqrt{c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}\middle | -1\right )}{8 \sqrt [4]{b} c^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(a)*b**(3/2)*x**3*(a/(b*x**4) + 1)**(3/4)*elliptic_f(atan(sqrt(a)/(sqrt(b)*x
**2))/2, 2)/(4*c*d*(a + b*x**4)**(3/4)) + x*(a + b*x**4)**(1/4)*(a*d - b*c)/(4*c
*d*(c + d*x**4)) + sqrt(a/(a + b*x**4))*sqrt(a + b*x**4)*(3*a*d + 2*b*c)*ellipti
c_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*d) + sqrt(a/(a + b*x**4))*sqrt(a + b*x**4)*(3*a*d + 2*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*d)

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Mathematica [C]  time = 0.646114, size = 440, normalized size = 1.48 \[ \frac{x \left (\frac{9 a c \left (5 a^2 d+a b \left (7 d x^4-5 c\right )-3 b^2 c x^4\right ) F_1\left (\frac{5}{4};\frac{3}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+5 x^4 \left (a+b x^4\right ) (b c-a d) \left (4 a d F_1\left (\frac{9}{4};\frac{3}{4},2;\frac{13}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+3 b c F_1\left (\frac{9}{4};\frac{7}{4},1;\frac{13}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )}{c \left (9 a c F_1\left (\frac{5}{4};\frac{3}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )-x^4 \left (4 a d F_1\left (\frac{9}{4};\frac{3}{4},2;\frac{13}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+3 b c F_1\left (\frac{9}{4};\frac{7}{4},1;\frac{13}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )\right )}-\frac{25 a^2 (3 a d+b c) F_1\left (\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{x^4 \left (4 a d F_1\left (\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+3 b c F_1\left (\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )-5 a c F_1\left (\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}\right )}{20 d \left (a+b x^4\right )^{3/4} \left (c+d x^4\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x*((-25*a^2*(b*c + 3*a*d)*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*A
ppellF1[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*(5*a^2*d - 3*b^2*c*x^4 + a*b*(-5
*c + 7*d*x^4))*AppellF1[5/4, 3/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + 5*(b*c -
a*d)*x^4*(a + b*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)]))/(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)])))))/(20*d*(a + b*x^4)^(3/4)*(c + d*x^4))

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( d{x}^{4}+c \right ) ^{2}} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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