3.100 \(\int \frac{\left (a+b x^4\right )^{9/4}}{c+d x^4} \, dx\)

Optimal. Leaf size=316 \[ \frac{\sqrt{a} b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} (6 b c-11 a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{12 d^2 \left (a+b x^4\right )^{3/4}}-\frac{b x \sqrt [4]{a+b x^4} (6 b c-11 a d)}{12 d^2}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (b c-a d)^2 \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^2}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (b c-a d)^2 \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^2}+\frac{b x \left (a+b x^4\right )^{5/4}}{6 d} \]

[Out]

-(b*(6*b*c - 11*a*d)*x*(a + b*x^4)^(1/4))/(12*d^2) + (b*x*(a + b*x^4)^(5/4))/(6*
d) + (Sqrt[a]*b^(3/2)*(6*b*c - 11*a*d)*(1 + a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCo
t[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(12*d^2*(a + b*x^4)^(3/4)) + ((b*c - a*d)^2*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*d^2) + ((b*c - a*d)^2*
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*d^2)

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Rubi [A]  time = 0.921635, antiderivative size = 316, 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 \[ \frac{\sqrt{a} b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} (6 b c-11 a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{12 d^2 \left (a+b x^4\right )^{3/4}}-\frac{b x \sqrt [4]{a+b x^4} (6 b c-11 a d)}{12 d^2}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (b c-a d)^2 \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^2}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (b c-a d)^2 \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^2}+\frac{b x \left (a+b x^4\right )^{5/4}}{6 d} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(9/4)/(c + d*x^4),x]

[Out]

-(b*(6*b*c - 11*a*d)*x*(a + b*x^4)^(1/4))/(12*d^2) + (b*x*(a + b*x^4)^(5/4))/(6*
d) + (Sqrt[a]*b^(3/2)*(6*b*c - 11*a*d)*(1 + a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCo
t[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(12*d^2*(a + b*x^4)^(3/4)) + ((b*c - a*d)^2*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*d^2) + ((b*c - a*d)^2*
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*d^2)

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Rubi in Sympy [A]  time = 112.517, size = 282, normalized size = 0.89 \[ - \frac{\sqrt{a} b^{\frac{3}{2}} x^{3} \left (11 a d - 6 b c\right ) \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{12 d^{2} \left (a + b x^{4}\right )^{\frac{3}{4}}} + \frac{b x \left (a + b x^{4}\right )^{\frac{5}{4}}}{6 d} + \frac{b x \sqrt [4]{a + b x^{4}} \left (11 a d - 6 b c\right )}{12 d^{2}} + \frac{\sqrt{\frac{a}{a + b x^{4}}} \sqrt{a + b x^{4}} \left (a d - b c\right )^{2} \Pi \left (- \frac{\sqrt{- a d + b c}}{\sqrt{b} \sqrt{c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}\middle | -1\right )}{2 \sqrt [4]{b} c d^{2}} + \frac{\sqrt{\frac{a}{a + b x^{4}}} \sqrt{a + b x^{4}} \left (a d - b c\right )^{2} \Pi \left (\frac{\sqrt{- a d + b c}}{\sqrt{b} \sqrt{c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}\middle | -1\right )}{2 \sqrt [4]{b} c d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**(9/4)/(d*x**4+c),x)

[Out]

-sqrt(a)*b**(3/2)*x**3*(11*a*d - 6*b*c)*(a/(b*x**4) + 1)**(3/4)*elliptic_f(atan(
sqrt(a)/(sqrt(b)*x**2))/2, 2)/(12*d**2*(a + b*x**4)**(3/4)) + b*x*(a + b*x**4)**
(5/4)/(6*d) + b*x*(a + b*x**4)**(1/4)*(11*a*d - 6*b*c)/(12*d**2) + sqrt(a/(a + b
*x**4))*sqrt(a + b*x**4)*(a*d - b*c)**2*elliptic_pi(-sqrt(-a*d + b*c)/(sqrt(b)*s
qrt(c)), asin(b**(1/4)*x/(a + b*x**4)**(1/4)), -1)/(2*b**(1/4)*c*d**2) + sqrt(a/
(a + b*x**4))*sqrt(a + b*x**4)*(a*d - b*c)**2*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*d**2)

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Mathematica [C]  time = 1.72467, size = 396, normalized size = 1.25 \[ \frac{x \left (-\frac{9 a b c x^4 \left (23 a^2 d^2-30 a b c d+12 b^2 c^2\right ) F_1\left (\frac{5}{4};\frac{3}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{\left (c+d x^4\right ) \left (x^4 \left (4 a d F_1\left (\frac{9}{4};\frac{3}{4},2;\frac{13}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+3 b c F_1\left (\frac{9}{4};\frac{7}{4},1;\frac{13}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )-9 a c F_1\left (\frac{5}{4};\frac{3}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )}-\frac{25 a^2 c \left (12 a^2 d^2-13 a b c d+6 b^2 c^2\right ) F_1\left (\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{\left (c+d x^4\right ) \left (x^4 \left (4 a d F_1\left (\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+3 b c F_1\left (\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )-5 a c F_1\left (\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )}+5 b \left (a+b x^4\right ) \left (13 a d-6 b c+2 b d x^4\right )\right )}{60 d^2 \left (a+b x^4\right )^{3/4}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x^4)^(9/4)/(c + d*x^4),x]

[Out]

(x*(5*b*(a + b*x^4)*(-6*b*c + 13*a*d + 2*b*d*x^4) - (25*a^2*c*(6*b^2*c^2 - 13*a*
b*c*d + 12*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)]))) - (9*a*b*c*(12*b^2*c^2 - 30*a*b*c*
d + 23*a^2*d^2)*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)])))))/(60*d^2*(a + b*x^4)^(3/4))

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Maple [F]  time = 0.098, size = 0, normalized size = 0. \[ \int{\frac{1}{d{x}^{4}+c} \left ( b{x}^{4}+a \right ) ^{{\frac{9}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(9/4)/(d*x^4+c),x)

[Out]

int((b*x^4+a)^(9/4)/(d*x^4+c),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{9}{4}}}{d x^{4} + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(9/4)/(d*x^4 + c),x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(9/4)/(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((b*x^4 + a)^(9/4)/(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((b*x**4+a)**(9/4)/(d*x**4+c),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{9}{4}}}{d x^{4} + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(9/4)/(d*x^4 + c),x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(9/4)/(d*x^4 + c), x)