3.336 \(\int \frac{1}{\left (a+b x^2\right )^{9/4} \left (c+d x^2\right )^2} \, dx\)

Optimal. Leaf size=371 \[ \frac{\sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} \left (-5 a^2 d^2-52 a b c d+12 b^2 c^2\right ) E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{10 a^{3/2} c \sqrt [4]{a+b x^2} (b c-a d)^3}-\frac{\sqrt [4]{a} d^{3/2} \sqrt{-\frac{b x^2}{a}} (11 b c-2 a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (a d-b c)^{7/2}}+\frac{\sqrt [4]{a} d^{3/2} \sqrt{-\frac{b x^2}{a}} (11 b c-2 a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (a d-b c)^{7/2}}-\frac{d x}{2 c \left (a+b x^2\right )^{5/4} \left (c+d x^2\right ) (b c-a d)}+\frac{b x (5 a d+4 b c)}{10 a c \left (a+b x^2\right )^{5/4} (b c-a d)^2} \]

[Out]

(b*(4*b*c + 5*a*d)*x)/(10*a*c*(b*c - a*d)^2*(a + b*x^2)^(5/4)) - (d*x)/(2*c*(b*c
- a*d)*(a + b*x^2)^(5/4)*(c + d*x^2)) + (Sqrt[b]*(12*b^2*c^2 - 52*a*b*c*d - 5*a
^2*d^2)*(1 + (b*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(10*a
^(3/2)*c*(b*c - a*d)^3*(a + b*x^2)^(1/4)) - (a^(1/4)*d^(3/2)*(11*b*c - 2*a*d)*Sq
rt[-((b*x^2)/a)]*EllipticPi[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSin[(a +
b*x^2)^(1/4)/a^(1/4)], -1])/(4*c*(-(b*c) + a*d)^(7/2)*x) + (a^(1/4)*d^(3/2)*(11
*b*c - 2*a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[(Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]
, ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(4*c*(-(b*c) + a*d)^(7/2)*x)

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Rubi [A]  time = 1.43189, antiderivative size = 371, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429 \[ \frac{\sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} \left (-5 a^2 d^2-52 a b c d+12 b^2 c^2\right ) E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{10 a^{3/2} c \sqrt [4]{a+b x^2} (b c-a d)^3}-\frac{\sqrt [4]{a} d^{3/2} \sqrt{-\frac{b x^2}{a}} (11 b c-2 a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (a d-b c)^{7/2}}+\frac{\sqrt [4]{a} d^{3/2} \sqrt{-\frac{b x^2}{a}} (11 b c-2 a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (a d-b c)^{7/2}}-\frac{d x}{2 c \left (a+b x^2\right )^{5/4} \left (c+d x^2\right ) (b c-a d)}+\frac{b x (5 a d+4 b c)}{10 a c \left (a+b x^2\right )^{5/4} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^2)^(9/4)*(c + d*x^2)^2),x]

[Out]

(b*(4*b*c + 5*a*d)*x)/(10*a*c*(b*c - a*d)^2*(a + b*x^2)^(5/4)) - (d*x)/(2*c*(b*c
- a*d)*(a + b*x^2)^(5/4)*(c + d*x^2)) + (Sqrt[b]*(12*b^2*c^2 - 52*a*b*c*d - 5*a
^2*d^2)*(1 + (b*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(10*a
^(3/2)*c*(b*c - a*d)^3*(a + b*x^2)^(1/4)) - (a^(1/4)*d^(3/2)*(11*b*c - 2*a*d)*Sq
rt[-((b*x^2)/a)]*EllipticPi[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSin[(a +
b*x^2)^(1/4)/a^(1/4)], -1])/(4*c*(-(b*c) + a*d)^(7/2)*x) + (a^(1/4)*d^(3/2)*(11
*b*c - 2*a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[(Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]
, ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(4*c*(-(b*c) + a*d)^(7/2)*x)

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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**2+a)**(9/4)/(d*x**2+c)**2,x)

[Out]

Timed out

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Mathematica [C]  time = 2.30074, size = 634, normalized size = 1.71 \[ \frac{x \left (\frac{18 a \left (5 a^3 d^3-30 a^2 b c d^2-26 a b^2 c^2 d+6 b^3 c^3\right ) F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{x^2 \left (4 a d F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-6 a c F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}+\frac{3 x^2 \left (5 a^4 d^3+10 a^3 b d^3 x^2+a^2 b^2 d \left (56 c^2+56 c d x^2+5 d^2 x^4\right )+4 a b^3 c \left (-4 c^2+9 c d x^2+13 d^2 x^4\right )-12 b^4 c^2 x^2 \left (c+d x^2\right )\right ) \left (4 a d F_1\left (\frac{5}{2};\frac{1}{4},2;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (\frac{5}{2};\frac{5}{4},1;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-5 a c \left (30 a^4 d^3+55 a^3 b d^3 x^2+a^2 b^2 d \left (336 c^2+284 c d x^2+25 d^2 x^4\right )+4 a b^3 c \left (-24 c^2+57 c d x^2+65 d^2 x^4\right )-12 b^4 c^2 x^2 \left (6 c+5 d x^2\right )\right ) F_1\left (\frac{3}{2};\frac{1}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{c \left (a+b x^2\right ) \left (10 a c F_1\left (\frac{3}{2};\frac{1}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )-x^2 \left (4 a d F_1\left (\frac{5}{2};\frac{1}{4},2;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (\frac{5}{2};\frac{5}{4},1;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )\right )}\right )}{30 a^2 \sqrt [4]{a+b x^2} \left (c+d x^2\right ) (b c-a d)^3} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((a + b*x^2)^(9/4)*(c + d*x^2)^2),x]

[Out]

(x*((18*a*(6*b^3*c^3 - 26*a*b^2*c^2*d - 30*a^2*b*c*d^2 + 5*a^3*d^3)*AppellF1[1/2
, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)])/(-6*a*c*AppellF1[1/2, 1/4, 1, 3/2, -
((b*x^2)/a), -((d*x^2)/c)] + x^2*(4*a*d*AppellF1[3/2, 1/4, 2, 5/2, -((b*x^2)/a),
-((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])) +
(-5*a*c*(30*a^4*d^3 + 55*a^3*b*d^3*x^2 - 12*b^4*c^2*x^2*(6*c + 5*d*x^2) + a^2*b^
2*d*(336*c^2 + 284*c*d*x^2 + 25*d^2*x^4) + 4*a*b^3*c*(-24*c^2 + 57*c*d*x^2 + 65*
d^2*x^4))*AppellF1[3/2, 1/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + 3*x^2*(5*a^4*
d^3 + 10*a^3*b*d^3*x^2 - 12*b^4*c^2*x^2*(c + d*x^2) + a^2*b^2*d*(56*c^2 + 56*c*d
*x^2 + 5*d^2*x^4) + 4*a*b^3*c*(-4*c^2 + 9*c*d*x^2 + 13*d^2*x^4))*(4*a*d*AppellF1
[5/2, 1/4, 2, 7/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[5/2, 5/4, 1, 7/2,
-((b*x^2)/a), -((d*x^2)/c)]))/(c*(a + b*x^2)*(10*a*c*AppellF1[3/2, 1/4, 1, 5/2,
-((b*x^2)/a), -((d*x^2)/c)] - x^2*(4*a*d*AppellF1[5/2, 1/4, 2, 7/2, -((b*x^2)/a)
, -((d*x^2)/c)] + b*c*AppellF1[5/2, 5/4, 1, 7/2, -((b*x^2)/a), -((d*x^2)/c)]))))
)/(30*a^2*(b*c - a*d)^3*(a + b*x^2)^(1/4)*(c + d*x^2))

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( d{x}^{2}+c \right ) ^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{9}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^2+a)^(9/4)/(d*x^2+c)^2,x)

[Out]

int(1/(b*x^2+a)^(9/4)/(d*x^2+c)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{9}{4}}{\left (d x^{2} + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(9/4)*(d*x^2 + c)^2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^(9/4)*(d*x^2 + 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^2 + a)^(9/4)*(d*x^2 + 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**2+a)**(9/4)/(d*x**2+c)**2,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{9}{4}}{\left (d x^{2} + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(9/4)*(d*x^2 + c)^2),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)^(9/4)*(d*x^2 + c)^2), x)