3.133 \(\int \frac{1}{\left (a-b x^2\right )^{4/3} \left (3 a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=807 \[ \frac{x}{12 a^3 \sqrt [3]{a-b x^2}}+\frac{x}{24 a^2 \sqrt [3]{a-b x^2} \left (b x^2+3 a\right )}+\frac{x}{12 a^3 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{16\ 2^{2/3} \sqrt{3} a^{17/6} \sqrt{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt{b} x}\right )}{16\ 2^{2/3} \sqrt{3} a^{17/6} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{48\ 2^{2/3} a^{17/6} \sqrt{b}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{16\ 2^{2/3} a^{17/6} \sqrt{b}}+\frac{\sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{8\ 3^{3/4} a^{8/3} b \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x}-\frac{\left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{6 \sqrt{2} \sqrt [4]{3} a^{8/3} b \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x} \]

[Out]

x/(12*a^3*(a - b*x^2)^(1/3)) + x/(24*a^2*(a - b*x^2)^(1/3)*(3*a + b*x^2)) + x/(1
2*a^3*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) + ArcTan[(Sqrt[3]*Sqrt[a])/(S
qrt[b]*x)]/(16*2^(2/3)*Sqrt[3]*a^(17/6)*Sqrt[b]) + ArcTan[(Sqrt[3]*a^(1/6)*(a^(1
/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(Sqrt[b]*x)]/(16*2^(2/3)*Sqrt[3]*a^(17/6)*Sqrt
[b]) - ArcTanh[(Sqrt[b]*x)/Sqrt[a]]/(48*2^(2/3)*a^(17/6)*Sqrt[b]) + ArcTanh[(Sqr
t[b]*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))]/(16*2^(2/3)*a^(17/6)*Sq
rt[b]) + (Sqrt[2 + Sqrt[3]]*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3
)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1
/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[
3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(8*3^(3/4)*a^(8/3)*b*x*Sqrt[
-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(
1/3))^2)]) - ((a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^
(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*Ellipt
icF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) -
(a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(6*Sqrt[2]*3^(1/4)*a^(8/3)*b*x*Sqrt[-((a^(
1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^
2)])

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Rubi [A]  time = 1.38853, antiderivative size = 807, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{x}{12 a^3 \sqrt [3]{a-b x^2}}+\frac{x}{24 a^2 \sqrt [3]{a-b x^2} \left (b x^2+3 a\right )}+\frac{x}{12 a^3 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{16\ 2^{2/3} \sqrt{3} a^{17/6} \sqrt{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt{b} x}\right )}{16\ 2^{2/3} \sqrt{3} a^{17/6} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{48\ 2^{2/3} a^{17/6} \sqrt{b}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{16\ 2^{2/3} a^{17/6} \sqrt{b}}+\frac{\sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{8\ 3^{3/4} a^{8/3} b \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x}-\frac{\left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{6 \sqrt{2} \sqrt [4]{3} a^{8/3} b \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x} \]

Warning: Unable to verify antiderivative.

[In]  Int[1/((a - b*x^2)^(4/3)*(3*a + b*x^2)^2),x]

[Out]

x/(12*a^3*(a - b*x^2)^(1/3)) + x/(24*a^2*(a - b*x^2)^(1/3)*(3*a + b*x^2)) + x/(1
2*a^3*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) + ArcTan[(Sqrt[3]*Sqrt[a])/(S
qrt[b]*x)]/(16*2^(2/3)*Sqrt[3]*a^(17/6)*Sqrt[b]) + ArcTan[(Sqrt[3]*a^(1/6)*(a^(1
/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(Sqrt[b]*x)]/(16*2^(2/3)*Sqrt[3]*a^(17/6)*Sqrt
[b]) - ArcTanh[(Sqrt[b]*x)/Sqrt[a]]/(48*2^(2/3)*a^(17/6)*Sqrt[b]) + ArcTanh[(Sqr
t[b]*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))]/(16*2^(2/3)*a^(17/6)*Sq
rt[b]) + (Sqrt[2 + Sqrt[3]]*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3
)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1
/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[
3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(8*3^(3/4)*a^(8/3)*b*x*Sqrt[
-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(
1/3))^2)]) - ((a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^
(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*Ellipt
icF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) -
(a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(6*Sqrt[2]*3^(1/4)*a^(8/3)*b*x*Sqrt[-((a^(
1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^
2)])

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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)**(4/3)/(b*x**2+3*a)**2,x)

[Out]

Timed out

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Mathematica [C]  time = 0.33416, size = 323, normalized size = 0.4 \[ \frac{x \left (\frac{27 a^2 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )}{2 b x^2 \left (F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )-F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )\right )+9 a F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )}-\frac{10 a b x^2 F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )}{2 b x^2 \left (F_1\left (\frac{5}{2};\frac{4}{3},1;\frac{7}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )-F_1\left (\frac{5}{2};\frac{1}{3},2;\frac{7}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )\right )+15 a F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )}+21 a+6 b x^2\right )}{72 a^3 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((a - b*x^2)^(4/3)*(3*a + b*x^2)^2),x]

[Out]

(x*(21*a + 6*b*x^2 + (27*a^2*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -(b*x^2)/(3*a
)])/(9*a*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -(b*x^2)/(3*a)] + 2*b*x^2*(-Appel
lF1[3/2, 1/3, 2, 5/2, (b*x^2)/a, -(b*x^2)/(3*a)] + AppellF1[3/2, 4/3, 1, 5/2, (b
*x^2)/a, -(b*x^2)/(3*a)])) - (10*a*b*x^2*AppellF1[3/2, 1/3, 1, 5/2, (b*x^2)/a, -
(b*x^2)/(3*a)])/(15*a*AppellF1[3/2, 1/3, 1, 5/2, (b*x^2)/a, -(b*x^2)/(3*a)] + 2*
b*x^2*(-AppellF1[5/2, 1/3, 2, 7/2, (b*x^2)/a, -(b*x^2)/(3*a)] + AppellF1[5/2, 4/
3, 1, 7/2, (b*x^2)/a, -(b*x^2)/(3*a)]))))/(72*a^3*(a - b*x^2)^(1/3)*(3*a + b*x^2
))

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( b{x}^{2}+3\,a \right ) ^{2}} \left ( -b{x}^{2}+a \right ) ^{-{\frac{4}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-b*x^2+a)^(4/3)/(b*x^2+3*a)^2,x)

[Out]

int(1/(-b*x^2+a)^(4/3)/(b*x^2+3*a)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + 3 \, a\right )}^{2}{\left (-b x^{2} + a\right )}^{\frac{4}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + 3*a)^2*(-b*x^2 + a)^(4/3)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + 3*a)^2*(-b*x^2 + a)^(4/3)), 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 + 3*a)^2*(-b*x^2 + a)^(4/3)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-b*x**2+a)**(4/3)/(b*x**2+3*a)**2,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + 3 \, a\right )}^{2}{\left (-b x^{2} + a\right )}^{\frac{4}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + 3*a)^2*(-b*x^2 + a)^(4/3)),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + 3*a)^2*(-b*x^2 + a)^(4/3)), x)